Int. J. Production Economics 145 (2013) 38–52

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Int. J. Production Economics

0925-52

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijpe

A hybrid genetic algorithm for the job shop scheduling problem withpractical considerations for manufacturing costs: Investigationsmotivated by vehicle production

Rui Zhang a,n, Pei-Chann Chang b, Cheng Wu c

a School of Economics and Management, Nanchang University, Nanchang 330031, PR Chinab Department of Information Management, Yuan Ze University, Chung-Li 32023, Taiwanc Department of Automation, Tsinghua University, Beijing 100084, PR China

a r t i c l e i n f o

Article history:

Received 3 February 2012

Accepted 20 October 2012Available online 8 November 2012

Keywords:

Shop scheduling

Manufacturing cost

Genetic algorithm

Local search

73/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.ijpe.2012.10.024

esponding author.

ail address: r.zhang@ymail.com (R. Zhang).

a b s t r a c t

This paper studies a job shop scheduling problem with two new objective functions based on the setup

and synergy costs besides the traditional total weighted tardiness criterion. The background is found in

the real-world situation of a commercial vehicle producer, where the reduction of manufacturing costs

has become a significant concern like in many heavy industry firms. The cost-related objective

functions have been modeled in a quite general way so that they can also be applied to other similar

types of production. To tackle this multi-objective scheduling problem, the paper presents a Pareto-

based genetic algorithm incorporating a local search module, which utilizes the neighborhood

properties specifically developed for each objective function. The computational experiments on both

real-world and randomly generated scheduling instances verify the effectiveness of the proposed

approach. The research presented in this paper could shed some light on the modeling and heuristic

solving of practical production scheduling problems.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Production scheduling is a crucial decision process which directlyaffects the operational efficiency of manufacturing firms. Amongothers, the job shop scheduling problem (JSSP) has long beenadopted as a basic model for the scheduling research (Ramasesh,1990; Ahmed and Fisher, 1992; Sabuncuoglu and Comlekci, 2002;Liu and Kozan, 2011). However, most variants of JSSP are NP-hardin the strong sense and thus defy ordinary solution methods.Enumerative approaches, such as the branch-and-bound algorithmof Carlier and Pinson (1989), can only conquer small-scale probleminstances. The most common heuristic methods devised in the earlydays include dispatching rules (Sculli, 1980; Green and Appel, 1981;Kanet and Hayya, 1982; Baker, 1984) and shifting bottleneck (Adamset al., 1988; Holtsclaw and Uzsoy, 1996; Balas and Vazacopoulos,1998; Liu and Kozan, 2012). In recent years, meta-heuristic algo-rithms, such as simulated annealing (SA) (Van Laarhoven et al., 1992;Zhang and Wu, 2011), genetic algorithm (GA) (Zobolas et al., 2009;Al-Hinai and ElMekkawy, 2011), scatter search (SS) (Sels et al., 2011),tabu search (TS) (Nowicki and Smutnicki, 1996; Zhu et al., 2010;Eshlaghy and Sheibatolhamdy, 2011), particle swarm optimization

ll rights reserved.

(PSO) (Moslehi and Mahnam, 2011), have clearly become theresearch focus in practical optimization methods for solving JSSPs.

In these traditional scheduling models, the objective functiononly includes measures on production efficiency (like makespanand due date related performances). However, in some manufac-turing sectors (especially in heavy industry), the firms are alsoconcerned with the minimization of production costs and energyconsumption. In this paper, we study a multi-objective produc-tion scheduling problem which is modeled on the basis of themanufacturing system of F Company, a listed company in Chinawhich specializes in the production of commercial vehicles.

The manufacturing system of F Company mainly consists ofthree workshops, i.e. the body shop, the coating shop and theassembly shop. The manufacturing process of a vehicle (pickuptrucks, middle buses or SUVs) passes through the three work-shops in order. However, these workshops do have quite differentcriteria to define ‘‘good’’ production schedules.

(1)

The body shop requires that the bodies of identical or similartypes should be processed as closely as possible, so that the costfor equipment switching and setup operations can be reduced.

(2)

The coating shop requires that the parts with identical orsimilar colors should be processed as closely as possible, sothat the cost for preparing and changing the paint can beminimized.

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 39

(3)

The assembly shop requires that the vehicles with differentlevels of complexity should be processed in a staggeredmanner (difficult jobs alternating with easy jobs), so thatthe production intensity can be smoothed and thus theundesirable accumulation or tardiness of jobs can be reduced.

Under such circumstances, traditional single-objective sche-duling models are incapable of guaranteeing a satisfactory solu-tion set. So we must resort to multi-objective optimizationmodels. Indeed, we will focus on a multi-objective job shopscheduling problem (MOJSSP) in this paper.

Multi-objective scheduling is quite different from single-objective scheduling in the following two aspects:

(1)

There usually exist direct conflicts between different objec-tives under consideration. This means, in order to obtainimprovement on one objective, some other objectives mayhave to worsen. Therefore, if the objective functions are notpositively correlated with each other, there will not exist asingle solution that is optimal with respect to each of theobjectives.

(2)

The users (production managers) require that the optimiza-tion algorithm output a set of satisfactory solutions (ratherthan one solution as in the single-objective case) which aresufficient in quantity and also distribute evenly. Then, thedecision-maker can choose the most suitable schedulingpolicy from this set based on the particular scenario that he/she faces (e.g., if delivery timeliness is the current bottleneck,it may be necessary to select the schedules that are biased infavor of tardiness-related objective functions).

The rest of the paper is organized as follows. Section 2provides a brief review on the production scheduling in auto-motive manufacturing and the concepts of multi-objective evolu-tionary algorithms. Section 3 gives a formal description of theproblem studied in this paper. Section 4 presents some neighbor-hood properties for minimizing each of the objective functionsunder investigation. Section 5 proposes a genetic local search(GLS) algorithm for solving the problem. The computationalresults are displayed in Sections 6 and 7. Finally, some conclu-sions are drawn in Section 8.

1 The ROADEF challenge is organized by the French Society of Operations

Research and Decision Aid every two years in order to allow theoretical

researchers to face up to a complex decisional problem occurring in industry.2 Genetic drift refers to the situation in which GA converges to a single

solution due to random selection errors. This is the most serious problem that

MOEA should avoid.3 Elitism refers to the usage of an external archive to record the elite

individuals of the current generation, some of which may be passed down to

the next generation.

2. Literature review

2.1. Scheduling in automotive manufacturing

The research on scheduling in the vehicle industry has mainlybeen focused on the car sequencing problem, which was firstdescribed by Parrello et al. (1986). The problem involves sequen-cing cars along an assembly line, where different options(e.g. sunroof, air-conditioning, among others) are installed onthe cars according to customer orders. Each option is installedby a different station, which can handle at most a certainproportion of cars passing consecutively through the assemblyline. Therefore, the cars requiring the same option have to bespaced (i.e., the ‘‘difficult’’ cars must be sufficiently far apart in theproduction sequence). Such restrictions are modeled as the ratio

constraints. For example, for the r-th option, the ratio constraintnr=pr ¼ 4=7 indicates that in any subsequence of 7 cars, thereshould be no more than 4 cars requiring this option. The decisionproblem is to decide whether it is possible to find a sequencewhich satisfies all the ratio constraints, while the optimizationproblem is to find a sequence with the minimum number ofconstraint violations.

Due to the high complexity of the car sequencing problem, theoptimization softwares based on constraint programming andinteger programming reach the limit when considering about 100cars with only a few options. Therefore, recent works on thisproblem have focused on the meta-heuristic algorithms. Inparticular, genetic algorithm (Zinflou et al., 2010) and ant colonyoptimization (Gravel et al., 2005; Morin et al., 2009) approacheshave been reported.

Remarkably, the ROADEF’2005 challenge1 proposed by theFrench car manufacturer Renault aroused a new round of researchon the car sequencing problem. In particular, the Renault sequen-cing problem differs from the above-mentioned standard carsequencing problem in the following aspects: (1) The ratioconstraints (violations of which are penalized in the objectivefunction) are further divided into two categories according to thecriticality level of the options. (2) The requirement of the coatingshop (i.e. minimizing the number of color switches) is considered.Based on these features and the fact that the coating andassembly shops in Renault process the same vehicle sequence, alexicographic multi-objective sequencing problem can be definedby assigning mutually distant weights to each objective. After thecompetition of this year, the European Journal of Operational

Research published a feature cluster dedicated to the leadingalgorithms addressing the challenge (Solnon et al., 2008). Becauseof the large size of the real-world instances and the strict runtimerequirement (10 min), all the successful algorithms reported arebased on heuristic approaches.

2.2. Multi-objective evolutionary algorithms

The potential of evolutionary algorithms for solving multi-objective optimization problems was first pointed out byRosenberg (1967). But the first practicable multi-objective evolu-tionary algorithm (MOEA) should be attributed to the VEGA(Vector Evaluation Genetic Algorithm) which was proposed bySchaffer (1984) for machine learning algorithms. Remarkably,there has appeared an increasing number of literature on MOEAsince the mid 1990s, especially after the launch of the biennialInternational Conference on Evolutionary Multi-Criterion Optimi-zation in 2001.

From the historical perspective, the existing algorithms can bedivided into two generations. The first generation is characterizedby the niching technique and Pareto ranking, while the secondgeneration is featured by the elitism strategy. The aim of nichingis to reduce the occurrence of genetic drift2 and guide GA toexplore multiple optimal areas in the solution space. Paretoranking is applied to sort the individuals in the populationaccording to their dominance relations. Typical algorithms thatbelong to first-generation MOEAs include NSGA (Non-dominatedSorting Genetic Algorithm) and NPGA (Niched-Pareto GeneticAlgorithm). The second generation of MOEAs introduce elitism3

and thus improve the original NSGA and NPGA to form NSGA-II(Deb et al., 2002) and NPGA 2 (Erickson et al., 2001), respectively.

In order to enhance the searching ability, the recent trend is tocombine GA with other efficient local search mechanisms, leading togenetic local search (GLS) algorithms, for example (Essafi et al., 2008).

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5240

GLS (sometimes referred to as memetic algorithms) utilizes theevolutionary framework of GA to perform large-scale explorationand relies on a finer-granular local search procedure to exploit smallsolution regions. Remarkably, GLS has also been applied for multi-objective optimization, leading to multi-objective genetic local search(MOGLS) algorithms. The representative works are the IM-MOGLSproposed by Ishibuchi and Murata (1998) and the J-MOGLS proposedby Jaszkiewicz (2002). The algorithm discussed in this paper alsobelongs to this category. It uses Pareto ranking as the first order offitness (same as NSGA-II) and then a spread metric (different thanNSGA-II) as the second order of fitness. Meanwhile, our algorithmincludes specially designed neighborhood search mechanisms to helppromote the overall optimization effect.

2.3. Differences between our work and the existing ones

From the previous introduction, the Renault car sequencingproblem has some similarity with the scheduling problem in thispaper. However, there exist noticeable differences between the two.

(1)

4

lows

subj

one

on.

The Renault problem is only a sequencing problem, where it isnecessary to decide a same production sequence for bothworkshops (coating and assembly). By contrast, the problemdiscussed in this paper is a job shop scheduling problem, wherethe aim is to decide a set of mutually interrelated sequences foreach workshop. The job shop model is appropriate for describingthe MTO (make-to-order) production environment with adiverse range of products. With the job shop model, the entiremanufacturing process could be managed in an integratedmanner, without neglecting the body shop. Meanwhile, sincejob shop includes flow shop as a special case, the modeling ofthe problem as job shop scheduling makes it easier to generalizethe research results to other industries where scheduling ismore complex than sequencing.

(2)

The objective functions are quite different:� The Renault problem assumes that the body shop is un-

important for making a production schedule, and thus thepreference of the body shop is completely neglected. How-ever, the adjustment of welding machines in the body shoprequires considerable human efforts and is prone to causedelays, which may affect the operations in the subsequentcoating shop. For this reason, the requirement of the bodyshop is explicitly considered in our scheduling model.� Meanwhile, the ratio constraints studied since 1986 are in

fact purely theoretical (the number of violations does nothave a practical implication on manufacturing cost). In themodern factories, the application of movable workstationsgreatly alleviates the pressure of processing successivecars with the same option. Thus, the preference of theassembly shop can be taken care of by the shop itself andis not incorporated into our scheduling model.� Up till now, most research on JSSP has been focused on the

makespan criterion. However, due date related performancesare becoming more significant in the fiercely competitivevehicle market. In this sense, the total weighted tardiness(TWT) measure reflects the critical factors related with thefirm’s profitability more accurately than makespan. Thus, weconsider the TWT objective in this paper.

Th

. Gi

ect t

opti

Obv

(3)

The Renault problem has been formulated as a lexicographic

multi-objective optimization problem4, which leads to a single

e lexicographic multi-objective optimization problem is defined as fol-

ven N objective functions f 1ðxÞ,f 2ðxÞ, . . . ,f NðxÞ, first one optimizes f 1ðxÞo xAX (the solution space), obtaining a set of optimal solutions Sn

1, next

mizes f 2ðxÞ subject to xASn

1, obtaining a set of optimal solutions Sn

2, and so

iously, Sn

1 +Sn

2 + � � �+Sn

N . If 9Sn

i 9¼ 1 for some i with 1r irN, the

(foot

rem

mult

obje

obje

output solution rather than a set of Pareto optimal solutions. Inother words, solving the Renault problem by whatever algo-rithm cannot produce a set of well compromised solutions(representing the tradeoffs between the coating shop and theassembly shop). By contrast, our problem is addressed by amulti-objective optimization algorithm in the real sense. A set ofPareto non-dominated solutions will be output by the algorithmfor the decision makers to choose from.

3. Problem description

In the body shop, the most dominant operation is welding,which is performed by a series of welding robots installed alongthe production line. Most welding robots have six programmableaxes, and the welding tool (e.g. a welding tong) is connected tothe end axis. All the axes are driven by AC servomotors. Particu-larly, axes 1, 2 and 3 are responsible for sending the welding toolto the desired position in space, and axes 4, 5 and 6 areresponsible for adjusting the attitude (orientation) of the toolfor a specific task. If the model of the current car body is differentfrom the previous one, some welding robots need to be ‘‘reset’’since the positions and technological requirements for weldingwill generally change. The reset procedure may include thedownload of new programs, the replacement of welding toolsand the operation of relevant servomotors, which need humanparticipation and increase the instantaneous power consumption.Therefore, the body shop desires to smooth the labor workloadand energy consumption, and to achieve this, car bodies of thesame or similar models should be processed contiguously wherepossible.

In the coating shop, the paint equipments (e.g. spray guns)have to be cleaned (using a special solvent) if the color of the nextcar to be painted is different from the previous car. In accordancewith the chemical engineering standards, the type of solvent andthe required amount are pre-specified for the switches betweenany two colors that are possible to occur in the workshop.In general, the larger the difference in color, the more solvent willbe consumed. Since the solvents must be bought from an outsidesupplier and most types of solvents are quite costly, a majorconcern of the coating shop is to minimize the consumption ofpaint solvents. In order to achieve this, frequent and radical colorchanges in the production process should be avoided.

The preference that the vehicles with different complexitylevels should be assembled in a staggered order is not reflected inthis study. In many car manufacturers (including F Company), theproduction sequence of the assembly shop can be different fromthe production sequence of the coating shop due to the existenceof buffers. In addition, since the assembly shop constitutes thefinal procedure in the entire manufacturing process, the shopactually has a sufficient degree of freedom to locally adjust (fine-tune) the job sequence given by the complete schedule in order tosuit its own needs, without severely affecting the overall produc-tion performance. Therefore, the requirement of the assemblyshop is not regarded as an objective within the schedulingproblem but is left to the independent treatment of the assemblyshop itself.

With the fast increase of product diversity and the risingrequirement of customization, most automotive companiesare adopting the MTO production strategy instead of the make-to-stock mode. Therefore, the job shop model is suitable for

note continued)

aining objective functions can be left out of consideration. The lexicographic

i-objective optimization problem can be converted to an equivalent single-

ctive optimization problem by assigning widely distant weights to each

ctive, i.e. letting f ðxÞ ¼PN

i ¼ 1 wif iðxÞ, such that w1 bw2 b � � �bwN .

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 41

characterizing the scheduling of automotive manufacturing pro-cesses, and the due-date-related performance measure is impor-tant for the companies emphasizing customer service quality.

Based on the above explanation and necessary mathematicalabstraction, in this paper we are dealing with a job shopscheduling problem with three objectives:

(1)

5

will

The total weighted tardiness of all jobs

f 1ðpÞ ¼Xn

j ¼ 1

wujTujðpÞ, ð1Þ

where p denotes a feasible schedule; wujis the weight of job j

and TujðpÞ is the tardiness of job j in the current schedule

(uj represents the last operation of job j). n is the number ofjobs in the problem.

(2)

The closeness in model types for the job sequence on machine k1

f 2ðpÞ ¼Xn

j ¼ 2

ðMTpðk1 ,jÞ�MTpðk1 ,j�1ÞÞ2, ð2Þ

where MTpðk1 ,jÞ denotes the model type of the j-th job processedby machine k1 (represented by an integer between 1 and 10).

(3)

The closeness in color types for the job sequence on machine k2

f 3ðpÞ ¼Xn

j ¼ 2

ðCTpðk2 ,jÞ�CTpðk2 ,j�1ÞÞ2, ð3Þ

where CTpðk2 ,jÞ denotes the color type of the j-th job processed bymachine k2 (also represented by integers from 1 to 10).

Minimizing f 2 means that machine k1 should process in anorder where jobs are clustered according to their model types, inthe hope of reducing the cost in tool switch. Minimizing f 3 meansthat the jobs on k2 should be processed in an order where similarcolor types are clustered so as to reduce the cost of preparing andswitching paints. In the formulae of f2 and f3, we use 10 numbersto describe the similarity/distance between the jobs with respectto either model or color. This is certainly a simplification policy inorder to define a universal objective function, which facilitatesthe generation of random instances for benchmarking and com-paring algorithms. Indeed, using such a definition makes theproblem more difficult to solve, because the objective value isnow very sensitive to slight changes in the schedule,5 and thus ahard compromise has to be made by the optimization algorithm.However, in the computational experiment for the real-worldinstance from F Company (Section 6), we will use the real costdata collected from the MES (Manufacturing Execution System)and ERP (Enterprise Resource Planning) systems of the firm tocompute these objective functions.

It is noticeable that the first objective (f 1 ¼ TWT) is a globalcriterion, which is influenced by the performance of each work-shop. If each workshop arranges the production sequence com-pletely according to its own criterion, then the overall TWT coulddeteriorate dramatically. Therefore, both f2 (based on the bodyshop preference) and f3 (based on the coating shop preference)have to be compromised with f1 in the optimization process,leading to a set of Pareto optimal solutions.

In sum, the optimization objectives considered in this paper are

min f ðpÞ ¼ ðf 1ðpÞ,f 2ðpÞ,f 3ðpÞÞ: ð4Þ

The motivation of considering f2 and f3 comes from practical needs.By contrast, the objectives studied in the existing literature are

By contrast, in the real-world situation, the possible values for switch cost

not be so widely different as ð10�1Þ2 : ð2�1Þ2 ¼ 81 : 1.

almost limited to theoretical functions, such as

�

Completion time based objective functions: makespan, aver-age completion time, maximum flow time, total flow time, etc. � Due date related objective functions: maximum tardiness,

number of tardy jobs, etc.

� Machine performance oriented objective functions: utilization

ratio of critical machines, average idle time, etc.

4. Neighborhood properties

Neighborhood properties which are closely based on thespecific characteristics of the pending problem is vital for theoptimization process. According to the No Free Lunch Theorem(NFLT) proven by Wolpert and Macready (1997), all algorithmshave identical performance when averaged across all possibleproblems. The NFLT implies that methods must incorporateproblem-specific information to improve performance on a subsetof problems. Otherwise, heuristic methods will, on average, not beable to achieve greater performance than other heuristics. Moti-vated by the findings, in this section we discuss relevant neigh-borhood properties that can be used by the local search procedurein the proposed GLS.

4.1. The neighborhood properties for minimizing f1

The objective f1 is the sum of weighted tardiness of all jobs. Soin this part, we consider the job shop scheduling problem withthe objective of minimizing total weighted tardiness (abbreviatedas JSSP–TWT).

4.1.1. The mathematical model and its duality

We utilize the concept of disjunctive graph for formulatingJSSP–TWT. A graph GðN,A,EÞ can be associated with a JSSP–TWTinstance which includes n jobs and m machines. N¼O [ f0g ¼0,1,2, . . . ,n�mf g is the set of nodes, where O¼ 1, . . . ,n�mf g

corresponds to the operation set of the JSSP instance. Node0 stands for a dummy operation which starts before all the realoperations, while the starting time and the processing time of thisdummy operation are both zero. A is the set of conjunctive arcs,and each conjunctive arc indicates the processing order of twooperations belonging to the same job. Meanwhile, node 0 isconnected with the first operation of each job with a separateconjunctive arc. If we use FðOÞ to denote the set of the firstoperations of all jobs, then 8f jAFðOÞ, ð0,f jÞAA. E¼

Smk ¼ 1 Ek is the

set of disjunctive arcs, where Ek represents the disjunctive arcsrelated with machine k. Each disjunctive arc connects two opera-tions that should be processed by the same machine, the proces-sing order of the two operations yet to be determined.

With the disjunctive graph, JSSP–TWT can be described as alinear disjunctive programming model

min TWTðSÞ ¼P

uj AUðOÞ

wujTuj

s:t:

si1þpi1rsi2 8ði1,i2ÞAA, ðaÞ

ðsi1þpi1 rsi2 Þ3ðsi2þpi2 rsi1 Þ 8/i1,i2SAEk, k¼ 1, . . . ,m, ðbÞ

TujZsuj

þpuj�duj

8ujAUðOÞ, ðcÞ

TujZ0 8ujAUðOÞ: ðdÞ

8>>>>>>>>>>><>>>>>>>>>>>:ð5Þ

In the above formulation, si denotes the starting time ofoperation i, and the decision variable S is a vector consisting ofthe starting times of all the operations. pi denotes the requiredprocessing time of operation i. For the dummy operation 0, we set

Fig. 1. Illustration of the neighborhood operation SWAP ða,bÞ.

Table 1A 3�3 JSSP–TWT instance.

Job no. j 1 2 3

Operation no. i 1 2 3 4 5 6 7 8 9

pi 2 4 1 2 5 4 1 5 3

Mi 1 2 3 2 1 3 3 1 2

wuj1 2 3

duj7 10 12

cuj ,0 ¼ puj�duj

�6 �6 �9

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5242

s0 ¼ p0 ¼ 0. uj represents the last operation of job j, and thusUðOÞ ¼ u1,u2, . . . ,unf g denotes the set of ultimate operations of allthe jobs. For any operation i, di and wi respectively denotes thedue date and the weight of the job to which operation i belongs.Associated with the ultimate operation of job j, Tuj

denotes thetardiness of job j (defined as maxfsuj

þpuj�duj

,0g).From the perspective of disjunctive graphs, finding a feasible

solution to JSSP is equivalent to determining the directions of allthe disjunctive arcs such that for any /i1,i2SAE, only one of thetwo disjunctive inequalities in constraint (b) is satisfied. Now, letus suppose the directions of all disjunctive arcs have beendetermined, and we use s to denote the set of directed disjunctivearcs. In this situation, the above problem (Eq. (5)) becomes alinear programming model, i.e. Eq. (6). It is easy to show that thesemi-active schedule is the optimal solution to this problem.

min TWTsðSÞ ¼P

uj AUðOÞ

wujTuj

s:t:

si2�si1 Zpi18ði1,i2ÞAA, ðaÞ

si2�si1 Zpi18ði1,i2ÞAs, ðbÞ

Tuj�suj

Zpuj�duj

8ujAUðOÞ, ðcÞ

TujZ0 8ujAUðOÞ: ðdÞ

8>>>>>>>>>>><>>>>>>>>>>>:ð6Þ

As suggested by Bontridder (2005), the dual problem of thelinear program (6) which minimizes TWTs is a maximum costflow problem

max TCsðFÞ ¼P

ði1 ,i2ÞAA[spi1

Fi1 ,i2þP

uj AUðOÞ

ðpuj�dujÞFuj ,0

s:t:

Fuj ,0rwuj8ujAUðOÞ, ðeÞP

x:ðx,iÞAA[sFx,i�

Pz:ði,zÞAA[s[U

Fi,z ¼ 0 8iAO, ðfÞ

Fi1 ,i2 Z0 8ði1,i2ÞAA [ s [ U: ðgÞ

8>>>>>>>>><>>>>>>>>>:ð7Þ

In the above formulation, Fi1 ,i2 represents the flow over the arc(i1,i2); U ¼ ðu1,0Þ,ðu2,0Þ, . . . ,ðun,0Þ

� �¼ fðuj,0Þg

nj ¼ 1 is a newly defined

arc set, each arc pointing from the last operation of job j to node 0;p0 is still assumed to be 0.

Theorem 1. For the maximum cost flow problem (7), there exists an

optimal solution Fn which satisfies the following condition: each node

(except node 0) has at most one incoming arc with nonzero flow.

Proof. See Appendix.

In the following, we will only consider the optimal flows thatsatisfy the condition of this theorem, i.e., each node except 0 hasat most one positive incoming flow.

4.1.2. A neighborhood property for the swap of adjacent operations

We know from the previous subsection that, given a feasible s(i.e. the directions of all disjunctive arcs), the current schedule isdetermined, and meanwhile, a network flow graph is associatedwith the current schedule. The task of neighborhood search is tomodify certain parts of s to get s0, so that the total weightedtardiness can be reduced, i.e. TWTmin

s0 oTWTmins (or equivalently,

TCmaxs0 oTCmax

s ).If we want to improve the schedule under the current s, we

should only consider modifying the disjunctive arcs that satisfythe two conditions: (i) the arc belongs to a certain block and(ii) the arc carries positive flow in the dual network. The lattercondition implies that this disjunctive arc is on the critical path ofat least one tardy job, so altering this arc can possibly reduce theTWT. Next, we will consider whether exchanging the positions oftwo adjacent operations in a block can really improve the TWT.

Under a given s, suppose the operations ð1,2, . . . ,zÞ constitute ablock in the corresponding optimal schedule, and the associatednetwork flows are partially shown in the upper part of Fig. 1. Theamount of flow on the outgoing disjunctive arc of operation i isnoted as xi and suppose xi40 8i¼ 1,2, . . . ,z�1. In this case, theinput flows of these nodes all come from the input disjunctivearcs. According to Theorem 1, the input conjunctive arcs of thesenodes must carry zero flow and thus they are not marked in thefigure. However, each of the nodes can still have an outgoing arcwith positive flow because (i) each node in O\UðOÞ has anoutgoing conjunctive arc which can carry positive flow, and (ii)for each node uj in UðOÞ, the flow on the arc ðuj,0ÞAU can bepositive. For simplicity, these outgoing arcs with possibly positiveflow are drawn in solid arrows in the figure, and the amount offlow is noted as FiZ0, respectively.

Theorem 2. Suppose a block with consecutive positive flows con-

tains the following operations: ð1,2, . . . ,a,b, . . . ,zÞ. If the condition

Fapa

ZFb

pbð8Þ

is satisfied, then swapping the two operations a and b will not lead to

improvement on the objective function (TWT).

Proof. See Appendix.

Example 1. Here we provide a 3�3 instance to illustrate thefunction of Theorem 2. The data of this instance is shown inTable 1. According to the technological constraints, A¼ fð1,2Þ,ð2,3Þ,ð4,5Þ,ð5,6Þ,ð7,8Þ,ð8,9Þg [ fð0,1Þ,ð0,4Þ,ð0,7Þg. The exogenouslydetermined directions of all the disjunctive arcs are given ass¼ fð1,5Þ,ð5,8Þ,ð4,2Þ,ð2,9Þ,ð7,6Þ,ð6,3Þg.

Under the given s, we can directly obtain the semi-active

schedule which is optimal to the primal problem, with an

objective value of TWTmins ¼ 16. Meanwhile, the optimal solution

to the dual problem can be constructed by the procedure listed in

the proof of Theorem 1, and the solution is shown in the left-hand

side of Fig. 2, where the unit cost and the flow on each arc are

marked on the network in the format of ‘‘unit cost, flow (capacity

upper bound, if any)’’. So the objective value of the dual solution

Fig. 2. The dual solution under s (a), and the effect of SWAP (b).

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 43

is TCmaxs ¼ 1� ð11�6Þþ2� ð7�6Þþ3� ð12�9Þ ¼ 16, which is

exactly equal to TWTmins .

Note that ð1,5,8Þ is a block which satisfies the condition of the

theorem. Now we are concerned whether this solution can be

improved by exchanging the processing order of operation 5 and

operation 8 (i.e. scheduling operation 8 before operation 5).

When applying Theorem 2, let a¼ 5, b¼ 8, and the new

network is shown in the right-hand side of Fig. 2. We find

F5=p5 ¼ 3=5¼ F8=p8. Therefore, according to the theorem, swap-

ping operations 8 and 5 cannot improve the solution.

If operations 8 and 5 are really swapped, the new semi-active

schedule has an objective value of TWTmins0 ¼ 22416, which

indeed deteriorates.

Therefore, the function of Theorem 2 is that it helps to excludesome non-improving moves in the local search process, so thatthe optimization efficiency can be improved. However, such agreedy mechanism must be combined with a large-scale rando-mized search (like GA) in order not to get trapped by local optima.

4.2. The neighborhood properties for minimizing f2 and f3

First, the following observations are clear to obtain:

(1)

f2 is minimized if and only if machine k1 processes the jobs inan increasing or decreasing order of the model types (MTj).

(2)

f 3 is minimized if and only if machine k2 processes the jobs inan increasing or decreasing order of the color types (CTj).

Then, the following neighborhood properties for f2 and f3 canbe shown. Similar to the case of f1, we focus on the SWAP operatorfor neighborhood modifications.

Theorem 3. Suppose that in the current schedule, the job sequence

on machine k1 is J½1�,J½2�, . . . ,J½n�, and the corresponding model types

are MT1,MT2, . . . ,MTn. If 1raobrn, then exchanging the posi-

tions of the a-th job and the b-th job will lead to improvement on the

objective f 2 if

ðMTa�MTbÞðMTa�1þMTaþ1�MTb�1�MTbþ1Þo0 b�a41,

ðMTa�MTbÞðMTa�1�MTbþ1Þo0 b�a¼ 1:

(ð9Þ

In the above formulae, let MT0 ¼MTnþ1 ¼ 0.

Proof. See Appendix.

Theorem 4. Suppose that in the current schedule, the job sequence

on machine k2 is J½1�,J½2�, . . . ,J½n�, and the corresponding color types are

CT1,CT2, . . . ,CTn. If 1raobrn, then exchanging the positions of

the a-th job and the b-th job will lead to improvement on the

objective f 3 if

ðCTa�CTbÞðCTa�1þCTaþ1�CTb�1�CTbþ1Þo0 b�a41,

ðCTa�CTbÞðCTa�1�CTbþ1Þo0 b�a¼ 1:

(ð10Þ

In the above formulae, let CT0 ¼ CTnþ1 ¼ 0.

Proof. Same as that of Theorem 3.

5. The GLS algorithm

GLS is a hybridization of GA (which performs large-scale explora-tion) and local search (which performs fine-scale exploitation). Theproposed GLS for the multi-objective JSSP is described as follows.

Step 1:

Initialize the population. Apply the composite dispatch-ing rules (described later) and random permutations togenerate the initial population P0 with a size of PS. LetTP0 ¼ P0, generation index g¼0, and the set of elitesolutions E¼ |.

Step 2:

Let Pg ¼ TPg [ E. Step 3: Perform crossover for ½PS=2� times: each time randomly

select two individuals from the current population Pg

and then execute crossover with probability pc. The setof offspring individuals produced by crossover isdenoted by Xg .

Step 4:

Perform mutation for the individuals in Xg with prob-ability pm.

Step 5:

Delete duplicate (redundant) individuals from Xg . Step 6: Perform Pareto sort to the remaining individuals in Xg

(within the same Pareto rank, the solutions in lower-density areas are positioned forward).

Step 7:

Perform a local search for each of the solutions thatrank in the first s%. The solutions obtained after thelocal search are denoted by Gg .

Step 8:

Update E with Gg . If the number of solutions in E

exceeds the upper limit (assumed to be ½PS=5�), thenremove the surplus.

Step 9:

Using the result of sorting in Step 6, select ðPS�9E9Þindividuals from Xg and insert them into TPgþ1.

Step 10:

Let g’gþ1. If goGN, then return to Step 2. Otherwise,terminate the algorithm and output E.

In the above procedure, s% is the proportion of individuals whichare selected for local search, and GN is the total number of evolvinggenerations. To reflect the algorithm framework more clearly, weprovide the flowchart of the proposed GLS in Fig. 3. In the followingwe will describe the main aspects of the proposed GLS.

Fig. 3. Flowchart of the proposed GLS.

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5244

5.1. Encoding and decoding

The encoding scheme is based on operation preferential lists.In particular, each machine is related with an operation sequencewith length n, which indicates the preferential order of processingthe relevant operations by this machine. Thus, the completeencoding of a solution is in the form of a matrix L¼ ðlk,jÞm�n,where the k-th row describes the preference (the scheduling rule)of machine k.

The decoding process concerns iteratively scheduling theready operations (whose preceding operations have been sched-uled) on each machine according to their priority order and asearly as possible. Only active schedules will be produced. Thedetailed decoding procedure is given as follows.

Step 1:

Let s be an empty matrix of size m�n. Step 2: If L becomes empty, output the schedule s (and the

corresponding objective values if necessary) and termi-nate the procedure. Otherwise, set k¼1 and continue thefollowing steps.

Step 3:

If the k-th row of L is empty, go to Step 5. Otherwise,continue the following steps.

Step 4:

6 The standard form of ATC is: IiðtÞ ¼ ðwj=piÞ� expf�½dj�t�pi �P

qA JSðiÞ

ðcW qþpqÞ�þ =ðK � pÞg, and the operation with the largest Ii will be selected at time

t. JSðiÞ denotes the job successors of operation i in job j, and p is the average

processing time of the current waiting operations. cW q is the estimated waiting

time of operation q, which must be given by experience.

Find the first schedulable operation in the k-th row of L,denoted as lk,#. Delete lk,# from the k-th row of L. Then,schedule the operation lk,# in the following way:(4.1) Scan the Gantt chart of machine k (which records

the processing information of the already scheduledoperations) from time zero and test whether lk,# canbe inserted into each idle period ½a,b�, i.e., whetherthe following condition is met: maxfa,CIJPðlk,#Þ

gþ

plk,#rb (where CIJPðlk,#Þ

denotes the completion timeof the immediate job predecessor of operation lk,#,and plk,#

the processing time of lk,#).(4.2) If the above inequality is satisfied for the idle

interval between operation o1 and o2 on machinek, then insert lk,# between o1 and o2 in the k-th rowof s. Otherwise (no idle intervals can hold lk,#),

insert it at the back of the k-th row of s. Update theGantt chart records for the starting time and com-pletion time of operation lk,#.

Step 5:

Let k’kþ1. If krm, go to Step 3. Otherwise, go back toStep 2.

5.2. Initialization

The following scheduling rules are involved in the generationof initial solutions:

(1)

ATC (apparent tardiness cost)6 as detailed in Vepsalainen andMorton (1987). The parameters are set as K¼2 andcW q ¼ 0:4� pq following the recommendation of Jensen et al.(1995).

(2)

CMT (closest model type): When machine k1 becomes idle,select from buffer the job whose model type is the closest tothe job processed last on this machine.

(3)

CCT (closest color type): When machine k2 becomes idle,select from buffer the job whose color type is the closest tothe job processed last on this machine.

The ATC rule is efficient for minimizing the total weightedtardiness (i.e. f1), while CMT and CCT are beneficial for minimiz-ing f 2 and f 3. In the simulation procedure for generating initialsolutions, the machines other than k1 and k2 directly adopt theATC rule, while machine k1 adopts the composite ATC–CMT ruleand machine k2 adopts the composite ATC–CCT rule. The lattertwo rules are detailed below.

In the ATC–CMT rule, when machine k1 finishes processing jobjp and becomes idle at time t, the following priority index Ik1

i ðt,jpÞ

is calculated for each operation i (belonging to job j) in the bufferand then the operation that maximizes Ik1

i ðt,jpÞwill be selected forprocessing.

Ik1

i ðt,jpÞ ¼wj

pi

� exp �½dj�t�pi�

PqA JSðiÞð

cW qþpqÞ�þ

K � p

( )

�exp �MTDðjp,jÞ

K 0 �MTD

� �, ð11Þ

where MTDðjp,jÞ ¼ 9MTj�MTjp9 represents the distance in model

types between job j and job jp, and MTD is the average value ofMTDðjp,jÞ for all the jobs currently waiting in machine k1’s buffer.K 0 is a scaling factor with a similar function as K, and here weassume K 0 ¼ 0:5 considering the relative sizes of MTD and p. Thenotation ðxÞþ is an abbreviation for maxfx,0g.

Likewise, the ATC–CCT rule adopted by machine k2 computesthe following priority index for operation i:

Ik2

i ðt,jpÞ ¼wj

pi

� exp �½dj�t�pi�

PqA JSðiÞð

cW qþpqÞ�þ

K � p

( )

�exp �CTDðjp,jÞ

K 0 � CTD

� �, ð12Þ

where CTDðjp,jÞ ¼ 9CTj�CTjp 9 denotes the distance between thecolor types of job j and job jp, CTD is the average of CTDðjp,jÞ forthe currently waiting jobs, and K 0 ¼ 0:5.

Once a solution has been constructed with these rules, theother initial solutions (needed to form an initial population) are

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 45

obtained by imposing a moderate level of random perturbationson this solution.

5.3. Pareto sorting

Pareto sorting is important in GLS because it is the procedureto differentiate the quality of different individuals. The sortingprocedure includes two steps: (a) the division of Pareto ranks;(b) the sorting of solutions within the same rank.

Pareto ranking divides the population into K subsets F 1,F2,. . . ,FK such that:

(1)

For any 1r io jrK , and for any solution z in F j, there mustexist a solution in F i which dominates z.

(2)

For each i, there exist no dominance relations between anytwo solutions in F i (neither dominates the other).

Clearly, F 1 is the set of Pareto non-dominated solutions in thecurrent population.

Pareto ranking is equivalent to the topological sorting pro-blem, and the algorithm is detailed in Deb et al. (2002).

Within the same Pareto rank F i, the sorting of solutionsdepends on the level of density. The solutions that reside inlow-density regions (in the objective space) should come beforethe solutions in crowded areas. The aim is to ensure that thesolutions ultimately found are diverse and distributed evenly.

In order to define a density measure sdi for solution xi, weconsider a non-dominated solution set F (none of the solutions inF dominates another), and two solutions x1, x2AF . The distancebetween x1 and x2 in the objective space can be calculated as follows:

dðx1,x2Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i ¼ 1

f iðx1Þ�f iðx2Þ

f maxi ðF Þ�f min

i ðF Þ

!2vuut , ð13Þ

where N is the number of objectives; f maxi ðF Þ ¼maxff iðxÞ9xAF g and

f mini ðF Þ ¼minff iðxÞ9xAF g respectively denote the maximum and

minimum value of the i-th objective in the scope of F . Such normal-ization ensures that the objectives under different dimensions canbe added.

Then, sort the solutions in F \fxig according to their distancesto xi. Let mdð1Þi denote the distance between xi and the nearestsolution to xi, i.e. mdð1Þi ¼min dðxi,xjÞ9xjAF \fxig

� �. Likewise, let

mdð2Þi denote the distance between xi and the second nearestsolution to xi. � � � Let mdðkÞi denote the distance between xi

and its k-th nearest neighbor solution. Finally, the density of xi

is defined as

sdi ¼1

LðF ÞXLðF Þk ¼ 1

mdðkÞi

!�1

, ð14Þ

where the truncation length LðF Þ is assumed to be 15� 9F 9� �

. Inother words, 20% of the solutions which are closest to xi in theobjective space are used to calculate the density of xi. The solutionswith smaller sdi values are given higher priority in sorting.

5.4. Elitism

Elitism refers to the strategy that preserves elite solutionsfound in earlier generations to later generations. This policy haslong been known to be very effective in accelerating theconvergence of GA.

The elitism mechanism adopted in our algorithm has thefollowing features:

(1)

An elitist archive E is maintained in the evolutionaryprocess of GA.

(2)

In each generation, the individuals in E are combined into thecurrent population and then participate in the crossover andmutation operation.

(3)

E is constantly updated by the (better) solutions found by theguided local search procedure.

Based on the consideration for population diversity, the upperlimit on the number of solutions in E is set as ½PS=5�. Meanwhile,the steps of using the new solution set G acquired by local searchto update E are given as follows.

Step 1:

Perform the following steps for each solution xG in G:Step 1.1: Check whether xG has been dominated by a

solution in E. If not, proceed to Step 1.2.Step 1.2: Delete from E all the solutions that are domi-

nated by xG. Let E’E [ fxGg.

Step 2: If 9E94 ½PS=5�, then sort all the solutions in E in an

increasing order of sdi, and delete the ð9E9�½PS=5�Þ solu-tions with the largest density.

5.5. Genetic operators

Crossover is performed using the LOX (linear order crossover)operator (Croce et al., 1995) for a randomly selected row of theencoding matrix. Mutation is performed using the simple SWAPoperator which exchanges two operations in a randomly selectedrow of the encoding matrix.

Selection is performed after the Pareto sorting, and ðPS�9E9Þindividuals should be selected to enter the temporary populationTP. In this process, selection probabilities are assigned in a linearway, that is, the probability of selecting the individual that ranksin the i-th position is

P½i�ð9X9Þ ¼ ð9X9þ1�iÞ9X9ð9X9þ1Þ

2

, ð15Þ

where 9X9 is the current size of the population. After oneindividual has been selected according to the above probability,delete the selected one from the sequence, let 9X9’9X9�1 and re-evaluate P½i�ð9X9Þ. Repeat this until the desired number of solutionshave been selected.

5.6. Deletion of redundant individuals

In the evolutionary process of GA, there usually appearduplicate individuals in the population, which can be identifiedas two cases:

(1)

The encodings are identical for some individuals. (2) For some other individuals, the corresponding N objective

values are all identical despite the different encodings.

In the proposed GLS, these redundant individuals will beremoved so as to achieve the highest possible level of diversitywithin a population with fixed size. This procedure iterativelychecks each individual in the population, and if the objectivevalues of the current solution are identical with a previoussolution, then perform random mutation to this individual untilthe objective values get different.

5.7. The embedded local search

After the traditional crossover and mutation, GLS performsa one-step local search for the best s% of solutions in thepopulation. The major steps of the guided local search is

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5246

described as follows.

Step 1:

Perform the following steps for each one of the top s%solutions (noted as Si ði¼ 1, . . . ,uÞ):

Step 1.1: Generate a random number xi, and letci ¼ xi mod 3. If ci ¼ 0, go to Step 1.2; ifci ¼ 1, go to Step 1.3; otherwise, go to Step 1.4.

Step 1.2: Select two operations from machine k1 whichsatisfy the condition described in Theorem 3and then exchange their positions. Do q1 suchtrials on Si, and the obtained new solutions arerespectively noted as Sð1Þi ,Sð2Þi , . . . ,Sðq1Þ

i .Step 1.3: Select two operations from machine k2 which

satisfy the condition described in Theorem 4and then exchange their positions. Do q2 suchtrials on Si, and the obtained new solutions arerespectively noted as Sð1Þi ,Sð2Þi , . . . ,Sðq2Þ

i .Step 1.3: Select two operations from any machine which

satisfy the condition described in Theorem 2 andthen exchange their positions. Do q3 such trialson Si, and the obtained new solutions are respec-tively noted as Sð1Þi ,Sð2Þi , . . . ,Sðq3Þ

i .

Step 2:

Let G0 ¼ fS1,Sð1Þ1 ,Sð2Þ1 , . . . ,Sðqc1 þ 1Þ

1 g[ fS2,Sð1Þ2 ,Sð2Þ2 , . . . ,Sðqc2 þ 1Þ

2 g

[ � � � [ fSu,Sð1Þu ,Sð2Þu , . . . ,Sðqcu þ 1Þ

u g. Find the non-dominated

solutions in G0 and insert them into G.

Without special preferences to f1, f2 or f3, we set q1 ¼ q2 ¼

q3 ¼ q. The output of this local search procedure (solution set G)will be used for updating E.

From the above description, it is clear that the aim of guidedlocal search is to further improve one of the three objectives forelite solutions in the current population: Step 1.2 tries to improve f 2,Step 1.3 tries to improve f 3 and Step 1.4 tries to improve f1. Ofcourse, improvement on one objective often implies deterioration ofthe other, and thus forming somewhat ‘‘extreme’’ solutions withrespect to a certain objective. However, when such solutions enterthe next generation, it is possible that they can help to producemore Pareto optimal solutions by crossover with other individuals.Thus, combining local search with GA is hopeful for promoting theoverall optimization efficiency.

6. Main computational experiments

6.1. The test instances and the parameter settings

In order to test the performance of the proposed GLS, we conductcomputational experiments on three sets of instances: (I) 11 JSSP

Fig. 4. Illustration of the manufacturing pro

instances adapted from the literature; (II) randomly generated large-scale instances; (III) a real-world scheduling instance.

For each instance in (I) and (II), the due dates of jobs are set inthe following manner: first, randomly generate a feasible sche-dule, and denote the completion time of job j under this scheduleby ~C j; then, the due date of job j is set as dj ¼

~C jþx, wherex� U½� ~C j=10, ~C j=10� is a random number uniformly distributedbetween � ~C j=10 and ~C j=10. The model type MTj and the colortype CTj of each job is an integer uniformly distributed between1 and 10, and the two values are mutually independent. k1 and k2

represent two different machines which are randomly deter-mined from the m machines. Meanwhile, the weight of each jobis also an integer following the uniform distribution U½1,10�.

The 11 different-sized JSSP instances taken from OR-Library(Beasley, 1990; Giffler and Thompson, 1960) have been adaptedto the MOJSSP instances as needed by the experiment. Amongthese instances, FT06, LA01, LA06 and LA11 are generally con-sidered as easy instances but the other 7 ones are relatively hardto optimize. To avoid confusion, we add a prefix ‘m’ to the namesof the modified instances used here.

For the completely random instances in set (II), the processingroute of each job is a random permutation of the m machines, andthe required processing time of each operation follows a uniformdistribution U½1,99� and also takes integer values.

In the real-world instance (III), we consider the scheduling of110 vehicles (with 18 model types and 9 color types) whichshould be manufactured in a week. There are totally 14 mainprocedures to be completed by the three workshops: 8 in thebody shop, 3 in the coating shop and 3 in the assembly shop (seeFig. 4). The production organization is actually a flow shop, butnot a permutation flow shop (i.e., the production sequences ineach shop can be different). So our algorithm has been veryeffective for the optimization of such schedules.

The proposed GLS is compared with another efficient geneticlocal search algorithm (noted as J-MOGLS) proposed byJaszkiewicz (2002). Based on extensive preliminary experiments,the parameters of GLS are set as follows: the population sizePS¼ 50, the maximum number of generations GN¼ 500 (ordetermined by the exogenous limit on computational time), thecrossover probability pc ¼ 0:7, the mutation probability pm ¼ 0:1,the proportion of individuals that undergo local search s%¼ 40%,and the number of local search steps q¼2. According to theoriginal paper and a specific fine-tuning for the studied problem,the parameters of J-MOGLS are set as follows: the size oftemporary population K¼15, the number of initial solutionsS¼50 and the maximum number of generations is 500. Thealgorithms have been implemented using Visual Cþþ 7 and tested

cedures for light buses in F Company.

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 47

on a platform of Intel Core2 Duo 2.2 GHz CPU, 2 GB RAM andWindows XP.

6.2. The adopted performance measures

The performances in three aspects should be considered in theevaluation of multi-objective algorithms:

(1)

The number of Pareto optimal solutions that the algorithmfinds. Users usually hope more solutions could be provided forthem to choose from.

(2)

The distance between the obtained Pareto front and the realPareto front (assumed to be known), i.e. the level ofoptimality.

(3)

The degree of dispersiveness of the obtained solutions alongthe Pareto front. It is usually desired that the solutions shouldbe distributed evenly, representing different combinations ofpreferences for each objective.

Fig. 5. The typical optimization output of GLS for instance m-FT10.

Table 2

Algorithm comparison (I) based on the general quality of solutions (NS and RN).

Instance Size (n�m) GLS J-MOGLS p-Value

In accordance with the above requirements, we adopt 4 per-formance measures: NS, RN, Dav (Dmax) and TS.

Suppose k different algorithms produce k non-dominatedsolution set for the same problem,i.e. SS1, . . . ,SSk. In order toevaluate the performance of algorithm a ðaAf1, . . . ,kgÞ,the per-formance measures are calculated in the following way:

NS RN NS RN pNS pRN

(1)

m-FT06 6�6 9.4 1.00 9.3 0.09 0.164 0.000

NSðaÞ ¼ 9SSa9,which is the number of Pareto non-dominatedsolutions obtained by algorithm a.

m-LA01 10�5 12.1 0.97 10.5 0.17 0.003 0.000

(2) m-LA06 15�5 10.7 0.81 10.2 0.41 0.084 0.000

m-LA11 20�5 11.8 0.65 9.6 0.44 0.002 0.007

m-FT10 10�10 14.3 0.89 11.5 0.36 0.001 0.003

m-FT20 20�5 10.8 0.82 10.9 0.40 0.287 0.000

m-LA16 10�10 15.2 1.00 10.6 0.11 0.000 0.000

m-LA21 15�10 16.5 0.99 14.9 0.29 0.012 0.000

m-LA26 20�10 13.2 0.84 11.9 0.21 0.007 0.000

RNðaÞ ¼ 9SSa\fxASSa9(yASS\SSas:t:y!xg9=NSðaÞ,where SS is theset of non-dominated solutions in SS1 [ SS2 [ � � � [ SSk. RNðaÞ ¼ 1indicates that none of the solutions produced by algorithm a isdominated by any solution in SS. On the other hand, RNðaÞ ¼ 0suggests that each solution in SSa has been dominated by somesolution in SS. Usually, RNð1Þþ � � � þRNðkÞ41. This measure isalso known as the C-metric.

m-LA31 30�10 18.4 0.90 16.3 0.17 0.000 0.000

m-LA36 15�15 11.1 0.96 11.1 0.35 0.622 0.001

(3) DavðaÞ¼ð1=9SS9Þ

PyASSminxA SSa

dðx,yÞ,DmaxðaÞ¼maxyASSfminxASSa

dðx,yÞg (Ulungu et al., 1998), where dðx,yÞ ¼maxNi ¼ 1

fðf iðxÞ�f iðyÞÞ=Dig, Di ¼maxyASS[SSaff iðyÞg�minyA SS[SSa

ff iðyÞg. Bydefinition, dðx,yÞ is a normalized distance measure for twosolutions x and y. For each solution in SS (which is assumed tobe the optimal Pareto front since the ‘‘real’’ Pareto front is simplyunknown), there exists a solution in SSa with the shortest distanceto that solution. Then, Dav measures the average value of suchdistances while Dmax measures the maximum value of suchdistances. Therefore, Dav and Dmax reflect the distance betweenthe Pareto front found by algorithm a and the real Pareto frontwhich is approximated by SS.

(4)

TSðaÞ ¼ ð1=DÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=9SSa9Þ

Pxi A SSa

ðDi�DÞ2q

(Tan et al., 2006),

where D ¼ ð1=9SSa9ÞP

xi A SSaDi and Di is the Euclidean distance

(in the objective space) between xi and its nearest neighborsolution in SSa. The smaller the value of TS, the more evenlythe solutions in SSa are distributed.

6.3. Results and discussions

Since the discussed MOJSSP involves three objectives whichare lowly correlated, the number of Pareto optimal solutions isfrequently larger than 10. For example, the Pareto front obtainedby GLS for m-FT10 is shown in Fig. 5, where 15 non-dominatedsolutions are found and they are distributed quite evenly.

In the computational experiments, GLS and J-MOGLS areexecuted 10 times independently for each instance. In order tomake fair comparisons, we exert the same computational timelimit on both algorithms. For instance set (I), the allowed time is

set as 40 s for m-FT06, m-LA01, m-LA06 and m-LA11, and 60 s forthe rest. For instance set (II), the time limit is 200 s. Whencalculating RN, Dav, Dmax with the above definitions, let SS

represent the non-dominated solutions in SSJ�MOGLS [ SSGLS. Thecomputational results for set (I) are displayed in Tables 2–4, andthe results for set (II) are shown in Tables 5–7 where the 15randomly generated instances are labeled as P-1–P-15. In thetables, NS, RN , Dav , Dmax and TS respectively denote the averagevalues of these indices in the 10 independent runs. We have alsoperformed a two-tailed t-test on the two groups of data obtainedfor each performance measure. The corresponding p-values(assuming homoscedasticity) are reported in Tables 2–8. Aninstance of 0.000 in the tables indicates that the p-value is lessthan 0:5� 10�3.

According to Tables 2–4 which record the computationalresults for smaller instances (set (I)), GLS obtains more Paretooptimal solutions than J-MOGLS on most instances (exceptm-FT20). Meanwhile, the RN index shows that a large part ofthe non-dominated solutions found by J-MOGLS are dominatedby the solutions found by GLS. The Dav (Dmax) and TS indices showthat the Pareto front obtained by GLS for most instances hashigher quality (in terms of both optimality and distributionalevenness).

Focusing on the results for large-scale instance set (II) (Tables5–7), although the number of non-dominated solutions obtainedby GLS is fewer than that of J-MOGLS, the solutions of GLS havehigher quality since they still dominate most of the solutionsfound by J-MOGLS. Table 7 suggests that the solutions of J-MOGLS

Table 3

Algorithm comparison (I) based on optimality of the obtained Pareto front (Dav

and Dmax ).

Instance Size (n�m) GLS J-MOGLS p-Value

Dav Dmax Dav Dmax pDavpDmax

m-FT06 6�6 0.014 0.029 0.047 0.081 0.007 0.000

m-LA01 10�5 0.003 0.018 0.028 0.095 0.000 0.000

m-LA06 15�5 0.035 0.084 0.040 0.127 0.070 0.000

m-LA11 20�5 0.052 0.131 0.044 0.163 0.183 0.036

m-FT10 10�10 0.021 0.095 0.041 0.105 0.009 0.117

m-FT20 20�5 0.033 0.117 0.036 0.150 0.086 0.044

m-LA16 10�10 0.001 0.010 0.039 0.216 0.000 0.000

m-LA21 15�10 0.014 0.083 0.057 0.184 0.000 0.000

m-LA26 20�10 0.101 0.285 0.241 0.331 0.001 0.000

m-LA31 30�10 0.126 0.317 0.142 0.379 0.023 0.026

m-LA36 15�15 0.114 0.219 0.255 0.357 0.000 0.006

Table 4

Algorithm comparison (I) based on solution distribution (TS).

Instance Size (n�m) GLS J-MOGLS p-Value

TS TS pTS

m-FT06 6�6 0.682 0.711 0.068

m-LA01 10�5 0.593 0.605 0.141

m-LA06 15�5 0.707 0.752 0.105

m-LA11 20�5 0.774 0.760 0.625

m-FT10 10�10 0.694 0.831 0.043

m-FT20 20�5 0.542 0.503 0.302

m-LA16 10�10 0.833 0.916 0.073

m-LA21 15�10 1.014 1.135 0.029

m-LA26 20�10 0.849 1.001 0.006

m-LA31 30�10 1.245 1.261 0.270

m-LA36 15�15 1.119 1.113 0.596

Table 5

Algorithm comparison (II) based on the general quality of solutions (NS and RN).

Instance Size (n�m) GLS J-MOGLS p-Value

NS RN NS RN pNS pRN

P-1 25�20 18.5 0.91 20.2 0.15 0.033 0.000

P-2 50�10 17.3 0.93 23.0 0.18 0.000 0.000

P-3 100�5 14.7 0.88 14.9 0.23 0.179 0.005

P-4 20�30 18.1 0.96 22.7 0.08 0.004 0.000

P-5 30�20 15.6 0.95 19.4 0.11 0.000 0.000

P-6 60�10 13.2 0.94 18.8 0.13 0.000 0.000

P-7 35�20 15.7 0.91 21.5 0.14 0.001 0.000

P-8 70�10 12.9 0.90 19.8 0.20 0.000 0.000

P-9 20�40 17.8 0.91 24.6 0.15 0.000 0.000

P-10 40�20 16.0 0.94 20.9 0.14 0.000 0.000

P-11 50�16 13.7 0.95 27.2 0.10 0.000 0.000

P-12 30�30 16.3 0.89 26.1 0.17 0.000 0.002

P-13 50�18 17.7 0.93 28.5 0.25 0.000 0.005

P-14 60�15 17.9 0.97 25.5 0.18 0.000 0.000

P-15 50�20 18.6 0.97 32.7 0.19 0.000 0.000

Table 6

Algorithm comparison (II) based on optimality of the obtained Pareto front (Dav

and Dmax ).

Instance Size (n�m) GLS J-MOGLS p-Value

Dav Dmax Dav Dmax pDavpDmax

P-1 25�20 0.103 0.217 0.141 0.336 0.008 0.015

P-2 50�10 0.078 0.204 0.136 0.315 0.003 0.007

P-3 100�5 0.058 0.253 0.153 0.378 0.000 0.001

P-4 20�30 0.026 0.174 0.139 0.382 0.000 0.004

P-5 30�20 0.066 0.194 0.127 0.349 0.000 0.000

P-6 60�10 0.046 0.226 0.152 0.418 0.000 0.000

P-7 35�20 0.057 0.236 0.108 0.345 0.001 0.001

P-8 70�10 0.063 0.292 0.104 0.316 0.011 0.003

P-9 20�40 0.052 0.284 0.117 0.350 0.000 0.019

P-10 40�20 0.037 0.186 0.181 0.442 0.000 0.005

P-11 50�16 0.039 0.205 0.175 0.410 0.000 0.000

P-12 30�30 0.096 0.406 0.128 0.438 0.027 0.210

P- 13 50�18 0.061 0.319 0.143 0.453 0.000 0.048

P-14 60�15 0.031 0.278 0.155 0.387 0.000 0.011

P-15 50�20 0.035 0.299 0.161 0.415 0.000 0.004

Table 7

Algorithm comparison (II) based on solution distribution (TS).

Instance Size (n�m) GLS J-MOGLS p-Value

TS TS pTS

P-1 25�20 0.826 1.005 0.009

P-2 50�10 1.023 0.968 0.415

P-3 100�5 1.218 1.250 0.208

P-4 20�30 0.951 1.124 0.017

P-5 30�20 0.774 0.935 0.000

P-6 60�10 1.057 1.034 0.311

P-7 35�20 1.033 1.152 0.034

P-8 70�10 1.075 0.910 0.037

P-9 20�40 1.232 1.119 0.062

P-10 40�20 1.226 1.018 0.055

P-11 50�16 1.328 1.261 0.091

P-12 30�30 1.453 0.967 0.000

P-13 50�18 0.886 1.039 0.002

P-14 60�15 1.129 1.117 0.574

P-15 50�20 1.340 1.094 0.001

7 In practice, the production managers in F Company use ATC to obtain the

initial schedule and then manually adjust the schedule in order to meet the

preference of each workshop. This process normally takes about half an hour

each day.

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5248

distribute more evenly for most instances, but these solutions arecomparatively far from the approximated true Pareto front(because of larger Dav values).

The difference in the results for smaller and larger instancescan be explained as follows. With the expansion of problem size,the total weighted tardiness objective (f1) becomes more difficultto optimize. In this case, utilization of the neighborhood proper-ties helps GLS to control f1 under a low level. When f1 iscontrolled, the number of possible combinations of f2 and f3 isrelatively small. By contrast, the J-MOGLS which does not

effectively make use of the problem information tends to findmore non-dominated combinations of the three objectives, but alarge part of these solutions will be dominated by GLS solutions.

This feature can also be observed from the results on the real-world instance (III) from F Company which consists of 110 jobsand 14 machines. In real-world scheduling, it is usually requiredby production managers that the due date related performance(i.e. TWT in this paper) should not be worse than a specifiedtolerance level. Based on such a motivation, we design thefollowing test.

First, we use the ATC rule (as introduced in Section 5.2) toconstruct a benchmark schedule.7 The reason of choosing ATC isthat, as a combination of the WSPT (weighted shortest processingtime) rule and the MS (minimum slack) rule, ATC is the mosteffective dispatching heuristic for minimizing the total weightedtardiness criterion (Vepsalainen and Morton, 1987; Jensen et al.,1995). Meanwhile, since the ATC rule only cares about the TWTobjective and neglects the other two objectives, it can generate asolution that is biased towards TWT and thus suitable as a

Fig. 6. The number of solutions within different TWT levels.

Table 8Algorithm comparison for the real-world instance (III).

Measures GLS J-MOGLS p-Value

NS 20.4 31.2 0.000

RN 0.93 0.19 0.000

Dav 0.014 0.113 0.000

Dmax 0.086 0.205 0.002

TS 0.716 0.688 0.259

Table 9TWT results on Singer & Pinedo’s benchmark instances.

Instance f¼1.6 f¼1.5 f¼1.3

GLS J-MOGLS GLS J-MOGLS GLS J-MOGLS

ABZ05 5.8 7.4 86.2 88.5 1840.3 1860.9

ABZ06 2.3 2.3 19.0 22.3 547.9 557.4

LA16 7.9 10.3 220.7 215.2 1547.1 1583.2

LA17 81.5 84.8 325.6 343.4 1140.3 1199.5

LA18 14.3 22.3 43.4 43.8 1193.5 1237.2

LA19 4.1 6.4 27.7 27.9 1214.5 1269.4

LA20 1.6 2.0 3.7 5.0 1069.4 1076.2

LA21 7.1 9.2 20.9 28.7 579.1 580.4

LA22 27.3 32.8 250.9 251.7 1362.6 1393.6

LA23 1.6 1.9 7.6 8.8 1117.8 1145.9

LA24 21.6 31.2 104.8 109.9 1080.5 1116.6

MT10 193.0 203.8 515.3 516.9 1737.0 1816.8

ORB01 776.7 811.3 1404.7 1448.0 3350.5 3381.3

ORB02 54.8 54.9 396.2 397.1 1779.2 1890.5

ORB03 584.6 620.3 1181.3 1217.2 2638.3 2896.9

ORB04 91.5 99.7 473.9 481.9 2056.1 2128.1

ORB05 215.1 242.2 547.8 572.1 2045.9 2086.3

ORB06 60.3 59.8 572.4 578.8 2260.7 2272.8

ORB07 12.5 16.0 73.0 78.5 765.2 769.5

ORB08 815.7 827.5 1369.5 1472.1 3181.4 3198.2

ORB09 99.7 110.5 374.8 404.5 1694.6 1704.8

ORB10 113.4 136.6 459.0 503.3 2165.4 2251.3

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 49

reference solution for this test. In our experiment, the ATC rule isapplied in a discrete event simulation procedure for making thedispatching decisions at each time when a machine becomesavailable.

If the total weighted tardiness achieved by applying the ATCrule alone is denoted by TWTref , then we can count the number ofsolutions with bounded TWT values, i.e. Q ðcÞ ¼ 9fxASSa9f 1ðxÞrc � TWTref g9, where SSa is the solution set obtained by algorithmaAfGLS; J�MOGLSg. In a typical run of both candidate algorithmson the real-world instance, GLS finds 19 non-dominated solutionswhile J-MOGLS finds 31, and the results (Q) are shown in Fig. 6.According to the results, 14 (74%) of 19 solutions found by GLShave a TWT of less than 0:75TWTref , while only 8 (26%) of 31solutions by J-MOGLS achieve the same requirement. The figuresuggests that the distribution of GLS solutions is bent towards alow level of TWT, and thus better satisfies the stricter require-ments on due date performances. This advantage is obtained byeffectively utilizing the neighborhood properties described inSection 4.

The comparisons regarding the other performance measuresfor the real-world instance are displayed in Table 8, which againdemonstrates the superiority of the proposed approach.

7. Supplementary computational experiments

7.1. The total weighted tardiness performance

To observe the TWT performance of the proposed GLS (whiletemporarily neglecting the other two objectives), we adopt the 66benchmark instances from Singer and Pinedo (1998). Three duedate tightness levels are involved: f¼1.6 for loose due dates,f¼1.5 for moderate due dates and f¼1.3 for tight due dates.Considering that all the instances have the same size of 10�10,the problem data related with vehicle production are additionally

generated such that MTj ¼ j, CTj ¼ 11�j (j¼ 1, . . . ,10) and k1 ¼ 4,k2 ¼ 7.

GLS and J-MOGLS are separately executed 10 times on eachinstance under the time constraint of 60 s for each execution. Theaverage value of the minimum TWT found in each of the 10obtained solution sets is reported in Table 9. According to theseresults, the relative percentage improvement of GLS overJ-MOGLS is 14.13% under f¼1.6, 6.23% under f¼1.5 and 2.64%under f¼1.3. GLS outperforms J-MOGLS (in terms of the TWTobjective) on all except two instances, which exactly demon-strates the efficacy of the proposed neighborhood property forminimizing TWT.

7.2. The tuning of parameters

We have applied a design of experiments (DOEs) approach todetermine the desired values for all the involved parameters ofGLS. In the DOE, we consider 5 parameters (PS, pc, pm, s and q),each with 3 levels. The full factorial design (including 35

¼ 243combinations) is almost unaffordable in this case. Therefore, weresort to the Taguchi design method Fowlkes et al. (1995) withthe orthogonal array L27ð3

5Þ, which means only 27 orthogonal

scenarios have to be tested.The m-LA26 instance (with 60 s time limit) is adopted for the

experiment described here. We first run J-MOGLS 10 times andobtain 10 reference solution sets, and then run GLS 10 timesunder each scenario of parameter settings, so that the RN indexcan be evaluated on a one-to-one basis. The results (S/N ratios,i.e. signal-to-noise ratios) based on the RN index are shown inFig. 7 (output by the Minitabs software). From the figure, weobtain the parameter values as recommended in Section 6.1.

7.3. The impact of local search intensity

As an example study on parameter influence, we examine theeffect of the local search procedure by varying the proportion (s%)of individuals that are modified by this step. The results forinstance set (I) are displayed in Tables 10 and 11. The samecomputational time limit is exerted, that is, 40 s for m-FT06,m-LA01, m-LA06, m-LA11 and 60 s for the others.

Fig. 7. Average S/N ratio at each level of the parameters.

Table 10

The impact of s on RN .

Instance Size (n�m) GLS: RN

s¼0 s¼20 s¼40 s¼60 s¼80 s¼100

m-FT06 6�6 0.81 0.87 0.95 0.93 0.90 0.92

m-LA01 10�5 0.50 0.58 0.89 0.95 0.81 0.73

m-LA06 15�5 0.68 0.69 0.76 0.82 0.87 0.85

m-LA11 20�5 0.57 0.62 0.88 0.78 0.91 0.85

m-FT10 10�10 0.83 0.90 0.96 0.80 0.80 0.78

m-FT20 20�5 0.75 0.86 0.84 0.89 0.83 0.81

m-LA16 10�10 0.69 0.77 0.85 0.87 0.94 0.92

m-LA21 15�10 0.73 0.86 0.88 0.99 0.93 0.80

m-LA26 20�10 0.80 0.93 0.96 0.89 0.88 0.74

m-LA31 30�10 0.76 0.84 0.98 0.81 0.87 0.90

m-LA36 15�15 0.71 0.91 0.89 0.90 0.78 0.82

Table 11

The impact of s on Dav .

Instance Size (n�m) GLS: Dav

s¼0 s¼20 s¼40 s¼60 s¼80 s¼100

m-FT06 6�6 0.024 0.015 0.011 0.017 0.020 0.011

m-LA01 10�5 0.029 0.025 0.015 0.012 0.017 0.017

m-LA06 15�5 0.033 0.031 0.022 0.019 0.018 0.019

m-LA11 20�5 0.120 0.111 0.070 0.063 0.074 0.086

m-FT10 10�10 0.107 0.082 0.034 0.069 0.073 0.122

m-FT20 20�5 0.085 0.073 0.051 0.066 0.062 0.068

m-LA16 10�10 0.104 0.094 0.053 0.046 0.041 0.043

m-LA21 15�10 0.115 0.104 0.039 0.023 0.047 0.045

m-LA26 20�10 0.059 0.050 0.058 0.044 0.048 0.062

m-LA31 30�10 0.094 0.089 0.008 0.074 0.068 0.077

m-LA36 15�15 0.099 0.085 0.074 0.066 0.072 0.060

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5250

When s is too small, GLS cannot fully utilize the advantage ofthe proposed neighborhood properties and thus the overallsolution quality deteriorates to certain extent. On the other hand,if s is set too large, the over-frequent local search reduces theexploration ability of GA. Because the total computational time islimited, too much local exploitation means the number of

generations actually realized by GA is decreased, which can alsoworsen the solution quality. According to Table 10, variations of s

can bring noticeable influences for the optimization of someinstances (like m-LA01, m-FT10 and m-LA21), but the quality ofsolutions is generally robust with respect to s. Summarizing theresults, the recommended value of s for most instances would be40–60. But if the computational time is not so sufficient relativeto the problem size (as in most real-world cases), then s should bedecreased to some extent: 30–40 would be fine to achieve abalance between GA and local search.

8. Conclusion

In this paper, we address a production scheduling problemarising from a local commercial vehicle company. The maincontributions can be summarized as follows:

(1)

Based on real-world production features in the vehicle indus-try, two new objective functions are introduced into the jobshop scheduling model. Therefore, the studied problem hasbeen formulated as a three-objective optimization model.

(2)

A multi-objective genetic local search algorithm has beendesigned to solve the problem. The local search procedureworks in a way that complements the search mechanism ofGA, and thus improves the overall efficiency of the entirealgorithm.

(3)

Effective neighborhood properties have been proposed forminimizing each of the three objectives. These propertiesserve as a connection between generic optimizers and char-acteristics of the specific problem at hand. Computationalresults show that the proposed GLS surpasses another hybridalgorithm which does not utilize problem information.

Our future research will be conducted in the following aspects:

(1)

This paper considers two special objective functions whichhave practical meaning to manufacturing processes. We willinvestigate more practical objectives in future research. Forexample, it is interesting to consider the interfaces between

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–52 51

production scheduling and other departments in the firm,such as supply chain management and financial perfor-mances. Multi-objective optimization is a powerful tool forhandling integrated decision problems.

(2)

Incorporation of problem information into optimization algo-rithms is always a meaningful and challenging research topic.This paper has provided some preliminary results on thisissue, especially with respect to the minimization of totalweighted tardiness in job shops. However, we call it ‘‘pre-liminary’’ because other sorts of neighborhood structures andproperties are worthy to be investigated. TWT is a morecomplex objective than makespan and thus much lessresearch has been conducted on it. We will continue ourresearch with this objective function.

Acknowledgments

The authors are grateful to four reviewers whose commentshave helped to improve the quality of the paper. This work issupported by the National Natural Science Foundation of China(61104176, 61273233), the Science and Technology Project ofJiangxi Provincial Education Department (GJJ12131) and theSocial Sciences Research Project of Jiangxi Provincial EducationDepartment (GL1236).

Appendix A. Proof of the Theorems

Proof of Theorem 1. We prove the theorem by constructing suchan optimal solution. First, because the unit cost of each arc in A [

s is equal to the processing time of the operation that isconnected to the tail of the arc, i.e. ci1 ,i2 ¼ pi1

, the maximum costpath from node 0 to node uj is exactly the critical path fromoperation 0 to operation uj. Then, because the total input flowshould equal the total output flow for every node, a feasiblesolution F to this problem must be composed of several cycleflows. In this case, it is clear that each cycle contains exactly onearc in U. Based on the above insights, we can construct an optimalsolution Fn with the following steps:

Step 1:

Initialize the flow on each arc to be 0. Let j¼1. Step 2: Find the unique critical path from operation 0 to opera-

tion uj and denote it as Pnð0,ujÞ (to guarantee uniqueness,

we attach priority to the machine predecessor over thejob predecessor in the backward search of critical pathoperations).

Step 3:

If the total unit cost (i.e. length) of the path Pnð0,ujÞ plus

the unit cost of arc ðuj,0Þ is greater than 0, then add a flowamount of wuj

to arc ðuj,0Þ and each arc in Pnð0,ujÞ.

Step 4:

Let j’jþ1. If jrn, then return to Step 2. Otherwise,terminate the procedure.

8 Without this assumption, we may have to adjust the flows throughout the

whole network, which is impossible because the other parts of the network are

unknown (i.e. dependent on the specific instance).

In such a constructed solution Fn, each node except 0 has at mostone positive-flow incoming arc. This is because the flows are onlydistributed on the arcs belonging to critical paths, and mean-while, only one critical path is considered from node 0 to anyother node i. &

Proof of Theorem 2. As Fig. 1 shows, the flows in the initialnetwork are denoted by xi (xi40), so we have the flow equili-brium condition

xa�1 ¼ Faþxa

xa ¼ Fbþxb

xb ¼ Fbþ1þxbþ1

8><>: : ð16Þ

Now we are swapping the operations a and b. After executing theSWAP operator, we can construct a new feasible solution to thedual problem by adjusting the amount of flows on each arc. In thenew network illustrated in the lower part of Fig. 1, the flow on theoutgoing disjunctive arc of operation i is noted as yi. In order toenable accurate analysis, we should keep all the Fi valuesconstant8 in the process of adjusting the flows within this block,so that the flow equilibrium outside the considered scope will notbe affected. Therefore, the following equilibrium equations mustbe satisfied:

ya�1 ¼ Fbþybyb ¼ Faþyaya ¼ Fbþ1þybþ1

8><>: : ð17Þ

In order to keep the equilibrium of the remaining nodes a�a

(1raoa) and bþb (1obrz�b), we have ya�a ¼ xa�a andybþb�1 ¼ xbþb�1. Then, solving Eq. (17) together with Eq. (16)yields

yb ¼ xa�1þxb�xa

ya ¼ xb

(: ð18Þ

The nonnegativity of yb is ensured because Theorem 1 impliesxa�1Zxa.

The difference in the total cost of the new flows fyig and the

original flows fxig is

D¼ TCðyÞ�TCðxÞ ¼Xbi ¼ a

piyi�Xbi ¼ a

pixi

¼ paxbþpbðxa�1þxb�xaÞ�paxa�pbxb

¼ pbðxa�1�xaÞ�paðxa�xbÞ

¼ pbFa�paFb: ð19Þ

If DZ0, then TCmaxs0 ZTCðyÞZTCðxÞ ¼ TCmax

s (s denotes the original

set of directed disjunctive arcs while s0 denotes the new set after

the execution of SWAP on ða,bÞ). Considering TCmaxs ¼ TWTmin

s and

TCmaxs0 ¼ TWTmin

s0 , the above inequality becomes TWTmins0 ZTWTmin

s .

Therefore, if DZ0 (i.e. Fa=paZFb=pb), swapping a and b will not

improve the current solution. &

Proof of Theorem 3. (I) If b�a¼ 1 (i.e. swapping two adjacentoperations), then the difference in objective value f2 resulted fromexchanging J½a� and J½b� is

f 02�f 2 ¼ ½ðMTb�MTa�1Þ2þðMTbþ1�MTaÞ

2��½ðMTa�MTa�1Þ

2

þðMTbþ1�MTbÞ2� ¼ �2MTa�1MTb�2MTaMTbþ1

þ2MTa�1MTaþ2MTbMTbþ1

¼ 2ðMTa�MTbÞðMTa�1�MTbþ1Þ: ð20Þ

Therefore, f 02o f 23ðMTa�MTbÞðMTa�1� MTbþ1Þo0.

(II) If b�a41, then the difference between f 02 after exchanging

J½a� and J½b� and the original f 2 is

f 02�f 2 ¼ ½ðMTb�MTa�1Þ2þðMTaþ1�MTbÞ

2þðMTa�MTb�1Þ

2

þðMTbþ1�MTaÞ2��½ðMTa�MTa�1Þ

2þðMTaþ1�MTaÞ

2

þðMTb�MTb�1Þ2þðMTbþ1�MTbÞ

2�

¼ �2MTa�1MTb�2MTaþ1MTb�2MTaMTb�1�2MTaMTbþ1

þ2MTa�1MTaþ2MTaMTaþ1þ2MTb�1MTbþ2MTbMTbþ1

¼ 2ðMTa�MTbÞðMTa�1þMTaþ1�MTb�1�MTbþ1Þ: ð21Þ

R. Zhang et al. / Int. J. Production Economics 145 (2013) 38–5252

Therefore, f 02o f 23ðMTa�MTbÞðMTa�1þMTaþ1�MTb�1 �MTbþ1Þ

o0. &

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