Review Article The Airport Gate Assignment Problem: A Surveydownloads.hindawi.com/journals/tswj/2014/923859.pdf · Review Article The Airport Gate Assignment Problem: A Survey AbdelghaniBouras,MageedA.Ghaleb,UmarS.Suryahatmaja,andAhmedM.Salem

Review ArticleThe Airport Gate Assignment Problem A Survey

Abdelghani Bouras Mageed A Ghaleb Umar S Suryahatmaja and Ahmed M Salem

Industrial Engineering Department College of Engineering King Saud University PO Box 800 Riyadh 11421 Saudi Arabia

Correspondence should be addressed to Abdelghani Bouras bourasksuedusa

Received 6 August 2014 Accepted 2 September 2014 Published 20 November 2014

Academic Editor Dehua Xu

Copyright copy 2014 Abdelghani Bouras et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The airport gate assignment problem (AGAP) is one of the most important problems operations managers face daily Manyresearches have been done to solve this problem and tackle its complexityThe objective of the task is assigning each flight (aircraft)to an available gate while maximizing both conveniences to passengers and the operational efficiency of airport This objectiverequires a solution that provides the ability to change and update the gate assignment data on a real time basis In this paper wesurvey the state of the art of these problems and the various methods to obtain the solution Our survey covers both theoreticaland real AGAP with the description of mathematical formulations and resolution methods such as exact algorithms heuristicalgorithms andmetaheuristic algorithmsWe also provide a research trend that can inspire researchers about new problems in thisarea

1 Introduction

The complexity of airport management has increased signif-icantly Flight delays or accidents might happen if operationswere not handled well and domino effect might happen toinfluence the whole operations of airport In airports thetasks related to gate assignment problem (AGAP) are one ofthe most important daily operations many researches havebeen published onwith the aimof solving the problem in spiteof its complexity The objective of the task is assigning eachflight (aircraft) to an available gate while maximizing bothconveniences to passengers and the operational efficiency ofairport Large airlines typically need tomanage different gatesacross an airport in the most efficient way in a dynamic oper-ational environment This requires a solution that providesthe ability to change and update the gate assignment data ona real time basis It should also provide robust and efficientdisruption management while maintaining safety securityand cost efficiency

Numerous methods have been developed to solve thisproblem since 1974 Steuart [1] proposed simple stochasticmodel to find the efficiency use of the gate positions Theresearch interest in this fieldwas slow in development becausethere were less than 15 publications within 25 years However

after 2000 the interest to develop solutions for this problemincreased until nowadays though with small growth Theobjective of this problem varied and depended on the point ofviewThefirst is as an airport owner which is the governmentThe objectives are to maximize the utilization of the availablegates and terminal [1ndash4] minimize the number of gateconflicts [5]minimize the number of ungated flights [3 6ndash9]and minimize the flights delay [10] Another point of view isas an airlines ownerTheir goals were to increase the customersatisfaction with minimizing the passenger walking distancebetween gates [3 6 7 11ndash18] and minimizing the travellingdistance from runway to the gate [19]

Dorndorf et al [20] divided the objectives into fiveparts which are reducing the number of the proceduresfor the costly aircraft towing minimizing the passengerstotal walking distance minimizing the deviations in theschedules minimizing the number of ungated aircrafts andmaximizing the preferences (ie certain aircrafts should gofor particular gates) They also defined three usually usedconstraints which are the fact that only one aircraft can begated in a defined amount of time the fulfillment of thespace restriction and service requirements and the assuranceof getting a minimum time between sequent aircrafts and aminimum ground time

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 923859 27 pageshttpdxdoiorg1011552014923859

2 The Scientific World Journal

The solution approaches and the solving techniques arevariedwith nomethods until nowadays that provide a robusttechnique for such problem This study focuses on assessingthe trend of solving gate assignment problem in light of thepreceding four points Specifically this study will address thefollowing research questions (1) Is this problemNP-hard (2)What formulation can be defined for such problem (3) Howeffective are the recent methods and techniques to solve theproblem (4) What recommendation can be made based onthe current findings with regard to research trends

From amathematical view AGAP has been formulated asinteger binary or mixed integer general linear or nonlinearmodels Specific formulation as binary or mixed binaryquadratic models has also been suggested Other well-knownrelated problems in combinatorial optimization such asquadratic assignment problem (QAP) clique partitioningproblem (CPP) and scheduling problem have been usedto formulate AGAP However few publications on AGAPtackled stochastic or robust optimization

While the goal of combinatorial optimization researchis to find an algorithm that guarantees an optimal solutionin polynomial time with respect to the problem size themain interest in practice is to find a nearly optimal or atleast good-quality solution in a reasonable amount of timeMany approaches to solve the GAP have been proposedvarying from Brand and Bound (BampB) to highly esotericoptimization methodsThemajority of these methods can bebroadly classified as either ldquoexactrdquo algorithms or ldquoheuristicrdquoalgorithms Exact algorithms are those that yield an optimalsolution As discussed in Section 31 different exact solutiontechniques have been used to solve the GAP and in someresearch the authors used some optimization programminglanguages like CPLEX and AMPL

Basically the GAP is a QAP and it is an NP-hardproblem as shown in Obata [21] Since the AGAP is NP-hard researchers have suggested various heuristic and meta-heuristics approaches for solving the GAP With heuristicalgorithms theoretically there is a chance to find an optimalsolution That chance can be remote because heuristics oftenreach a local optimal solution and get stuck at that point Butmetaheuristics or ldquomodern heuristicsrdquo introduce systematicrules to deal with this problem The systematic rules avoidlocal optima or give the ability of moving out of localoptima The common characteristic of these metaheuristicsis the use of some mechanisms to avoid local optimaMetaheuristics succeed in leaving the local optimum bytemporarily accepting moves that cause worsening of theobjective function value Sections 32 and 33 addressed theheuristic and metaheuristics approaches for solving the GAPSome papers presenting good overviews as well as annotatedbibliographies on the topic of GAP and a good literatureon the AGAP and the use of metaheuristics for AGAP areDorndorf et al [20 22] and Cheng et al [23]

This paper surveys a large number of models and tech-niques developed to deal with GAP In Section 2 we detailthe models formulations of the problem In Section 3 weaddressed the resolution methods used to solve the problemWe conclude in Section 4 and we represent the researchtrends

2 Formulations of AGAP andRelated Problems

Many researchers formulated the AGAP as an integer binaryor mixed integer linear or nonlinear model and some ofthem formulated it as binary or mixed binary quadraticmodels whereas some of the researchers have formulatedthe AGAP as well-known related problems in combinatorialoptimization such as quadratic assignment problem (QAP)clique partitioning problem (CPP) and scheduling problemor even as a network representation However some of theresearchers formulated the AGAP as a robust optimizationmodel In this section according to the way of how theresearchers deal with the gate assignment problem a classifi-cation for the AGAP has been made as follows

21 Selected AGAP Formulations

211 Integer Linear Programming Formulations (IP) Lim etal [24] formulated the AGAP as an integer programmingmodel and developed two models with time windows Thefirst model was devoted to minimization of the passengerwalking distance (travel time)

Minimize119899

sum119894=1

119899

sum119895=1

119898

sum119896=1

119898

sum119897=1

119891119894119895119908119896119897119911119894119895119896119897

+

119899

sum119894=1

119901119894(119888119894minus 119886

119894) (1)

while the second model optimized the gate assignments withcargo handling costs

Minimize119899

sum119894=1

119899

sum119895=1

119898

sum119896=1

119898

sum119897=1

[119891119860

119894119895(119888119896119897

+ 119888119860

) + 119891119861

119894119895(119888119896119897

+ 119888119861

)] 119911119894119895119896119897

+

119899

sum119894=1

119901119894(119888119894minus 119886

119894)

(2)

Both of these objectives put a penalty function due to a delayThese two objectives used the constraints as follows

119898

sum119896=1

119909119894119896

= 1 1 le 119894 le 119899 (3)

119911119894119895119896119897

le 119909119894119896 1 le 119894 119895 le 119899 1 le 119896 119897 le 119898 (4)

119911119894119895119896119897

le 119909119895119897 1 le 119894 119895 le 119899 1 le 119896 119897 le 119898 (5)

119909119894119896

+ 119909119895119897

minus 1 le 119911119894119895119896119897

1 le 119894 119895 le 119899 1 le 119896 119897 le 119898 (6)

119888119894ge 119886

119894 1 le 119894 le 119899 (7)

119888119894le 119887

119894minus 119889

119894 1 le 119894 le 119899 (8)

(119888119894+ 119889

119894) minus 119888

119894+ 119910

119894119895119872 gt 0 1 le 119894 119895 le 119899 (9)

(119888119894+ 119889

119894) minus 119888

119894minus (1 minus 119910

119894119895)119872 le 0 1 le 119894 119895 le 119899 (10)

119910119894119895

+ 119910119895119894

le 119911119894119895119896119897

1 le 119894 119895 le 119899 119894 = 119895 1 le 119896 le 119898 (11)

where 119909119894119896 119910

119894119895 and 119911

119894119895119896119897are binary and 119888

119894is integer

The Scientific World Journal 3

Constraint (3) ensures that each flight must be assignedto exactly one gate Constraints (4)-(5) state that a binaryvariable 119911

119894119895119896119897can be equal to one if flight 119894 is assigned to gate 119896

(119909119894119896

= 1) and flight 119895 is assigned to gate 119897 (119909119895119897

= 1) Constraint(6) further specifies the necessary condition that 119911

119894119895119896119897must

be equal to one if 119909119894119896

= 1 and 119909119895119897

= 1 Constraints (7) and(8) ensure that the flight must land and depart within thespecified time window Constraint (9) indicates that 119910

119894119895= 1 =

1 if (119888119894+ 119889

119894) le 119888

119895 which means 119910

119894119895= 1 when flight 119894 departs

before or right at the time when some gate opens for flight 119895Constraint (10) states that 119910

119894119895= 0 if (119888

119894+119889

119894) gt 119888

119895 whichmeans

119910119894119895

= 0when flight 119894 departs after some gate opens for flight 119895Constraint (11) specifies that one gate cannot be occupied bytwo different flights simultaneously

In the first model and according to the linearity of theobjective function and constraints they used a standard IPsolver (CPLEX) to find the optimal solution whereas inthe second model authors used several heuristic algorithmsnamely the ldquoInsert Move Algorithmrdquo the ldquoInterval ExchangeMove Algorithmrdquo and a ldquoGreedy Algorithmrdquo to generatesolutions The generated solutions then have been improvedusing a tabu search (TS) and memetic algorithm The resultsshowed that the used heuristics performed better than the IPsolver (CPLEX) in both CPU time and solutions quality

Diepen et al [25 26] formulated the AGAP as integerlinear programming model with a relaxation for the inte-grality After relaxing the integrality the resulting relaxed LPwas exploited to obtain solutions of ILP by using columngeneration (CG) The problem was divided into two phasesplanning and attaching The first phase was the planningsection and it was easier to model and calculate Theirobjective is tominimize the cost of a gate planThey proposedthe following model

Minimize119899

sum119895=1

119888119895119909119895 (12)

subject to

119899

sum119895=1

119892119894119895119909119895+ 119880

119894= 1 for 119894 = 1 119898 (13)

where

119892119894119895

= 1 if flight 119894 is in gate plan 119895

0 otherwise

119880119894ge 0 for 119894 = 1 119898 119880

119894is a penalty variable

(14)

This constraint defined the high price penalty (119880119894) when the

flights were not assigned to the gates This penalty appearedsince the planner should do the assignment manually Inaddition they added another constraint regarding the assign-ment since there was possibility that a long stay flight couldbe split into two parts The extra flights 119894

119886and 119894

119887that refer to

the arrival and departure of flight 119894were added to the previousconstraints

119899

sum119895=1

(119892119894119895

+ 119892119894119860119895

) 119909119895+ 119880

119894119860= 1

119899

sum119895=1

(119892119894119895

+ 119892119894119861119895

) 119909119895+ 119880

119894119861= 1

(15)

119899

sum119895=1

119905119895ℎ

119909119895= 119879

ℎfor ℎ = 1 119867 (16)

119909119895isin 0 1 for 119895 = 1 119899 (17)

where

119909119895=

1 if gate plan 119895 is selected0 otherwise

119905119895ℎ

= 1 if gate plan 119895 is type ℎ

0 otherwise

(18)

119899

sum119895=1

119898

sum119894=1

119867

sumℎ=1

119901119894ℎ119903

119905119895ℎ

119892119894119895119909119895ge 119875

119903for 119903 = 1 119877 (19)

where

119901119894ℎ119903

=

1 if flight 119894 has preference on gate type ℎ

in preference 119903

05 if the unsplit version of flight 119894 haspreference on gate type ℎ in preference 119903

0 otherwise(20)

and 119877 denotes the total number of preferencesThis constraint defined the flight preferences for exam-

ple a flight should be assigned to the same gate due to theownership or security The coefficient 05 refers to the extraflight defined in constraint (15)

They checked the solutionrsquos optimality using pricing prob-lem (minimum reduced cost) since they had dual multipliers120587119894 120582

ℎ and 120595

119903for constraints (15) (16) and (19) respectively

119888119895minus

119867

sumℎ=1

119905119895ℎ

120582ℎminus

119898

sum119894=1

(119892119894119895120587119894+

119877

sum119903=1

119867

sumℎ=1

(119892119894119895119905119895ℎ

119901119894ℎ119903

120595119903)) (21)

The second phase was amatter of assignment in physical gateThey made the rules to solve this phase as follows

(i) Sort the gates based upon the quality(ii) Sort the gate plans from the highest on the total

number of departing passengers that are on the flightsin that gate plan

(iii) Assign the gate plan to the best gate considering thehighest number of departing passengers assign thenext gate plan to the next-best gate and so on


In [26] Diepen et al used the solution obtained from theirassignment of gates as an input to solve the bus-planningproblem in the same airport

212 Binary Integer Programming In 2009 Tang et al [27]formulated the AGAP as a binary integer programing modelas below The output model was used to generate a lowerbound to their original problem

Minimize 119885 = sum119894isin119868

sum119895isin119864119894

sum119896isin119863119894119895

119889119894119896119909119894119895119896

+ sum119894isin119868

sum119895isin119864119894

sum119896isin119863119894119895

119908119894119895119909119894119895119896

(22)

subject to

sum119895isin119864119894

sum119896=119863119894119895

119909119894119895119896

= 1 forall119894 isin 119868 (23)

sum119894isin119865119895119904

sum119896=119867119894119904

119909119894119895119896

le 1 forall119895 isin 119866 forall119904 isin 119878 (24)

sum119894isin119871119905119902

sum119895isin119873119902

sum119896isin119867119894119904

119909119894119895119896

le 1 forall119905 isin 119879119902 forall119902 isin 119876 forall119904 isin 119878 (25)

119909119894119895119896

=

1 if flight 119894 is assign to gate 119895

at the time point (starting point) 119896

0 otherwise

forall119896 isin 119863119894119895 forall119895 isin 119864

119894 forall119894 isin 119868

(26)

Parameter Variables

119889119894119896 time inconsistency value indicating that the 119894th flightis assigned at the 119896th time point (starting time) if 119896 isequal to the original time point then 119889

119894119896= 0

119908119894119895 space inconsistency value indicating that the 119894th flightis assigned to the 119895th gate if 119895 equals the original gatethen 119908

119894119895= 0

The following sets have been defined

119868 considered flights119866 available gates119864119894 gates that the 119894th flight can be assigned to

119863119894119895 time points in which the 119894th flight can be assigned tothe 119895th gate

119865119895119904 flights that can be assigned to the 119895th gate so that theirtime windows will cover the 119904th time point

119878 all time points (ie the time points from the planningtime at each stage to the end of daily operations)

119867119894119904 time points (starting times) assigned to the 119894th flightwhere the resulting time windows cover the 119904th timepoint

119879119902 conflicting flight pairs for the 119892th adjacent gate pair

119871119905119902 flights included in the 119905th conflicting flight pair for the119902th adjacent gate pair

119876 adjacent gate pairs

Equation (23) is the flight constraint indicating that everyflight is exactly assigned to a gate Equation (24) is the gateconstraint ensuring that every gate is assigned to at most oneflight at any time Constraint (25) is related gate adjacencydenoting that two conflicting flights cannot be concurrentlyassigned to an adjacent gate pair Constraint (26) indicatesthat the assignment variables are either zero or one

Kumar et al [18] presented a binary integer programingmodel that produced a feasible gate plan in the light of all thebusiness constraints

119909119894119896

= 1 if turn 119894 is assigned to gate 119896

0 otherwise

119910119894=

1 if turn 119894 is not assigned to any gate0 otherwise

119908119894=

1 if long turn 119905 is towed0 otherwise

Maximize sum119894isin119879

sum119896isin119870

119862119894119896119909119894119896

minus 1198621sum119905isin119871

119908119905minus 119862

2sum119894isin119879

119910119894

(27)

subject to

sum119896isin119870

119909119894119896

+ 119910119894= 1 119894 isin 119879 (28)

sum119896isin119870119890119894isin119864119896

119909119894119896

+ 119910119894= 1 119894 isin 119879 (29)

sum119894isin119879

119910119894le 120585 (30)

119910119894119896

+ 119910119895119896

le 1 119894 119895 isin 119879 119896 isin 119870119886119894lt 119887

119895+ 120572

119886119895lt 119887

119894+ 120572 119894 = 119895

(31)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119897 isin 119870

(119896 119897) isin 119869119886119894lt 119887

119895 119886

119895lt 119887

119894

119894 = 119895 119890119894isin 119864

1

119896 119890

119895isin 119864

1

119897

(32)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119896

119877

119871119894119865119900) isin 119871119865119886

119895le 119886

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(33)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119897119877

119871119894119865119900) isin 119871119865119886

119895le 119887

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(34)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119875119861

119897119875119861

) isin 119869119875119861

119887119894minus 120573 lt 119887

119895le 119887

119894+ 120573

119894 = 119895 119890119894isin 119864

119896 119890

119895isin 119864

119897

(35)


119908119905le 120591 119905 isin 119879

119871(36)

1199101198941119896

minus 1199101198942119896

le 119908119905 119894

1 1198942isin 119879 119905 isin 119879

119871

119896 isin 1198701198941

= 1198942 119901

1198941= 119901

1198942= 119905

(37)

1199101198941119896

minus 1199101198943119896

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

119896 isin 1198701198861198941

lt 1198861198943 119887

1198943lt 119886

1198942 119901

1198941= 119897 119901

1198942= 119897

(38)

11991011989411198961

minus 11991011989431198962

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

1198961 119896

2isin 119870 119895 isin 119869119886

1198941lt 119886

1198943 119887

1198943lt 119887

1198942 119901

1198941= 119897

1199011198942

= 119897 1198961= 119902

1

119895 119896

2= 119902

2

119895 119890

1198941isin 119864

1

119895 119890

1198942isin 119864

2

119895

(39)

Constraint (28) ensures that turn 119894 is assigned to at most onegate Constraint (29) states that turn 119894 is assigned to a gate onlyif its equipment type is among the types which the assignedgate can accommodate Constraint (30) restricts the numberof ungated turns to less than or equal to the allowed number120585 Constraint (31) shows that at any given time at most oneturn is assigned to one gate Constraint (32) ensures thatadjacency constraints are observed Constraints (33)-(34)enforce LIFO restrictions Constraint (35) guarantees thatpushback restrictions are observed Constraint (36) confirmsthat no turn is towed if towing is not allowed Finallyconstraints (37)ndash(39) certify that if a long turn 119905 is towed the119908119905variable is set to be 1Mangoubi and Mathaisel [11] also developed a binary

integer model to minimize the passenger total walking dis-tance and proposed a heuristic method to find the solutionThe heuristic method result has been compared with theresults from a standard IP solver and the comparison resultsshowed that the heuristic method was superior to the LPsolver the average walking distance using the LP is 527 feetwhile heuristic is 558 feetThe developedmodel is introducedas follows

Minimize 119885 =

119872

sum119894=1

119872

sum119894=1

(119901119886

119894119889119886

119895+ 119901

119889

119894119889119889

119895+ 119901

119905

119894119889119905

119895) 119909

119894119895 (40)

where

119909119894119895

= 1 if flight 119894 is assigned to gate 119895

0 otherwise(41)

Transfer passenger walking distances are determined froma uniform probability distribution of all intergate walkingdistancesThe expected walking distance if119908

119895119896is the distance

between gate 119895 and gate 119896 is

119889119905

119895=

1

119873

119873

sum119896=1

119908119895119896

forall119895 = 1 119873 (42)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 = 1 119872 (43)

sumℎisin119871(119894)

119909ℎ119895

+ 119909119894119895

le 1 forall119894 = 1 119872 forall119895 = 1 119873 (44)

sumℎisin119871(119901+1)

119909ℎ119895

+ 119909119901+1119895

= 119909119901minus3119895

+ 119909119901minus2119895

+ 119909119901minus1119895

+ 119909119901119895

+ 119909119901+1119895

le 1

(45)

119871 (119901) sub 119871 (119901 + 1) sub sdot sdot sdot sub 119871 (119901 + 119896) (46)

119873

sum119911=1

119873

sum119904=1

119909119892119911

119908119911119904119909ℎ119904

le 119863max119892ℎ

(47)

119901119886

119891119889119886

119904+ 119901

119889

119891119889119889

119904+ 119901

119905

119891119889119905

119904= min

119895isin119878

119901119886

119891119889119886

119895+ 119901

119889

119891119889119889

119895+ 119901

119905

119891119889119905

119895 (48)

Constraint (43) shows that each flight is assigned to atmost one gate Constraint (44) ensures that no two planesare assigned to the same gate concurrently Constraint (45)determines the conflict constraint for each gate 119895 Constraint(46) is written to consider only the constraint generatedby the last plane of two or more flights arriving with nodeparture in between Constraint (47) ensures that flightsare assigned to nearby gates Constraint (48) assigns flight 119891to gate 119904 isin 119878 where 119904 is the gate with the minimum totalpassenger walking distance for flight 119891

Vanderstraeten and Bergeron [28] formulated the GAP asa binary integer model but with the objective of minimizingthe off-gate events and they developed a new heuristic whichis the ldquoAffectation Directe des Avions aux Portes (ADAP)rdquo tosolve the developed model A real case has been studied inan Air Canada terminal A new heuristic was applied to realdata at Toronto International Airport The developed modelwas as follows

Maximize 119885 = sum119894isin119868

sum119895isin119869

119881119894119895 (49)

subject to

sum119895isin119869119894

119881119894119895

le 1 119894 isin 119868 (50)

119881119894119895

+ sum119896isin119860119894119895

119881119896119895

le 1 119894 isin 119868 119895 isin 119869119894 (51)

119881119894119895

+ sum119896isin119861119894119895119903

119881119896119903

+ sum119896isin119861119894119895119897

119881119896119897

le 1 119894 isin 119868 119895 isin 119869119894 (52)

119881119894119895

= 0 1 119894 isin 119868 119895 isin 119869119894 (53)

Constraint (50) ensures that each flight is assigned to atmost one gate Constraint (51) defines the occupation timeat any gate Constraint (52) determines the neighboringconstraint Constraint (53) expresses the binary constraintfor all decision variables The results showed that using the


developed method resulted in no more than 30 events everbeing handled off gate while the manual procedure obtainedevents up to 50 of the 300 events being handled off gate

Bihr [12] developed a binary integer model to minimizethe passenger walking distance and applied this model tosolve a sample problem using primal-dual simplex algorithmAs a result he obtained a total walking distance of 22640Thedeveloped model is introduced as follows

Minimize 120579 = sum119862119894119895119883119894119895 119894 = 1 119898 119895 = 1 119896 (54)

where the 119862119894119895are the elements of the matrix product of

PAX(119894 119895) lowast DIST(119894 119895)119879 and

PAX(119894 119895) = number of passengers arriving on flight 119894

and departing from gate 119895

DIST(119894 119895) = number of passengers minus distance unitsfrom gate 119894 to gate 119895

119883119894119895

= 0 or 1

subject to

sum119883119894119895

= 1

sum119883119895119894

= 1

(55)

In 2002 Yan et al [29] formulated the static GAP as a binaryinteger programing model to serve as a basis of real timegate assignments in a simulation framework developed toanalyze the effects of stochastic flight delays on static gateassignments The presented model is as follows

Minimize 119885 =

119872

sum119894=1

119873

sum119895=1

119888119894119895119909119894119895

(56)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 (57)

sum119894isin119871119904

119909119894119895

le 1 forall119904 forall119895 (58)

119909119894119895

= 0 or 1 forall119894 forall119895 (59)

Constraint (57) ensures that each flight is assigned to at mostone gate Constraint (58) certifies that no two planes areassigned to the same gate concurrently Constraint (59) isrelated to the binary constraint for all decision variables Twogreedy heuristics were used to solve the model and theirresults were compared with the insights of the optimizationmethod The simulation framework was tested to solvecertain real case instances from CKS airport The results ofthe usedmethodswere 24562588 for the optimizationmodeland 27833552 and 30166809 (meters) for the two greedyheuristics

213 Mixed Integer Linear Programming (MILP) Bolat [30]formulated a mixed integer program for the AGAP with theobjective of minimizing the range of slack times (slack timeis an idle time between two successive utilizations of thegate) Certain instances with more than 20 gates have beenconsidered according to airplane types gate types terminaltypes and utilization levels

Minimize V+ (Γ) minus Vminus (Γ) (60)

subject to

V+ (Γ) = max V119894(Γ) 119894 = 1 119873 + 119872 (61)

Vminus (Γ) = min V119894(Γ) 119894 = 1 119873 + 119872 (62)

V119894(Γ) = 119860

119894minus 119864

119894Γ119894 119894 = 1 119873 (63)

119864119894Γ119894

= 119863119896lowast where 119896lowast = max 119896 Γ

119896= Γ

119894 119896 = 1 119894 minus 1

119861Γ if no such 119896 exist

(64)

V119873+119895

(Γ) = 119879119895minus 119864

119873+1119895 119895 = 1 119872 (65)

The results related to expected average utilizations wererespectively 8854 6713 and 4557over heavily utilizednormally utilized and underutilized problems Concerningthe average number of flights results were 10 759 and515 per gate

In 2001 Bolat [31] presented a framework for the GAPthat transformed the nonlinear binary models (it will bediscussed in Section 214 according to our classification)into an equivalent linear binary model with the objective ofminimizing the range or the variance of the idle times Theframework consists of five mathematical models where twoof the five models were formulated as a mixed integer linearprogramming and the others as a mixed integer nonlinearprogrammingModels P1 to P4were defined for homogenousgate while model P5 was defined for heterogeneous gate

119883119895119896


0 otherwise

119875119895119896

=

1 if the assignment of flight 119895 119905119900 gate 119896 cansatisfy all considerations

0 otherwise(66)

Using the presented framework nonlinear model P1 (modelP1 will be discussed in Section 214 according to our classifi-cation) was transformed to the followingmixed integer linearmodel which is model P2

Model P2 Consider

Minimize 119878max minus 119878min (67)


subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)

Similarly for model P3 (Section 214) the resultant modelwas model P4 that is a mixed binary model as in model P2but with two additional real variables as follows

Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)

Different instances have been studied according to the num-ber of the gates small (five gates) medium (10 gates) andlarge (20 gates) Instances with more than 20 gates were notconsidered The results were as follows average numbersof flight were 26125 5225 and 105417 and the averageutilizationswere 45725 66548 and 88871 according to thegate size respectively

Seker and Noyan [9] formulate the GAP as a mixedinteger programwith the objective ofminimizing the numberof conflicts and at the same time minimizing the totalsemideviation between idle time and buffer time

Minimize sum119894isin1198731198880

sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)

Anothermodel was developed as amixed integer program forthe same objective function The model was the same as theprevious model but with some differences


sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=

1 if idle time of flight 119894 isless than the buffer time

0 otherwise(90)

These two models have the same constraints properties

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)

All remaining variables ge 0 (100)

while objective (89) has the following additional constraints

119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)

214 Mixed Integer Nonlinear Programming Li [5] formu-lated the GAP as a nonlinear binary mixed integer modelhybrid with a constraint programing in order tominimize thenumber of gate conflicts of any two adjacent aircrafts assignedto the same gate The developed model has been solved usingCPLEX software


sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)

where 119886 scheduled arriving time 119889 scheduled departuretime and 119887 buffer time (constant) Consider

119909119894119896

= 1 if and only if aircraft 119891

119894is assigned to gate 119888

119896

0 otherwise (1 le 119894 le 119899 1 le 119896 le 119888)

119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)

In another work Li [32] defined the objective as

Minimize sum119894119895isin119873119894 =119895

119910119894119895

119886119894minus 119889

119895+ 2119887

(106)

These two models have the same constraints all constraintsare as follows

sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)

forall1 le 119894 119895 le 119899 119894 = 119895 forall1 le 119896 le 119888 (111)

Constraint (107) indicates that each aircraft is assigned to atmost only one gate Constraint (108) represents a method tocompute the auxiliary variable 119910

119894119895from 119909

119894119896 Constraint (109)

ensures that one gate can only be assigned atmost one aircraftat the same time Some additional constraints in the realoperations are ignored Constraint (110) represents binaryvalue of the decision variables

As mentioned in Section 213 Bolat [31] proposed twomodels formulated as a mixed integer linear program whichhave been transformed from a mixed integer nonlinearprogram The proposed mixed integer nonlinear programwas as follows

119883119895119896


0 otherwise

119875119895119896

=

1 if the assignment of flight 119895 to gate 119896

can satisfy all considerations0 otherwise

(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)

Bolat [31] also proposed two alternative formulationsfor homogenous and heterogeneous gates The proposedextended formulation for the homogenous gates was asfollows

119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894

119885 otherwise(122)

1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)

In addition the proposed extended formulation for theheterogeneous gates was as follows

119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)

As mentioned in Section 213 different instances have beenstudied according to the number of the gates small (fivegates) medium (10 gates) and large (20 gates) Instances withmore than 20 gates were not considered The results wereas follows average numbers of flight were 26125 5225 and105417 and the average utilizations were 45725 66548 and88871 according to the gate size respectively

215 Quadratic Programming

(1) Quadratic Mixed Binary Programming Zheng et al [33]formulated the GAP as a mixed binary quadratic programwith minimizing the slack time overall variance as theobjective function an assumption has been stated such thatthe flights are sequenced with the smallest arrival time Theproposed mixed binary quadratic model was as follows

Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896

ge sum119895isin119873

119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=

1 if flight 119894 and flight 119895 are bothassigned to gate 119896

0 otherwise(153)

where the indices 119894 119895 119896 in (143)ndash(151) denote 119894 119895 isin 119873 119896 isin 119872Equation (143) represents the objective function with the aimofminimizing overall variance of slack time Constraint (144)imposes the assignment of every flight to one gate Constraint(145) obliges every flight to have at most one immediateprecedent flight Constraint (146) enforces every flight to haveat most one immediate succeeding flight Constraints (147)and (148) define the first and last slack time of each gateand constraint (149) defines the other slack times Constraint(150) stipulates that the flight can be assigned to the gate whenthe preceding flight has departed for dwell time Constraint(151) indicates that the different type of gate allows parkingdifferent type of flight

Solutions were obtained using tabu search based on someinitial (starting) solutions the results were compared withthose of a random algorithm developed in the literatureUsing data from Beijing International Airport (10 gates and100 of flights between 600 and 1600) the initial solutionsusing metaheuristic and random algorithm were 9821 and15775 respectively

Bolat [34] formulated the AGAP as a mixed binaryquadratic programming model to minimize the varianceof idle times and used branch and bound algorithm andproposed two heuristics which were ldquosingle pass heuristicrdquo(SPH) and ldquoheuristic branch and boundrdquo (HBB) for solvingthe proposed model The proposed mixed binary quadraticmodel was stated as follows

minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)

Real instances from King Khalid International Airport (72generated sets) were used During the initial phase theproposed heuristic methods gave an average improvement of8739 on the number of remote assigned flights whereasthe average improvement on the number of towed aircraftsduring the real time phase was 7619

Xu and Bailey [14] formulated the GAP as a mixedbinary quadratic programming model (Model 1) and theobjective was to minimize the passenger connection timeThe proposedmodel (Model 1) was reformulated (linearized)into anothermodel (Model 2) in which the objective functionand the constraints have been linearized (the resultant modelwas a mixed binary integer model) Model 1 and Model 2 arelisted below

Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896

forall119894isin 119873 forall

119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=

1 if flight 119894 119895 are both assigned to gate 119896

flight 119894 immediately precedes flight 119895

0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)

where objective function (163) seeks to minimize the totalconnection times by passengers Constraint (164) specifiesthat every flight must be assigned to one gate Constraint(165) indicates that every flight can have at most one flightimmediately followed at the same gate Constraint (166)indicates that every flight can have at most one precedingflight at the same gate Constraints (167) and (168) stipulatethat a gate must open for boarding on a flight during thetime between its arrival and departure and also must allowsufficient time for handling the passenger boarding whichis assumed to be proportional to the number of passengersgoing on board Constraint (169) establishes the precedencerelationship for the binary variable 119911

119894119895119896and the time variables

119905119894and 119905

119895and is only effective when 119911

119894119895119896= 1 It stipulates

that if flight 119894 is assigned immediately before flight 119895 to thesame gate 119896 the gate must open for flight 119894 earlier than forflight 119895 Therefore it ensures each gate only serves one flightat any particular time Constraint (170) further states that theaircraft can only arrive at the gate when the previous flighthas departed for certain time

Model 2 Consider

119909119894119895119896119897

=

1 if and only if flight 119894 is assigned to gate 119896

and flight 119895 is assigned to gate 119897

0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)

where constraints (186) and (187) state that a binary variable119909119894119895119896119897

can be equal to one if flight 119894 is assigned to gate 119896 (119910119894119896

= 1)and flight 119895 is assigned to gate 1 (119910

119895119897= 1) Constraint (188)

further gives the necessary condition which is that 119909119894119895119896119897

mustbe equal to one if 119910

119894119896= 1 and 119910

119895119897= 1

The BampB and tabu search algorithm were used to solvethe generated instances (seven instances up to 400 flightsand 50 gates for 5 consecutive working days) The resultsof the analyzed instances showed an average saving of theconnection time of 247

(2) Binary Quadratic Programming Ding et al [6 35]developed a binary quadratic programming model for theoverconstrained AGAP to minimize the number of ungatedflights A greedy algorithm was designed to obtain an initialsolution which has been improved using tabu search (TS)The developed model was stated as follows

Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=

1 if flight 119894 is assigned to gate 119896

(0 lt 119896 le 119898 + 1)

0 otherwise(195)

forall119894 1 le 119894 le 119899 forall119896 1 le 119896 le 119898 + 1 (196)

where constraint (192) ensures that every flight must beassigned to one and only one gate or assigned to the apronConstraint (193) specifies that the departure time of eachflight is later than its arrival time Constraint (194) says that anassigned gate cannot admit overlapping the schedule of twoflights

In 2005 Ding et al [7] developed a binary quadratic pro-grammingmodel for the overconstrainedAGAP tominimizethe number of ungated flights The developed model was asfollows

Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)


where constraint (199) ensures that every flight must beassigned to one and only one gate or assigned to the apron andconstraint (200) requires that flights cannot overlap if they areassigned to the same gate

Using the same case study by Ding et al [6 35] a greedyalgorithm was designed to obtain an initial solution whichhas been improved using simulated annealing (SA) and ahybrid of simulated annealing and tabu search (SA-TS)

216 Multiple Objective AGAP Formulations Hu and DiPaolo [36] mathematically formulated the multiobjectiveGAP (MOGAP) as a minimization problem and solvedthis problem using a new genetic algorithm with uniformcrossover The developed MOGAP model was presented asfollows

Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)

119869MOGAP = 120572119869TPWD + 120573119869TBTD + (1 minus 120572 minus 120573) 120593119869TPWT (210)

120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)

Wei and Liu [16] considered the AGAP as a fuzzy model andadopted a hybrid genetic algorithm to solve the developedmodel The main objectives were minimizing passengersrsquototal walking distance and gates idle times variance Theydeveloped the following model

minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896

isin 0 1 forall119894 isin 119873 forall119896 isin 119870 (216)

where objective function (212) reflects the total walkingdistance of passengers 119910

119894119896is 0-1 variable 119910

119894119896= 1 if flight

119894 is assigned to gate 119896 otherwise it is 0 119891119894119895describes the

number of passengers transferring from flight 119894 to 119895 and 119879119896119897

is walking distance for passenger from gate 119896 to 119897 Objectivefunction (213) is used as a surrogate for the variance ofidle times The actual number of assignments is 119873 and thenumber of nondummy idle times is 119873 + 119872 Constraint


(214) indicates that every flight must be assigned to one gateConstraint (215) shows that flights that have overlap schedulecannot be assigned to the same gate where 120572 is the least safetime between continuous aircrafts assigned to the same gateConstraint (216) denotes that 119910

119894119896is a binary variable

In 2001 Yan andHuo [2] formulated theAGAPas amodelwith two objectives minimizing (3) the walking distanceand (4) the waiting time for the passengers The proposedmathematical model is binary integer linear programming

Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896

le 1 forall119895 forall119878 (217d)

119909119894119895119896

= 0 or 1 forall119894 forall119895 forall119896 (217e)

where objective (217a) represents the minimum total pas-senger walking distance Objective (217b) represents theminimum total passenger waiting time Constraint (217c)denotes that every flight must be assigned to one and onlyone gate Constraint (217d) ensures that at most one aircraftis assigned to every gate in every time window

Column generation approach simplex method and BampBalgorithm were used to solve the proposed problem whichwas a case study in Chiang Kai-Shek Airport Taiwan Theproblem consisted of 24 gates (of which two were temporaryeight out of 24 gates were only available for the wide type ofaircrafts whereas the rest were available for the other types)and 145 flights The results showed that the obtained solution(7300660 s the best feasible solution found so far) was awayfrom the optimal one by 0077 (5595s)

Wipro Technologies [17] proposed a binary multipleobjective integer quadratic programming model for theAGAP with a quadratic objective function The proposedmodel was reformulated into a mixed binary integer linearprogramming model (linear objective functions and con-straints) The proposed model has been solved using greedyheuristic SA and TS (MIP solvers based BampB cannot solvethe proposedmodel within a reasonable time)Thedevelopedmodel was represented as follows

Generic Model Consider

Minimizesum119894isin119873

119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896

= 0 forall119894 isin 119873 forall119896 isin 119870119894 (221)

119910119894119896

gt 119910119895119897 forall119894 isin 119873 forall119896 isin 119870 forall119895 isin 119873

119894 forall119897 isin 119870

119896 (222)

(119910119894119896

+ 119910119895119897) le 1 forall119894 isin 119873 forall119896 isin 119870 forall119895 isin 119873

119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896

forall119894 isin 119873 forall119896 isin 119870 (224)

119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911

119894119895119896) lowast 119872) forall119894 119895 isin 119873

(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889

119894+ 120573) forall119894 119895 isin 119873 forall119896 isin 119870

(229)

119910119894119896

119911119894119895119896

isin 0 1 forall119894 119895 isin 119873 forall119896 119897 isin 119870 (230)

119905119894ge 0 forall119894 isin 119873 (231)

where objective function (218) aims at minimizing the num-ber of flights that must be assigned to the apron that isthose left ungatedObjective function (219) seeks tominimizethe total connection times by passengers Constraint (220)specifies that every flight must be assigned to one gateConstraint (221) shows the equipment restriction on certaingates Constraints (222) and (223) restrict the assignment ofspecific adjacent flights to adjacent gates Constraints (224)and (225) indicate that every flight can have at most one flightimmediately following and at most one flight immediatelypreceding at the same gate Constraints (226) and (227)stipulate that a gate must open for boarding on a flightduring the time between its arrival and departure and it alsomust allow sufficient time for handling the passengerluggageboarding which is assumed to be proportional to the numberof passengers going on board Constraint (228) ensures thateach gate only serves one flight at any particular time (ieif flight 119894 is assigned immediately before flight 119895 to thesame gate 119896 the gate must open for flight 119894 earlier thanfor flight 119895) Constraint (229) further states the aircraftcan only arrive at the gate when the previous flight hasdeparted while also including the buffer time between the


flights Constraints (230) and (231) specify the binary andnonnegative requirements for the decision variables

The Reformulated Model Minimizing (218) will remain thesame


sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)

subject to Constraints (220)ndash(231) from the genericmodel

119909119894119895119896119897

le 119910119894119896 forall119894 119895 isin 119873 forall119896 119897 isin 119870 (233)

119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894 119895 isin 119873 forall119896 119897 isin 119870 (235)119909119894119895119896119897


where constraints (233) (234) and (235) specify that 119909119894119895119896119897

canbe equal to one if and only if flight 119894 is assigned to gate 119896 andflight 119895 is assigned to gate 119897 Constraint (236) expresses thebinary requirement for the decision variable 119909

119894119895119896119897

Kaliszewski and Miroforidis [37] considered agap withthe objective of assigning incoming flights to airport gateswith some assumptions those assumptions were as followsgate assignment has no significant impact on passengerwalking distance and no restrictions on the gates (all gates cantake any type of airplanes) and neighboring gate operationscan be carried out without any constraints The model wasstated as follows

minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)

217 Stochastic Models Yan and Tang [10] designed a frame-work for a stochastic AGAP (flight delays are stochastic)Theframework included three main parts the gate assignmentmodel a rule for the reassignments and two adjustmentmethods for penalties The performance of the developedframework has been evaluated using simulation-based eval-uation method

The formulation of the stochastic gate assignment model(the objective was to minimize the total waiting time of thepassengers) was addressed as follows


sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904

ge 0 forall119904 isin Ω (248)

119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)

where function (243) denotes the minimization of the totalpassenger waiting time the expected penalty value for all119899 scenarios and the expected semideviation risk measure(SRM) for all 119899 scenarios multiplied by the weighting vector119908 Constraint (244) is the flow conservation constraint atevery node in each network Constraint (245) denotes thatevery flight is assigned to only one gate and one time windowConstraint (246) ensures that the number of gates used ineach network does not exceed its available number of gatesConstraints (247) and (248) are used to calculate the SRMConstraint (249) ensures that the cycle arc flows are integersConstraint (250) indicates that except for the cycle arcs allother arc flows are either zero or one

The value of the performance measure (the objectiveminimizing the total waiting time of the passengers) for eachscenario in the real time stage was calculated as follows

119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)


For each iteration the penalties were calculated using thedeveloped two adjustment methods for penalties as follows

Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896

forall119896 isin 119870 forall119904 isin Ω

(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)

The data was taken from the Chiang Kai-Shek (CKS) air-port (172 flights 2 gate types and 14 aircraft types) thedistributions for the flight delays were obtained from theactual data taken from the CKS airport The obtained resultswere 197 minutes which was the longest solution time of theframeworkwhichwas efficient in the planning stage but after40 scenarios the solution times increased significantly butthe solution results were more stable

Genc et al [38] developed a stochastic model for AGAPwith the objective of minimizing the gate duration gateduration defined as the total time of the allocated gates (forall flights in a day)

119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)

Seker and Noyan [9] also developed a stochastic model con-sidering the minimization of the number of conflicts and theexpected variance of the idle times as a performancemeasurethe proposed performance measure was a part of the mixedinteger programming model presented in Section 213


119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)


22 AGAP Related Problems In some of the publicationson the GAP the researchers have formulated the AGAP as


well-known related problems such as quadratic assignmentproblem (QAP) clique partitioning problem (CPP) andscheduling problem or even as a network representationHowever some of the researchers formulated the AGAP asa robust optimization model In this section we will presentthe work that has been done on the GAP as a well-knownrelated problem

221 Quadratic Assignment Problem (QAP) Drexl andNikulin [3]modeled themulticriteria airport gate assignmentas quadratic assignment problem (QAP) and solved theproblem using Pareto simulated annealing The performancemeasures were as follows minimizing connection times ortotal passenger walking distances maximizing the prefer-ences of total gate assignment and minimizing the numberof ungated flights

min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)

where objective (279a) addresses the number of flights thatare not assigned to any terminal gate (ie to the apron)Objective (279b) represents the total passenger walkingdistance It consists of three terms the walking distance oftransfer passengers originating departure passengers anddisembarking arrival passengers Objective (279c) representsthe total value for flight gate assignment preference Con-straint (280) ensures that every flight must be assignedto exactly one gate including the apron Constraint (281)prohibits schedule overlapping of two flights if they areassigned to the same terminal gate Constraint (282) definesthe variables to be Boolean

Haghani and Chen [13] modeled the AGAP as QAP withminimizing the total passenger walking distances (transfer

passengers and local passenger) as a performance measureThe QAP model was expressed as follows

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)

According to the simplicity of solving linear models theprevious model was transformed into a linear model asfollows

11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)

The results and conclusions were as follows the proposedapproach was efficient which provided results close to theoptimal solution according to the percent of improvementfrom the starting solution (initial solution) in the case of 10flights and 10 gates 20 flights and 5 gates and 30 flights and7 gates

222 Scheduling Problems In 2010 Li [39] formulated theGAP as a parallel machines scheduling problem and usedthe dynamic scheduling and the direct graph model to solvethe proposed model BampB was used in solving the small sizeproblems while the large size problems have been solvedusing dynamic scheduling

223 Clique Partitioning Problem (CPP) Dorndorf et al[8] developed an optimization model for the GAP andtransformed that model into a CPP model the two modelsare written below A heuristic approach that was developed


by Dorndorf and Pesch (1994) based on the ejection chainalgorithm has been used to solve the transformed model(CPP model)

The Optimization Model for the AGAP Consider

Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)

The CPP Transformation of the Problem Consider

maximize sum1le119894lt119895le119886

119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)

minus1205723sdot max 119905max minus 119905

119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=

minusinfin if 119895 notin 119872 (119894)

1205721sdot 119901lowast

119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895

= minusinfin forall119894 119895 gt 119899 (302)

224 Network Representation Maharjan and Matis [40]formulated the GAP as a binary integer multicommoditynetwork flow model with minimizing the passengers com-fort and aircraft fuel burn as a performance measure Forpassengers comfort and with arguments of distance and timefor connection a penalty function in three dimensions wasspecified For large size problem and based on amethodologyof zoning a decomposition approach was provided andcompared with the assignments made by the airline and theresults showed that the developed methodology was shownto be computationally efficient

TheMathematical Formulation Consider

119883119896

119878119894 binary variable representing initial assignment ofgate 119896 isin K to aircraft 119894 isin F

119883119896

119895119894 binary variable representing assignment of gate 119896 isin

K to aircraft 119895 isin D followed by 119894 isin A119883119896

119895119879 binary variable representing last assignment of gate119896 isin K to aircraft 119895 isin D

119883119896

119878119879 binary variable representing no assignment of gate119896 isin K to any aircraft


sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)

where (303) represents the expected taxi in and out fuelburn cost of assigning a plane to a particular gate based onthe expected runway distance corresponding to arrival anddeparture cities for the flight Equation (304) is referred toas a flow-in constraint because it deals with the gate flowfrom the source node to the arrival flight node Equation(305) is referred to as conservation of flow at the arrivalnode Equation (306) is conservation of flow at the departurenode Equation (307) is the flow-out constraint that forcesall the flow to leave the departure node to the terminalnode Equation (308) is referred to as a unit flow servingarc constraint as it allows only one unit gate 119896 isin 119870 to flowthrough serving arc Equation (309) is the binary constraints

The above model was with a quadratic objective functionand a linearization for that objective was made as follows

Minimize1198852= sum

(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)

The linearization has been made by replacing the quadraticterm (119883119896

1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840

1198941198941015840 defined as follows

1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)

where inequalities (313) and (314) indicate that variable 1198841198961198961015840

1198941198941015840

is equal to 1 if and only if binary variables 119883119896

119894119895and 119883119896

1015840

11989410158401198951015840 are

equal to 1 Equation (315) specifies that1198841198961198961015840

1198941198941015840 cannot be greater

than 1 and (316) further specifies that1198841198961198961015840

1198941198941015840 cannot be less than

zero Due to the binary nature of 119883119896

119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840

1198941198941015840 is forced to be a binary variable

According to the passengers comfort a cost function(119862119896119896

1015840

1198941198941015840 ) was represented as follows

1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)

where (317) is a surface plot of the cost function

Mathematical Formulation Using the Methodology of ZoningConsider



sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)

subject to the followingThe constraints of (304) and (308) are modified and

replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)

In order to solve the developed model a code for thedeveloped model was written using an AMPLCPLEX 112package and as mentioned before the results showed that thedeveloped methodology was shown to be computationallyefficient

23 Robust Optimization Diepen et al [41] formulated acompletely new integer linear programming formulation forthe GAP with a robust objective function that is based onthe so-called gate plans The objective was to maximizethe robustness of a solution which can be expressed as anallocation of amaximumpossible idle time between each pairof consecutive flights to guarantee that each flight can affordto land with some slight earliness or tardiness without theneed for re-planning the schedule

119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where

119892V119894 = 1 if flight V is in gate plan 119894

0 otherwise

119890119894119886

= 1 if flight 119894 is of type 119886

0 otherwise

(325)

In addition to add the preferences to the ILP model thefollowing constraints have been added to the model

119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where

(i) 119901V119886119896 = 1 if flight V has preference for gate type 119886 in preference 1198960 otherwise

(ii) 119875119896denotes the minimum number of flights that have

to be assigned to a given gate type(iii) according to preference 119896(iv) 119870 denotes the total number of preferences

119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)

where 119880119860119865V ge 0 for V = 1 V and 119880119860119865V is a penaltyvariable

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=

1 if flight V has preference on gate type ain preference 119896

05 if the split version of flight V has preferenceon gate type a in preference 119896

0 otherwise(328)

The integrality of the developed model has been relaxed forthe integrality and the resulting relaxed LP was exploited toobtain solutions of ILP by using column generation (CG)Table 1 summarizes all the above mathematical formulationsused recently for the AGAP

3 Resolution Methods

As mentioned before in Section 2 most of the solutiontechniques presented in Sections 32 and 33 have been usedconcurrently with complex mathematical formulations that


Table 1 Formulations of AGAP and related problems

Formulation References Criterion (comments) Problem type

Integer linearprogramming (IP)

Lim et al [24] (i) Minimizing the sum of the delay penalties(ii) Minimizing the total walking distance Theoretical

Diepen et al [25](i) Minimizing the deviation of arrival and departuretime(ii) Minimizing replanning the schedule

Real case (AmsterdamAirport Schiphol)

Diepen et al [26] Minimizing the deviations from the expected arrivaland departure times


Binary integerprogramming

Mangoubi and Mathaisel[11] Yan et al [29] Minimizing passenger walking distances

Real case (TorontoInternational Airport) Real

case (Chiang Kai-ShekAirport)

Vanderstraeten andBergeron [28] Minimizing the number off-gate event Theoretical

Bihr [12] Minimizing of the total passenger distance Theoretical

Tang et al [27] Developing a gate reassignment framework and asystematic computerized tool

Real case (TaiwanInternational Airport)

Prem Kumar and Bierlaire[18]

(i) Maximizing the gate rest time between two turns(ii) Minimizing the cost of towing an aircraft with along turn(iii) Minimizing overall costs that include penalizationfor not assigning preferred gates to certain turns

Theoretical

Mixed integer linearprogramming (MILP)

Bolat [30] Minimizing the range of slack times Real case (King KhaledInternational Airport)

Bolat [31] Minimizing the variance or the range of gate idle time Real case (King KhaledInternational Airport)

Mixed integernonlinearprogramming

Li [5 32] Minimizing the number of gate conflicts of any twoadjacent aircrafts assigned to the same gate

Real case (ContinentalAirlines Houston GorgeBush Intercontinental

Airport)


Multiple objectiveGAP formulations

Hu and Di Paolo [36]Minimize passenger walking distance baggagetransport distance and aircraft waiting time on theapron

Theoretical

Wei and Liu [16] (i) Minimizing the total walking distance for passengers(ii) Minimizing the variance of gates idle times Theoretical

BACoEB Team andAICoE Team [17]

(i) Minimizing walking distance(ii) Maximizing the number of gated flights(iii) Minimizing flight delays

Theoretical

Yan and Huo [2] (i) Minimizing passenger walking distances(ii) Minimizing the passenger waiting time

Real case (Chiang Kai-ShekAirport)

Kaliszewski andMiroforidis [37]

Finding gate assignment efficiency which representsrational compromises between waiting time for gateand apron operations

Theoretical

Stochastic modelYan and Tang [10] Minimizing the total passenger waiting time Real case (Taiwan

International Airport)

Genc et al [38] Maximizing gate duration which is total time of thegates allocated

Theoretical and real case(Ataturk Airport ofIstanbul Turkey)

Seker and Noyan [9] Minimizing the expected variance of the idle time Theoretical

Quadratic assignmentproblem (QAP)

Drexl and Nikulin [3]

(i) Minimizing the number of ungated flights(ii) Minimizing the total passenger walking distances orconnection times(iii) Maximizing the total gate assignment preferences

Theoretical

Haghani and Chen [13] Minimizing the total passenger walking distances Theoretical


Table 1 Continued


Scheduling problems Li [39]

(i) Maximizing the sum of the all products of the flighteigenvalue(ii) Maximizing the gate eigenvalue that the flightassigned

Theoretical

Quadratic mixedbinary programming

Bolat [34] Minimizing the variance of idle times Real case (King KhaledInternational Airport)

Zheng et al [33] Minimizing the overall variance of slack timeReal case (Beijing

International AirportChina)

Xu and Bailey [14] Minimizing the passenger connection time TheoreticalBinary quadraticprogramming Ding et al [6 7 35] Minimize the number of ungated flights and the total

walking distances or connection times Theoretical

Clique partitioningproblem (CPP) Dorndorf et al [8]

(i) Maximizing the total assignment preference score(ii) Minimizing the number of unassigned flights(iii) Minimizing the number of tows(iv) Maximizing the robustness of the resultingschedule

Theoretical

Networkrepresentation Maharjan and Matis [40] Minimizing both fuel burn of aircraft and the comfort

of connecting passengers

Real case (ContinentalAirlines at George W BushIntercontinental Airport in

Houston (IAH))

Robust optimization Diepen et al [41] Maximizing the robustness of a solution to the gateassignment problem


led to very high computing time Section 31 addressed theexact solution techniques and the optimization programminglanguage used to solve the proposed models to their opti-mality Sections 31 32 and 33 include the research workthat has been done on the exact heuristic and metaheuristicapproaches for solving the AGAP

31 Exact Algorithms Exact algorithms are those that yieldan optimal solution According to the literature differentexact solution techniques have been used to solve the GAPAs an example branch and bound was used as well as columngeneration and other methods and in some research theauthors used some optimization programming languages likeCPLEX and AMPL In this section only the research workthat has been done on the exact solution techniques forsolving the AGAP is presented

Li [5 32] solved the proposed hybridmathematicalmodelusing CPLEX software Mangoubi and Mathaisel [11] relaxedthe integrality of the developed ILP model and solved therelaxed ILP model using CG an optimal solution has beenobtained for minimizing the total walking distance Bihr[12] proposed a primal-dual simplex algorithm to find thesolution and found the optimal solution Yan and Huo[2] used simplex algorithm with column generation andweighting method to solve the provided model Bolat [3034] Li [39] and Yan and Huo [2] used branch and boundalgorithm to solve themodels they have developed Reference[14] used branch and bound algorithm and compared theresult with tabu search algorithm

32 Heuristic Algorithms Basically theGAP is aQAP and it isan NP-hard problem as shown in Obata [21] Since the AGAPis NP-hard researchers have suggested various heuristic andmetaheuristics approaches for solving the GAP This sectionis for the heuristic algorithms with heuristic algorithmstheoretically there is a chance to find an optimal solutionThat chance can be remote because heuristics often reach alocal optimal solution and get stuck at that point so it wasnecessary to have modern heuristics called metaheuristicThis approach will be presented in the following part in thissection the research work that has been done on the heuristicapproaches for solving the AGAP is presented

Yan and Tang [10] developed a framework designedto deal with the GAP which has stochastic flight delaysthe developed framework was with a heuristic approachembedded in it Genc [42] used several heuristics whichare the ldquoGround Time Maximization Heuristicrdquo ldquoIdle TimeMinimization algorithmrdquo and ldquoPrime Time Heuristicrdquo tosolve the GAP with minimizing the idle gate time (ormaximizing the number of assigned flights) as a performancemeasure Ding et al [6 35] designed a greedy algorithmfor solving the GAP with the objective of minimizing thenumber of ungated flights Lim et al [24] used severalsolution approaches which are the ldquoInsert Move Algorithmrdquothe ldquoInterval Exchange Move Algorithmrdquo and a ldquoGreedyAlgorithmrdquo to solve the developed model for the GAP Yanet al [29] proposed a simulation framework and developedan optimization model (Section 212) and then solved themodel using two greedy heuristics the first was related tothe number of passengers and the second was related to the


Table 2 Resolution methods

Method References Approachresults Problem type

Exact algorithms

Mangoubi and Mathaisel[11] Linear programming relaxation Real case (Toronto International

Airport)Bihr [12] Primal-dual simplex Theoretical

Yan and Huo [2] Simplexbranch and bound Real case (Chiang Kai-Shek Airport)

Bolat [30 34] Xu andBailey [14] Li [39] Branch and bound Real case (King Khaled International

Airport KSA) theoretical

Heuristic algorithms

Thengvall et al [43] Bundle algorithm approach Theoretical

Yan and Tang [10] Heuristic approach embedded in aframework designed

Real case (Taiwan InternationalAirport)

Ding et al [6 35] Greedy algorithm Theoretical

Lim et al [24]The Insert Move Algorithm theInterval Exchange Move Algorithmand a Greedy Algorithm

Theoretical

Diepen et al [25] Column generation Real case (Amsterdam AirportSchiphol)

Dorndorf et al [8] Heuristic based on the ejection chainalgorithm Theoretical

Mangoubi and Mathaisel[11] Heuristic approach Real case (Toronto International

Airport)Vanderstraeten andBergeron [28] ADAP Theoretical

Yan et al [29] Greedy heuristics Real case (Chiang Kai-Shek Airport)

Bolat [30] Heuristic branch and trim Real case (King Khaled InternationalAirport KSA)

Bolat [34] Heuristic branch and bound SPHheuristic

Real case (King Khaled InternationalAirport KSA)

Haghani and Chen [13] Heuristic approach Theoretical

Genc [42] Ground time maximization heuristicand idle time minimization heuristic

Theoretical and real case (AtaturkAirport of Istanbul Turkey)


A hybrid heuristics algorithm guidedby simulated annealing and greedyheuristic

Theoretical

Bouras et al [45] Heuristic approach Theoretical

Metaheuristic algorithms

Ding et al [6 35] Tabu search Theoretical

Ding et al [7] Simulated annealing hybrid ofsimulated annealing and tabu search Theoretical

Lim et al [24] TS algorithm and a memetic algorithm Theoretical

Hu and Di Paolo [36] New genetic algorithm with uniformcrossover Theoretical

Drexl and Nikulin [3] Pareto simulated annealing TheoreticalXu and Bailey [14] Tabu search Theoretical

Bolat [31] Genetic algorithm Real case (King Khaled InternationalAirport KSA)

Seker and Noyan [9] Tabu search algorithms Theoretical

Zheng et al [33] A tabu search algorithm andmetaheuristic method

Real case (Beijing InternationalAirport China)

Wei and Liu [16] A hybrid genetic algorithm TheoreticalGu and Chung [44] Genetic algorithms approach Theoretical

Cheng et al [23]Genetic algorithm (GA) tabu search(TS) simulated annealing (SA) and ahybrid approach based on SA and TS

Real case (Incheon InternationalAirport South Korea)

Bouras et al [45] Genetic algorithm (GA) tabu search(TS) and simulated annealing (SA) Theoretical


Table 2 Continued


OPL

Li [5 32] Optimization programming language(CPLEX)

Real case (Continental AirlinesHouston Gorge Bush IntercontinentalAirport)

Tang et al [27] Using CPLEX 100 solver concert withC language



Optimizationprogramming language (OPL) Theoretical

Maharjan and Matis [40] AMPLCPLEX 112Real case (Continental Airlines atGeorge W Bush IntercontinentalAirport in Houston (IAH))

arrival time of the flight at the gateThedistance result showedthat the optimization model outperformed the heuristic butthe heuristic simulation time outperformed the optimizationmodel Diepen et al [25] solved the resulting LP-relaxationand the original ILP model using column generation

Thengvall et al [43] represented a heuristic approachfor the problem of schedules recovery in airports duringhub closures the proposed approach was a bundle algorithmapproach Dorndorf et al [8] used a heuristic approachthat was developed by Dorndorf and Pesch (1994) whichwas based on the ejection chain algorithm to solve thetransformed model (CPP model) Mangoubi and Mathaisel[11] also used heuristic approach to solve the GAP with theobjective of minimizing walking distance for the passengersand the obtained solutions were compared with the optimalsolution using LP to obtain the deviation in the resultsHaghani and Chen [13] used a heuristic approach to solve theGAP with the objective of minimizing walking distance forthe passengers Vanderstraeten and Bergeron [28] developeda direct assignment of flights to gates algorithm namedADAP the developed algorithmwas an implicit enumerationwhich has been faster by carefully applying some variablesselection criteria which are the concept of ldquomain chainrdquothe assigned weights to variables and the single assignmentconstraints Bolat [30] used branch and trim heuristic tosolve the GAP with the objective of minimizing the slacktimes range Bolat [34] used the HBB and SPH heuristics tosolve the models that he has developed for the GAP The firstapproach (HBB) is a BampB approach that has been utilized bysome restrictions on the number of the nodes that has to bebranched in a search tree while the second approach (SPH)developed a heuristic that after 119873 iterations builds only onesolutionThe used heuristics assigned flights one at a time byconsidering all available gates and for determining the mostpermissible gate a priority function was utilized

33 Metaheuristic Algorithms As mentioned before inSection 32 heuristics often get stuck in a local optimal solu-tion but metaheuristics or ldquomodern heuristicsrdquo introducesystematic rules to deal with this problem The systematicrules avoid local optima or give the ability of movingout of local optima The common characteristic of thesemetaheuristics is the use of some mechanisms to avoid localoptima Metaheuristics succeed in leaving the local optimumby temporarily accepting moves that cause worsening of the

objective function value In this section the research workthat has been done on the metaheuristic approaches forsolving the AGAP is presented

Gu and Chung [44] introduced a genetic algorithmmodel to solve the AGAP The developed model has beenimplemented in a high-level programming language theeffectiveness of the developed model has been validated bytesting different scenarios and the results showed that theperformance of the developed model was efficient Seker andNoyan [9] developed stochastic programming models thedeveloped model has been formulated as a mixed integerprogramming but in large scale The developed models weresolved using tabu search (TS) algorithm and the obtainedresults were with high quality

Cheng et al [23] studied the performance of severalmetaheuristics in solving the GAP The metaheuristics weregenetic algorithm (GA) tabu search (TS) simulated anneal-ing (SA) and a hybrid of SA and TS Tabu search (TS) out-performs SA and GA but the hybrid approach outperformsTS in terms of solution quality Ding et al [6 35] used atabu search algorithm to solve the GAP the starting (initial)solution was obtained using a designed greedy algorithm Xuand Bailey [14] developed a tabu search algorithm to solvethe GAP and compared the result of the developed algorithmwith a branch and bound algorithm The results showed thatthe two approaches provide optimal solutions for the studiedproblems but TS obtained was better in the CPU time

Zheng et al [33] developed a model for solving the GAP(see Section 2) and used a TS algorithm to obtain solutionsfor the developed model Ding et al [7] used a simulatedannealing and a hybrid of SA and TS to solve the GAPmodel that they have developed the starting (initial) solutionwas obtained using a designed greedy algorithm Lim etal [24] proposed TS and memetic algorithms to solve theGAP Drexl and Nikulin [3] solved the multicriteria airportgate assignment using Pareto simulated annealing Hu andDi Paolo [36] solved the multiobjective gate assignmentproblem (MOGAP) using a new genetic algorithm withuniform crossover Bolat [31] used genetic algorithm (GA)to minimize the variance or the range of gate idle time Weiand Liu [16] modified a hybrid genetic algorithm to solvethe fuzzy AGAP model Table 2 summarizes all the abovesolution techniques used recently for the AGAP

Recently Bouras et al [45] approached the AGAP as aparallel machine-scheduling problem with some priority and


Table 3 Number of publications per year

Year Number of publications References1974 1 Steuart [1]1985 1 Mangoubi and Mathaisel [11]1988 1 Vanderstraeten and Bergeron [28]1990 1 Bihr [12]1998 1 Haghani and Chen [13]1999 2 Bolat [30] Gu and Chung [44]2000 1 Bolat [34]2001 3 Bolat [31] Xu and Bailey [14] Yan and Huo [2]2002 2 Lam et al [15] Yan et al [29]2003 1 Thengvall et al [43]2004 2 Ding et al [6 35]2005 3 Ding et al [7] Lim et al [24] Al-Khalifah [19]2007 4 Diepen et al [25] Dorndorf et al [20 22] Yan and Tang [10]2008 4 Diepen et al [26] Drexl and Nikulin [3] Li [5 32]

2009 4 Wei and Liu [16] Hu and Di Paolo [36] Tang et al [27] BACoEB Team and AICoE Team[17]

2010 4 Dorndorf et al [8] Genc [42] Zheng et al [33] Li [39]2011 2 Prem Kumar and Bierlaire [18] Genc et al [38]

2012 5 Li [39] Seker and Noyan [9] Diepen et al [41] Maharjan and Matis [40] Kaliszewski andMiroforidis [37] Cheng et al [23]

2014 1 Bouras et al [45]

eligibility They solved the problem with the aim of mini-mizing the following objectives total cost total tardinessandmaximum tardinessThey developed three heuristics andused three metaheuristics (simulated annealing tabu searchand genetic algorithms) The evaluation was conducted over238 generated instances but only 50 instances were presentedin the report The results showed that simulated annealingwas the most efficient metaheuristic to solve the problem

4 Conclusion and Research Trends

In this survey we have presented the very recent publicationsabout the airport gate assignment problem The collectedliterature has the aim of identifying the contributions and thetrends in the research using exact or approximate methods

An abundant literature is listed to describe mathematicalformulation on AGAP or other related problems They havebeen grouped in such a way that the user is guided toidentify each problem specification For single objectiveintegerbinary models are described along with mixed inte-ger ones Nonlinear formulations are also described formixedinteger models Rare are the authors who really cameout with exact solutions using existing commercial opti-mization software or their own exact methods (branch andbound ) Heuristics were suggested to build feasible solu-tions and improve the latter solutions using metaheuristics

For multiobjective optimization several models havebeen formulated as nonlinear objectives with little success insolving such problems with exact methods in a reasonabletime

11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014

Number of publications

Figure 1 Number of publications per year

Related problems to AGAP have also been introduced inthe survey Some cases of AGAP have been formulated assome well-known combinatorial optimization problems suchas QAP

Since 2005 (Table 3 Figures 1-2) most of the peoplestarted to consider using heuristicsmetaheuristics as tools tosolve AGAP since the problem is NP-hard and the issue oftime solving was still unresolved by the existing tools

In practice major airlines may have more than 1000 dailyflights to handle at more than 50 gates which results inbillions of binary variables in formulation BampB based MIPsolvers (ie CPLEX) will not be able to handle such huge sizeproblems within a reasonable time bound


Table 4 Number of publications per research area

Area Number ofpublications References

Integer linearprogramming (IP) 3 Lim et al [24] Diepen et al [25] Diepen et al [26]

Binary integerprogramming 6 Mangoubi and Mathaisel [11] Yan et al [29] Vanderstraeten and Bergeron [28]

Bihr [12] Tang et al [27] Prem Kumar and Bierlaire [18]Mixed integer linearprogramming (MILP) 3 Bolat [30] Bolat [31] Kaliszewski and Miroforidis [37]

Mixed integer nonlinearprogramming 3 Li [5 32] Bolat [31]

Quadratic programming(QP) 6 Bolat [34] Zheng et al [33] Xu and Bailey [14] Ding et al [6 7 35]

Multiple objective GAPformulations 5 Hu and Di Paolo [36] Wei and Liu [16] BACoEB Team and AICoE Team

[17] Yan and Huo [2] Kaliszewski and Miroforidis [37]Stochastic models 3 Yan and Tang [10] Genc et al [38] Seker and Noyan [9]Quadratic assignmentproblem (QAP) 2 Drexl and Nikulin [3] Haghani and Chen [13]

Scheduling problems 1 Li [39]Clique partitioningproblem (CPP) 1 Dorndorf et al [8]

Network representation 1 Diepen et al [41]Robust optimization 1 Maharjan and Matis [40]

Exact algorithms 7 Mangoubi and Mathaisel [11] Bihr [12] Yan and Huo [2] Bolat [30 34] Xu andBailey [14] Li [39]

Heuristic algorithms 16

Thengvall et al [43] Yan and Tang [10] Ding et al [6 35] Lim et al [24] Diepen etal [25] Dorndorf et al [8] Mangoubi and Mathaisel [11] Vanderstraeten andBergeron [28] Yan et al [29] Bolat [30 34] Haghani and Chen [13] Genc [42]

BACoEB Team and AICoE Team [17] Bouras et al [45]

Metaheuristic algorithms 14Ding et al [6 35] Ding et al [7] Lim et al [24] Hu and Di Paolo [36] Drexl andNikulin [3] Xu and Bailey [14] Bolat [31] Seker and Noyan [9] Zheng et al [33]

Wei and Liu [16] Gu and Chung [44] Cheng et al [23] Bouras et al [45]OPL 4 Li [5 32] Tang et al [27] Prem Kumar and Bierlaire [18] Maharjan and Matis [40]

10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18


1970ndash19751976ndash19801981ndash19851986ndash19901991ndash19951996ndash20002001ndash20052006ndash20102011ndash2014

Figure 2 Number of publications per 5 years

A growing interest in metaheuristics (Table 4 Figure 3)has been observed in the recent papers on AGAP TS SAand GA are the most used improvement methods Somehybridizedmethods combining thesemethods have also beensuggested

It is quite understandable that researches move towardsthe use of such methods Airport managers usually facechanges on their plans and need to change their plans dueto uncertainties

With a lack of studies on robust methods or stochasticmethods where only few papers appeared on these subjectsresearchers have at their disposal a battery of methods suchas evolutionary methods parallel metaheuristics and self-tuning metaheuristics to apply on this interesting problem

With the actual trend of heuristics use it would beinteresting to work on a combination of these algorithms

In the framework of a hyperheuristic approach manyhard combinatorial optimization problems have already beentackled using heuristics (Burke et al [46]) which motivatesour recommendation of this approach Strengthening weak-nesses is the essence of hyperheuristics since they smartlywork with search spaces of heuristics The idea is during aprocess of exploring new solutions to choose the adequatemetaheuristic where the currently used one is failing toimprove or to generate new heuristics by using the compo-nents of existing ones (Soubeiga [47] Burke et al [48])


0 2 4 6 8 10 12 14 16 18Integer linear programming (IP)

Binary integer programmingMixed integer linear programming (MILP)

Mixed integer nonlinear programmingQuadratic assignment problem (QAP)

Scheduling problemsQuadratic mixed binary programming

Binary quadratic programmingClique partitioning problem (CPP)

Multiple objective GAP formulationsExact algorithms

Heuristic algorithmsMetaheuristic algorithms

OPL


Figure 3 Number of publications per research area

There is an alternative to these approximate methodswhich could be explored the use of efficient exact methods(branch and bound ) with the introduction of tight lowerbounds In our review we found that only Tang et al[27] developed a lower bound and used a classical branchand bound algorithm while other authors combined specialheuristics with their method (Bolat [30 34] Li [39] and Yanand Huo [2])

We have also noticed the absence of a data set for AGAP Itcould be interesting to have a set of instances of different sizesthat can be shared by researchers with benchmarks (optimaland best-known values) and CPU times to help comparingmethods as it is the case for known problems quadraticassignment problem and travelling salesman problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors extend their appreciation to the Deanship ofScientific Research Center of the College of Engineering atKing Saud University for supporting this work

References

[1] G N Steuart ldquoGate position requirements at metropolitanairportsrdquo Transportation Science vol 8 no 2 pp 169ndash189 1974

[2] S Yan and C-M Huo ldquoOptimization of multiple objective gateassignmentsrdquo Transportation Research A Policy and Practicevol 35 no 5 pp 413ndash432 2001

[3] A Drexl and Y Nikulin ldquoMulticriteria airport gate assignmentand Pareto simulated annealingrdquo IIE Transactions vol 40 no4 pp 385ndash397 2008

[4] W Li and X Xu ldquoOptimized assignment of airport gateconfiguration based on immune genetic algorithmrdquo in Mea-suring Technology and Mechatronics Automation in ElectricalEngineering pp 347ndash355 Springer 2012

[5] C Li ldquoAirport gate assignment A hybrid model and implemen-tationrdquo httparxivorgabs09032528

[6] H Ding A Lim B Rodrigues and Y Zhu ldquoNew heuristicsfor over-constrained flight to gate assignmentsrdquo Journal of theOperational Research Society vol 55 no 7 pp 760ndash768 2004

[7] H Ding A Lim B Rodrigues and Y Zhu ldquoThe over-constrained airport gate assignment problemrdquo Computers ampOperations Research vol 32 no 7 pp 1867ndash1880 2005

[8] UDorndorf F Jaehn and E Pesch ldquoFlight gate schedulingwithrespect to a reference schedulerdquo Annals of Operations Researchvol 194 pp 177ndash187 2012

[9] M Seker and N Noyan ldquoStochastic optimization models forthe airport gate assignment problemrdquo Transportation ResearchE Logistics and Transportation Review vol 48 no 2 pp 438ndash459 2012

[10] S Yan and C-H Tang ldquoA heuristic approach for airport gateassignments for stochastic flight delaysrdquo European Journal ofOperational Research vol 180 no 2 pp 547ndash567 2007

[11] R S Mangoubi and D F X Mathaisel ldquoOptimizing gateassignments at airport terminalsrdquo Transportation Science vol19 no 2 pp 173ndash188 1985

[12] R A Bihr ldquoA conceptual solution to the aircraft gate assignmentproblem using 0 1 linear programmingrdquo Computers and Indus-trial Engineering vol 19 no 1ndash4 pp 280ndash284 1990

[13] A Haghani and M-C Chen ldquoOptimizing gate assignmentsat airport terminalsrdquo Transportation Research A Policy andPractice vol 32 no 6 pp 437ndash454 1998

[14] J Xu and G Bailey ldquoThe airport gate assignment problemmathematical model and a tabu search algorithmrdquo in Proceed-ings of the IEEE 34th Annual Hawaii International ConferenceDelta Technology Inc Atlanta Ga USA 2001

[15] S Lam J Cao andH Fan ldquoDevelopment of an intelligent agentfor airport gate assignmentrdquo Journal of Airport Transportationvol 7 no 2 pp 103ndash114 2002

[16] D-X Wei and C-Y Liu ldquoFuzzy model and optimization forairport gate assignment problemrdquo in Proceedings of the IEEEInternational Conference on Intelligent Computing and IntelligentSystems (ICIS rsquo09) pp 828ndash832 Shanghai China November2009

[17] BACoEB Team and AICoE Team Gate Assignment Solu-tion (GAM) Using Hybrid Heuristics Algorithm Wipro Tech-nologies (WIT) 2009


[18] V Prem Kumar and M Bierlaire ldquoMulti-objective airport gateassignment problem in planning and operationsrdquo Journal ofAdvanced Transportation vol 48 no 7 pp 902ndash926 2014

[19] H S Al-Khalifah Dynamic Gate Assignment at an AirportDepartment of Industrial Engineering King Saud UniversityRiyadh Saudi Arabia 2005

[20] U Dorndorf A Drexl Y Nikulin and E Pesch ldquoFlight gatescheduling state-of-the-art and recent developmentsrdquo Omegavol 35 no 3 pp 326ndash334 2007

[21] T Obata Quadratic Assignment Problem Evaluation of ExactandHeuristic Algorithms Rensselaer Polytechnic Institute TroyNY USA 1979

[22] U Dorndorf F Jaehn C Lin H Ma and E Pesch ldquoDisruptionmanagement in flight gate schedulingrdquo Statistica NeerlandicaJournal of the Netherlands Society for Statistics and OperationsResearch vol 61 no 1 pp 92ndash114 2007

[23] C-H Cheng S C Ho and C-L Kwan ldquoThe use of meta-heuristics for airport gate assignmentrdquo Expert Systems withApplications vol 39 no 16 pp 12430ndash12437 2012

[24] A Lim B Rodrigues and Y Zhu ldquoAirport gate scheduling withtime windowsrdquo Artificial Intelligence Review vol 24 no 1 pp5ndash31 2005

[25] G Diepen J M van den Akker J A Hoogeveen and JW Smeltink Using Column Generation for Gate Planning atAmsterdam Airport Schiphol Department of Information andComputing Sciences Utrecht University 2007

[26] G Diepen J M V D Akker and J A Hoogeveen Integrationof Gate Assignment and Platform Bus Planning Department ofInformation andComputing SciencesUtrechtUniversity 2008

[27] C Tang S Yan andYHou ldquoA gate reassignment framework forreal time flight delaysrdquo 4OR A Quarterly Journal of OperationsResearch vol 8 no 3 pp 299ndash318 2010

[28] G Vanderstraeten and M Bergeron ldquoAutomatic assignmentof aircraft to gates at a terminalrdquo Computers and IndustrialEngineering vol 14 no 1 pp 15ndash25 1988

[29] S Yan C-Y Shieh and M Chen ldquoA simulation framework forevaluating airport gate assignmentsrdquo Transportation ResearchA Policy and Practice vol 36 no 10 pp 885ndash898 2002

[30] A Bolat ldquoAssigning arriving flights at an airport to the availablegatesrdquo Journal of the Operational Research Society vol 50 no 1pp 23ndash34 1999

[31] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001

[32] C Li ldquoAirport gate assignment new model and implementa-tionrdquo httparxivorgabs08111618

[33] P Zheng S Hu and C Zhang ldquoAirport gate assignmentsmodeland algorithmrdquo in Proceedings of the 3rd IEEE InternationalConference on Computer Science and Information Technology(ICCSIT rsquo10) vol 2 pp 457ndash461 IEEE Chengdu China July2010

[34] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000

[35] H Ding A Lim B Rodrigues and Y Zhu ldquoAircraft andgate scheduling optimization at airportsrdquo in Proceedings of theProceedings of the 37th Annual Hawaii International Conferenceon System Sciences pp 1185ndash1192 January 2004

[36] X B Hu and E Di Paolo ldquoAn efficient Genetic Algorithmwith uniform crossover for the multi-objective airport gateassignment problemrdquo in Proceedings of the IEEE Congress on

Evolutionary Computation (CEC rsquo07) pp 55ndash62 September2007

[37] I Kaliszewski and JMiroforidis ldquoOn interfacingmultiobjectiveoptimisation models the case of the airport gate assignmentproblemrdquo in Proceedings of the International Conference onApplication andTheory of Automation in Command and ControlSystems London UK 2012

[38] H M Genc O K Erol I Eksin M F Berber and B OGuleryuz ldquoA stochastic neighborhood search approach forairport gate assignment problemrdquo Expert Systems with Appli-cations vol 39 no 1 pp 316ndash327 2012

[39] W Li ldquoOptimized assignment of civil airport gaterdquo in Proceed-ings of the International Conference on Intelligent System Designand Engineering Application (ISDEA rsquo10) pp 33ndash38 ChangshaChina October 2010

[40] B Maharjan and T I Matis ldquoMulti-commodity flow networkmodel of the flight gate assignment problemrdquo Computers andIndustrial Engineering vol 63 no 4 pp 1135ndash1144 2012

[41] G Diepen J M van den Akker and J A a HoogeveenldquoFinding a robust assignment of flights to gates at AmsterdamAIRport Schipholrdquo Journal of Scheduling vol 15 no 6 pp 703ndash715 2012

[42] H M Genc ldquoA new solution approach for the airport gateassignment problem for handling of uneven gate demandsrdquoin Proceedings of the Word Conference of Transport Research(WCTR rsquo10) Lisbon Portugal 2010

[43] B G Thengvall J F Bard and G Yu ldquoA bundle algorithmapproach for the aircraft schedule recovery problem during hubclosuresrdquo Transportation Science vol 37 no 4 pp 392ndash4072003

[44] Y Gu and C A Chung ldquoGenetic algorithm approach toaircraft gate reassignment problemrdquo Journal of TransportationEngineering vol 125 no 5 pp 384ndash389 1999

[45] A Bouras M A Ghaleb U S Suryahatmaja and A M SalemldquoThe Airport Gate Assignment Problem (AGAP) as parallelmachine scheduling with eligibility metaheuristic approachesrdquoIndustrial Engineering Technical Report KSUENG-IED-TR-012014-01 Department of Industrial Engineering King SaudUniversity Riyadh Saudi Arabia 2014

[46] E K Burke M Gendreau M Hyde et al ldquoHyper-heuristics asurvey of the state of the artrdquo Journal of the Operational ResearchSociety vol 64 no 12 pp 1695ndash1724 2013

[47] E Soubeiga Development and Application of Hyperheuristics toPersonnel Scheduling University of Nottingham 2003

[48] E K Burke M Hyde G Kendall G Ochoa E Ozcan andR Qu ldquoA survey of hyper-heuristicsrdquo Tech Rep NOTTCS-TR-SUB-0906241418-2747 School of Computer Science andInformation Technology University of Nottingham 2009

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


The solution approaches and the solving techniques arevariedwith nomethods until nowadays that provide a robusttechnique for such problem This study focuses on assessingthe trend of solving gate assignment problem in light of thepreceding four points Specifically this study will address thefollowing research questions (1) Is this problemNP-hard (2)What formulation can be defined for such problem (3) Howeffective are the recent methods and techniques to solve theproblem (4) What recommendation can be made based onthe current findings with regard to research trends

From amathematical view AGAP has been formulated asinteger binary or mixed integer general linear or nonlinearmodels Specific formulation as binary or mixed binaryquadratic models has also been suggested Other well-knownrelated problems in combinatorial optimization such asquadratic assignment problem (QAP) clique partitioningproblem (CPP) and scheduling problem have been usedto formulate AGAP However few publications on AGAPtackled stochastic or robust optimization

While the goal of combinatorial optimization researchis to find an algorithm that guarantees an optimal solutionin polynomial time with respect to the problem size themain interest in practice is to find a nearly optimal or atleast good-quality solution in a reasonable amount of timeMany approaches to solve the GAP have been proposedvarying from Brand and Bound (BampB) to highly esotericoptimization methodsThemajority of these methods can bebroadly classified as either ldquoexactrdquo algorithms or ldquoheuristicrdquoalgorithms Exact algorithms are those that yield an optimalsolution As discussed in Section 31 different exact solutiontechniques have been used to solve the GAP and in someresearch the authors used some optimization programminglanguages like CPLEX and AMPL

Basically the GAP is a QAP and it is an NP-hardproblem as shown in Obata [21] Since the AGAP is NP-hard researchers have suggested various heuristic and meta-heuristics approaches for solving the GAP With heuristicalgorithms theoretically there is a chance to find an optimalsolution That chance can be remote because heuristics oftenreach a local optimal solution and get stuck at that point Butmetaheuristics or ldquomodern heuristicsrdquo introduce systematicrules to deal with this problem The systematic rules avoidlocal optima or give the ability of moving out of localoptima The common characteristic of these metaheuristicsis the use of some mechanisms to avoid local optimaMetaheuristics succeed in leaving the local optimum bytemporarily accepting moves that cause worsening of theobjective function value Sections 32 and 33 addressed theheuristic and metaheuristics approaches for solving the GAPSome papers presenting good overviews as well as annotatedbibliographies on the topic of GAP and a good literatureon the AGAP and the use of metaheuristics for AGAP areDorndorf et al [20 22] and Cheng et al [23]

This paper surveys a large number of models and tech-niques developed to deal with GAP In Section 2 we detailthe models formulations of the problem In Section 3 weaddressed the resolution methods used to solve the problemWe conclude in Section 4 and we represent the researchtrends

2 Formulations of AGAP andRelated Problems

Many researchers formulated the AGAP as an integer binaryor mixed integer linear or nonlinear model and some ofthem formulated it as binary or mixed binary quadraticmodels whereas some of the researchers have formulatedthe AGAP as well-known related problems in combinatorialoptimization such as quadratic assignment problem (QAP)clique partitioning problem (CPP) and scheduling problemor even as a network representation However some of theresearchers formulated the AGAP as a robust optimizationmodel In this section according to the way of how theresearchers deal with the gate assignment problem a classifi-cation for the AGAP has been made as follows

21 Selected AGAP Formulations

211 Integer Linear Programming Formulations (IP) Lim etal [24] formulated the AGAP as an integer programmingmodel and developed two models with time windows Thefirst model was devoted to minimization of the passengerwalking distance (travel time)

Minimize119899

sum119894=1

119899

sum119895=1

119898

sum119896=1

119898

sum119897=1

119891119894119895119908119896119897119911119894119895119896119897

+

119899

sum119894=1

119901119894(119888119894minus 119886

119894) (1)

while the second model optimized the gate assignments withcargo handling costs

Minimize119899

sum119894=1

119899

sum119895=1

119898

sum119896=1

119898

sum119897=1

[119891119860

119894119895(119888119896119897

+ 119888119860

) + 119891119861

119894119895(119888119896119897

+ 119888119861

)] 119911119894119895119896119897

+

119899

sum119894=1

119901119894(119888119894minus 119886

119894)

(2)

Both of these objectives put a penalty function due to a delayThese two objectives used the constraints as follows

119898

sum119896=1

119909119894119896

= 1 1 le 119894 le 119899 (3)

119911119894119895119896119897

le 119909119894119896 1 le 119894 119895 le 119899 1 le 119896 119897 le 119898 (4)

119911119894119895119896119897

le 119909119895119897 1 le 119894 119895 le 119899 1 le 119896 119897 le 119898 (5)

119909119894119896

+ 119909119895119897

minus 1 le 119911119894119895119896119897

1 le 119894 119895 le 119899 1 le 119896 119897 le 119898 (6)

119888119894ge 119886

119894 1 le 119894 le 119899 (7)

119888119894le 119887

119894minus 119889

119894 1 le 119894 le 119899 (8)

(119888119894+ 119889

119894) minus 119888

119894+ 119910

119894119895119872 gt 0 1 le 119894 119895 le 119899 (9)

(119888119894+ 119889

119894) minus 119888

119894minus (1 minus 119910

119894119895)119872 le 0 1 le 119894 119895 le 119899 (10)

119910119894119895

+ 119910119895119894

le 119911119894119895119896119897

1 le 119894 119895 le 119899 119894 = 119895 1 le 119896 le 119898 (11)

where 119909119894119896 119910

119894119895 and 119911

119894119895119896119897are binary and 119888

119894is integer




(119909119894119896



119894119895119896119897must

be equal to one if 119909119894119896

= 1 and 119909119895119897


119894119895= 1 =

1 if (119888119894+ 119889

119894) le 119888




119894119895= 0 if (119888

119894+119889

119894) gt 119888

119895 whichmeans

119910119894119895




Minimize119899

sum119895=1

119888119895119909119895 (12)

subject to

119899

sum119895=1

119892119894119895119909119895+ 119880

119894= 1 for 119894 = 1 119898 (13)

where

119892119894119895


0 otherwise

119880119894ge 0 for 119894 = 1 119898 119880


(14)



119886and 119894

119887that refer to


119899

sum119895=1

(119892119894119895

+ 119892119894119860119895

) 119909119895+ 119880

119894119860= 1

119899

sum119895=1

(119892119894119895

+ 119892119894119861119895

) 119909119895+ 119880

119894119861= 1

(15)

119899

sum119895=1

119905119895ℎ

119909119895= 119879

ℎfor ℎ = 1 119867 (16)

119909119895isin 0 1 for 119895 = 1 119899 (17)

where

119909119895=


119905119895ℎ


0 otherwise

(18)

119899

sum119895=1

119898

sum119894=1

119867

sumℎ=1

119901119894ℎ119903

119905119895ℎ

119892119894119895119909119895ge 119875

119903for 119903 = 1 119877 (19)

where

119901119894ℎ119903

=




0 otherwise(20)




ℎ and 120595


119888119895minus

119867

sumℎ=1

119905119895ℎ

120582ℎminus

119898

sum119894=1

(119892119894119895120587119894+

119877

sum119903=1

119867

sumℎ=1

(119892119894119895119905119895ℎ

119901119894ℎ119903

120595119903)) (21)









sum119895isin119864119894

sum119896isin119863119894119895

119889119894119896119909119894119895119896

+ sum119894isin119868

sum119895isin119864119894

sum119896isin119863119894119895

119908119894119895119909119894119895119896

(22)

subject to

sum119895isin119864119894

sum119896=119863119894119895

119909119894119895119896

= 1 forall119894 isin 119868 (23)

sum119894isin119865119895119904

sum119896=119867119894119904

119909119894119895119896


sum119894isin119871119905119902

sum119895isin119873119902

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896

=



0 otherwise


119894 forall119894 isin 119868

(26)

Parameter Variables


119894119896= 0


119894119895= 0












119909119894119896


0 otherwise

119910119894=


119908119894=



sum119896isin119870

119862119894119896119909119894119896

minus 1198621sum119905isin119871

119908119905minus 119862

2sum119894isin119879

119910119894

(27)

subject to

sum119896isin119870

119909119894119896

+ 119910119894= 1 119894 isin 119879 (28)

sum119896isin119870119890119894isin119864119896

119909119894119896

+ 119910119894= 1 119894 isin 119879 (29)

sum119894isin119879

119910119894le 120585 (30)

119910119894119896

+ 119910119895119896

le 1 119894 119895 isin 119879 119896 isin 119870119886119894lt 119887

119895+ 120572

119886119895lt 119887

119894+ 120572 119894 = 119895

(31)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119897 isin 119870

(119896 119897) isin 119869119886119894lt 119887

119895 119886

119895lt 119887

119894

119894 = 119895 119890119894isin 119864

1

119896 119890

119895isin 119864

1

119897

(32)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119896

119877

119871119894119865119900) isin 119871119865119886

119895le 119886

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(33)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119897119877

119871119894119865119900) isin 119871119865119886

119895le 119887

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(34)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119875119861

119897119875119861

) isin 119869119875119861

119887119894minus 120573 lt 119887

119895le 119887

119894+ 120573

119894 = 119895 119890119894isin 119864

119896 119890

119895isin 119864

119897

(35)


119908119905le 120591 119905 isin 119879

119871(36)

1199101198941119896

minus 1199101198942119896

le 119908119905 119894

1 1198942isin 119879 119905 isin 119879

119871

119896 isin 1198701198941

= 1198942 119901

1198941= 119901

1198942= 119905

(37)

1199101198941119896

minus 1199101198943119896

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

119896 isin 1198701198861198941

lt 1198861198943 119887

1198943lt 119886

1198942 119901

1198941= 119897 119901

1198942= 119897

(38)

11991011989411198961

minus 11991011989431198962

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

1198961 119896

2isin 119870 119895 isin 119869119886

1198941lt 119886

1198943 119887

1198943lt 119887

1198942 119901

1198941= 119897

1199011198942

= 119897 1198961= 119902

1

119895 119896

2= 119902

2

119895 119890

1198941isin 119864

1

119895 119890

1198942isin 119864

2

119895

(39)



Minimize 119885 =

119872

sum119894=1

119872

sum119894=1

(119901119886

119894119889119886

119895+ 119901

119889

119894119889119889

119895+ 119901

119905

119894119889119905

119895) 119909

119894119895 (40)

where

119909119894119895


0 otherwise(41)




119889119905

119895=

1

119873

119873

sum119896=1

119908119895119896

forall119895 = 1 119873 (42)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 = 1 119872 (43)

sumℎisin119871(119894)

119909ℎ119895

+ 119909119894119895

le 1 forall119894 = 1 119872 forall119895 = 1 119873 (44)

sumℎisin119871(119901+1)

119909ℎ119895

+ 119909119901+1119895

= 119909119901minus3119895

+ 119909119901minus2119895

+ 119909119901minus1119895

+ 119909119901119895

+ 119909119901+1119895

le 1

(45)


119873

sum119911=1

119873

sum119904=1

119909119892119911

119908119911119904119909ℎ119904

le 119863max119892ℎ

(47)

119901119886

119891119889119886

119904+ 119901

119889

119891119889119889

119904+ 119901

119905

119891119889119905

119904= min

119895isin119878

119901119886

119891119889119886

119895+ 119901

119889

119891119889119889

119895+ 119901

119905

119891119889119905

119895 (48)




sum119895isin119869

119881119894119895 (49)

subject to

sum119895isin119869119894

119881119894119895

le 1 119894 isin 119868 (50)

119881119894119895

+ sum119896isin119860119894119895

119881119896119895

le 1 119894 isin 119868 119895 isin 119869119894 (51)

119881119894119895

+ sum119896isin119861119894119895119903

119881119896119903

+ sum119896isin119861119894119895119897

119881119896119897

le 1 119894 isin 119868 119895 isin 119869119894 (52)

119881119894119895

= 0 1 119894 isin 119868 119895 isin 119869119894 (53)





Minimize 120579 = sum119862119894119895119883119894119895 119894 = 1 119898 119895 = 1 119896 (54)






119883119894119895

= 0 or 1

subject to

sum119883119894119895

= 1

sum119883119895119894

= 1

(55)


Minimize 119885 =

119872

sum119894=1

119873

sum119895=1

119888119894119895119909119894119895

(56)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 (57)

sum119894isin119871119904

119909119894119895


119909119894119895

= 0 or 1 forall119894 forall119895 (59)




subject to

V+ (Γ) = max V119894(Γ) 119894 = 1 119873 + 119872 (61)

Vminus (Γ) = min V119894(Γ) 119894 = 1 119873 + 119872 (62)

V119894(Γ) = 119860

119894minus 119864

119894Γ119894 119894 = 1 119873 (63)

119864119894Γ119894


119896= Γ

119894 119896 = 1 119894 minus 1


(64)

V119873+119895

(Γ) = 119879119895minus 119864

119873+1119895 119895 = 1 119872 (65)



119883119895119896


0 otherwise

119875119895119896

=


0 otherwise(66)


Model P2 Consider



subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)


Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)




sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)



sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=


0 otherwise(90)


sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119909119894119896



119894119895119896119897must

be equal to one if 119909119894119896

= 1 and 119909119895119897


119894119895= 1 =

1 if (119888119894+ 119889

119894) le 119888




119894119895= 0 if (119888

119894+119889

119894) gt 119888

119895 whichmeans

119910119894119895




Minimize119899

sum119895=1

119888119895119909119895 (12)

subject to

119899

sum119895=1

119892119894119895119909119895+ 119880

119894= 1 for 119894 = 1 119898 (13)

where

119892119894119895


0 otherwise

119880119894ge 0 for 119894 = 1 119898 119880


(14)



119886and 119894

119887that refer to


119899

sum119895=1

(119892119894119895

+ 119892119894119860119895

) 119909119895+ 119880

119894119860= 1

119899

sum119895=1

(119892119894119895

+ 119892119894119861119895

) 119909119895+ 119880

119894119861= 1

(15)

119899

sum119895=1

119905119895ℎ

119909119895= 119879

ℎfor ℎ = 1 119867 (16)

119909119895isin 0 1 for 119895 = 1 119899 (17)

where

119909119895=


119905119895ℎ


0 otherwise

(18)

119899

sum119895=1

119898

sum119894=1

119867

sumℎ=1

119901119894ℎ119903

119905119895ℎ

119892119894119895119909119895ge 119875

119903for 119903 = 1 119877 (19)

where

119901119894ℎ119903

=




0 otherwise(20)




ℎ and 120595


119888119895minus

119867

sumℎ=1

119905119895ℎ

120582ℎminus

119898

sum119894=1

(119892119894119895120587119894+

119877

sum119903=1

119867

sumℎ=1

(119892119894119895119905119895ℎ

119901119894ℎ119903

120595119903)) (21)









sum119895isin119864119894

sum119896isin119863119894119895

119889119894119896119909119894119895119896

+ sum119894isin119868

sum119895isin119864119894

sum119896isin119863119894119895

119908119894119895119909119894119895119896

(22)

subject to

sum119895isin119864119894

sum119896=119863119894119895

119909119894119895119896

= 1 forall119894 isin 119868 (23)

sum119894isin119865119895119904

sum119896=119867119894119904

119909119894119895119896


sum119894isin119871119905119902

sum119895isin119873119902

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896

=



0 otherwise


119894 forall119894 isin 119868

(26)

Parameter Variables


119894119896= 0


119894119895= 0












119909119894119896


0 otherwise

119910119894=


119908119894=



sum119896isin119870

119862119894119896119909119894119896

minus 1198621sum119905isin119871

119908119905minus 119862

2sum119894isin119879

119910119894

(27)

subject to

sum119896isin119870

119909119894119896

+ 119910119894= 1 119894 isin 119879 (28)

sum119896isin119870119890119894isin119864119896

119909119894119896

+ 119910119894= 1 119894 isin 119879 (29)

sum119894isin119879

119910119894le 120585 (30)

119910119894119896

+ 119910119895119896

le 1 119894 119895 isin 119879 119896 isin 119870119886119894lt 119887

119895+ 120572

119886119895lt 119887

119894+ 120572 119894 = 119895

(31)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119897 isin 119870

(119896 119897) isin 119869119886119894lt 119887

119895 119886

119895lt 119887

119894

119894 = 119895 119890119894isin 119864

1

119896 119890

119895isin 119864

1

119897

(32)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119896

119877

119871119894119865119900) isin 119871119865119886

119895le 119886

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(33)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119897119877

119871119894119865119900) isin 119871119865119886

119895le 119887

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(34)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119875119861

119897119875119861

) isin 119869119875119861

119887119894minus 120573 lt 119887

119895le 119887

119894+ 120573

119894 = 119895 119890119894isin 119864

119896 119890

119895isin 119864

119897

(35)


119908119905le 120591 119905 isin 119879

119871(36)

1199101198941119896

minus 1199101198942119896

le 119908119905 119894

1 1198942isin 119879 119905 isin 119879

119871

119896 isin 1198701198941

= 1198942 119901

1198941= 119901

1198942= 119905

(37)

1199101198941119896

minus 1199101198943119896

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

119896 isin 1198701198861198941

lt 1198861198943 119887

1198943lt 119886

1198942 119901

1198941= 119897 119901

1198942= 119897

(38)

11991011989411198961

minus 11991011989431198962

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

1198961 119896

2isin 119870 119895 isin 119869119886

1198941lt 119886

1198943 119887

1198943lt 119887

1198942 119901

1198941= 119897

1199011198942

= 119897 1198961= 119902

1

119895 119896

2= 119902

2

119895 119890

1198941isin 119864

1

119895 119890

1198942isin 119864

2

119895

(39)



Minimize 119885 =

119872

sum119894=1

119872

sum119894=1

(119901119886

119894119889119886

119895+ 119901

119889

119894119889119889

119895+ 119901

119905

119894119889119905

119895) 119909

119894119895 (40)

where

119909119894119895


0 otherwise(41)




119889119905

119895=

1

119873

119873

sum119896=1

119908119895119896

forall119895 = 1 119873 (42)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 = 1 119872 (43)

sumℎisin119871(119894)

119909ℎ119895

+ 119909119894119895

le 1 forall119894 = 1 119872 forall119895 = 1 119873 (44)

sumℎisin119871(119901+1)

119909ℎ119895

+ 119909119901+1119895

= 119909119901minus3119895

+ 119909119901minus2119895

+ 119909119901minus1119895

+ 119909119901119895

+ 119909119901+1119895

le 1

(45)


119873

sum119911=1

119873

sum119904=1

119909119892119911

119908119911119904119909ℎ119904

le 119863max119892ℎ

(47)

119901119886

119891119889119886

119904+ 119901

119889

119891119889119889

119904+ 119901

119905

119891119889119905

119904= min

119895isin119878

119901119886

119891119889119886

119895+ 119901

119889

119891119889119889

119895+ 119901

119905

119891119889119905

119895 (48)




sum119895isin119869

119881119894119895 (49)

subject to

sum119895isin119869119894

119881119894119895

le 1 119894 isin 119868 (50)

119881119894119895

+ sum119896isin119860119894119895

119881119896119895

le 1 119894 isin 119868 119895 isin 119869119894 (51)

119881119894119895

+ sum119896isin119861119894119895119903

119881119896119903

+ sum119896isin119861119894119895119897

119881119896119897

le 1 119894 isin 119868 119895 isin 119869119894 (52)

119881119894119895

= 0 1 119894 isin 119868 119895 isin 119869119894 (53)





Minimize 120579 = sum119862119894119895119883119894119895 119894 = 1 119898 119895 = 1 119896 (54)






119883119894119895

= 0 or 1

subject to

sum119883119894119895

= 1

sum119883119895119894

= 1

(55)


Minimize 119885 =

119872

sum119894=1

119873

sum119895=1

119888119894119895119909119894119895

(56)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 (57)

sum119894isin119871119904

119909119894119895


119909119894119895

= 0 or 1 forall119894 forall119895 (59)




subject to

V+ (Γ) = max V119894(Γ) 119894 = 1 119873 + 119872 (61)

Vminus (Γ) = min V119894(Γ) 119894 = 1 119873 + 119872 (62)

V119894(Γ) = 119860

119894minus 119864

119894Γ119894 119894 = 1 119873 (63)

119864119894Γ119894


119896= Γ

119894 119896 = 1 119894 minus 1


(64)

V119873+119895

(Γ) = 119879119895minus 119864

119873+1119895 119895 = 1 119872 (65)



119883119895119896


0 otherwise

119875119895119896

=


0 otherwise(66)


Model P2 Consider



subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)


Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)




sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)



sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=


0 otherwise(90)


sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sum119895isin119864119894

sum119896isin119863119894119895

119889119894119896119909119894119895119896

+ sum119894isin119868

sum119895isin119864119894

sum119896isin119863119894119895

119908119894119895119909119894119895119896

(22)

subject to

sum119895isin119864119894

sum119896=119863119894119895

119909119894119895119896

= 1 forall119894 isin 119868 (23)

sum119894isin119865119895119904

sum119896=119867119894119904

119909119894119895119896


sum119894isin119871119905119902

sum119895isin119873119902

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896

=



0 otherwise


119894 forall119894 isin 119868

(26)

Parameter Variables


119894119896= 0


119894119895= 0












119909119894119896


0 otherwise

119910119894=


119908119894=



sum119896isin119870

119862119894119896119909119894119896

minus 1198621sum119905isin119871

119908119905minus 119862

2sum119894isin119879

119910119894

(27)

subject to

sum119896isin119870

119909119894119896

+ 119910119894= 1 119894 isin 119879 (28)

sum119896isin119870119890119894isin119864119896

119909119894119896

+ 119910119894= 1 119894 isin 119879 (29)

sum119894isin119879

119910119894le 120585 (30)

119910119894119896

+ 119910119895119896

le 1 119894 119895 isin 119879 119896 isin 119870119886119894lt 119887

119895+ 120572

119886119895lt 119887

119894+ 120572 119894 = 119895

(31)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119897 isin 119870

(119896 119897) isin 119869119886119894lt 119887

119895 119886

119895lt 119887

119894

119894 = 119895 119890119894isin 119864

1

119896 119890

119895isin 119864

1

119897

(32)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119896

119877

119871119894119865119900) isin 119871119865119886

119895le 119886

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(33)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119865

119871119894119865119900 119897119877

119871119894119865119900) isin 119871119865119886

119895le 119887

119894le 119887

119895

119894 = 119895 119890119894isin 119864

119865

119896119871119894119865119900 119890

119895isin 119864

119877

119897119871119894119865119900

(34)

119910119894119896

+ 119910119895119897

le 1 119894 119895 isin 119879 119896 119897 isin 119870

(119896119875119861

119897119875119861

) isin 119869119875119861

119887119894minus 120573 lt 119887

119895le 119887

119894+ 120573

119894 = 119895 119890119894isin 119864

119896 119890

119895isin 119864

119897

(35)


119908119905le 120591 119905 isin 119879

119871(36)

1199101198941119896

minus 1199101198942119896

le 119908119905 119894

1 1198942isin 119879 119905 isin 119879

119871

119896 isin 1198701198941

= 1198942 119901

1198941= 119901

1198942= 119905

(37)

1199101198941119896

minus 1199101198943119896

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

119896 isin 1198701198861198941

lt 1198861198943 119887

1198943lt 119886

1198942 119901

1198941= 119897 119901

1198942= 119897

(38)

11991011989411198961

minus 11991011989431198962

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

1198961 119896

2isin 119870 119895 isin 119869119886

1198941lt 119886

1198943 119887

1198943lt 119887

1198942 119901

1198941= 119897

1199011198942

= 119897 1198961= 119902

1

119895 119896

2= 119902

2

119895 119890

1198941isin 119864

1

119895 119890

1198942isin 119864

2

119895

(39)



Minimize 119885 =

119872

sum119894=1

119872

sum119894=1

(119901119886

119894119889119886

119895+ 119901

119889

119894119889119889

119895+ 119901

119905

119894119889119905

119895) 119909

119894119895 (40)

where

119909119894119895


0 otherwise(41)




119889119905

119895=

1

119873

119873

sum119896=1

119908119895119896

forall119895 = 1 119873 (42)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 = 1 119872 (43)

sumℎisin119871(119894)

119909ℎ119895

+ 119909119894119895

le 1 forall119894 = 1 119872 forall119895 = 1 119873 (44)

sumℎisin119871(119901+1)

119909ℎ119895

+ 119909119901+1119895

= 119909119901minus3119895

+ 119909119901minus2119895

+ 119909119901minus1119895

+ 119909119901119895

+ 119909119901+1119895

le 1

(45)


119873

sum119911=1

119873

sum119904=1

119909119892119911

119908119911119904119909ℎ119904

le 119863max119892ℎ

(47)

119901119886

119891119889119886

119904+ 119901

119889

119891119889119889

119904+ 119901

119905

119891119889119905

119904= min

119895isin119878

119901119886

119891119889119886

119895+ 119901

119889

119891119889119889

119895+ 119901

119905

119891119889119905

119895 (48)




sum119895isin119869

119881119894119895 (49)

subject to

sum119895isin119869119894

119881119894119895

le 1 119894 isin 119868 (50)

119881119894119895

+ sum119896isin119860119894119895

119881119896119895

le 1 119894 isin 119868 119895 isin 119869119894 (51)

119881119894119895

+ sum119896isin119861119894119895119903

119881119896119903

+ sum119896isin119861119894119895119897

119881119896119897

le 1 119894 isin 119868 119895 isin 119869119894 (52)

119881119894119895

= 0 1 119894 isin 119868 119895 isin 119869119894 (53)





Minimize 120579 = sum119862119894119895119883119894119895 119894 = 1 119898 119895 = 1 119896 (54)






119883119894119895

= 0 or 1

subject to

sum119883119894119895

= 1

sum119883119895119894

= 1

(55)


Minimize 119885 =

119872

sum119894=1

119873

sum119895=1

119888119894119895119909119894119895

(56)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 (57)

sum119894isin119871119904

119909119894119895


119909119894119895

= 0 or 1 forall119894 forall119895 (59)




subject to

V+ (Γ) = max V119894(Γ) 119894 = 1 119873 + 119872 (61)

Vminus (Γ) = min V119894(Γ) 119894 = 1 119873 + 119872 (62)

V119894(Γ) = 119860

119894minus 119864

119894Γ119894 119894 = 1 119873 (63)

119864119894Γ119894


119896= Γ

119894 119896 = 1 119894 minus 1


(64)

V119873+119895

(Γ) = 119879119895minus 119864

119873+1119895 119895 = 1 119872 (65)



119883119895119896


0 otherwise

119875119895119896

=


0 otherwise(66)


Model P2 Consider



subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)


Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)




sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)



sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=


0 otherwise(90)


sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119908119905le 120591 119905 isin 119879

119871(36)

1199101198941119896

minus 1199101198942119896

le 119908119905 119894

1 1198942isin 119879 119905 isin 119879

119871

119896 isin 1198701198941

= 1198942 119901

1198941= 119901

1198942= 119905

(37)

1199101198941119896

minus 1199101198943119896

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

119896 isin 1198701198861198941

lt 1198861198943 119887

1198943lt 119886

1198942 119901

1198941= 119897 119901

1198942= 119897

(38)

11991011989411198961

minus 11991011989431198962

minus 1 le 119908119905 119894

1 1198942 1198943isin 119879 119905 isin 119879

119871

1198961 119896

2isin 119870 119895 isin 119869119886

1198941lt 119886

1198943 119887

1198943lt 119887

1198942 119901

1198941= 119897

1199011198942

= 119897 1198961= 119902

1

119895 119896

2= 119902

2

119895 119890

1198941isin 119864

1

119895 119890

1198942isin 119864

2

119895

(39)



Minimize 119885 =

119872

sum119894=1

119872

sum119894=1

(119901119886

119894119889119886

119895+ 119901

119889

119894119889119889

119895+ 119901

119905

119894119889119905

119895) 119909

119894119895 (40)

where

119909119894119895


0 otherwise(41)




119889119905

119895=

1

119873

119873

sum119896=1

119908119895119896

forall119895 = 1 119873 (42)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 = 1 119872 (43)

sumℎisin119871(119894)

119909ℎ119895

+ 119909119894119895

le 1 forall119894 = 1 119872 forall119895 = 1 119873 (44)

sumℎisin119871(119901+1)

119909ℎ119895

+ 119909119901+1119895

= 119909119901minus3119895

+ 119909119901minus2119895

+ 119909119901minus1119895

+ 119909119901119895

+ 119909119901+1119895

le 1

(45)


119873

sum119911=1

119873

sum119904=1

119909119892119911

119908119911119904119909ℎ119904

le 119863max119892ℎ

(47)

119901119886

119891119889119886

119904+ 119901

119889

119891119889119889

119904+ 119901

119905

119891119889119905

119904= min

119895isin119878

119901119886

119891119889119886

119895+ 119901

119889

119891119889119889

119895+ 119901

119905

119891119889119905

119895 (48)




sum119895isin119869

119881119894119895 (49)

subject to

sum119895isin119869119894

119881119894119895

le 1 119894 isin 119868 (50)

119881119894119895

+ sum119896isin119860119894119895

119881119896119895

le 1 119894 isin 119868 119895 isin 119869119894 (51)

119881119894119895

+ sum119896isin119861119894119895119903

119881119896119903

+ sum119896isin119861119894119895119897

119881119896119897

le 1 119894 isin 119868 119895 isin 119869119894 (52)

119881119894119895

= 0 1 119894 isin 119868 119895 isin 119869119894 (53)





Minimize 120579 = sum119862119894119895119883119894119895 119894 = 1 119898 119895 = 1 119896 (54)






119883119894119895

= 0 or 1

subject to

sum119883119894119895

= 1

sum119883119895119894

= 1

(55)


Minimize 119885 =

119872

sum119894=1

119873

sum119895=1

119888119894119895119909119894119895

(56)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 (57)

sum119894isin119871119904

119909119894119895


119909119894119895

= 0 or 1 forall119894 forall119895 (59)




subject to

V+ (Γ) = max V119894(Γ) 119894 = 1 119873 + 119872 (61)

Vminus (Γ) = min V119894(Γ) 119894 = 1 119873 + 119872 (62)

V119894(Γ) = 119860

119894minus 119864

119894Γ119894 119894 = 1 119873 (63)

119864119894Γ119894


119896= Γ

119894 119896 = 1 119894 minus 1


(64)

V119873+119895

(Γ) = 119879119895minus 119864

119873+1119895 119895 = 1 119872 (65)



119883119895119896


0 otherwise

119875119895119896

=


0 otherwise(66)


Model P2 Consider



subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)


Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)




sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)



sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=


0 otherwise(90)


sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Minimize 120579 = sum119862119894119895119883119894119895 119894 = 1 119898 119895 = 1 119896 (54)






119883119894119895

= 0 or 1

subject to

sum119883119894119895

= 1

sum119883119895119894

= 1

(55)


Minimize 119885 =

119872

sum119894=1

119873

sum119895=1

119888119894119895119909119894119895

(56)

subject to

119873

sum119895=1

119909119894119895

= 1 forall119894 (57)

sum119894isin119871119904

119909119894119895


119909119894119895

= 0 or 1 forall119894 forall119895 (59)




subject to

V+ (Γ) = max V119894(Γ) 119894 = 1 119873 + 119872 (61)

Vminus (Γ) = min V119894(Γ) 119894 = 1 119873 + 119872 (62)

V119894(Γ) = 119860

119894minus 119864

119894Γ119894 119894 = 1 119873 (63)

119864119894Γ119894


119896= Γ

119894 119896 = 1 119894 minus 1


(64)

V119873+119895

(Γ) = 119879119895minus 119864

119873+1119895 119895 = 1 119872 (65)



119883119895119896


0 otherwise

119875119895119896

=


0 otherwise(66)


Model P2 Consider



subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)


Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)




sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)



sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=


0 otherwise(90)


sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










subject to

119878max ge 119878119895119896

119895 = 1 119873 119896 = 1 119872 (68)

119878min le 119878119895119896

+ (1 minus 119883119895119896

)119885 119895 = 1 119873 119896 = 1 119872

(69)

119878max 119878min ge 0 (70)

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (71)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (72a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(72b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (73)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (74a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873

119896 = 1 119872(74b)

119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (75)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (76)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(77)


Model P4 Consider


subject to

119878max ge 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (79)

119878min le 119868119894119895119884119894119895

119894 = 0 119873 119895 = 119894 + 1 119873 + 1 (80)

119878max 119878min ge 0 (81)

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (82)

119873+1

sum119895=1

1198840119895

le 119872 (83)

119873

sum119894=0

119884119894119873+1

le 119872 (84)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (85)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (86)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (87)




sum119904isin119878

119877119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (88)



sum119904isin119878

119867119894119904119901119904+ Λ sum

119894isin119873119888

sum119895isin119873119888

sum119904isin119878

119888119894119895119904

119901119904 (89)

where

119867119894119904

=


0 otherwise(90)


sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (91)

1199090119896

= 1 119896 isin 119872 (92)

119909119899+119896119896

= 1 119896 isin 119872 (93)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119888

119896 isin 119872 119904 = 0 (94)


119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119888119894119895119904

ge 119909119894119896

+ 119909119895119896

minus 1 119894 isin 119873119888

119896 isin 119872 119904 isin 119878 119895 isin 119871119894119904

(95)

119877119894119904

ge 119887 minus 119868119894119904 119894 isin 119873

119888

0 119904 isin 119878 (96)

119863119894119904

ge (119909119894119896

+ 119909119895119896

minus 2)119885 + 119889119895119904

119894 isin 119873119888

0 119896 isin 119872

119904 isin 119878 119895 isin 119876119894119904

(97)

119868119894119904

le 119886119894119904

minus 119863119894119904

+ 119887 sum119895isin119873119888

119888119894119895119904

119894 isin 119873119888

0 119904 isin 119878 (98)

119909119894119896

isin 0 119894 isin 119873119888

119896 isin 119872 (99)



119867119894119904119887 ge 119887 minus 119868

119894119904 119894 isin 119873

119888

0 119904 isin 119878 (101)

119867119894119904

isin 0 1 119894 isin 119873119888

0 119904 isin 119878 (102)



sum119895lt119894119895isin119873

119910119894119895

lowast 119864 (119901 (119894 119895)) (103)

where

119864 (119901 (119894 119895)) =1

119886119894minus 119889

119895+ 2119887

(104)


119909119894119896



119896


119910119894119895

= 1 if exist119896 119909

119894119896= 119909

119895119896= 1 (1 le 119896 le 119888)

0 otherwise (1 le 119894 119895 le 119899)

(105)



119910119894119895

119886119894minus 119889

119895+ 2119887

(106)


sum119894isin119873

sum119896isin119862

119909119894119896

= 1 (107)

sum119894isin119873

sum119895lt119894119895isin119873

sum119896isin119862

(119909119894119896

lowast 119909119895119896

) = 119910119894119895

(108)

119910119894119896

lowast 119910119895119896

lowast (119889119894minus 119886

119895) lowast (119889

119895minus 119886

119894) le 0 (109)

119909119894119896

isin 0 1 (110)



119894119895from 119909

119894119896 Constraint (109)



119883119895119896


0 otherwise

119875119895119896

=



(112)

Model P1 Consider

Minimize119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(113)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (114)

1198641119896

= Maximize 11986011198831119896

119861119896 119896 = 1 119872 (115a)

119864119895119896

= Maximize 119860119895119883119895119896

119871119895minus1119896

119895 = 2 119873

119896 = 1 119872

(115b)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (116)

1198781119896

= 1198641119896

minus 119861119896

119896 = 1 119872 (117a)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 2 119873 119896 = 1 119872 (117b)


119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119878119873+1119896

= 119865119896minus 119871

119873119896119896 = 1 119872 (118)

119883119895119896

= 0 or 1 119895 = 1 119873 119896 = 1 119872 (119)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873 119896 = 1 119872

(120)


119885 = max119896=1119872

119865119896minus 119861

119896 (121)

119868119894119895

= 119860119895minus 119863

119894 if 119860

119895ge 119863

119894


1198680119895

= 119860119895

119895 = 1 119873 (123)

119868119894119873+1

= 119867 minus 119863119894

119894 = 1 119873 (124)

Model P3 Consider

Minimize119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119884119894119895

(125)

subject to119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895

= 119873 + 119872 (126)

119873+1

sum119895=1

1198840119895

le 119872 (127)

119873

sum119894=0

119884119894119873+1

le 119872 (128)

119895minus1

sum119894=0

119884119894119895

= 1 119895 = 1 119873 (129)

119873+1

sum119895=119894+1

119884119894119895

= 1 119894 = 1 119873 (130)

119884119894119895

= 0 or 1 119894 = 0 1 119873 119895 = 119894 + 1 119873 + 1 (131)


119868119894119895119896

=

119860119895minus 119863

119894 if 119860

119894gt 119861

119896 119860

119895ge 119863

119894

119875119894119896

= 119875119895119896

= 1

119885 otherwise

(132)

1198680119895119896

= 119860119895minus 119861

119896 if 119860

119895ge 119861

119896


119868119894119873+1119896

= 119865119896minus 119863

119894 if 119863

119894le 119865

119896


Model P5 Consider

Minimize119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

1198682

119894119895119896119884119894119895119896

(135)

subject to

119872

sum119896=1

119873

sum119894=0

119873+1

sum119895=119894+1

119884119894119895119896

= 119873 + 119872 (136)

119873+1

sum119895=1

1198840119895119896

le 1 119896 = 1 119872 (137)

119872

sum119896=1

119895minus1

sum119894=0

119884119894119895119896

= 1 119895 = 1 119873 (138)

119872

sum119896=1

119873+1

sum119895=119894+1

119884119894119895119896

= 1 119895 = 1 119873 (139)

119884119894119895119896

+

119872

sum119905=1

119905 =119896

119873+1

sumV=119895+1

119884119895V119905 le 1 119894 = 0 119873 minus 1

119895 = 119894 + 1 119873 119896 = 1 119872

(140)

119884119894119895119896

= 0 or 1 119894 = 0 119873 119895 = 119894 + 1 119873 + 1

119896 = 1 119872(141)

119884119886119887119888

+

119872

sum119905=1

119905 =119888

119873+1

sumV=119887+1

119884119887V119905 le 1 (142)




Minimize 119891 =

119873+1

sum119894=1

119872

sum119896=1

(119877119894119896

minus 119877)2

(143)


subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










subject to

sum119896isin119872

119910119894119896

= 1 (144)

119910119894119896


119911119894119895119896

(145)

119910119895119896


119911119894119895119896

(146)

119877119895119896

= 119886119895minus 119878119879

119896 119895 = min (119895) 119895 isin (119895 | 119910

119895119896= 1) (147)

119877119873+1119896

= 119864119879119896minus 119889

119894 119894 = max (119894) 119894 isin (119894 | 119910

119895119896= 1) (148)

119877119894119896

= 119886119895minus 119889

119894 forall (119894 119895 119896) isin ((119894 119895 119896) | 119911

119894119895119896= 1) (149)

119886119895+ (1 minus 119911

119894119895119896) 119878 ge 119889

119894+ 119868 (150)

V119894le 119890

119896+ (1 minus 119910

119894119896) 119878 (151)

119910119894119896


0 otherwise(152)

119911119894119895119896

=


0 otherwise(153)




minimize 119885 =

119872

sum119896=1

119873+1

sum119895=1

1198782

119895119896(154)

subject to

119872

sum119896=1

119875119895119896

119883119895119896

= 1 119895 = 1 119873 (155)

119864119895119896

ge 119860119895119883119895119896

119895 = 1 119873 119896 = 1 119872 (156)

119864119895119896

ge 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (157)

119871119895119896

= 119864119895119896

+ 119866119895119883119895119896

119895 = 1 119873 119896 = 1 119872

(158)

119878119895119896

= 119864119895119896

minus 119871119895minus1119896

119895 = 1 119873 119896 = 1 119872 (159)

119878119873+1119896

= 119871119873+1119896

minus 119871119873119896

119896 = 1 119872 (160)

119883119895119896


0 otherwise

119895 = 1 119873 119896 = 1 119872

(161)

119864119895119896

119871119895119896

119878119895119896

119878119873+1119896

ge 0 119895 = 1 119873

119896 = 1 119872(162)



Model 1 Consider


sum119896119897isin119870

119891119894119895119888119896119897119910119894119896119910119895119897 (163)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (164)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (165)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (166)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (167)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (168)


119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(169)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (170)

119910119894119896


0 otherwise(171)

119911119894119895119896

=



0 otherwise(172)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (173)

119905119894ge 0 forall

119894isin 119873 (174)



119905119894and 119905


119894119895119896= 1 It stipulates


Model 2 Consider

119909119894119895119896119897

=



0 otherwise(175)


sum119896119897isin119870

119891119894119895119888119896119897119909119894119895119896119897 (176)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894isin 119873 (177)

119910119894119896

= sum119895isin119873

119911119894119895119896


119896isin 119870 (178)

119910119895119896

= sum119894isin119873

119911119894119895119896


119896isin 119870 (179)

119905119894ge 119886

119894+ prop forall

119894isin 119873 (180)

119905119894le 119889

119894minus 120579

119894sum119895isin119873

119891119895119894 forall

119894isin 119873 (181)

119905119894+ 120579

119894sum119895isin119873

119891119895119894

le 119905119895+ (1 minus 119911

119894119895119896)119872 forall

119894119895isin 119873 forall

119896isin 119870

(182)

119886119895+ (1 minus 119911

119894119895119896)119872 ge 119889

119894+ 120573 forall

119894119895isin 119873 forall

119896isin 119870 (183)

119911119894119895119896

119911119894119895119896

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (184)

119905119894ge 0 forall

119894isin 119873 (185)

119909119894119895119896119897

le 119910119894119896 forall

119894119895isin 119873 forall

119896119897isin 119870 (186)

119909119894119895119896119897

le 119910119895119897 forall

119894119895isin 119873 forall

119896119897isin 119870 (187)

119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897

forall119894119895

isin 119873 forall119896119897

isin 119870 (188)

119909119894119895119896119897

isin 0 1 forall119894119895

isin 119873 forall119896119897

isin 119870 (189)




119895119897= 1) Constraint (188)



119894119896= 1 and 119910

119895119897= 1



Minimize119899

sum119898+1

119910119894119898+1

(190)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(191)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (192)

119886119894lt 119889

119894(forall119894 1 le 119894 le 119899) (193)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(194)


119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(195)




Minimize119899

sum119898+1

119910119894119898+1

(197)

Minimize119899

sum119894=1

119899

sum119895=1

119898+1

sum119894=1

119891119894119895

119908119896119897

119910119894119896

119910119895119897

+

119899

sum119894=1

1198910119894

1199080119894

+

119899

sum119894=1

1198911198940

1199081198940

(198)

subject to

119898+1

sum119896=1

119910119894119896

= 1 (forall119894 1 le 119894 le 119899) (199)

119910119894119896

119910119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0

(forall119894 1 le 119894 119895 le 119899 119896 = 119898 + 1)

(200)

119910119894119896

=


(0 lt 119896 le 119898 + 1)

0 otherwise(201)





Minimize1198761 119876119873119866

119869MOGAP (203)

subject to

119873119866

sum119892=1

119867119892

= 119873119860119862

(204)

119864119876119892(119895)

=

119875119876119892(119895)

119895 = 1

max (119875119876119892(119895)

119864119876119892(119895minus1)

119866119876119892(119895minus1)

) 119895 gt 1

119895 = 1 119867119892 119892 = 1 119873

119866

(205)

119882119894= 119864

119894minus 119875

119894 119894 = 1 119873

119860119862 (206)

119869TPWD =

119873119866+1

sum119892=1

119867119892

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PWD (119892 V

119894) (207)

119869TBTD =

119873119866+1

sum119892=1

119883119899

sum119895=1

119873119860119862+1

sum119894=1

119872119875(119876

119892(119895) 119894)119872PTD (119892 V

119894) (208)

119869TPWT =

119873119860119862

sum119892=1

119882119894

119873119860119862+1

sum119894=1

(119872119875(119894 119895) + 119872

119875(119895 119894)) (209)


120572 + 120573 le 1 0 le 120572 le 1 0 le 120573 le 1 (211)


minimize 1198851=

119873

sum119894=1

119873

sum119895=1

119872+1

sum119896=1

119872+1

sum119897=1

119891119894119895119879119896119897119910119894119896119910119894119897 (212)

minimize 1198852=

119873

sum119894=1

1198782

119894+

119872

sum119896=1

1198782

119896(213)

subject to

119872

sum119896=1

119910119894119896

= 1 forall119894 isin 119873 (214)

(119889119895minus 119886

119894+ 120572) (119889

119894minus 119886

119895+ 120572) 119910

119894119896119910119895119896

le 0

forall119894 119895 = 1 119873 forall119896 = 1 119872

(215)

119910119894119896



119894119896is 0-1 variable 119910

119894119896= 1 if flight








Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Minimize 1198851=

119872

sum119894=1

119873

sum119895=1

119871 119894

sum119896=119861119894

(119888119894119895

minus 119909119894119895119896

) (217a)

Minimize 1198852=

119872

sum119894=1

(119875119894(

119873

sum119895=1

119871 119894

sum119896=119861119894

119896119909119894119895119896

minus 119861119894)) (217b)

subject to

119873

sum119895=1

119871 119894

sum119896=119861119894

119909119894119895119896

= 1 forall119894 (217c)

sum119894isin119865119895

sum119896isin119867119894119904

119909119894119895119896


119909119894119895119896







119910119894(119898+1)

(218)


sum119896119897isin119870

119891119894119895119888119888119897119910119894119896119910119895119897 (219)

subject to

sum119896isin119870

119910119894119896

= 1 forall119894 isin 119873 (220)

119910119894119896


119910119894119896


119894 forall119897 isin 119870

119896 (222)

(119910119894119896


119894 forall119897 isin 119870

119896

(223)

119910119894119896

= sum119895isin119873

119911119894119895119896


119910119894119896

= sum119894isin119873

119911119894119895119896


119905119894ge 119886

119894+ 120572 forall119894 isin 119873 (226)

119905119894le 119889

119894minus 120579

119894lowast sum

119895isin119873

119891119895119894 forall119894 isin 119873 (227)

(119905119894+ 120579

119894lowast sum

119895isin119873

119891119895119894) le (119905

119895+ (1 minus 119911


(228)

(119886119895+ (1 minus 119911

119894119895119896) lowast 119872) ge (119889


(229)

119910119894119896

119911119894119895119896


119905119894ge 0 forall119894 isin 119873 (231)






sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sum119896119897isin119870

119891119894119895119888119888119897119909119894119895119896119897 (232)


119909119894119895119896119897


119909119894119895119896119897


119910119894119896

+ 119910119895119897

minus 1 le 119909119894119895119896119897





119894119895119896119897


minimize 1198911(119909)

= 120575max119895

(

119898

sum119894=1

119909119886119895+1

119895119894+ 2

119898

sum119894=1

119909119886119895+2

119895119894+ sdot sdot sdot + 120572

119898

sum119894=1

119909120572

119895119894)

(237)

minimize 1198912(119910) =

119899

sum119895=1

119910119895 (238)

subject to

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 for 119895 = 1 119899 (239)

119909119905

119895119894le 119910

119906

119894for 119906 = 119905 119905 + 1 119905 + 119892

119895 (240)

119899

sum119895=1

119909119905

119895119894le 1 119905 = 1 Δ 119894 = 1 119898 (241)

119898

sum119894=1

Δ

sum119905=119886119895

119909119905

119895119894le 1 minus 119910

119895 for 119895 = 1 119899 (242)




sum

119894119895isin119860119896

119888119896

119894119895119909119896

119894119895

+ sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895+ 119908sum

119904isinΩ

119901119904

ℎ119904

(243)

subject to

sum

119895isin119873119896

119909119896

119894119895minus sum

119903isin119873119896

119909119896

119903119894= 0 forall119894 isin 119873

119896

forall119896 isin 119870 (244)

sum119896isin119870

sum119894119895isin119865119905

119909119896

119894119895= 1 forall119905 isin 119860119865 (245)

0 le 119909119896

119894119895le 119892

119896

forall (119894 119895) isin 119862119860119896

forall119896 isin 119870 (246)

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895minus ℎ

119904

le sum119904isinΩ

119901119904

sum119896isin119870

sum

119894119895isin119876119860119896

119906119904119896

119894119895119909119896

119894119895forall119904 isin Ω

(247)

ℎ119904


119909119896

119894119895isin 119885

+forall (119894 119895) isin 119862119860

119896

forall119896 isin 119870 (249)

119909119896

119894119895= 0 1 forall (119894 119895) isin 119860

119896

minus 119862119860119896

forall119896 isin 119870 (250)



119911119903119904

= 119908119905119904

+ 119894119905119904

(251)

119885119877 = 119860119882119879 + 119860119868119879 + 119908 times 119878119877119872119877

= sum119904=Ω

119901119904

119908119905119904

+ sum119904=Ω

119901119904

119894119905119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

= sum119904=Ω

119901119904

119911119903119904

+ 119908sum119904=Ω

119901119904

(max(0 119911119903119904

minus sum119904=Ω

119901119904

119911119903119904

))

(252)



Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Method 1 Consider

(119906119904119896

119894119895)119898+1

= (119906119904119896

119894119895)119898

+ (119889119908119904119896

119894119895+ 119889119894

119904119896

119894119895)119898

(253)

Method 2 Consider

(119889V119904119896119894119895

)119898

= 1 if 119889119908

119904119896

119894119895gt 0 or 119889119894

119904119896

119894119895gt 0

0 if 119889119908119904119896

119894119895= 0 119889119894

119904119896

119894119895= 0

forall (119894 119895) isin 119876119860119896


(254)

119887119898

= max

0minussum (119889V119904119896

119894119895)119898

(119889119904119896

119894119895)119898minus1

100381710038171003817100381710038171003817(119889119904119896

119894119895)119898minus1100381710038171003817100381710038171003817

2

(255)

(119889119904119896

119894119895)119898

= (119889V119904119896119894119895

)119898

+ 119887119898

(119889119904119896

119894119895)119898minus1

forall (119894 119895) isin 119876119860119896


(256)

119905119898

=120582

1003816100381610038161003816119885119877119898 minus 119885119875119898100381610038161003816100381610038171003817100381710038171003817119889V119904119896

119894119895

10038171003817100381710038171003817

2 0 lt 120582 le 2 (257)

(119906119904119896

119894119895)1198981

= (119906119904119896

119894119895)119898

+ 119905119898

(119889119904119896

119894119895)119898

forall (119894 119895) isin 119876119860119896


(258)



119865fitness =

119873119892

sum119896=1

119873119905

sum119897=1

any (119872119888(119896 119897)) (259)

subject to

any (119872119888(119896 119897)) =

1 if 119872119888(119896 119897) = 0

0 otherwise(260)



119881119904119901119904+ Λ sum

119894isin119873119887

sum119896isin119872

sum119904isin119878

(119888119901

119894119896119904+ 119888

119891

119894119896119904) 119901

119904 (261)

subject to

sum119896isin119872

119909119894119896

= 1 119894 isin 119873 (262)

1199090119896

= 1 119896 isin 119872 (263)

119909119899+119896119896

= 1 119896 isin 119872 (264)

sum119895isin119871 119894119904

119909119895119896

+ 119909119894119896

le 1 119894 isin 119873119887

119896 isin 119872 119904 = 0 (265)

119888119901

119894119896119904ge sum

119895isin119871119901

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(266)

119888119891

119894119896119904ge sum

119895isin119871119891

119894119904

119909119895119896

+ (119909119894119896

minus 1) (119899 + 119898)

119894 isin 119873119887

119896 isin 119872 119904 = 119878

(267)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 1198711015840

119894119904 119896 isin 119872 119904 = 119878

(268)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(269)

119860119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 + 119886119895119904

119894 isin 119873119887

0 119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(270)

sum

119894isin1198731198870

119860119894119904

= sum119894isin119873

119886119894119904

+ 1198860119904

119898 119904 = 119878 (271)

119868119894119904

= 119860119894119904

minus 119889119894119904

+ 119904119901

119894119904+ 119904

119891

119894119904 119894 isin 119873

119887

0 119904 = 119878 (272)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885

119894 isin 119873119887

0 119895 isin 119871119901

119894119904 119896 isin 119872 119904 = 119878

(273)

119868119894119904

le (2 minus 119909119894119896

minus 119909119895119896

)119885 119894 isin 119873119887

0

119895 isin 119871119891

119894119904 119896 isin 119872 119904 = 119878

(274)

119904119901

119894119904le 119885 sum

119896isin119872

119888119901

119894119896119904 119894 isin 119873

119887

119904 = 119878 (275)

119904119891

119894119904le 119885 sum

119896isin119872

119888119891

119894119896119904 119894 isin 119873

119887

119904 = 119878 (276)

119909119894119896

isin 0 1 119894 isin 119873119887

119896 isin 119872 (277)






min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












min 1199111=

119899

sum119894=1

120587119894119898+1

(279a)

min 1199112=

119899

sum119894=1

119899

sum119895=1

119898+1

sum119896=1

119898+1

sum119897=1

119891119894119895

119908119896119897

120587119894119896

120587119895119897

+

119899

sum119894=1

119898+1

sum119896=1

1198910119894

1199080119896

120587119894119896

+

119899

sum119894=1

119898+1

sum119896=1

1198911198940

1199081198960

120587119894119896

(279b)

min 1199113=

119899

sum119894=1

119898+1

sum119896=1

V119894119906119894119896

120587119894119896

(279c)

subject to

119898+1

sum119896=1

120587119894119896

= 1 1 le 119894 le 119899 (280)

120587119894119896

120587119895119896

(119889119895minus 119886

119894) (119889

119894minus 119886

119895) le 0 1 le 119894 119895 le 119899 119896 = 119898 + 1

(281)

120587119894119896

isin 0 1 1 le 119894 le 119899 1 le 119896 le 119898 + 1 (282)




Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840

+ sum119894

sum

1198941015840

(11987511989401198631198950

+ 11987501198941198630119895

)119883119894119895119905

(283)

1198751198941198941015840119863

1198951198951015840 larr997888 119875

11989401198631198950

+ 11987501198941198630119895

forall119894 119895 (284)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119883

119894119895119905119883119894101584011989510158401199051015840 (285)


11988411989411989511989410158401198951015840 = 119883

119894119895119905119883119894101584011989510158401199051015840 (286)

11988411989411989511989410158401198951015840 =



0 otherwise(287)

Minimize sum119894

sum

1198941015840

sum119895

sum

1198951015840

1198751198941198941015840119863

1198951198951015840119884

11989411989511989410158401198951015840 (288)

subject to

sum119895

119883119894119895119905

= 1 forall119894 119905119886

119894le 119905 le 119905

119889

119894 (289)

sum119894

119883119894119895119905

le 1 forall119895 119905 (290)

119883119894119895119905

le 119883119894119895(119905+1)

forall119905 119905119886

119894le 119905 le 119905

119889

119894minus 1 (291)

119883119886

119894119895119905119894

+ 119883119886

1198941198951199051198941015840minus 2119884

119886

11989411989511989410158401198951015840 ge 0 forall119894 lt 119894

1015840

119895 1198951015840

(292)

sum119895

sum

1198951015840

11988411989411989511989410158401198951015840 = 1 forall119894 = 119894

1015840

(293)







Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Minimize119891|119891119873rarr119872

12057211199111(119891) + 120572

21199112(119891) + 120572

31199113(119891) (294)

subject to

119891 (119894) isin 119872 (119894) forall119894 isin 119873

119891 (119894) = 119891 (119895) forall119905119894119895

lt 0

119891 (119894) = 119899 + 119898 forall119894 119895 isin 119873

119891 (119894) = 119896 or 119891 (119894) = 119897 forall (119894 119896 119895 119897) isin 119878

(295)

1199111= minus

119899

sum119894=1

119901lowast

119894119891(119894)

1199112=

1003816100381610038161003816119894 isin 119873 | 119880 (119894) = 0 and 119891 (119894) = 119891 (119880 (119894))1003816100381610038161003816

1199113= sum

(119894119895)|119894lt119895119891(119894)=119891(119895) =119899+119898

max 119905max minus 119905119894119895 0

(296)



119908119894119895119909119894119895 (297)

subject to

119909119894119895

+ 119909119895119896

minus 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

minus 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

minus 119909119894119895

+ 119909119895119896

+ 119909119894119896

le 1 for 1 le 119894 lt 119895 lt 119896 le 119886

119909119894119895

isin 0 1 for 1 le 119894 lt 119895 le 119886

(298)

119881 = 1 2 119899 + 119898 minus 1 (299)

119908119894119895

=

minusinfin if 119905119894119895

lt 0

1205722

if 119905119894119895

ge 0

and (119880 (119894) = 119895 or 119880 (119895) = 119894)


119894119895 0 if 119905

119894119895ge 0 and 119880 (119894) = 119895

and 119880 (119895) = 119894 forall119894 119895 lt 119899

(300)

119908119894119895

=



119894119895if 119895 isin 119872 (119894)

forall119894 le 119899 119895 gt 119899

(301)

119908119894119895




119883119896


119883119896




119883119896



sum119896isin119870

(119862119896+

2119891119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(303)


subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










subject to

sum119894

119883119896

119904119894+ 119883

119896

119878119879= 1 forall119896 isin 119870 119894 isin 119865 (304)

119883119896

119904119894+ sum

119894

ln119894119895119883119896

119895119894= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(305)

sum119894

ln119894119895119883119896

119895119894+ 119883

119896

119895119879= 119883

119896

119898119899forall119894 119898 isin 119860 119895 119899 isin 119863

119894 = 119895 119898 = 119899

(306)

sum119894

119883119896

119895119879+ sum

119894

119883119896

119878119879= 1 119894 isin 119865 119896 isin 119870 (307)

sum119894

119883119896

119894119895= 1 forall119894 isin 119860 119895 isin 119863 119894 = 119895 (308)

119883119896

119904119894 119883

119896

119895119894 119883

119896

119898119899 119883

119896

119878119879 119883

119896

119895119879= 0 1 (309)

ln119894119895

=

1 if (119860119894minus 119863

119895) ge 119901 forall119894 = 119895

0 otherwise(310)




(119894isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=1198951198941015840=1198951015840 119894 =1198941015840119895 =1198951015840)

sum119896isin119870

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840 (311)


1198941198951198831198961015840

11989410158401198951015840) by a new variable 119884119896119896

1015840


1198841198961198961015840

1198941198941015840 =

1 if (119883119896

119894119895= 1 119883119896

1015840

11989410158401198951015840 = 1 forall119894 = 119895

1198941015840

= 1198951015840

119894 = 1198941015840

119895 = 1198951015840

119896 = 1198961015840

)

0 otherwise

(312)

subject to

1198841198961198961015840

1198941198941015840 minus 119883

119896

119894119895le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(313)

1198841198961198961015840

1198941198941015840 minus 119883

1198961015840

11989410158401198951015840 le 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(314)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 le 1 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(315)

119883119896

119894119895+ 119883

1198961015840

11989410158401198951015840 minus 119884

1198961198961015840

1198941198941015840 ge 0 forall119894 = 119895 119894

1015840

= 1198951015840

119894 = 1198951015840

119896 = 1198961015840

(316)


1198941198941015840


119894119895and 119883119896

1015840

11989410158401198951015840 are






119894119895and119883119896

1015840

11989410158401198951015840 with the above

constraints 1198841198961198961015840



1015840


1198621198961198961015840

1198941198941015840 =

119904radic1198891198961198961015840 (2 minus Δ119905

1198941198941015840)2

forall0 lt Δ1199051198941198941015840 le 119905max

0 otherwise(317)





sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum119896isin119870119911

(2119891

119894119891119888

119891119904119889

1198961+ 119889

1198962)(119883

119896

119904119894+ sum

119895isin119863

119883119896

119895119894)

+ sum

(119894isin119865119894=1198951198941015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum

(1198941015840isin119865119894=119895119894

1015840=1198951015840119894 =1198941015840119895 =1198951015840)

sum119896isin119870119911

sum

1198961015840isin119870

1198731198941198941015840119862

1198941198941015840

1198961198961015840119883

119896

1198941198951198831198961015840

11989410158401198951015840

(318)


replaced with

sum119894

119883119896

119904119894+ 119883

119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119896 isin 119870119911 119894 isin 119865 (319)

sum119895

119883119896

119895119879+ sum

119896

119883119896

119878119879=

1003816100381610038161003816119870119911

1003816100381610038161003816 forall119894 isin 119865 119896 isin 119870119911 (320)



119909119895=


Minimize119873

sum119894=1

119888119894119909119894

(321)

subject to

119873

sum119894=1

119892V119894119909119894 = 1 for V = 1 119881 (322)

119873

sum119894=1

119890119894119886119909119894= 119878

119886for 119886 = 1 119860 (323)

119909119895isin 0 1 for 119895 = 1 119899 (324)

where


0 otherwise

119890119894119886


0 otherwise

(325)


119873

sum119894=1

119881

sumV=1

119860

sum119886=1

119901V119886119896119890119894119886119892V119894119909119894 ge 119875119896

for 119896 = 1 119870 (326)

where




119873

sum119894=1

119892V119894119909119894 + 119880119860119865V = 1 for V = 1 119881 (327)


119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119873

sum119894=1

(119892V119894 + 119892V119860119894) 119909119894+ 119880119860119865V119860 = 1

119901119894ℎ119903

=



0 otherwise(328)























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Theoretical







Airport)




Theoretical




Theoretical





Theoretical









Theoretical



Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 Continued




Theoretical









Theoretical




Houston (IAH))











Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Exact algorithms












Theoretical














Theoretical

















Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 Continued


OPL




























11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















11111 21

3212 3

4444

25

1

0 1 2 3 4 5 61974198519881990199819992000200120022003200420052007200820092010201120122014
























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























10

12

04

1116

8

0 2 4 6 8 10 12 14 16 18

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















OPL







Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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