Research ArticleIntegrated Scheduling of Production and Distribution withRelease Dates and Capacitated Deliveries

Xueling Zhong1 and Dakui Jiang2

1Department of Internet Finance and Information Engineering Guangdong University of Finance Guangzhou 510520 China2College of Management and Economics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Xueling Zhong zhongxuelhotmailcom

Received 17 November 2015 Accepted 13 March 2016

Academic Editor Alexandre B Dolgui

Copyright copy 2016 X Zhong and D Jiang 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

This paper investigates an integrated scheduling of production and distribution model in a supply chain consisting of a singlemachine a customer and a sufficient number of homogeneous capacitated vehicles In this model the customer places a set oforders each of which has a given release date All orders are first processed nonpreemptively on the machine and then batchdelivered to the customer Two variations of the model with different objective functions are studied one is to minimize the arrivaltime of the last order plus total distribution cost and the other is to minimize total arrival time of the orders plus total distributioncost For the former one we provide a polynomial-time exact algorithm For the latter one due to its NP-hard property we providea heuristic with a worst-case ratio bound of 2

1 Introduction

Integrated scheduling of production and distribution prob-lem coordinates the scheduling batching and delivery deci-sions at the detailed scheduling level of the supply chain soas to achieve optimal operational performance in a supplychainTheproblemhas been attracting considerable attentionfrom scheduling researchers as well as operationmanagers inthe past few decades In this paper we study a static offlinemodel on scheduling a set of 119899 orders119873 = 1 2 119899 on asingle machine and batch delivering them to the customerIn the production stage each order 119895 isin 119873 associatedwith a release date 119903119895 and a processing time 119901119895 needs to beprocessed once without interruption Order 119895 isin 119873 cannot beprocessed before 119903119895 and is required to process119901119895 time units onthe machine In the distribution stage completed orders aredelivered to the customer in batches by a sufficient numberof homogeneous vehicles so that each vehicle will be used atmost once and each delivery shipment will be transported bya dedicated vehicleWithout loss of generality we assume that119899 vehicles (1198611 1198612 119861119899) are available at time 0 Each order ispackaged into a standard-size pallet for delivery convenience

regardless of its size and each vehicle can deliver at most119887 orders The delivery cost incurred by each shipment is 119891Since all orders are delivered to the same customer and thedelivery time of any vehicle is the same we assume withoutloss of generality that the delivery time is 0 For a givensolution we use 119862119895 119863119895 and 119879 to denote the processingcompletion time of order 119895 isin 119873 the arrival time of order119895 isin 119873 and total distribution cost respectively Additionallydefine 119863max = max119863119895 | 119895 isin 119873 and 119863total = sum119895isin119873119863119895The problem is to find a feasible solution such that one ofthe following two objective functions is minimized (1) thearrival time of the last order plus total distribution cost thatis 119863max + 119879 (2) the total arrival time of orders plus totaldistribution cost that is 119863total + 119879 In the following let 1198751and 1198752 represent the two variations of model to minimize (1)and (2) respectively

The researches on integrated scheduling of productionand distribution models have been studied extensively inrecent years and the majority of them assume that all ordersare available at time 0 (eg Hall and Potts [1] Hall and Potts[2] Chen and Vairaktarakis [3] Chen and Pundoor [4] Jiet al [5] Li andOu [6]Wang andCheng [7] Armstrong et al

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9315197 5 pageshttpdxdoiorg10115520169315197

2 Mathematical Problems in Engineering

[8] Jiang and Li [9] Jiang et al [10] Li et al [11] and Viergutzand Knust [12]) However in some applications orders havearbitrary release dates Hence some related models involvingrelease dates were also studied Potts [13] was the first tostudy the model with single machine in which each order isdelivered individually and immediately upon its completionA heuristic with a worst-case ratio bound of 32 was proposedto minimize the arrival time of the last order After thatHall and Shmoys [14] proposed a better heuristic for themodel with a worst-case ratio bound of 43 Zdrzałka [15]provided a heuristic with a worst-case ratio bound of 32 for asimilar model with constraints that preemption is permittedand a sequence-independent setup time is incurred beforeprocessing a job Gharbi and Haouari [16] studied a modelwith identical-parallel-machine configuration and proposedan exact branch-and-bound algorithm to solve the modelIn addition a polynomial-time approximation scheme waspresented for this model by Mastrolilli [17] Moreover thesingle machine online model was also studied by Seiden [18]Hoogeveen and Vestjens [19] and van den Akker et al [20]respectively In recent years batch delivery and delivery costare involved in the models Averbakh and Xue [21] and Aver-bakh [22] assumed that the orders are released online andbatch delivered to the customer To minimize the sum of thetotal flow time and total delivery cost they proposed onlinealgorithms for the single machine model with uncapacitatedand capacitated deliveries respectively Fan [23] presented anapproximation algorithm for the offline version of the modelwith uncapacitated deliveries proposed by Averbakh and Xue[21] Lu et al [24] studied an offline single machine modelwith capacitated deliveries that assumed that only one vehicleis available to deliver orders They showed that the problemof minimizing the arrival time of the last order is stronglyNP-hard and provided a 53-approximation algorithm Liuand Lu [25] proposed an improved approximation algorithmfor the model which improved the performance ratio from53 to 32 However to the best of our knowledge littleattention is paid on the offline model with release dates inwhich the delivery batches are capacitated and delivery cost isconsidered in the objective functions For more informationwe refer the reader to Chen [26] for a comprehensive reviewon integrated production-distribution scheduling models

The model we study in this paper can be viewed as anextension of production scheduling model with release datesIn production scheduling models the following two func-tions are analogous to two widely used functions for measur-ing customer service total completion time 119862total = sum119895isin119873119862119895andmaximumcompletion time119862max = max119895isin119873119862119895 In the fol-lowing we review some articles on singlemachine schedulingmodels with release dates For ease of presentation we adoptthe well-known three-field notation 120572|120573|120574 introduced byGraham et al [27] where 120572 describes the machine config-uration 120573 specifies restrictions and constraints associatedwith the jobs and 120574 describes the objective function Baker[28] showed that the preemptive model 1|119903119895 119901119898119905119899|119862total ispolynomially solvable whereas Lenstra et al [29] showed thatthe nonpreemptive model 1|119903119895|119862total is NP-hard in the strongsense In addition themodel 1|119903119895 119901119898119905119899|119862max is polynomially

solvable Lawler [30] pointed out that the nonpreemptivemodel 1|119903119895|119862max is also polynomially solvable Since bothproduction scheduling with release dates and batch deliveryscheduling are considered our model will be more complexThis leads us to be interested in the computational complexityof 1198751 and 1198752

In this paper we analyze the computational complexityof the problems and provide efficient algorithms for theproblems respectively The paper is organized as followsIn Sections 2 and 3 we study the problems with differentobjective functions respectively In Section 4 we concludethe paper

2 P1 Minimizing Dmax + T

In this section we first introduce some optimality propertiessatisfied for problem 1198751 Then we provide a polynomial-time exact algorithm for itWe nowpresent some preliminaryresults about the structure of an optimal solution of 1198751 asfollows

Lemma 1 There exists an optimal solution for problem 1198751 inwhich all of the following hold

(1) The orders are delivered in the increasing sequence oftheir processing completion time

(2) The orders are delivered in 119911 = lceil119899119887rceil shipments wherelceil119899119887rceil is the smallest integer no less than 119899119887 The firstshipment contains 119899 minus (119911 minus 1)119887 orders and each of theremaining 119911 minus 1 shipments contains 119887 orders

(3) The departure time of each shipment is the time whenall the orders in it complete processing

The above optimality properties are straightforward weomit the proofs

Further we define an Earliest Release Date first (ERD)rule That is a set of customer orders is in ERD order if thecustomer orders are sequenced in a nondecreasing order oftheir release dates and the customer orders with equal releasedates are sequenced in the arbitrary order

Lemma 2 There exists an optimal solution for problem 1198751 inwhich the customer orders are scheduled in ERD order

Proof By Lemma 1 the orders are delivered by 119911 = lceil119899119887rceil

vehicles in an optimal solution of problem 1198751 which isindependent of how the orders are scheduled on themachineHence problem 1198751 is reduced to minimizing 119863max Becausethe delivery time of each shipment is assumed to be 0 wehave 119863max = 119862max for a given solution in which 119862maxis the maximum processing completion time of orders Asa result the optimal solution of single machine makespanscheduling problemwith release dates 1|119903119895|119862max is an optimalorder processing schedule of problem 1198751 Note that ERDrule provides an optimal solution for problem 1|119903119895|119862max[30] Hence ERD rule provides an optimal order processingschedule for problem 1198751

Mathematical Problems in Engineering 3

Based on the results stated in Lemmas 1 and 2 an exactalgorithm for problem 1198751 is given as follows

Algorithm EA

Step 1 Place all orders in a list in ERD order Reindex theorders by their position in the list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Deliver the orders such that they satisfy Lemma 1

In algorithm EA the majority of computation time isused for sorting orders in Step 1 which takes119874(119899 log 119899) timeHence we have the following

Theorem 3 Algorithm EA finds an optimal solution of prob-lem 1198751 in 119874(119899 log 119899) time

3 P2 Minimizing Dtotal + T

Whendelivery cost of each shipment is ignored that is119891 = 0problem 1198752 is reduced to problem 1|119903119895|119862total which is knownto be NP-hard in the strong sense [29] Thus problem 1198752 isalsoNP-hard in the strong sense To solve problem1198752 we firststudy an auxiliary problem 119875aux Everything else in problem119875aux is the same as in problem 1198752 except that preemptionis allowed that is order processing can be preempted andresumed later We first introduce an optimality property of119875aux related to the Shortest Remaining Processing Time first(SRPT) rule which prescribes to process at each instant 119905the job with the smallest remaining processing time amongall already released unfinished jobs with some unambiguousrule for breaking ties For example the jobs can be indexedand in case of a conflict between jobs with equal remainingprocessing times the job with the smallest index wouldbe given priority such rule is called lexicographic [22]According to this definition there is exactly one SRPT orderfor a given set of jobs Moreover for two classical preemptiveproblems 1|119903119895 119901119898119905119899|119862total and 1|119903119895 119901119898119905119899|119862max the SRPTrule provides an optimal solution [28] Pruhs et al [31] alsopresented the following result

Lemma4 In the SRPT job schedule at any instant the numberof processed jobs is not smaller than in any other job schedule

According to Lemma 4 we obtain the following resultsatisfied for problem 119875aux

Lemma 5 There exists an optimal solution for problem119875119886119906119909 inwhich the orders are processed in the sequence ordered by SRPTrule on the machine

Proof Suppose that there exists an optimal solution in whichthe order processing schedule does not follow the SRPT ruleDenote the number of vehicles used in this solution by 119896lowastLet 120591119896 be the arrival time of vehicle 119896 and 119899119896 the number ofthe finished orders in time 120591119896 in this solution 119896 = 1 119896lowast

Clearly 119899119896lowast = 119899 Then the total arrival time of orders in thissolution is equal to

11989911205911 +

119896lowastminus1

sum

119896=1

(119899119896+1 minus 119899119896) 120591119896+1 =

119896lowastminus1

sum

119896=1

119899119896 (120591119896 minus 120591119896+1) + 119899120591119896lowast (1)

By Lemma 4 at any instant 119896 the number of processedorders 119899119896 in the SRPT order processing schedule is the biggestone in all order processing schedules In addition 120591119896 minus 120591119896+1 lt0 for 119896 = 1 119896

lowastminus 1 Hence we can rearrange the jobs

in the SRPT order without increasing sum119896lowastminus1119896=1 119899119896(120591119896 minus 120591119896+1)

Also because the SRPT rule provides an optimal solution for1|119903119895 119901119898119905119899|119862max the SRPT order processing schedule is theone in which 120591119896lowast is minimized Therefore we can rearrangethe customer orders in the SRPT order without increasing theobjective value

Without loss of generality we assume that the customerorders are indexed in SRPT order so that 1198621 le sdot sdot sdot le 119862119899 Thefollowing dynamic programming algorithm finds an optimalsolution for 119875aux

Algorithm DP Define value function 119881(119895) = minimum totalcost contribution of a solution for the first 119895 orders 1 119895

Initial condition 119881(0) = 0Recursive relation for 119895 = 1 119899

119881 (119895) = min 119881 (119895 minus ℎ) + ℎ sdot 119862119895 + 119891 | ℎ

= 1 min (119887 119895) (2)

Optimal solution value it is 119881(119899)

Theorem 6 Algorithm DP finds an optimal solution of prob-lem 119875119886119906119909 in 119874(119899 log 119899 + 119899119887) time

Proof By Lemma 5 in a solution for the first 119895 order1 119895 where the production schedule follows SRPT rulethe departure time of the last vehicle is always119862119895This impliesthat the delivery time of all the orders in the last vehicle is119862119895 In the recursive relation the value function is computedby trying every possible size of the last vehicle Given thatthe size of the last vehicle is ℎ the total cost contributed bythe last vehicle is ℎ sdot 119862119895 + 119891 where ℎ sdot 119862119895 is the contributionto the total delivery time and 119891 is the distribution cost Thisproves the correctness of the recursive relation and hence theoptimality of the algorithmThere are 119899 states in this DP andit takes nomore than119874(119887) time to calculate the value for eachstate Sorting the orders by SRPT rule takes 119874(119899 log 119899) timeTherefore the overall time complexity of DP is bounded by119874(119899 log 119899 + 119899119887)

Corollary 7 The value 119881(119899) generated by algorithm DP is alower bound of the optimal objective value of 1198752

Proof Because the optimal solution of 1198752 is a feasiblesolution of 119875aux the optimal objective value of 119875aux is nolarger than that of 1198752 By Theorem 6 the solution obtained

4 Mathematical Problems in Engineering

by algorithm DP is an optimal one for problem 119875aux Hencethe value 119881(119899) generated by algorithm DP is a lower boundof the optimal objective value of 1198752

In the remainder of this section we present a polynomial-time heuristic for problem 1198752 and analyze its worst-caseperformance

Heuristic HA

Step 1 Run algorithm DP to obtain a basic solution 119878lowastauxDenote the number of vehicles used in this solution by 119896lowastLet 119862aux(119895) be denoted as the processing completion time oforder 119895 for 119895 isin 119873 Place all orders in a list sorted by anincreasing order of their processing completion times119862aux(119895)in this basic solution Reindex the orders by their position inthe list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Use 119896lowast vehicles to deliver orders Each vehicle deliversthe same orders as in 119878lowastaux The departure time of each vehicleis the time when all the orders in it complete processingDenote the resulting solution by 119878lowast

Clearly the time complexity of heuristic HA is119874(119899 log 119899+119899119887) because the majority of its computation time is to runalgorithm DP Next we analyze its worst-case performanceDenote the optimal objective value of problems 1198752 and 119875auxby 119865lowast1198752 and 119865

lowastaux respectively Let 1198651198752(119878

lowast) be denoted as the

contribution of solution 119878lowast to the objective value of problem1198752

Theorem 8 1198651198752(119878lowast) le 2119865lowast1198752 that is the worst-case ratio of

heuristic HA is bounded by 2

Proof In 119878lowast let 119896 be the order in vehicle 119861119896 with largestprocessing completion time 119862119895 119896 = 1 119896

lowast Clearly 119862119896=

119863119861119896

Because there is no idle time between instant max119895isin119861119896

119903119895

and 119862119896 we have

119862119896= 119863119861

119896

le max119895isin119861119896

119903119895 +

119896

sum

ℎ=1

sum

119895isin119861ℎ

119901119895 (3)

Denote the arrival time of order 119895 in 119878lowastaux by 119863aux(119895) for119895 isin 119873 Let 119863aux(119861119896) be denoted as the arrival time of vehicle119861119896 in 119878

lowastaux 119896 = 1 119896

lowast For each order 119895 in vehicle 119861119896 wehave 119903119895 le 119862aux(119895) le 119862aux(119896) This implies that max119895isin119861

119896

119903119895 le

119862aux(119896) In addition sum119896ℎ=1sum119895isin119861ℎ

119901119895 le 119862aux(119896) Hence wehave 119862

119896le 2119862aux(119896) Consequently |119861119896|119863119861

119896

le 2|119861119896|119863aux(119861119896)Then

sum

119895isin119873

119863119895 =

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863119861119896

le 2

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863aux (119861119896) (4)

Since the numbers of vehicles used in 119878lowastaux and 119878lowast areequal the distribution cost of these solutions is also identical

Hence 1198651198752(119878lowast) le 2119865

lowastaux By Corollary 7 we have 119865

lowastaux le 119865

lowast1198752

Hence 1198651198752(119878lowast) le 2119865

lowast1198752

4 Conclusions

In this paper we have studied two variations of the integratedscheduling of production and distributionmodel with releasedates and capacitated deliveries each of which has a differentobjective function The first variation is to minimize thearrival time of the last order plus total distribution cost wefirst presented some optimality properties satisfied for thevariation Adopting these optimality properties we proposeda polynomial-time exact algorithm which implies that thisproblem is easy The second variation is to minimize totalarrival time of the orders plus total distribution cost dueto its NP-hard property we developed a heuristic for it Inthis heuristic we first construct an auxiliary problem whichis the problem with preemptions and develop a dynamicprogramming algorithm for the auxiliary problem Then wepropose the heuristic based on the solution obtained bythe dynamic programming algorithm Finally we analyze itsworse-case performance and show that its worst-case ratio isbounded by 2

For future research it will be worth extending the existingmodel to parallel machines and developing effective andefficient algorithms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by the National NaturalScience Foundation of China (71501051) the Humanitiesand Social Sciences Research Foundation of Ministry ofEducation of China (13YJC630239) and the Foundationfor Distinguished Young Teachers in Higher Education ofGuangdong Province (YQ201403)

References

[1] N G Hall and C N Potts ldquoSupply chain scheduling batchingand deliveryrdquo Operations Research vol 51 no 4 pp 566ndash5842003

[2] N G Hall and C N Potts ldquoThe coordination of scheduling andbatch deliveriesrdquo Annals of Operations Research vol 135 no 1pp 41ndash64 2005

[3] Z-L Chen and G L Vairaktarakis ldquoIntegrated scheduling ofproduction and distribution operationsrdquo Management Sciencevol 51 no 4 pp 614ndash628 2005

[4] Z-L Chen and G Pundoor ldquoOrder assignment and schedulingin a supply chainrdquo Operations Research vol 54 no 3 pp 555ndash572 2006

[5] M Ji Y He and T C E Cheng ldquoBatch delivery scheduling withbatch delivery cost on a single machinerdquo European Journal ofOperational Research vol 176 no 2 pp 745ndash755 2007

Mathematical Problems in Engineering 5

[6] C-L Li and J Ou ldquoCoordinated scheduling of customerorders with decentralized machine locationsrdquo IIE Transactions(Institute of Industrial Engineers) vol 39 no 9 pp 899ndash9092007

[7] X Wang and T C E Cheng ldquoMachine scheduling with anavailability constraint and job delivery coordinationrdquo NavalResearch Logistics vol 54 no 1 pp 11ndash20 2007

[8] R Armstrong S Gao and L Lei ldquoA zero-inventory productionand distribution problem with a fixed customer sequencerdquoAnnals of Operations Research vol 159 pp 395ndash414 2008

[9] D K Jiang and B Li ldquoSupply chain scheduling based on hybridtaboo search algorithmrdquo Journal of Mechanical Engineering vol47 no 20 pp 53ndash59 2011

[10] D-K Jiang B Li and J-Y Tan ldquoIntegrated optimizationapproach for order assignment and scheduling problemrdquo Con-trol and Decision vol 28 no 2 pp 217ndash222 2013

[11] S Li X Zhong H Li and S Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014

[12] C Viergutz and S Knust ldquoIntegrated production and distribu-tion scheduling with lifespan constraintsrdquo Annals of OperationsResearch vol 213 no 1 pp 293ndash318 2014

[13] C N Potts ldquoAnalysis of a heuristic for one machine sequencingwith release dates and delivery timesrdquoOperations Research vol28 no 6 pp 1436ndash1441 1980

[14] L A Hall and D B Shmoys ldquoJacksonrsquos rule for single-machinescheduling making a good heuristic betterrdquo Mathematics ofOperations Research vol 17 no 1 pp 22ndash35 1992

[15] S Zdrzałka ldquoPreemptive scheduling with release dates deliverytimes and sequence independent setup timesrdquoEuropean Journalof Operational Research vol 76 no 1 pp 60ndash71 1994

[16] A Gharbi and M Haouari ldquoMinimizing makespan on parallelmachines subject to release dates and delivery timesrdquo Journal ofScheduling vol 5 no 4 pp 329ndash355 2002

[17] MMastrolilli ldquoEfficient approximation schemes for schedulingproblems with release dates and delivery timesrdquo Journal ofScheduling vol 6 no 6 pp 521ndash531 2003

[18] S Seiden ldquoRandomized online scheduling with delivery timesrdquoJournal of Combinatorial Optimization vol 3 no 4 pp 399ndash416 1999

[19] J A Hoogeveen and A P Vestjens ldquoA best possible determin-istic on-line algorithm for minimizing maximum delivery timeon a single machinerdquo SIAM Journal on Discrete Mathematicsvol 13 no 1 pp 56ndash63 2000

[20] M van den Akker H Hoogeveen and N Vakhania ldquoRestartscan help in the on-line minimization of the maximum deliverytime on a single machinerdquo Journal of Scheduling vol 3 no 6pp 333ndash341 2000

[21] I Averbakh and Z Xue ldquoOn-line supply chain schedulingproblems with preemptionrdquo European Journal of OperationalResearch vol 181 no 1 pp 500ndash504 2007

[22] I Averbakh ldquoOn-line integrated production-distributionscheduling problems with capacitated deliveriesrdquo EuropeanJournal of Operational Research vol 200 no 2 pp 377ndash3842010

[23] J Fan ldquoSupply chain scheduling with jobrsquos release times on asingle machinerdquo Journal of Systems Science and MathematicalSciences vol 31 no 11 pp 1439ndash1443 2011

[24] L Lu J Yuan and L Zhang ldquoSingle machine scheduling withrelease dates and job delivery to minimize the makespanrdquoTheoretical Computer Science vol 393 no 1ndash3 pp 102ndash1082008

[25] P Liu and X Lu ldquoAn improved approximation algorithmfor single machine scheduling with job deliveryrdquo TheoreticalComputer Science vol 412 no 3 pp 270ndash274 2011

[26] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[27] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 4 pp 287ndash326 1979

[28] K R Baker Introduction to Sequencing and Scheduling JohnWiley amp Sons New York NY USA 1974

[29] J K Lenstra AH R RinnooyKan andP Brucker ldquoComplexityofmachine scheduling problemsrdquoAnnals of DiscreteMathemat-ics vol 1 pp 343ndash362 1977

[30] E L Lawler ldquoOptimal sequencing of a single machine subjectto precedence constraintsrdquo Management Science vol 19 no 5pp 544ndash546 1973

[31] K Pruhs J Sgall and E Torng ldquoOn-line schedulingrdquo inHandbook of Scheduling Algorithms Models and PerformanceAnalysis J Y-T Leung Ed CRC Press New York NY USA2004

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

[8] Jiang and Li [9] Jiang et al [10] Li et al [11] and Viergutzand Knust [12]) However in some applications orders havearbitrary release dates Hence some related models involvingrelease dates were also studied Potts [13] was the first tostudy the model with single machine in which each order isdelivered individually and immediately upon its completionA heuristic with a worst-case ratio bound of 32 was proposedto minimize the arrival time of the last order After thatHall and Shmoys [14] proposed a better heuristic for themodel with a worst-case ratio bound of 43 Zdrzałka [15]provided a heuristic with a worst-case ratio bound of 32 for asimilar model with constraints that preemption is permittedand a sequence-independent setup time is incurred beforeprocessing a job Gharbi and Haouari [16] studied a modelwith identical-parallel-machine configuration and proposedan exact branch-and-bound algorithm to solve the modelIn addition a polynomial-time approximation scheme waspresented for this model by Mastrolilli [17] Moreover thesingle machine online model was also studied by Seiden [18]Hoogeveen and Vestjens [19] and van den Akker et al [20]respectively In recent years batch delivery and delivery costare involved in the models Averbakh and Xue [21] and Aver-bakh [22] assumed that the orders are released online andbatch delivered to the customer To minimize the sum of thetotal flow time and total delivery cost they proposed onlinealgorithms for the single machine model with uncapacitatedand capacitated deliveries respectively Fan [23] presented anapproximation algorithm for the offline version of the modelwith uncapacitated deliveries proposed by Averbakh and Xue[21] Lu et al [24] studied an offline single machine modelwith capacitated deliveries that assumed that only one vehicleis available to deliver orders They showed that the problemof minimizing the arrival time of the last order is stronglyNP-hard and provided a 53-approximation algorithm Liuand Lu [25] proposed an improved approximation algorithmfor the model which improved the performance ratio from53 to 32 However to the best of our knowledge littleattention is paid on the offline model with release dates inwhich the delivery batches are capacitated and delivery cost isconsidered in the objective functions For more informationwe refer the reader to Chen [26] for a comprehensive reviewon integrated production-distribution scheduling models

The model we study in this paper can be viewed as anextension of production scheduling model with release datesIn production scheduling models the following two func-tions are analogous to two widely used functions for measur-ing customer service total completion time 119862total = sum119895isin119873119862119895andmaximumcompletion time119862max = max119895isin119873119862119895 In the fol-lowing we review some articles on singlemachine schedulingmodels with release dates For ease of presentation we adoptthe well-known three-field notation 120572|120573|120574 introduced byGraham et al [27] where 120572 describes the machine config-uration 120573 specifies restrictions and constraints associatedwith the jobs and 120574 describes the objective function Baker[28] showed that the preemptive model 1|119903119895 119901119898119905119899|119862total ispolynomially solvable whereas Lenstra et al [29] showed thatthe nonpreemptive model 1|119903119895|119862total is NP-hard in the strongsense In addition themodel 1|119903119895 119901119898119905119899|119862max is polynomially

solvable Lawler [30] pointed out that the nonpreemptivemodel 1|119903119895|119862max is also polynomially solvable Since bothproduction scheduling with release dates and batch deliveryscheduling are considered our model will be more complexThis leads us to be interested in the computational complexityof 1198751 and 1198752

In this paper we analyze the computational complexityof the problems and provide efficient algorithms for theproblems respectively The paper is organized as followsIn Sections 2 and 3 we study the problems with differentobjective functions respectively In Section 4 we concludethe paper

2 P1 Minimizing Dmax + T

In this section we first introduce some optimality propertiessatisfied for problem 1198751 Then we provide a polynomial-time exact algorithm for itWe nowpresent some preliminaryresults about the structure of an optimal solution of 1198751 asfollows

Lemma 1 There exists an optimal solution for problem 1198751 inwhich all of the following hold

(1) The orders are delivered in the increasing sequence oftheir processing completion time

(2) The orders are delivered in 119911 = lceil119899119887rceil shipments wherelceil119899119887rceil is the smallest integer no less than 119899119887 The firstshipment contains 119899 minus (119911 minus 1)119887 orders and each of theremaining 119911 minus 1 shipments contains 119887 orders

(3) The departure time of each shipment is the time whenall the orders in it complete processing

The above optimality properties are straightforward weomit the proofs

Further we define an Earliest Release Date first (ERD)rule That is a set of customer orders is in ERD order if thecustomer orders are sequenced in a nondecreasing order oftheir release dates and the customer orders with equal releasedates are sequenced in the arbitrary order

Lemma 2 There exists an optimal solution for problem 1198751 inwhich the customer orders are scheduled in ERD order

Proof By Lemma 1 the orders are delivered by 119911 = lceil119899119887rceil

vehicles in an optimal solution of problem 1198751 which isindependent of how the orders are scheduled on themachineHence problem 1198751 is reduced to minimizing 119863max Becausethe delivery time of each shipment is assumed to be 0 wehave 119863max = 119862max for a given solution in which 119862maxis the maximum processing completion time of orders Asa result the optimal solution of single machine makespanscheduling problemwith release dates 1|119903119895|119862max is an optimalorder processing schedule of problem 1198751 Note that ERDrule provides an optimal solution for problem 1|119903119895|119862max[30] Hence ERD rule provides an optimal order processingschedule for problem 1198751

Mathematical Problems in Engineering 3

Based on the results stated in Lemmas 1 and 2 an exactalgorithm for problem 1198751 is given as follows

Algorithm EA

Step 1 Place all orders in a list in ERD order Reindex theorders by their position in the list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Deliver the orders such that they satisfy Lemma 1

In algorithm EA the majority of computation time isused for sorting orders in Step 1 which takes119874(119899 log 119899) timeHence we have the following

Theorem 3 Algorithm EA finds an optimal solution of prob-lem 1198751 in 119874(119899 log 119899) time

3 P2 Minimizing Dtotal + T

Whendelivery cost of each shipment is ignored that is119891 = 0problem 1198752 is reduced to problem 1|119903119895|119862total which is knownto be NP-hard in the strong sense [29] Thus problem 1198752 isalsoNP-hard in the strong sense To solve problem1198752 we firststudy an auxiliary problem 119875aux Everything else in problem119875aux is the same as in problem 1198752 except that preemptionis allowed that is order processing can be preempted andresumed later We first introduce an optimality property of119875aux related to the Shortest Remaining Processing Time first(SRPT) rule which prescribes to process at each instant 119905the job with the smallest remaining processing time amongall already released unfinished jobs with some unambiguousrule for breaking ties For example the jobs can be indexedand in case of a conflict between jobs with equal remainingprocessing times the job with the smallest index wouldbe given priority such rule is called lexicographic [22]According to this definition there is exactly one SRPT orderfor a given set of jobs Moreover for two classical preemptiveproblems 1|119903119895 119901119898119905119899|119862total and 1|119903119895 119901119898119905119899|119862max the SRPTrule provides an optimal solution [28] Pruhs et al [31] alsopresented the following result

Lemma4 In the SRPT job schedule at any instant the numberof processed jobs is not smaller than in any other job schedule

According to Lemma 4 we obtain the following resultsatisfied for problem 119875aux

Lemma 5 There exists an optimal solution for problem119875119886119906119909 inwhich the orders are processed in the sequence ordered by SRPTrule on the machine

Proof Suppose that there exists an optimal solution in whichthe order processing schedule does not follow the SRPT ruleDenote the number of vehicles used in this solution by 119896lowastLet 120591119896 be the arrival time of vehicle 119896 and 119899119896 the number ofthe finished orders in time 120591119896 in this solution 119896 = 1 119896lowast

Clearly 119899119896lowast = 119899 Then the total arrival time of orders in thissolution is equal to

11989911205911 +

119896lowastminus1

sum

119896=1

(119899119896+1 minus 119899119896) 120591119896+1 =

119896lowastminus1

sum

119896=1

119899119896 (120591119896 minus 120591119896+1) + 119899120591119896lowast (1)

By Lemma 4 at any instant 119896 the number of processedorders 119899119896 in the SRPT order processing schedule is the biggestone in all order processing schedules In addition 120591119896 minus 120591119896+1 lt0 for 119896 = 1 119896

lowastminus 1 Hence we can rearrange the jobs

in the SRPT order without increasing sum119896lowastminus1119896=1 119899119896(120591119896 minus 120591119896+1)

Also because the SRPT rule provides an optimal solution for1|119903119895 119901119898119905119899|119862max the SRPT order processing schedule is theone in which 120591119896lowast is minimized Therefore we can rearrangethe customer orders in the SRPT order without increasing theobjective value

Without loss of generality we assume that the customerorders are indexed in SRPT order so that 1198621 le sdot sdot sdot le 119862119899 Thefollowing dynamic programming algorithm finds an optimalsolution for 119875aux

Algorithm DP Define value function 119881(119895) = minimum totalcost contribution of a solution for the first 119895 orders 1 119895

Initial condition 119881(0) = 0Recursive relation for 119895 = 1 119899

119881 (119895) = min 119881 (119895 minus ℎ) + ℎ sdot 119862119895 + 119891 | ℎ

= 1 min (119887 119895) (2)

Optimal solution value it is 119881(119899)

Theorem 6 Algorithm DP finds an optimal solution of prob-lem 119875119886119906119909 in 119874(119899 log 119899 + 119899119887) time

Proof By Lemma 5 in a solution for the first 119895 order1 119895 where the production schedule follows SRPT rulethe departure time of the last vehicle is always119862119895This impliesthat the delivery time of all the orders in the last vehicle is119862119895 In the recursive relation the value function is computedby trying every possible size of the last vehicle Given thatthe size of the last vehicle is ℎ the total cost contributed bythe last vehicle is ℎ sdot 119862119895 + 119891 where ℎ sdot 119862119895 is the contributionto the total delivery time and 119891 is the distribution cost Thisproves the correctness of the recursive relation and hence theoptimality of the algorithmThere are 119899 states in this DP andit takes nomore than119874(119887) time to calculate the value for eachstate Sorting the orders by SRPT rule takes 119874(119899 log 119899) timeTherefore the overall time complexity of DP is bounded by119874(119899 log 119899 + 119899119887)

Corollary 7 The value 119881(119899) generated by algorithm DP is alower bound of the optimal objective value of 1198752

Proof Because the optimal solution of 1198752 is a feasiblesolution of 119875aux the optimal objective value of 119875aux is nolarger than that of 1198752 By Theorem 6 the solution obtained

4 Mathematical Problems in Engineering

by algorithm DP is an optimal one for problem 119875aux Hencethe value 119881(119899) generated by algorithm DP is a lower boundof the optimal objective value of 1198752

In the remainder of this section we present a polynomial-time heuristic for problem 1198752 and analyze its worst-caseperformance

Heuristic HA

Step 1 Run algorithm DP to obtain a basic solution 119878lowastauxDenote the number of vehicles used in this solution by 119896lowastLet 119862aux(119895) be denoted as the processing completion time oforder 119895 for 119895 isin 119873 Place all orders in a list sorted by anincreasing order of their processing completion times119862aux(119895)in this basic solution Reindex the orders by their position inthe list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Use 119896lowast vehicles to deliver orders Each vehicle deliversthe same orders as in 119878lowastaux The departure time of each vehicleis the time when all the orders in it complete processingDenote the resulting solution by 119878lowast

Clearly the time complexity of heuristic HA is119874(119899 log 119899+119899119887) because the majority of its computation time is to runalgorithm DP Next we analyze its worst-case performanceDenote the optimal objective value of problems 1198752 and 119875auxby 119865lowast1198752 and 119865

lowastaux respectively Let 1198651198752(119878

lowast) be denoted as the

contribution of solution 119878lowast to the objective value of problem1198752

Theorem 8 1198651198752(119878lowast) le 2119865lowast1198752 that is the worst-case ratio of

heuristic HA is bounded by 2

Proof In 119878lowast let 119896 be the order in vehicle 119861119896 with largestprocessing completion time 119862119895 119896 = 1 119896

lowast Clearly 119862119896=

119863119861119896

Because there is no idle time between instant max119895isin119861119896

119903119895

and 119862119896 we have

119862119896= 119863119861

119896

le max119895isin119861119896

119903119895 +

119896

sum

ℎ=1

sum

119895isin119861ℎ

119901119895 (3)

Denote the arrival time of order 119895 in 119878lowastaux by 119863aux(119895) for119895 isin 119873 Let 119863aux(119861119896) be denoted as the arrival time of vehicle119861119896 in 119878

lowastaux 119896 = 1 119896

lowast For each order 119895 in vehicle 119861119896 wehave 119903119895 le 119862aux(119895) le 119862aux(119896) This implies that max119895isin119861

119896

119903119895 le

119862aux(119896) In addition sum119896ℎ=1sum119895isin119861ℎ

119901119895 le 119862aux(119896) Hence wehave 119862

119896le 2119862aux(119896) Consequently |119861119896|119863119861

119896

le 2|119861119896|119863aux(119861119896)Then

sum

119895isin119873

119863119895 =

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863119861119896

le 2

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863aux (119861119896) (4)

Since the numbers of vehicles used in 119878lowastaux and 119878lowast areequal the distribution cost of these solutions is also identical

Hence 1198651198752(119878lowast) le 2119865

lowastaux By Corollary 7 we have 119865

lowastaux le 119865

lowast1198752

Hence 1198651198752(119878lowast) le 2119865

lowast1198752

4 Conclusions

In this paper we have studied two variations of the integratedscheduling of production and distributionmodel with releasedates and capacitated deliveries each of which has a differentobjective function The first variation is to minimize thearrival time of the last order plus total distribution cost wefirst presented some optimality properties satisfied for thevariation Adopting these optimality properties we proposeda polynomial-time exact algorithm which implies that thisproblem is easy The second variation is to minimize totalarrival time of the orders plus total distribution cost dueto its NP-hard property we developed a heuristic for it Inthis heuristic we first construct an auxiliary problem whichis the problem with preemptions and develop a dynamicprogramming algorithm for the auxiliary problem Then wepropose the heuristic based on the solution obtained bythe dynamic programming algorithm Finally we analyze itsworse-case performance and show that its worst-case ratio isbounded by 2

For future research it will be worth extending the existingmodel to parallel machines and developing effective andefficient algorithms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by the National NaturalScience Foundation of China (71501051) the Humanitiesand Social Sciences Research Foundation of Ministry ofEducation of China (13YJC630239) and the Foundationfor Distinguished Young Teachers in Higher Education ofGuangdong Province (YQ201403)

References

[1] N G Hall and C N Potts ldquoSupply chain scheduling batchingand deliveryrdquo Operations Research vol 51 no 4 pp 566ndash5842003

[2] N G Hall and C N Potts ldquoThe coordination of scheduling andbatch deliveriesrdquo Annals of Operations Research vol 135 no 1pp 41ndash64 2005

[3] Z-L Chen and G L Vairaktarakis ldquoIntegrated scheduling ofproduction and distribution operationsrdquo Management Sciencevol 51 no 4 pp 614ndash628 2005

[4] Z-L Chen and G Pundoor ldquoOrder assignment and schedulingin a supply chainrdquo Operations Research vol 54 no 3 pp 555ndash572 2006

[5] M Ji Y He and T C E Cheng ldquoBatch delivery scheduling withbatch delivery cost on a single machinerdquo European Journal ofOperational Research vol 176 no 2 pp 745ndash755 2007

Mathematical Problems in Engineering 5

[6] C-L Li and J Ou ldquoCoordinated scheduling of customerorders with decentralized machine locationsrdquo IIE Transactions(Institute of Industrial Engineers) vol 39 no 9 pp 899ndash9092007

[7] X Wang and T C E Cheng ldquoMachine scheduling with anavailability constraint and job delivery coordinationrdquo NavalResearch Logistics vol 54 no 1 pp 11ndash20 2007

[8] R Armstrong S Gao and L Lei ldquoA zero-inventory productionand distribution problem with a fixed customer sequencerdquoAnnals of Operations Research vol 159 pp 395ndash414 2008

[9] D K Jiang and B Li ldquoSupply chain scheduling based on hybridtaboo search algorithmrdquo Journal of Mechanical Engineering vol47 no 20 pp 53ndash59 2011

[10] D-K Jiang B Li and J-Y Tan ldquoIntegrated optimizationapproach for order assignment and scheduling problemrdquo Con-trol and Decision vol 28 no 2 pp 217ndash222 2013

[11] S Li X Zhong H Li and S Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014

[12] C Viergutz and S Knust ldquoIntegrated production and distribu-tion scheduling with lifespan constraintsrdquo Annals of OperationsResearch vol 213 no 1 pp 293ndash318 2014

[13] C N Potts ldquoAnalysis of a heuristic for one machine sequencingwith release dates and delivery timesrdquoOperations Research vol28 no 6 pp 1436ndash1441 1980

[14] L A Hall and D B Shmoys ldquoJacksonrsquos rule for single-machinescheduling making a good heuristic betterrdquo Mathematics ofOperations Research vol 17 no 1 pp 22ndash35 1992

[15] S Zdrzałka ldquoPreemptive scheduling with release dates deliverytimes and sequence independent setup timesrdquoEuropean Journalof Operational Research vol 76 no 1 pp 60ndash71 1994

[16] A Gharbi and M Haouari ldquoMinimizing makespan on parallelmachines subject to release dates and delivery timesrdquo Journal ofScheduling vol 5 no 4 pp 329ndash355 2002

[17] MMastrolilli ldquoEfficient approximation schemes for schedulingproblems with release dates and delivery timesrdquo Journal ofScheduling vol 6 no 6 pp 521ndash531 2003

[18] S Seiden ldquoRandomized online scheduling with delivery timesrdquoJournal of Combinatorial Optimization vol 3 no 4 pp 399ndash416 1999

[19] J A Hoogeveen and A P Vestjens ldquoA best possible determin-istic on-line algorithm for minimizing maximum delivery timeon a single machinerdquo SIAM Journal on Discrete Mathematicsvol 13 no 1 pp 56ndash63 2000

[20] M van den Akker H Hoogeveen and N Vakhania ldquoRestartscan help in the on-line minimization of the maximum deliverytime on a single machinerdquo Journal of Scheduling vol 3 no 6pp 333ndash341 2000

[21] I Averbakh and Z Xue ldquoOn-line supply chain schedulingproblems with preemptionrdquo European Journal of OperationalResearch vol 181 no 1 pp 500ndash504 2007

[22] I Averbakh ldquoOn-line integrated production-distributionscheduling problems with capacitated deliveriesrdquo EuropeanJournal of Operational Research vol 200 no 2 pp 377ndash3842010

[23] J Fan ldquoSupply chain scheduling with jobrsquos release times on asingle machinerdquo Journal of Systems Science and MathematicalSciences vol 31 no 11 pp 1439ndash1443 2011

[24] L Lu J Yuan and L Zhang ldquoSingle machine scheduling withrelease dates and job delivery to minimize the makespanrdquoTheoretical Computer Science vol 393 no 1ndash3 pp 102ndash1082008

[25] P Liu and X Lu ldquoAn improved approximation algorithmfor single machine scheduling with job deliveryrdquo TheoreticalComputer Science vol 412 no 3 pp 270ndash274 2011

[26] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[27] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 4 pp 287ndash326 1979

[28] K R Baker Introduction to Sequencing and Scheduling JohnWiley amp Sons New York NY USA 1974

[29] J K Lenstra AH R RinnooyKan andP Brucker ldquoComplexityofmachine scheduling problemsrdquoAnnals of DiscreteMathemat-ics vol 1 pp 343ndash362 1977

[30] E L Lawler ldquoOptimal sequencing of a single machine subjectto precedence constraintsrdquo Management Science vol 19 no 5pp 544ndash546 1973

[31] K Pruhs J Sgall and E Torng ldquoOn-line schedulingrdquo inHandbook of Scheduling Algorithms Models and PerformanceAnalysis J Y-T Leung Ed CRC Press New York NY USA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Based on the results stated in Lemmas 1 and 2 an exactalgorithm for problem 1198751 is given as follows

Algorithm EA

Step 1 Place all orders in a list in ERD order Reindex theorders by their position in the list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Deliver the orders such that they satisfy Lemma 1

In algorithm EA the majority of computation time isused for sorting orders in Step 1 which takes119874(119899 log 119899) timeHence we have the following

Theorem 3 Algorithm EA finds an optimal solution of prob-lem 1198751 in 119874(119899 log 119899) time

3 P2 Minimizing Dtotal + T

Whendelivery cost of each shipment is ignored that is119891 = 0problem 1198752 is reduced to problem 1|119903119895|119862total which is knownto be NP-hard in the strong sense [29] Thus problem 1198752 isalsoNP-hard in the strong sense To solve problem1198752 we firststudy an auxiliary problem 119875aux Everything else in problem119875aux is the same as in problem 1198752 except that preemptionis allowed that is order processing can be preempted andresumed later We first introduce an optimality property of119875aux related to the Shortest Remaining Processing Time first(SRPT) rule which prescribes to process at each instant 119905the job with the smallest remaining processing time amongall already released unfinished jobs with some unambiguousrule for breaking ties For example the jobs can be indexedand in case of a conflict between jobs with equal remainingprocessing times the job with the smallest index wouldbe given priority such rule is called lexicographic [22]According to this definition there is exactly one SRPT orderfor a given set of jobs Moreover for two classical preemptiveproblems 1|119903119895 119901119898119905119899|119862total and 1|119903119895 119901119898119905119899|119862max the SRPTrule provides an optimal solution [28] Pruhs et al [31] alsopresented the following result

Lemma4 In the SRPT job schedule at any instant the numberof processed jobs is not smaller than in any other job schedule

According to Lemma 4 we obtain the following resultsatisfied for problem 119875aux

Lemma 5 There exists an optimal solution for problem119875119886119906119909 inwhich the orders are processed in the sequence ordered by SRPTrule on the machine

Proof Suppose that there exists an optimal solution in whichthe order processing schedule does not follow the SRPT ruleDenote the number of vehicles used in this solution by 119896lowastLet 120591119896 be the arrival time of vehicle 119896 and 119899119896 the number ofthe finished orders in time 120591119896 in this solution 119896 = 1 119896lowast

Clearly 119899119896lowast = 119899 Then the total arrival time of orders in thissolution is equal to

11989911205911 +

119896lowastminus1

sum

119896=1

(119899119896+1 minus 119899119896) 120591119896+1 =

119896lowastminus1

sum

119896=1

119899119896 (120591119896 minus 120591119896+1) + 119899120591119896lowast (1)

By Lemma 4 at any instant 119896 the number of processedorders 119899119896 in the SRPT order processing schedule is the biggestone in all order processing schedules In addition 120591119896 minus 120591119896+1 lt0 for 119896 = 1 119896

lowastminus 1 Hence we can rearrange the jobs

in the SRPT order without increasing sum119896lowastminus1119896=1 119899119896(120591119896 minus 120591119896+1)

Also because the SRPT rule provides an optimal solution for1|119903119895 119901119898119905119899|119862max the SRPT order processing schedule is theone in which 120591119896lowast is minimized Therefore we can rearrangethe customer orders in the SRPT order without increasing theobjective value

Without loss of generality we assume that the customerorders are indexed in SRPT order so that 1198621 le sdot sdot sdot le 119862119899 Thefollowing dynamic programming algorithm finds an optimalsolution for 119875aux

Algorithm DP Define value function 119881(119895) = minimum totalcost contribution of a solution for the first 119895 orders 1 119895

Initial condition 119881(0) = 0Recursive relation for 119895 = 1 119899

119881 (119895) = min 119881 (119895 minus ℎ) + ℎ sdot 119862119895 + 119891 | ℎ

= 1 min (119887 119895) (2)

Optimal solution value it is 119881(119899)

Theorem 6 Algorithm DP finds an optimal solution of prob-lem 119875119886119906119909 in 119874(119899 log 119899 + 119899119887) time

Proof By Lemma 5 in a solution for the first 119895 order1 119895 where the production schedule follows SRPT rulethe departure time of the last vehicle is always119862119895This impliesthat the delivery time of all the orders in the last vehicle is119862119895 In the recursive relation the value function is computedby trying every possible size of the last vehicle Given thatthe size of the last vehicle is ℎ the total cost contributed bythe last vehicle is ℎ sdot 119862119895 + 119891 where ℎ sdot 119862119895 is the contributionto the total delivery time and 119891 is the distribution cost Thisproves the correctness of the recursive relation and hence theoptimality of the algorithmThere are 119899 states in this DP andit takes nomore than119874(119887) time to calculate the value for eachstate Sorting the orders by SRPT rule takes 119874(119899 log 119899) timeTherefore the overall time complexity of DP is bounded by119874(119899 log 119899 + 119899119887)

Corollary 7 The value 119881(119899) generated by algorithm DP is alower bound of the optimal objective value of 1198752

Proof Because the optimal solution of 1198752 is a feasiblesolution of 119875aux the optimal objective value of 119875aux is nolarger than that of 1198752 By Theorem 6 the solution obtained

4 Mathematical Problems in Engineering

by algorithm DP is an optimal one for problem 119875aux Hencethe value 119881(119899) generated by algorithm DP is a lower boundof the optimal objective value of 1198752

In the remainder of this section we present a polynomial-time heuristic for problem 1198752 and analyze its worst-caseperformance

Heuristic HA

Step 1 Run algorithm DP to obtain a basic solution 119878lowastauxDenote the number of vehicles used in this solution by 119896lowastLet 119862aux(119895) be denoted as the processing completion time oforder 119895 for 119895 isin 119873 Place all orders in a list sorted by anincreasing order of their processing completion times119862aux(119895)in this basic solution Reindex the orders by their position inthe list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Use 119896lowast vehicles to deliver orders Each vehicle deliversthe same orders as in 119878lowastaux The departure time of each vehicleis the time when all the orders in it complete processingDenote the resulting solution by 119878lowast

Clearly the time complexity of heuristic HA is119874(119899 log 119899+119899119887) because the majority of its computation time is to runalgorithm DP Next we analyze its worst-case performanceDenote the optimal objective value of problems 1198752 and 119875auxby 119865lowast1198752 and 119865

lowastaux respectively Let 1198651198752(119878

lowast) be denoted as the

contribution of solution 119878lowast to the objective value of problem1198752

Theorem 8 1198651198752(119878lowast) le 2119865lowast1198752 that is the worst-case ratio of

heuristic HA is bounded by 2

Proof In 119878lowast let 119896 be the order in vehicle 119861119896 with largestprocessing completion time 119862119895 119896 = 1 119896

lowast Clearly 119862119896=

119863119861119896

Because there is no idle time between instant max119895isin119861119896

119903119895

and 119862119896 we have

119862119896= 119863119861

119896

le max119895isin119861119896

119903119895 +

119896

sum

ℎ=1

sum

119895isin119861ℎ

119901119895 (3)

Denote the arrival time of order 119895 in 119878lowastaux by 119863aux(119895) for119895 isin 119873 Let 119863aux(119861119896) be denoted as the arrival time of vehicle119861119896 in 119878

lowastaux 119896 = 1 119896

lowast For each order 119895 in vehicle 119861119896 wehave 119903119895 le 119862aux(119895) le 119862aux(119896) This implies that max119895isin119861

119896

119903119895 le

119862aux(119896) In addition sum119896ℎ=1sum119895isin119861ℎ

119901119895 le 119862aux(119896) Hence wehave 119862

119896le 2119862aux(119896) Consequently |119861119896|119863119861

119896

le 2|119861119896|119863aux(119861119896)Then

sum

119895isin119873

119863119895 =

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863119861119896

le 2

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863aux (119861119896) (4)

Since the numbers of vehicles used in 119878lowastaux and 119878lowast areequal the distribution cost of these solutions is also identical

Hence 1198651198752(119878lowast) le 2119865

lowastaux By Corollary 7 we have 119865

lowastaux le 119865

lowast1198752

Hence 1198651198752(119878lowast) le 2119865

lowast1198752

4 Conclusions

In this paper we have studied two variations of the integratedscheduling of production and distributionmodel with releasedates and capacitated deliveries each of which has a differentobjective function The first variation is to minimize thearrival time of the last order plus total distribution cost wefirst presented some optimality properties satisfied for thevariation Adopting these optimality properties we proposeda polynomial-time exact algorithm which implies that thisproblem is easy The second variation is to minimize totalarrival time of the orders plus total distribution cost dueto its NP-hard property we developed a heuristic for it Inthis heuristic we first construct an auxiliary problem whichis the problem with preemptions and develop a dynamicprogramming algorithm for the auxiliary problem Then wepropose the heuristic based on the solution obtained bythe dynamic programming algorithm Finally we analyze itsworse-case performance and show that its worst-case ratio isbounded by 2

For future research it will be worth extending the existingmodel to parallel machines and developing effective andefficient algorithms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by the National NaturalScience Foundation of China (71501051) the Humanitiesand Social Sciences Research Foundation of Ministry ofEducation of China (13YJC630239) and the Foundationfor Distinguished Young Teachers in Higher Education ofGuangdong Province (YQ201403)

References

[1] N G Hall and C N Potts ldquoSupply chain scheduling batchingand deliveryrdquo Operations Research vol 51 no 4 pp 566ndash5842003

[2] N G Hall and C N Potts ldquoThe coordination of scheduling andbatch deliveriesrdquo Annals of Operations Research vol 135 no 1pp 41ndash64 2005

[3] Z-L Chen and G L Vairaktarakis ldquoIntegrated scheduling ofproduction and distribution operationsrdquo Management Sciencevol 51 no 4 pp 614ndash628 2005

[4] Z-L Chen and G Pundoor ldquoOrder assignment and schedulingin a supply chainrdquo Operations Research vol 54 no 3 pp 555ndash572 2006

[5] M Ji Y He and T C E Cheng ldquoBatch delivery scheduling withbatch delivery cost on a single machinerdquo European Journal ofOperational Research vol 176 no 2 pp 745ndash755 2007

Mathematical Problems in Engineering 5

[6] C-L Li and J Ou ldquoCoordinated scheduling of customerorders with decentralized machine locationsrdquo IIE Transactions(Institute of Industrial Engineers) vol 39 no 9 pp 899ndash9092007

[7] X Wang and T C E Cheng ldquoMachine scheduling with anavailability constraint and job delivery coordinationrdquo NavalResearch Logistics vol 54 no 1 pp 11ndash20 2007

[8] R Armstrong S Gao and L Lei ldquoA zero-inventory productionand distribution problem with a fixed customer sequencerdquoAnnals of Operations Research vol 159 pp 395ndash414 2008

[9] D K Jiang and B Li ldquoSupply chain scheduling based on hybridtaboo search algorithmrdquo Journal of Mechanical Engineering vol47 no 20 pp 53ndash59 2011

[10] D-K Jiang B Li and J-Y Tan ldquoIntegrated optimizationapproach for order assignment and scheduling problemrdquo Con-trol and Decision vol 28 no 2 pp 217ndash222 2013

[11] S Li X Zhong H Li and S Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014

[12] C Viergutz and S Knust ldquoIntegrated production and distribu-tion scheduling with lifespan constraintsrdquo Annals of OperationsResearch vol 213 no 1 pp 293ndash318 2014

[13] C N Potts ldquoAnalysis of a heuristic for one machine sequencingwith release dates and delivery timesrdquoOperations Research vol28 no 6 pp 1436ndash1441 1980

[14] L A Hall and D B Shmoys ldquoJacksonrsquos rule for single-machinescheduling making a good heuristic betterrdquo Mathematics ofOperations Research vol 17 no 1 pp 22ndash35 1992

[15] S Zdrzałka ldquoPreemptive scheduling with release dates deliverytimes and sequence independent setup timesrdquoEuropean Journalof Operational Research vol 76 no 1 pp 60ndash71 1994

[16] A Gharbi and M Haouari ldquoMinimizing makespan on parallelmachines subject to release dates and delivery timesrdquo Journal ofScheduling vol 5 no 4 pp 329ndash355 2002

[17] MMastrolilli ldquoEfficient approximation schemes for schedulingproblems with release dates and delivery timesrdquo Journal ofScheduling vol 6 no 6 pp 521ndash531 2003

[18] S Seiden ldquoRandomized online scheduling with delivery timesrdquoJournal of Combinatorial Optimization vol 3 no 4 pp 399ndash416 1999

[19] J A Hoogeveen and A P Vestjens ldquoA best possible determin-istic on-line algorithm for minimizing maximum delivery timeon a single machinerdquo SIAM Journal on Discrete Mathematicsvol 13 no 1 pp 56ndash63 2000

[20] M van den Akker H Hoogeveen and N Vakhania ldquoRestartscan help in the on-line minimization of the maximum deliverytime on a single machinerdquo Journal of Scheduling vol 3 no 6pp 333ndash341 2000

[21] I Averbakh and Z Xue ldquoOn-line supply chain schedulingproblems with preemptionrdquo European Journal of OperationalResearch vol 181 no 1 pp 500ndash504 2007

[22] I Averbakh ldquoOn-line integrated production-distributionscheduling problems with capacitated deliveriesrdquo EuropeanJournal of Operational Research vol 200 no 2 pp 377ndash3842010

[23] J Fan ldquoSupply chain scheduling with jobrsquos release times on asingle machinerdquo Journal of Systems Science and MathematicalSciences vol 31 no 11 pp 1439ndash1443 2011

[24] L Lu J Yuan and L Zhang ldquoSingle machine scheduling withrelease dates and job delivery to minimize the makespanrdquoTheoretical Computer Science vol 393 no 1ndash3 pp 102ndash1082008

[25] P Liu and X Lu ldquoAn improved approximation algorithmfor single machine scheduling with job deliveryrdquo TheoreticalComputer Science vol 412 no 3 pp 270ndash274 2011

[26] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[27] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 4 pp 287ndash326 1979

[28] K R Baker Introduction to Sequencing and Scheduling JohnWiley amp Sons New York NY USA 1974

[29] J K Lenstra AH R RinnooyKan andP Brucker ldquoComplexityofmachine scheduling problemsrdquoAnnals of DiscreteMathemat-ics vol 1 pp 343ndash362 1977

[30] E L Lawler ldquoOptimal sequencing of a single machine subjectto precedence constraintsrdquo Management Science vol 19 no 5pp 544ndash546 1973

[31] K Pruhs J Sgall and E Torng ldquoOn-line schedulingrdquo inHandbook of Scheduling Algorithms Models and PerformanceAnalysis J Y-T Leung Ed CRC Press New York NY USA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

by algorithm DP is an optimal one for problem 119875aux Hencethe value 119881(119899) generated by algorithm DP is a lower boundof the optimal objective value of 1198752

In the remainder of this section we present a polynomial-time heuristic for problem 1198752 and analyze its worst-caseperformance

Heuristic HA

Step 1 Run algorithm DP to obtain a basic solution 119878lowastauxDenote the number of vehicles used in this solution by 119896lowastLet 119862aux(119895) be denoted as the processing completion time oforder 119895 for 119895 isin 119873 Place all orders in a list sorted by anincreasing order of their processing completion times119862aux(119895)in this basic solution Reindex the orders by their position inthe list Set current instant 119905 = 1199031

Step 2 Choose the first unscheduled order 119895 of the listSchedule it to process at instant 1199051015840 = max119905 119903119895 Set 119905 = 119905

1015840+119901119895

Repeat Step 2 until all orders are scheduled

Step 3 Use 119896lowast vehicles to deliver orders Each vehicle deliversthe same orders as in 119878lowastaux The departure time of each vehicleis the time when all the orders in it complete processingDenote the resulting solution by 119878lowast

Clearly the time complexity of heuristic HA is119874(119899 log 119899+119899119887) because the majority of its computation time is to runalgorithm DP Next we analyze its worst-case performanceDenote the optimal objective value of problems 1198752 and 119875auxby 119865lowast1198752 and 119865

lowastaux respectively Let 1198651198752(119878

lowast) be denoted as the

contribution of solution 119878lowast to the objective value of problem1198752

Theorem 8 1198651198752(119878lowast) le 2119865lowast1198752 that is the worst-case ratio of

heuristic HA is bounded by 2

Proof In 119878lowast let 119896 be the order in vehicle 119861119896 with largestprocessing completion time 119862119895 119896 = 1 119896

lowast Clearly 119862119896=

119863119861119896

Because there is no idle time between instant max119895isin119861119896

119903119895

and 119862119896 we have

119862119896= 119863119861

119896

le max119895isin119861119896

119903119895 +

119896

sum

ℎ=1

sum

119895isin119861ℎ

119901119895 (3)

Denote the arrival time of order 119895 in 119878lowastaux by 119863aux(119895) for119895 isin 119873 Let 119863aux(119861119896) be denoted as the arrival time of vehicle119861119896 in 119878

lowastaux 119896 = 1 119896

lowast For each order 119895 in vehicle 119861119896 wehave 119903119895 le 119862aux(119895) le 119862aux(119896) This implies that max119895isin119861

119896

119903119895 le

119862aux(119896) In addition sum119896ℎ=1sum119895isin119861ℎ

119901119895 le 119862aux(119896) Hence wehave 119862

119896le 2119862aux(119896) Consequently |119861119896|119863119861

119896

le 2|119861119896|119863aux(119861119896)Then

sum

119895isin119873

119863119895 =

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863119861119896

le 2

119896lowast

sum

119896=1

10038161003816100381610038161198611198961003816100381610038161003816119863aux (119861119896) (4)

Since the numbers of vehicles used in 119878lowastaux and 119878lowast areequal the distribution cost of these solutions is also identical

Hence 1198651198752(119878lowast) le 2119865

lowastaux By Corollary 7 we have 119865

lowastaux le 119865

lowast1198752

Hence 1198651198752(119878lowast) le 2119865

lowast1198752

4 Conclusions

In this paper we have studied two variations of the integratedscheduling of production and distributionmodel with releasedates and capacitated deliveries each of which has a differentobjective function The first variation is to minimize thearrival time of the last order plus total distribution cost wefirst presented some optimality properties satisfied for thevariation Adopting these optimality properties we proposeda polynomial-time exact algorithm which implies that thisproblem is easy The second variation is to minimize totalarrival time of the orders plus total distribution cost dueto its NP-hard property we developed a heuristic for it Inthis heuristic we first construct an auxiliary problem whichis the problem with preemptions and develop a dynamicprogramming algorithm for the auxiliary problem Then wepropose the heuristic based on the solution obtained bythe dynamic programming algorithm Finally we analyze itsworse-case performance and show that its worst-case ratio isbounded by 2

For future research it will be worth extending the existingmodel to parallel machines and developing effective andefficient algorithms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by the National NaturalScience Foundation of China (71501051) the Humanitiesand Social Sciences Research Foundation of Ministry ofEducation of China (13YJC630239) and the Foundationfor Distinguished Young Teachers in Higher Education ofGuangdong Province (YQ201403)

References

[1] N G Hall and C N Potts ldquoSupply chain scheduling batchingand deliveryrdquo Operations Research vol 51 no 4 pp 566ndash5842003

[2] N G Hall and C N Potts ldquoThe coordination of scheduling andbatch deliveriesrdquo Annals of Operations Research vol 135 no 1pp 41ndash64 2005

[3] Z-L Chen and G L Vairaktarakis ldquoIntegrated scheduling ofproduction and distribution operationsrdquo Management Sciencevol 51 no 4 pp 614ndash628 2005

[4] Z-L Chen and G Pundoor ldquoOrder assignment and schedulingin a supply chainrdquo Operations Research vol 54 no 3 pp 555ndash572 2006

[5] M Ji Y He and T C E Cheng ldquoBatch delivery scheduling withbatch delivery cost on a single machinerdquo European Journal ofOperational Research vol 176 no 2 pp 745ndash755 2007

Mathematical Problems in Engineering 5

[6] C-L Li and J Ou ldquoCoordinated scheduling of customerorders with decentralized machine locationsrdquo IIE Transactions(Institute of Industrial Engineers) vol 39 no 9 pp 899ndash9092007

[7] X Wang and T C E Cheng ldquoMachine scheduling with anavailability constraint and job delivery coordinationrdquo NavalResearch Logistics vol 54 no 1 pp 11ndash20 2007

[8] R Armstrong S Gao and L Lei ldquoA zero-inventory productionand distribution problem with a fixed customer sequencerdquoAnnals of Operations Research vol 159 pp 395ndash414 2008

[9] D K Jiang and B Li ldquoSupply chain scheduling based on hybridtaboo search algorithmrdquo Journal of Mechanical Engineering vol47 no 20 pp 53ndash59 2011

[10] D-K Jiang B Li and J-Y Tan ldquoIntegrated optimizationapproach for order assignment and scheduling problemrdquo Con-trol and Decision vol 28 no 2 pp 217ndash222 2013

[11] S Li X Zhong H Li and S Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014

[12] C Viergutz and S Knust ldquoIntegrated production and distribu-tion scheduling with lifespan constraintsrdquo Annals of OperationsResearch vol 213 no 1 pp 293ndash318 2014

[13] C N Potts ldquoAnalysis of a heuristic for one machine sequencingwith release dates and delivery timesrdquoOperations Research vol28 no 6 pp 1436ndash1441 1980

[14] L A Hall and D B Shmoys ldquoJacksonrsquos rule for single-machinescheduling making a good heuristic betterrdquo Mathematics ofOperations Research vol 17 no 1 pp 22ndash35 1992

[15] S Zdrzałka ldquoPreemptive scheduling with release dates deliverytimes and sequence independent setup timesrdquoEuropean Journalof Operational Research vol 76 no 1 pp 60ndash71 1994

[16] A Gharbi and M Haouari ldquoMinimizing makespan on parallelmachines subject to release dates and delivery timesrdquo Journal ofScheduling vol 5 no 4 pp 329ndash355 2002

[17] MMastrolilli ldquoEfficient approximation schemes for schedulingproblems with release dates and delivery timesrdquo Journal ofScheduling vol 6 no 6 pp 521ndash531 2003

[18] S Seiden ldquoRandomized online scheduling with delivery timesrdquoJournal of Combinatorial Optimization vol 3 no 4 pp 399ndash416 1999

[19] J A Hoogeveen and A P Vestjens ldquoA best possible determin-istic on-line algorithm for minimizing maximum delivery timeon a single machinerdquo SIAM Journal on Discrete Mathematicsvol 13 no 1 pp 56ndash63 2000

[20] M van den Akker H Hoogeveen and N Vakhania ldquoRestartscan help in the on-line minimization of the maximum deliverytime on a single machinerdquo Journal of Scheduling vol 3 no 6pp 333ndash341 2000

[21] I Averbakh and Z Xue ldquoOn-line supply chain schedulingproblems with preemptionrdquo European Journal of OperationalResearch vol 181 no 1 pp 500ndash504 2007

[22] I Averbakh ldquoOn-line integrated production-distributionscheduling problems with capacitated deliveriesrdquo EuropeanJournal of Operational Research vol 200 no 2 pp 377ndash3842010

[23] J Fan ldquoSupply chain scheduling with jobrsquos release times on asingle machinerdquo Journal of Systems Science and MathematicalSciences vol 31 no 11 pp 1439ndash1443 2011

[24] L Lu J Yuan and L Zhang ldquoSingle machine scheduling withrelease dates and job delivery to minimize the makespanrdquoTheoretical Computer Science vol 393 no 1ndash3 pp 102ndash1082008

[25] P Liu and X Lu ldquoAn improved approximation algorithmfor single machine scheduling with job deliveryrdquo TheoreticalComputer Science vol 412 no 3 pp 270ndash274 2011

[26] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[27] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 4 pp 287ndash326 1979

[28] K R Baker Introduction to Sequencing and Scheduling JohnWiley amp Sons New York NY USA 1974

[29] J K Lenstra AH R RinnooyKan andP Brucker ldquoComplexityofmachine scheduling problemsrdquoAnnals of DiscreteMathemat-ics vol 1 pp 343ndash362 1977

[30] E L Lawler ldquoOptimal sequencing of a single machine subjectto precedence constraintsrdquo Management Science vol 19 no 5pp 544ndash546 1973

[31] K Pruhs J Sgall and E Torng ldquoOn-line schedulingrdquo inHandbook of Scheduling Algorithms Models and PerformanceAnalysis J Y-T Leung Ed CRC Press New York NY USA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

[6] C-L Li and J Ou ldquoCoordinated scheduling of customerorders with decentralized machine locationsrdquo IIE Transactions(Institute of Industrial Engineers) vol 39 no 9 pp 899ndash9092007

[7] X Wang and T C E Cheng ldquoMachine scheduling with anavailability constraint and job delivery coordinationrdquo NavalResearch Logistics vol 54 no 1 pp 11ndash20 2007

[8] R Armstrong S Gao and L Lei ldquoA zero-inventory productionand distribution problem with a fixed customer sequencerdquoAnnals of Operations Research vol 159 pp 395ndash414 2008

[9] D K Jiang and B Li ldquoSupply chain scheduling based on hybridtaboo search algorithmrdquo Journal of Mechanical Engineering vol47 no 20 pp 53ndash59 2011

[10] D-K Jiang B Li and J-Y Tan ldquoIntegrated optimizationapproach for order assignment and scheduling problemrdquo Con-trol and Decision vol 28 no 2 pp 217ndash222 2013

[11] S Li X Zhong H Li and S Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014

[12] C Viergutz and S Knust ldquoIntegrated production and distribu-tion scheduling with lifespan constraintsrdquo Annals of OperationsResearch vol 213 no 1 pp 293ndash318 2014

[13] C N Potts ldquoAnalysis of a heuristic for one machine sequencingwith release dates and delivery timesrdquoOperations Research vol28 no 6 pp 1436ndash1441 1980

[14] L A Hall and D B Shmoys ldquoJacksonrsquos rule for single-machinescheduling making a good heuristic betterrdquo Mathematics ofOperations Research vol 17 no 1 pp 22ndash35 1992

[15] S Zdrzałka ldquoPreemptive scheduling with release dates deliverytimes and sequence independent setup timesrdquoEuropean Journalof Operational Research vol 76 no 1 pp 60ndash71 1994

[16] A Gharbi and M Haouari ldquoMinimizing makespan on parallelmachines subject to release dates and delivery timesrdquo Journal ofScheduling vol 5 no 4 pp 329ndash355 2002

[17] MMastrolilli ldquoEfficient approximation schemes for schedulingproblems with release dates and delivery timesrdquo Journal ofScheduling vol 6 no 6 pp 521ndash531 2003

[18] S Seiden ldquoRandomized online scheduling with delivery timesrdquoJournal of Combinatorial Optimization vol 3 no 4 pp 399ndash416 1999

[19] J A Hoogeveen and A P Vestjens ldquoA best possible determin-istic on-line algorithm for minimizing maximum delivery timeon a single machinerdquo SIAM Journal on Discrete Mathematicsvol 13 no 1 pp 56ndash63 2000

[20] M van den Akker H Hoogeveen and N Vakhania ldquoRestartscan help in the on-line minimization of the maximum deliverytime on a single machinerdquo Journal of Scheduling vol 3 no 6pp 333ndash341 2000

[21] I Averbakh and Z Xue ldquoOn-line supply chain schedulingproblems with preemptionrdquo European Journal of OperationalResearch vol 181 no 1 pp 500ndash504 2007

[22] I Averbakh ldquoOn-line integrated production-distributionscheduling problems with capacitated deliveriesrdquo EuropeanJournal of Operational Research vol 200 no 2 pp 377ndash3842010

[23] J Fan ldquoSupply chain scheduling with jobrsquos release times on asingle machinerdquo Journal of Systems Science and MathematicalSciences vol 31 no 11 pp 1439ndash1443 2011

[24] L Lu J Yuan and L Zhang ldquoSingle machine scheduling withrelease dates and job delivery to minimize the makespanrdquoTheoretical Computer Science vol 393 no 1ndash3 pp 102ndash1082008

[25] P Liu and X Lu ldquoAn improved approximation algorithmfor single machine scheduling with job deliveryrdquo TheoreticalComputer Science vol 412 no 3 pp 270ndash274 2011

[26] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[27] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 4 pp 287ndash326 1979

[28] K R Baker Introduction to Sequencing and Scheduling JohnWiley amp Sons New York NY USA 1974

[29] J K Lenstra AH R RinnooyKan andP Brucker ldquoComplexityofmachine scheduling problemsrdquoAnnals of DiscreteMathemat-ics vol 1 pp 343ndash362 1977

[30] E L Lawler ldquoOptimal sequencing of a single machine subjectto precedence constraintsrdquo Management Science vol 19 no 5pp 544ndash546 1973

[31] K Pruhs J Sgall and E Torng ldquoOn-line schedulingrdquo inHandbook of Scheduling Algorithms Models and PerformanceAnalysis J Y-T Leung Ed CRC Press New York NY USA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of