Research Article A Branch and Bound Approach to Solve the …downloads.hindawi.com/journals/ijme/2013/930920.pdf · 2017. 7. 17. · A Branch and Bound Approach to ... strained project

Hindawi Publishing CorporationInternational Journal of Manufacturing EngineeringVolume 2013 Article ID 930920 7 pageshttpdxdoiorg1011552013930920

Research ArticleA Branch and Bound Approach toSolve the Preemptive Resource Leveling Problem

Behrouz Afshar-Nadjafi Zeinab Khalaj and Esmaeil Mehdizadeh

Faculty of Industrial and Mechanical Engineering Islamic Azad University Qazvin Branch PO Box 34185-14161 Qazvin Iran

Correspondence should be addressed to Behrouz Afshar-Nadjafi afsharnbalumsharifedu

Received 10 March 2013 Accepted 5 September 2013

Academic Editors A Che G Dessein A Lockamy and G Onwubolu

Copyright copy 2013 Behrouz Afshar-Nadjafi et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We study resource constrained project scheduling problem with respect to resource leveling as objective function and allowanceof preemption in activities The branch and bound algorithms proposed in previous researches on resource leveling problem donot consider preemption So representing a model for the problem a branch and bound algorithm is proposed This algorithmcan handle preemption in resource leveling problem Comparing the resource leveling problem and the preemptive resourceleveling problem it is observed that considering preemption in the problem leads to better results in the objective function Thisimprovement imposes additional time to solve the problem Coding the algorithm in MATLAB and checking it on the projectswith 8 and 10 activities results show that the proposed algorithm is efficient

1 Introduction

Increasing international competition enforces utilization ofvery expensive resources (eg heavy machines) by most ofthe companies Thus resource constrained project schedul-ing problem (RCPSP) has attracted more attention Severalobjective functions are studied in RCPSP Resource leveling isone of them inwhich the variation of resource utilization is tobe minimized In addition to resource constraints minimumand maximum time lags between different activities have tobe observed in general

Several studies have been done for RCPSPThe first math-ematical formulation of the RCPSPwas given by Pritsker et al[1] Then Kaplan [2] Olaguıbel and Goerlich [3] and Kleinand Scholl [4] continued their job In addition Blazewicz etal proved that RCPSP is NP-hard [5] Kastor and Sirakoulisanalyzed the effectiveness of three resource leveling problemtools on RCPSP Primavera p60 Microsoft Project 2007 andOpen Workbench 116 [6]

For resource leveling problems with minimum time lagsseveral exact and heuristic solution procedures have beenstudied

Exact algorithms contain enumeration integer program-ming or dynamic programming techniques Tavares repre-sented the effectiveness of resource leveling problem in costs

[7] Ahuja [8] Easa [9] Bandelloni et al [10] Demeule-meester [11] and Younis and Saad [12] presented exactprocedures for resource leveling problem Nubel developeda branch and bound procedure (BB) upon minimal delay-ing alternative and disjunction precedence constraints [13]Demeulemeester andHerroelen proposed amodel for projectscheduling problem with resource constraints and preemp-tion and a model for resource leveling problem and solved itwith some methods such as BB [14] Gather et al proposedexact methods for solving the resource leveling problem [15]

Some studies contain both exact and heuristic methodsNeumann and Zimmermann introduced three groups ofobjective functions for resource leveling problem [16 17]They proved that the issue of resource levelingwith each stud-ied objective function is NP-hard and proposed an exact anda heuristic procedure for the problem expressed in the paperIn their paper BB and truncated BB procedures with andwithout resource limitations are discussed Coughlan et alpresented a branch-and-price approach together with a newheuristic to solve resource leveling problem [18]

There are some works to solve resource leveling problemusing heuristics To solve resource leveling problemwith gen-eral objective function and maximum time lags Brinkmannand Neumann [19] and Neumann and Zimmermann [20]proposed heuristic methods upon priority rules Savin et al

2 International Journal of Manufacturing Engineering

used neural network to solve resource leveling problem [2122]Moder and Philips [23]Moder et al [24] Harris [25 26]and Takamoto et al [27] presented heuristic procedures tosolve resource leveling problem

Preemption is an important factor in our daily projectsand jobs Stopping a job when the night comes interrupt-ing a course as the class time is out putting the call onhold to answer someone else drinking coffee while readingnewspaper and so forth are some examples of preemption inour life By the aspect of preemption there are three types ofactivities activities which will be continued after preemption(preempt-resume type) activities which should be restartedafter preemption (preempt-repeat type) and activities forwhich preemption is not allowed Because of its role in ourlife preemption has attracted some researchers to work on itusing both exact methods and heuristics

Several studies on RCPSP use exact methods to solvethe problem Demeulemeester and Herroelen proposed anoptimal solution procedure for the preemptive resource con-strained project scheduling problem to minimize the projectmakespan [28] The maximum number of preemptions foreach activity was considered specific for the activity A modeland a BB algorithm were presented for solving the problemPatterson et al [29] developed a BB algorithm upon prece-dence tree which was reviewed by Sprecher [30] Stinson etal [31] Demeulemeester and Herroelen [32] Igelmund andRadermacher [33] and Reyck and Herroelen [34] studiedBB procedures for the RCPSP with generalized precedencerelations Afshar-Nadjafi et al have provided a branch andbound procedure for resource leveling problem inmultimodeRCPSP [35]

As an example for heuristics Ballestın et al studied pre-emption in project scheduling with resource constraints [3637] They show that one preemption to be allowed for eachactivity creates a proper state and more than one preemptionjust increases computational time with a slight improvementin the final solution In their model the number of preemp-tions allowed for each activity is specific for the activityThenan evolutionary algorithm with a coding and a crossoveroperator for the model has been presented Vanhoucke et alinvestigate the effect of variable activity durations under fixedwork content the possibility of allowing activity preemptionthe use of fast tracking on the total lead time and the totalresource utilization of resource constrained projects [38]

To the best of our knowledge no research has studiedthe usefulness of preemption in resource leveling problem Inthis paper we study effectiveness of preemption in resourceleveling problem We consider the preempt-resume type ofactivities Here resources are considered renewable and con-strained We will propose a newmodel for the problem and aBampB procedure to schedule activities in order to achieve themost leveled project

The rest of the paperwill be as follows In Section 2wewillintroduce the new model for preemptive resource levelingproblem In Section 3 the proposed BB algorithm will bedescribed for the problem Finally computational results willbe represented in Section 4

2 Problem Definition

In this paper we are supposed to schedule activities withmaximum 119901 preemptions allowed for each activity withrespect to resource leveling as the objective function If pre-emption is justified for the activity we will detect the properperiod in which it occurs In this paragraph we will studythe components of the model In Section 21 we will studyobjective function In Section 22 problem assumptions arestated Constraints are described in Section 23 Parametersand variables are proposed in Section 24 Finally we will getfamiliar with the model of the problem in Section 25

21 Objective Function Resource leveling problem deter-mines a sequence for the activities of a project which willminimize the amount of variation This issue is important indetermining the optimal sequence of activities to minimizethe level of hiring and firing workers or that of utilizationand unemployment of machines Leveling limited resourcesor unlimited resources prevents the excessive fluctuations ofthe resources and reduces their cost The chosen objectivefunction for this study is to level the cumulative amountof resources used in each moment to the former one Weconsider the resource amount used in both the first dayand the last day If the project is finished a day before thedeadline the resource used in the last day is considered bythe formula But if it is finished on the deadline we must addthe consumption to the objective function

22 Problem Assumptions In the project scheduling prob-lem 119899 activities which use 119896 resources are intended Finish-start relationships (FS) are the most commonly used Theactivity on arc network can easily be converted to the activityon node network so we use the second oneThe assumptionsof this problem are as follows

(i) The project involves 119899 activities of which the initialand the final ones are dummies

(ii) The project is presented by an activity on node net-work

(iii) The number of resources is 119870(iv) General relations of activities are FS0(v) Each activity can be preempted for at most 119901 times(vi) Preemption is free of time and cost(vii) Resources are renewable

23 Constraints In RCPSP activities should be scheduledso that the time constraints and resource constraints arerespected Obviously the model should consider precedencerelations between activities Hence the problem constraintsof the model are considered as follows

(i) Maximum resource 119896 availability in each period is 119877119896

units(ii) Project deadline is at the end of the period 119879(iii) Precedence relations should always be respected

International Journal of Manufacturing Engineering 3

24 Variables and Parameters In this model we use theseparameters and variables

119899 number of activities

119877119896 maximum availability of resource 119896

119879 deadline of the project

119870 number of resources

119901 maximum number of preemptions

119903119894119896 requirement of activity 119894 to the resource type 119896

119889119894 duration of activity 119894

119864 the set of precedence relations between activities

119890119904119894 earliest start time of activity 119894

119897119891119894 latest finish time of activity 119894

119910119894119905 1 activity 119894 is in progress at period 1199050 else

119909119894119905 1 activity 119894 is preempted at period 1199050 else

119891119905 the set of activities which can be in progress at

period 119905

119891119905= 119894 | exist119894 isin 2 119899 minus 1 119890119904

119894le 119905 lt 119897119891

119894 (1)

Note that when you work on the job 119894 at period 119905 119910119894119905

gets the value one but if the job is interrupted before 119905 andis not restarted yet 119910

119894119905gets the zero one Again if the job 119894 is

preempted just at period 119905 then 119909119894119905gets the value one But if

it is still interrupted at the next period 1199051015840 1199091198941199051015840 gets zero It is

seen that119909119894119905and119910119894119905cannot get the value one simultaneously

25 The Proposed Model The model proposed in this paperis as follows

min 119911 =119879

sum

119905=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894isin119891119905

119903119894119896119910119894119905minus sum

119894isin119891119905minus1

119903119894119896119910119894119905minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(2)

subject to

sum

119894isin1198911

1199101198941gt 0 (3)

119897119891119894

sum

119905=119890119904119894

119910119894119905= 119889119894 119894 = 1 119899 (4)

119910119895119905le1

119889119894

min(119905minus1119897119891119894)

sum

1199051015840=119890119904119894

1199101198941199051015840 (119894 119895) isin 119864

119905 = 119890119904119895 119897119891

119895 119894 = 1

(5)

119897119891119894minus1

sum

119905=119890119904119894+1

119909119894119905le 119901 119894 = 2 119899 minus 1 (6)

sum

119894isin119891119905

119903119894119896119910119894119905le 119877119896 119905 = 1 119879 119896 = 1 119870 (7)

119909119894119905

=

1 119910119894119905= 0 119910

119894119905minus1= 1

119905

sum

1199051015840=119890119904119894

1199101198941199051015840 lt 119889119894

0 else

119894 = 2 119899 minus 1 119905 = 119890119904119894+ 1 119897119891

119894minus 1

(8)

119909119894119905 119910119894119905= 0 1 119894 = 1 119899 119905 = 119890119904

119894 119897119891

119894 (9)

Equation (2) presents an objective function that mini-mizes the resource utility period to period Note that theobjective function includes the resource used in the first dayand the last one too So we should add the resource usage ofthe first day to the objective function Again if the projectis finished on the deadline we must add the amount ofresources used in the day to the function By restriction (3)the earliest start time of zero is assigned to the project (at leastone activity is scheduled at the first day) and (4) guarantiesthat each activity is in progress for 119889

119894time units Restriction

(5) considers generalized precedence relations of activities(while the preceding activity is not completely scheduled thepredecessor will not be scheduled) while (6) stipulates thepreemption constraints Restriction (7) specifies the resourceconstraints and (8) has formulated the period in whichpreemption occurs for each activityThe variables are definedas integers in (9)

3 Branch and Bound Procedure

Here a BB algorithm is proposed to solve the preemptiveresource leveling problem In Section 31 we will have somedefinitions used in the algorithm In Section 32 calculationof the function value at each node is stated Upper boundand dominance rules are considered in Sections 33 and 34Finally we will illustrate the algorithm with a numericalexample in Section 35

31 Definitions Used in the Algorithm Some definitions usedin the BB algorithm should be clearedHere we define ldquonoderdquoldquobranchrdquo and ldquocritical pathrdquo in the algorithm

Node in this algorithm each node represents the activi-ties scheduled at the current day

Branch here a branch means a partial schedulingIndeed a branch contains activities which are scheduled fromthe first day up to now

Here when an activity has not been started its criticalpath contains its duration But when an activity has beenstarted and is scheduled for a while the critical path containsits remaining duration

The algorithm presented in this paper is a depth-first oneThe process of branching continues until all of the nodes arechecked In fact in branching all the activities which can be


scheduled are checked (if the preceding activities are sched-uled before and the activity has not been completed yet)nodes contain all combinations of these activities For eachnode all the constraints (resource requirement andmaximumpreemption) are checked If the constraints are satisfied wewill use the dominance rules Searching all nodes the processends Components of the algorithm are proposed as follows

32 Calculating the Function Value at Each Node Functionvalue at each node is the total variance in resource utility fromday to day (this amount includes the amount of resourcesused on the first day) It is notable that the final value of theobjective function of a branch which is returned at the endof the branch applies the resource used in the last day of theproject (the final node of the branch)

33 Upper Bound At the first level the upper bound is doublethe total amount of resource requirements of all activitiesThis amount will be achieved if all the activities are scheduledone by one without parallel execution with at least one-daybreak between them (Figure 1) So in any feasible solutionwithout a break the function value will not be worse Con-tinuously the objective function of the first feasible solutionachieved is accepted to be the upper bound Achieving abetter solution it will be considered as the upper bound

34 Dominance Rules As mentioned before the dominancerules are needed to close branches without a better solutionbut they should not remove a promising solution Only thebranches with nonoptimal or unfeasible solution shall bediscontinued at the earliest time possible For all the nodesresource and preemption constraints are checked If theconstraints are satisfied then the dominance rules will beused if not we will continue with another node Now therules for this algorithm are as follows

(a) If the cumulative time of the current node and thecritical path of activities scheduled in the node (max-imum critical path of the activities) exceed thedeadline of the project the node is interrupted Forexample if in second day of a project an activitywith a four-day-long critical path is scheduled andthe deadline is five then the branch is interruptedWithout the rule you will continue branching and itwill be finished at the fifth day instead of the second

(b) Obviously if in the current node there is an activity forwhich more than 119901 preemption occurred the node isbounded

(c) If the cumulative amount of the objective function ata node and the resource used at the node exceed theupper bound the node is interrupted As an examplewhen the upper bound is six units a node by objectivefunction amount of four units that uses three units ofthe resource exceeds the upper bound (4 + 3 gt 6) soit is interrupted

(d) It is obviously seen that if the resource utility and theobjective function of the current node and a node in

12

3

Reso

urce

util

izat

ion

Time

Hiring resources

Firing resources

Figure 1 Upper bound

1

0

0

5

0

0

2

2

3

3

1

3

4

3

5

Figure 2 Project example

a different branch are the same and if the activitiesscheduled in the current branch are a subset of theothers the current branch will not lead to a bettersolution So the current node will be interruptedand will not be continued As an example considera branch that has scheduled at least a part of activities1198941 1198942 1198943 1198944 with objective function of five units and

resource utility of 4 units in the node 119897 Now supposethat the current node (119898) has scheduled at least a partof activities 119894

1 1198942 1198943 with objective function of five

units and resource utility of 4 units Then the node119898is not better than node 119897 and will not be continued

35 Numerical Example Here to better understand theproposed algorithm an example is given For this purposewe choose the project network as depicted in Figure 2 Thenumbers above the nodes show the activity durations and thenumbers below show the amount of resources required forthat activity The first activity and the last one are dummyactivities as shown in Figure 2

Here we assume that the amount of available resources is6 units and the project deadline is at the end of the fifth dayThe first upper bound is 22 2 lowast (0 + 3 + 3 + 5 + 0)

The first node of the branch and bound procedure putsup activity 1 In Level 2 activities 2 and 3 have the abilityto be scheduled three nodes are created put up activity2 alone activity 3 alone or both 2 and 3 The proposedalgorithm selects the first node and the branching continuesThis process is also repeated for the second day On day 4 only


110 0

0

223 3

13

33 3

14

2 36 6

1

523 3

26

33 3

27

2 36 6

2

833 3

39

45 7

3

1045 7

4

1145 7

5

Node

Flowed activities

Utilized resource

Objective function

Period

Figure 3 Branch and bound tree for project example

activity 3 can be scheduled But then the project will be donein 6 days So the first rule closes the branch (node 8) Thennode 6 is selected But using the fourth rule it is boundedby node 8 Branching continues from node 7 A solution isachieved at node 11 So the new upper bound is 12 (7 + 5)The last nodes are respectively interrupted nodes 6 and 7 bythe fourth ruleThe only solution achieved is the optimal one(Figure 3)

4 Computational Results

Thebranch and bound algorithm of Section 3 has been codedin MATLAB version 7100499 (R2010a) We have tested thealgorithm on a computer with an Intel Core 2 Duo processor4GB RamWe have used PROGEN as the problem generatorThe deadline of each project is considered the critical pathplus 20 of it We have tested the algorithm for projectswith 8 and 10 activities For each group 50 examples aretested simultaneously in both problems resource levelingproblem and preemptive resource leveling problem Foreasier computation we have considered one resource andmaximum one preemption for each activity Considering 8activities resource leveling problem could not find a solutionfor 8 examples And considering 10 activities it could notfind a solution for 13 examples So comparing the results withthose of nonpreemptive resource leveling problem we notice35 improvement in number of solved problems with 10activities (a scheduling is returned for 37 projects in resourceleveling problem and for 50 projects in preemptive resourceleveling problem) and 19 improvement in number of solvedproblems with 8 activities (a scheduling is returned for 42

Table 1 Number of solved examples for projects with 8 and 10 ac-tivities

Number ofactivities

Resourceleveling problem

Preemptiveresource

leveling problem improvement

8 42 50 1910 37 50 35

projects in resource leveling problem and for 50 projectsin preemptive resource leveling problem) (Table 1) By thesolved problem we mean the program has found a solutionwhich is finished on time or maybe in time with respect toconstraints In fact see Table 1

Table 2 contains computational results As seen in thetable for projects with 8 activities in resource levelingproblem an average objective function of 2614 is gained witha deviation of 403 These results are reached in an averagecomputational time of 052 seconds with a deviation of 100seconds For projects with 10 activities in resource levelingproblem the average objective function is 2946 with a devia-tion of 480 Here the average computational time achieved is609 seconds with a deviation of 722 seconds In preemptiveresource leveling problem for projects with 8 activities anaverage objective function of 2421 with a deviation of 316seconds is gained with an average computational time of 300secondswith deviation of 488 seconds Finally in preemptiveresource leveling problem for projects with 10 activities theaverage objective function reached is 2616 with a deviationof 477 Computational time has the average of 1603 seconds


Table 2 Computational results for resource leveling problem and preemptive resource leveling problem with 8 and 10 activities

Number of activities Objective function average CPU time average (sec)Resource leveling Preemptive resource leveling Resource leveling Preemptive resource leveling

8 2614 plusmn 403 2421 plusmn 316 052 plusmn 100 300 plusmn 488


Table 3 Efficiency of each dominance rule

Number ofactivities

Amount of nodesbounded by rule

119886


119888

Amount ofnodes bounded

by rule 1198898 03418 03983 0259910 04534 02903 02563

with a deviation of 1827 seconds Because of recursion limitof 500 in MATLAB more than 10 activities in a project couldnot be done for most of the time

The results verify the improvement of optimal solutionsobtained when preemption is applied into the time it isnot allowed With the increasing number of activities timeto solve the problem dramatically increased To comparethe results each rule is excluded in all the examples Com-paring the nonpromising branches bounded with rules theirefficiency is checked Efficiency of the dominance rules isreported in Table 3 Results show that bounding rule 119887 ismoreefficient for problemswith 8 activities whereas bounding rule119886 is more efficient for problems with 10 activities Also noneof the rules are useless

The algorithm presented by Neumann and Zimmer-mann solved resource constrained resource leveling probleminstants with 15 activities in less than 10 seconds and 992 ofthem in less than 100 seconds [17] For 10 activities it solved980 of the instants in less than 10 seconds As reported ouralgorithm solved the problem in the average of 609 secondsAlso our algorithm taking into account the preemptions ispreferable

5 Conclusions

In this paper we proposed a model for preemptive resourceleveling problemThe objective of the problem is tominimizethe variance in resource utilization as is traditional for theresource leveling problem We defined a limited maximumnumber of preemptions per activity as one of the modelconstraints A branch and bound algorithm is represented tosolve the problem Finally the program is used for solvingthe problem with and without allowance of preemption Itis obvious that considering preemption leads to a better valuefor the objective function but this solution is reached in moretime

The comparison results showed that the proposed branchand bound algorithm is competitive with the algorithmpresented by Neumann and Zimmermann to solve resourceconstrained resource leveling problem without preemption[17]

For future research metaheuristic algorithms can be usedas a method to solve the problem with large scales problems

Also one can consider nonrenewable resources and general-ized precedence relations (GPRs) to tackle real-life situationsThe results of this work have two fundamental usages Firstit can be used as benchmark for comparison with futuredeveloped methods especially metaheuristics Second thesolutions obtained from proposed method can be used byproject managers looking formore leveled schedule using theopportunity of preemption

References

[1] A A B Pritsker L J Watters and P M Wolfe ldquoMulti-projectscheduling with limited resources a zero-one programmingapproachrdquoManagement Science vol 16 no 1 pp 93ndash108 1969

[2] L A Kaplan Resource-constrained project scheduling with pre-emption of jobs [PhD thesis] University of Michigan 1988

[3] R A-V Olaguıbel and J T Goerlich ldquoThe project schedulingpolyhedron dimension facets and lifting theoremsrdquo EuropeanJournal of Operational Research vol 67 no 2 pp 204ndash220 1993

[4] R Klein and A Scholl ldquoComputing lower bounds by destruc-tive improvement an application to resource-constrainedproject schedulingrdquo European Journal of Operational Researchvol 112 no 2 pp 322ndash346 1999

[5] J Blazewicz J K Lenstra and A H G R Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983

[6] A Kastor and K Sirakoulis ldquoThe effectiveness of resourcelevelling tools for Resource Constraint Project SchedulingProblemrdquo International Journal of Project Management vol 27no 5 pp 493ndash500 2009

[7] L V Tavares ldquoA review of the contribution of OperationalResearch to Project Managementrdquo European Journal of Opera-tional Research vol 136 no 1 pp 1ndash18 2002

[8] H N Ahuja Construction Performance Control By NetworksJohn Wiley amp Sons 1976

[9] S M Easa ldquoRL in construction by optimizationrdquo Journal ofConstruction Engineering and Management vol 115 pp 302ndash316 1989

[10] M Bandelloni M Tucci and R Rinaldi ldquoOptimal resourceleveling using non-serial dyanamic programmingrdquo EuropeanJournal of Operational Research vol 78 no 2 pp 162ndash177 1994

[11] E Demeulemeester ldquoMinimizing resource availability costs intime-limited project networksrdquoManagement Science vol 41 pp1590ndash1598 1995

[12] M A Younis and B Saad ldquoOptimal resource leveling of multi-resource projectsrdquo Computers and Industrial Engineering vol31 no 1-2 pp 1ndash4 1996

[13] H Nubel ldquoA branch-and-bound procedure for the resourceinvestment problem with generalized precedence constraintsrdquoTech Rep WIOR-516 1998

[14] E Demeulemeester and W Herroelen Project Scheduling AResearch Handbook 2002


[15] T Gather J Zimmermann and J-H Bartels ldquoExact methodsfor the resource levelling problemrdquo Journal of Scheduling vol14 no 6 pp 557ndash569 2011

[16] K Neumann and J Zimmermann ldquoHeuristic methods forresource-constrained project scheduling with regular and non-regular objective functions and schedule-dependent time win-dowsrdquo in Project Scheduling Recent Models Algorithms andApplications J Weglarz Ed pp 261ndash287 Kluwer AcademicPublishers Boston Mass USA 1999

[17] KNeumann and J Zimmermann ldquoProcedures for resource lev-eling and net present value problems in project scheduling withgeneral temporal and resource constraintsrdquo European Journal ofOperational Research vol 127 no 2 pp 425ndash443 2000

[18] E Coughlan M Lubbecke and J Schulz ldquoA branch-and-pricealgorithm for multi-mode resource levelingrdquo in ExperimentalAlgorithms vol 6049 of Lecture Notes in Computer Science pp226ndash238 2010

[19] K Brinkmann and K Neumann ldquoHeuristic procedures forresource-constrained project scheduling with minimal andmaximal time lags the resource-leveling and minimum projectduration problemsrdquo Journal of Decision Systems vol 5 pp 129ndash155 1996

[20] K Neumann and J Zimmermann ldquoResource levelling forprojects with schedule-dependent time windowsrdquo EuropeanJournal of Operational Research vol 117 no 3 pp 591ndash605 1999

[21] D Savin S Alkass and P Fazio ldquoConstruction resource levelingusing neural networksrdquo Canadian Journal of Civil Engineeringvol 23 no 4 pp 917ndash925 1996

[22] D Savin S Alkass and P Fazio ldquoA procedure for calculatingthe weight-matrix of a neural network for resource levelingrdquoAdvances in Engineering Software vol 28 no 5 pp 277ndash2831997

[23] J J Moder and C R Phillips Project Management with CPMand PERT Van Nostrand Reinhold 1970

[24] J J Moder C R Phillips and E W Davis Project ManagementWith CPM PERT and Project Diagramming Van NostrandReinhold 1983

[25] R B Harris Precedence and Arrow Networking Techniques forConstruction John Wiley amp Sons 1978

[26] R B Harris ldquoPacking method for resource leveling (pack)rdquoJournal of Construction Engineering and Management vol 116no 2 pp 331ndash350 1990

[27] M Takamoto N Yamada Y Kobayashi H Nonaka andS Okoshi ldquoZero-one quadratic programming algorithm forresource leveling of manufacturing process schedulesrdquo Systemsand Computers in Japan vol 26 no 10 pp 68ndash76 1995

[28] E LDemeulemeester andW SHerroelen ldquoAn efficient optimalsolution procedure for the preemptive resource-constrainedproject scheduling problemrdquo European Journal of OperationalResearch vol 90 no 2 pp 334ndash348 1996

[29] J H Patterson R Slowinski F B Talbot and J Weglarz ldquoAnalgorithm for a general class of precedence and resource con-strained scheduling problemsrdquo Advances in Project Schedulingvol 187 pp 3ndash28 1989

[30] A Sprecher Resource-Constrained Project SchedulingmdashExactMethods For the Multi-Mode Case Lecture Notes in Economicsand Mathematical Systems 1994

[31] J P Stinson E W Davis and B M Khumawala ldquoMulti-ple resource-constrained scheduling using branch-and-boundrdquoAIIE Trans vol 10 no 3 pp 252ndash259 1978

[32] E Demeulemeester andW S Herroelen ldquoA branch-and-boundprocedure for the multiple resource-constrained project sched-uling problemrdquo Management Science vol 38 pp 1803ndash18181992

[33] G Igelmund and F J Radermacher ldquoPreselective strategies forthe optimization of stochastic project networks under resourceconstraintsrdquo Networks vol 13 no 1 pp 1ndash28 1983

[34] B De Reyck and W Herroelen ldquoA branch-and-bound pro-cedure for the resource-constrained project scheduling prob-lem with generalized precedence relationsrdquo European Journalof Operational Research vol 111 no 1 pp 152ndash174 1998

[35] B Afshar-Nadjafi H Najjarbashi and E Mehdizadeh ldquoAbranch-and-bound procedure for RL in multi-mode resourceconstraint project scheduling problemrdquo Research Journal ofRecent Sciences vol 1 no 7 pp 33ndash38 2012

[36] F Ballestın V Valls and S Quintanilla ldquoPre-emption inresource-constrained project schedulingrdquo European Journal ofOperational Research vol 189 no 3 pp 1136ndash1152 2008

[37] F Ballestın V Valls and S Quintanilla ldquoScheduling projectswith limited number of preemptionsrdquo Computers and Opera-tions Research vol 36 no 11 pp 2913ndash2925 2009

[38] M Vanhoucke and D Debels ldquoThe impact of various activityassumptions on the lead time and resource utilization ofresource-constrained projectsrdquo Computers and Industrial Engi-neering vol 54 no 1 pp 140ndash154 2008

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DistributedSensor Networks



used neural network to solve resource leveling problem [2122]Moder and Philips [23]Moder et al [24] Harris [25 26]and Takamoto et al [27] presented heuristic procedures tosolve resource leveling problem

Preemption is an important factor in our daily projectsand jobs Stopping a job when the night comes interrupt-ing a course as the class time is out putting the call onhold to answer someone else drinking coffee while readingnewspaper and so forth are some examples of preemption inour life By the aspect of preemption there are three types ofactivities activities which will be continued after preemption(preempt-resume type) activities which should be restartedafter preemption (preempt-repeat type) and activities forwhich preemption is not allowed Because of its role in ourlife preemption has attracted some researchers to work on itusing both exact methods and heuristics

Several studies on RCPSP use exact methods to solvethe problem Demeulemeester and Herroelen proposed anoptimal solution procedure for the preemptive resource con-strained project scheduling problem to minimize the projectmakespan [28] The maximum number of preemptions foreach activity was considered specific for the activity A modeland a BB algorithm were presented for solving the problemPatterson et al [29] developed a BB algorithm upon prece-dence tree which was reviewed by Sprecher [30] Stinson etal [31] Demeulemeester and Herroelen [32] Igelmund andRadermacher [33] and Reyck and Herroelen [34] studiedBB procedures for the RCPSP with generalized precedencerelations Afshar-Nadjafi et al have provided a branch andbound procedure for resource leveling problem inmultimodeRCPSP [35]

As an example for heuristics Ballestın et al studied pre-emption in project scheduling with resource constraints [3637] They show that one preemption to be allowed for eachactivity creates a proper state and more than one preemptionjust increases computational time with a slight improvementin the final solution In their model the number of preemp-tions allowed for each activity is specific for the activityThenan evolutionary algorithm with a coding and a crossoveroperator for the model has been presented Vanhoucke et alinvestigate the effect of variable activity durations under fixedwork content the possibility of allowing activity preemptionthe use of fast tracking on the total lead time and the totalresource utilization of resource constrained projects [38]

To the best of our knowledge no research has studiedthe usefulness of preemption in resource leveling problem Inthis paper we study effectiveness of preemption in resourceleveling problem We consider the preempt-resume type ofactivities Here resources are considered renewable and con-strained We will propose a newmodel for the problem and aBampB procedure to schedule activities in order to achieve themost leveled project

The rest of the paperwill be as follows In Section 2wewillintroduce the new model for preemptive resource levelingproblem In Section 3 the proposed BB algorithm will bedescribed for the problem Finally computational results willbe represented in Section 4

2 Problem Definition

In this paper we are supposed to schedule activities withmaximum 119901 preemptions allowed for each activity withrespect to resource leveling as the objective function If pre-emption is justified for the activity we will detect the properperiod in which it occurs In this paragraph we will studythe components of the model In Section 21 we will studyobjective function In Section 22 problem assumptions arestated Constraints are described in Section 23 Parametersand variables are proposed in Section 24 Finally we will getfamiliar with the model of the problem in Section 25

21 Objective Function Resource leveling problem deter-mines a sequence for the activities of a project which willminimize the amount of variation This issue is important indetermining the optimal sequence of activities to minimizethe level of hiring and firing workers or that of utilizationand unemployment of machines Leveling limited resourcesor unlimited resources prevents the excessive fluctuations ofthe resources and reduces their cost The chosen objectivefunction for this study is to level the cumulative amountof resources used in each moment to the former one Weconsider the resource amount used in both the first dayand the last day If the project is finished a day before thedeadline the resource used in the last day is considered bythe formula But if it is finished on the deadline we must addthe consumption to the objective function

22 Problem Assumptions In the project scheduling prob-lem 119899 activities which use 119896 resources are intended Finish-start relationships (FS) are the most commonly used Theactivity on arc network can easily be converted to the activityon node network so we use the second oneThe assumptionsof this problem are as follows

(i) The project involves 119899 activities of which the initialand the final ones are dummies

(ii) The project is presented by an activity on node net-work

(iii) The number of resources is 119870(iv) General relations of activities are FS0(v) Each activity can be preempted for at most 119901 times(vi) Preemption is free of time and cost(vii) Resources are renewable

23 Constraints In RCPSP activities should be scheduledso that the time constraints and resource constraints arerespected Obviously the model should consider precedencerelations between activities Hence the problem constraintsof the model are considered as follows

(i) Maximum resource 119896 availability in each period is 119877119896

units(ii) Project deadline is at the end of the period 119879(iii) Precedence relations should always be respected
















period 119905

119891119905= 119894 | exist119894 isin 2 119899 minus 1 119890119904

119894le 119905 lt 119897119891

119894 (1)








min 119911 =119879

sum

119905=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894isin119891119905

119903119894119896119910119894119905minus sum

119894isin119891119905minus1

119903119894119896119910119894119905minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(2)

subject to

sum

119894isin1198911

1199101198941gt 0 (3)

119897119891119894

sum

119905=119890119904119894

119910119894119905= 119889119894 119894 = 1 119899 (4)

119910119895119905le1

119889119894

min(119905minus1119897119891119894)

sum

1199051015840=119890119904119894

1199101198941199051015840 (119894 119895) isin 119864

119905 = 119890119904119895 119897119891

119895 119894 = 1

(5)

119897119891119894minus1

sum

119905=119890119904119894+1

119909119894119905le 119901 119894 = 2 119899 minus 1 (6)

sum

119894isin119891119905

119903119894119896119910119894119905le 119877119896 119905 = 1 119879 119896 = 1 119870 (7)

119909119894119905

=

1 119910119894119905= 0 119910

119894119905minus1= 1

119905

sum

1199051015840=119890119904119894

1199101198941199051015840 lt 119889119894

0 else

119894 = 2 119899 minus 1 119905 = 119890119904119894+ 1 119897119891

119894minus 1

(8)

119909119894119905 119910119894119905= 0 1 119894 = 1 119899 119905 = 119890119904

119894 119897119891

119894 (9)




















12

3

Reso

urce

util

izat

ion

Time

Hiring resources

Firing resources


1

0

0

5

0

0

2

2

3

3

1

3

4

3

5










110 0

0

223 3

13

33 3

14

2 36 6

1

523 3

26

33 3

27

2 36 6

2

833 3

39

45 7

3

1045 7

4

1145 7

5

Node

Flowed activities

Utilized resource

Objective function

Period






Number ofactivities


Preemptiveresource


8 42 50 1910 37 50 35









Number ofactivities


119886


119888


by rule 1198898 03418 03983 0259910 04534 02903 02563




5 Conclusions





References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























period 119905

119891119905= 119894 | exist119894 isin 2 119899 minus 1 119890119904

119894le 119905 lt 119897119891

119894 (1)








min 119911 =119879

sum

119905=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894isin119891119905

119903119894119896119910119894119905minus sum

119894isin119891119905minus1

119903119894119896119910119894119905minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(2)

subject to

sum

119894isin1198911

1199101198941gt 0 (3)

119897119891119894

sum

119905=119890119904119894

119910119894119905= 119889119894 119894 = 1 119899 (4)

119910119895119905le1

119889119894

min(119905minus1119897119891119894)

sum

1199051015840=119890119904119894

1199101198941199051015840 (119894 119895) isin 119864

119905 = 119890119904119895 119897119891

119895 119894 = 1

(5)

119897119891119894minus1

sum

119905=119890119904119894+1

119909119894119905le 119901 119894 = 2 119899 minus 1 (6)

sum

119894isin119891119905

119903119894119896119910119894119905le 119877119896 119905 = 1 119879 119896 = 1 119870 (7)

119909119894119905

=

1 119910119894119905= 0 119910

119894119905minus1= 1

119905

sum

1199051015840=119890119904119894

1199101198941199051015840 lt 119889119894

0 else

119894 = 2 119899 minus 1 119905 = 119890119904119894+ 1 119897119891

119894minus 1

(8)

119909119894119905 119910119894119905= 0 1 119894 = 1 119899 119905 = 119890119904

119894 119897119891

119894 (9)




















12

3

Reso

urce

util

izat

ion

Time

Hiring resources

Firing resources


1

0

0

5

0

0

2

2

3

3

1

3

4

3

5










110 0

0

223 3

13

33 3

14

2 36 6

1

523 3

26

33 3

27

2 36 6

2

833 3

39

45 7

3

1045 7

4

1145 7

5

Node

Flowed activities

Utilized resource

Objective function

Period






Number ofactivities


Preemptiveresource


8 42 50 1910 37 50 35









Number ofactivities


119886


119888


by rule 1198898 03418 03983 0259910 04534 02903 02563




5 Conclusions





References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















12

3

Reso

urce

util

izat

ion

Time

Hiring resources

Firing resources


1

0

0

5

0

0

2

2

3

3

1

3

4

3

5










110 0

0

223 3

13

33 3

14

2 36 6

1

523 3

26

33 3

27

2 36 6

2

833 3

39

45 7

3

1045 7

4

1145 7

5

Node

Flowed activities

Utilized resource

Objective function

Period






Number ofactivities


Preemptiveresource


8 42 50 1910 37 50 35









Number ofactivities


119886


119888


by rule 1198898 03418 03983 0259910 04534 02903 02563




5 Conclusions





References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










110 0

0

223 3

13

33 3

14

2 36 6

1

523 3

26

33 3

27

2 36 6

2

833 3

39

45 7

3

1045 7

4

1145 7

5

Node

Flowed activities

Utilized resource

Objective function

Period






Number ofactivities


Preemptiveresource


8 42 50 1910 37 50 35









Number ofactivities


119886


119888


by rule 1198898 03418 03983 0259910 04534 02903 02563




5 Conclusions





References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















Number ofactivities


119886


119888


by rule 1198898 03418 03983 0259910 04534 02903 02563




5 Conclusions





References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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Research Article A Branch and Bound Approach to Solve the …downloads.hindawi.com/journals/ijme/2013/930920.pdf · 2017. 7. 17. · A Branch and Bound Approach to ... strained project