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J. Ind. Eng. Int., 7(13), 74-83, Spring 2011ISSN: 1735-5702 IAU, South Tehran Branch

*Corresponding Author Email: [email protected].: +98 9111131380

I. Mahdavi 1* ; V. Zarezadeh 2 ; P. Shahnazari-Shahrezaei 3

1 Associate Professor, Dep. of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran 2 M.Sc., Dep. of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran

3Ph.D. Student, Dep. of Industrial Engineering, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Received: 18 July 2008; Revised: 31 December 2008; Accepted: 12 January 2009

Abstract: This article addresses a multi-stage flowshop scheduling problem with equal number of unrelatedparallel machines. The objective is to minimize the makespan for a given set of jobs in the system. Thisproblem class is NP-hard in the strong sense, so a hybrid heuristic method for sequencing and then

allocating operations of jobs to machines is developed. A number of test problems are randomly generatedand results obtained by proposed heuristic are compared with optimal solutions reported by the Lingo 8.0package applying the branch & bound approach. The results show that the proposed hybrid method is moreefficient when the problem sizes have been increased.

Keywords: Flexible flowshop; Makespan; Unrelated machines

1. Introduction

Production scheduling can be defined as theallocation of available production resources overtime to perform a collection of tasks (Baker,1974). In most manufacturing environments likeprocess industries; e.g. the chemical andpetrochemical, rubber, steel, textile and food, a setof tasks is sequentially performed by resources inseveral stages to complete a job. Such a system isreferred to as the flowshop environment andbelongs to the class of quantitative combinatorialoptimization problems.

This paper considers a flexible flowshopscheduling problem, where each production stageis made up of equal number of unrelated parallelmachines with the objective of minimizing

makespan. The considered problem generalizestwo other scheduling problems; namely, theflowshop problem and the single-stage parallelmachines problem allowing considerablereduction in makespan and the delays caused bybottleneck stages.

The main characteristic of the consideredproblem is the differences among the machines.The processing time of the jobs on the differentmachines, correspond to the three classicalparallel machines, may be identical, uniform orunrelated. The multiple machines are identical if

they do not differ in speed. The multiple machinesare purported uniformly in that they differ in

speed, but they differ by some constant speedfactors. Specifically, the machines are unrelated ifthere are no relationships between machines'speed but a hierarchy of the machines does exist.In better words, the machines are not necessarilyuniform to a speed factor, but the machines can beranked from the highest to the lowest speed.

On the other hand, the considered problem isprimarily concerned with industrial scheduling,where jobs have to be assigned to scarce resources(machines) at each stage first and then sequencedon each resource (machine) over time to optimizethe performance measure. Since the flowshop andthe single-stage parallel machines problems areknown to be NP-hard, our problem is stronglyNP-hard (Kis and Pesch, 2005). Therefore there isno escape from applying simple dispatching rules,

heuristics and improving meta-heuristics to solveit.The flowshop scheduling problem and its

generalizations is a very controversial issue andhas been the target of researches since Johnsonsseminal paper in 1954. A comprehensive reviewof flowshop scheduling problems over the last 50years is provided by Gupta and Stafford Jr. (2006)and a makespan review by Hejazi and Saghafian(2005) and also by Framinan et al. (2005).

A detailed survey for the flexible flowshopproblem has been given by Linn and Zhang

(1999) and Wang (2005). Most of theaforementioned works explore three different

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I. Mahdavi et al. / Journal of Industrial Engineering International 7(13) (2011) 74-83 75

issues: processing complexity, performancemeasures and solution methods. Although theflexible flowshop problem has been widelystudied in the literature, most of the studies relatedto flexible flowshop problems are concentrated onproblems with identical machines. While this

assumption may be true in some benchmarks, inmany real world cases the multiple machines arenot identical because multiple machines at aprocessing stage typically have small differencesin processing speed; the addition of a machine at abottleneck stage to relieve it is usually with anewer (faster) processor and the multiplemachines may be obtained from different vendors(and thus have different processing speed) (Roaand Santos, 2000). Yet a few number ofresearches considered the real life assumption thatthe multiple machines that may exist at a stage are

uniform or unrelated in processing speed.Brah et al. (1991) formulated a mixed integer

linear programming for flexible flowshop modelswith the general case of unrelated parallelmachines. Roa and Santos (2000) studied the twostage flowshops with multiple uniform machineswith the objective of minimizing makespan. Theyapplied dispatching procedures such as FIFO,LIFO, SPT, LPT etc. and also two heuristicsbased on modification of the famous Johnsonsalgorithm and based on a new application of aheuristic presented by Sule (1997) respectively.They evaluated their proposed dispatching rulesand heuristics against a makespan lower bounddeveloped by themselves and reported that theheuristic based on Johnsons algorithmoutperforms other dispatching rules and heuristicsin the two stage flexible flowshop. Soewandi andElmaghraby (2003) also considered the sameproblem and developed a heuristic (SE heuristic)and derived a machine speed-dependent worst-case ratio bound for it, but Kyparisis andKoulamas (2006) observed that the worst-casebound derived by Soewandi et al. for theirheuristic is not indicative of the expectedperformance of SE heuristic when the machinespeeds vary significantly. They addressed thisissue by deriving alternative tight speed-dependent bounds for the SE heuristic andreported that this new bound facilitates thenarrowing of the gap between averageexperimental performance and worst-caseperformance for the SE heuristic. In anotherresearch, Koulamas and Kyparisis (2007) studiedthe problem of minimizing the makespan in an

alike two-stage flowshop scheduling problem withuniform parallel machines which is ageneralization of the assembly flowshop problem

with concurrent operations in the first stage and asingle assembly operation in the second stage.They proposed a heuristic with an absoluteperformance bound which became asymptoticallyoptimal as the number of jobs became very largeas they showed. The aforementioned team;

namely Kyparisis and Koulamas (2006) alsostudied the multistage flexible flowshopscheduling problem with uniform parallelmachines in each stage and the objective ofminimizing makespan and developed a generalclass of heuristics which extend several well-known heuristics for the serial flowshop problemsuch as the slope index method developed byPalmer, the CDS heuristic of Campbell et al., andthe Dannenbring heuristic (Gupta and Stafford,2006). They obtained absolute performanceguarantees for their heuristics based on a similar

absolute performance guarantees for thecorresponding serial flowshop heuristics.

Low (2005) has developed a mathematicalmodel for the flowshop with unrelated parallelmachines and independent setup and dependentremoval times and proposed a simulated annealingheuristic for minimizing total flow time of jobs inthe system. Then Low et al. (2008) reported atwo-stage flowshop scheduling problem withunrelated alternative machines to minimize themakespan that focused on the functions of thealternative machines. In better words, their modelhas m unrelated alternative machines at the firstmachine center followed by a second machinecenter with a common processing machine in thesystem. For the processing of any job, it isassumed that the operation can be partiallysubstituted by other machines in the first center,depending on its machining constraints. 16combinations of heuristic algorithms anddispatching rules were applied by them and theassociated computational experiments indicatedthat the performance of the modified Johnsonsrule combined with the FF dispatching rule is thebest heuristic among all proposed algorithms forthe considered model.

In a new research, Jungwattanakit et al. (2009)has formulated a 0-1 mixed integer program forthe flexible flowshop problem with unrelatedparallel machines and sequence and machinedependent setup times to minimize a convexcombination of makespan and the number of tardy

jobs. They have investigated both constructiveand iterative (SA, TS and GA-based algorithms)approaches based on developing a job sequence

for the first stage by a constructive procedure andimproving it later iteratively, by sequencing the jobs for the remaining stages by both the

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76 I. Mahdavi et al. / Journal of Industrial Engineering International 7(13) (2011) 74-83

permutation and FIFO rules and by assigning the jobs at the stages to a particular machine using agreedy algorithm. From their computationalexperiences found that among the constructivealgorithms the insertion-based approach issuperior to the others, whereas the proposed SA

algorithms are better than TS and geneticalgorithms among the iterative metaheuristicalgorithms.

Ruiz and Marato (2006) proposed a heuristicbased on a genetic algorithm to solve the flexibleflowshop with sequence dependent setup times,unrelated parallel machines at each stage andmachine eligibility constraints to bridge theexisting gap between the theory of scheduling andits applications in real industrial settings. RecentlyRuiz et al. (2008) have presented a completeformulation as well as a mixed integer

programming mathematical model and someheuristics for a complex and realistic flowshopproblem. In this problem several realisticcharacteristics such as release dates for machines,existence of unrelated parallel machines at eachstage of the flowshop, machine eligibility,possibility for jobs to skip stages, sequencedependent setup times, possibility for setup timesto be both anticipatory as well as non-anticipatory,positive and/or negative time lags betweenoperations and generalized precedence relation-ships between jobs are jointly considered. Theyhave solved a comprehensive benchmark andcarried out statistical analysis by means ofdecision trees which have allowed identifyingsome counter-intuitive interactions among manydifferent characteristics of the realistic problemconsidered. Furthermore, they have also proposedsimple dispatching rules and an adaptation of theNEH algorithm.

This research focuses on the flexible flowshopproblem with equal number of unrelated machinesat each stage and development of good makespanschedules. In Section 2, the production modelunder study is described down to the last detail. InSection 3, a hybrid heuristic method is proposedand will be illustrated by an example in Section 4.In Section 5, experiments for verifying theperformance of the proposed heuristic algorithmare described. Finally, conclusions are presentedin Section 6.

2. Problem definition and notations

The flexible flowshop scheduling problem withunrelated parallel machines under considerationhas the following characteristics:

1. A set of N jobs denoted by { } N ii I ,...,2,1== is available at time zero and no job may becancelled before completion.

2. The production model consists of L consecutive stages. The set of stages is denoted by

{ } L j j J ,...,2,1== .3. Each stage J j is equipped with

1> M nonidentical machines. The set of machinesat stage j is denoted by

J , j ,...,M ,k e E jk j == 21 .

4. All jobs have to be processed serially throughall stages. Thus job I i consists of a sequenceof L operations, each of them corresponding to theprocessing of job i at stage j on machine jk e

during an uninterrupted processing time 0>ijk P .

5. The machines are continuously available fromtime zero onwards and may remain idle.

6. Each machine can process one job at a time.Furthermore, a job can be processed by any of themachines and will be processed by a singlemachine at each stage.

7. Setup and removal times are assumed to be apart of the processing time and are independent ofthe job sequence.

8. The jobs can wait in between stages and theintermediate storage is unlimited.

To maximizing system utilization, the objectiveis to develop a schedule that minimizes themakespan. In a flowshop based model, a schedulethat minimizes the makespan also minimizes thesum of job waiting times and the sum of machineidle times.

For the sake of completeness, a mixed integerlinear programming formulation for the

considered production model is included in thissection. The framework of mathematical modelinitially developed by Brah et al. (1991).

2.1. Input parameters

N The number of jobs.

M The number of parallel machines at eachstage.

J The index number of stages; j=1,,L .

P ijk The processing time of job i at stage j onmachine jk e .

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L The number of stages.

i The index number of jobs; i=1,,N .

k The index number of machines at eachstage; k=1,,M .

2.2. Decision variables

ijC The completion time of job i at stage j

maxC The makesapan

otherwise,0

machineon

stageat jobprecedes jobif ,1

jk irj e

jr i

X

otherwise,0

machinetoassignedisstageat jobif ,1 ik ijk

e jiY

2.3. Mathematical formulation

Min C max

Subject to:

C max C iL i (1)

11

==

M

k ijk Y ji, (2)

=

=

2and1

1

1111

jiPY C C

i PY C

M

k ijk ijk i,jij

M

k k ik ii

(3)

( )( )


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Table 1: Processing times for the six jobs.

Jobs

Stages1 2 3 4

Machines Machines Machines Machines1 2 3 1 2 3 1 2 3 1 2 3

1 25 30 32 45 54 58 52 62 68 40 48 52

2 7 9 10 41 51 56 22 28 30 66 82 903 41 47 52 55 63 70 33 38 42 21 24 274 74 88 100 12 14 16 24 28 33 48 57 675 7 9 11 15 20 25 72 96 120 52 69 866 12 15 16 14 16 20 22 25 28 32 35 38

(3) Dealing with each flowshop F k (k=1,2,3):

(3-1) For each job I i , ik is calculated asshown in Table 2.

(3-2) The subgroups of jobs U k (k=1,2,3) areformed such that { }

iLk k ik PP I iU


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Table 2: Calculation of ik for each job at each flowshop.

i

Flowshop F 1 Flowshop F 2 Flowshop F 3

( )11113

11 min )i(jij

ji PP +

=+= ( )212

13

12 min )i(jij

ji PP +

=+= ( )313

13

13 min )i(jij

ji PP +

=+=

1 min(25+45, 45+52, 52+40)=70 min(30+54, 54+62, 62+48)=84 min(32+58, 58+68, 68+52)=90

2 min(7+41, 41+22, 22+66)=48 min(9+51, 51+28, 28+82)=60 min(10+56, 56+30, 30+90)=663 min(41+55, 55+33, 33+21)=54 min(47+63, 63+38, 38+24)=62 min(52+70, 70+42, 42+27)=694 min(74+12, 12+24, 24+48)=36 min(88+14, 14+28, 28+57)=42 min(100+16, 16+33, 33+67)=495 min(7+15, 15+72, 72+52)=22 min(9+20, 20+96, 96+69)=29 min(11+25, 25+120, 120+86)=366 min(12+14, 14+22, 22+32)=26 min(15+16, 16+25, 25+35)=31 min(16+20, 20+28, 28+38)=36

Table 3: The effects of elimination of the jobs in NS from F 1 ondecreasing C 1 in the first iteration.

NS i C 1-T C 1

1 402 663 214 485 806 32

(10) According to step 10, J c=J 5is deleted from NS . Because NS , steps 6 to 10 of theproposed algorithm is repeated until = NS . Theiterations are shown in Table 4.

The final solution includes the jobs assigned toeach flowshop and sequence of them in order ofthe established priorities of jobs at each flowshopin step 3. The results are shown in Table 5.

At each flowshop, the sequence of jobs do notchange, so as a matter of fact, the assignment ofoperations of each job to machines at each stageand sequencing operations on each machine aredetermined and the final schedule in theconsidered flexible flowshop is obtained.

5. Computational experiments and results

To determine the quality of the proposedheuristic, a spreadsheet model of the consideredproblem was developed and a number of test

problems were randomly generated and solved byproposed heuristic coded in Microsoft VisualBasic for Applications. The results werecompared to the optimal solutions obtained via theimplementation of the mixed integer linearprogramming formulation (presented in section 2)

by the Lingo 8.0. All experimental tests wereimplemented on a personal computer with an IntelPentium III 633 GHz CPU and 256 MB of RAM.

Four sets of problems were tested, respectivelyfor 3 to 6 jobs. Each job has three operations andeach stage has two nonidentical machines. Theprocessing time of each operation was randomlygenerated and each set of problems was executedfor 15 tests. The makespans for problems of threeto six jobs by proposed heuristic and by thebranch & bound approach are shown respectivelyin Figures 1 to 4.

The mean relative error between the optimalmakespans and those obtained by proposedheuristic for each set of test problems is shown inTable 6. Over the entire collection of instances,the average relative error of the proposed heuristicis 6.08%. The average execution times for solvingproblems of three to six jobs by the Lingo 8.0 isshown in Figure 5. From the figures, it is easilyseen that the proposed heuristic got a little largermakespan than the branch & bound approach did,but the computational time needed by the Lingo8.0 was however much larger than that needed byproposed algorithm.

Figure 1: Makespans of 15 tests for three jobs.

M a k e s p a m

Problem #

Proposed heuristicBranch & Bound

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Figure 2: Makespans of 15 tests for four jobs.

Figure 3: Makespans of 15 tests for five jobs.

Figure 4: Makespans of 15 tests for six jobs.

Figure 5: The average CPU times for processing different numbers of jobs.

M a k e s p a m

Problem #

Problem #

M a k e s p a m

M a k e s p a m

Problem #

E x e c u

t i o n

t i m e s e c o n d s

Branch & bound

Branch & bound

Branch & bound

Branch & bound

Proposed heuristic

Proposed heuristic

Proposed heuristic

Proposed heuristic

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Table 4: The iterations of steps 6 to 10 of the proposed heuristic solving the illustrative example.

I t e

r a

t i o n

1

C max=353 NS={J 6 , J 2 , J 1 , J 3 , J 4 } J c={J 5 }

F k O k T k C

F 1 {J 6 , J 2 , J 1 , J 3 , J 4 } 273

F 2 {J 5 } 194

F 3 {J 5 } 242

T k c

C 194

{ }T T k C C c 1,max 273 Decision 273225

Remaining J 2 to F 1

I t e

r a

t i o n

2

C max=273 NS={J 6 , J 2 , J 1 , J 3 } J c={J 4 }

F k O k T k C

F 1 {J 6 , J 2 , J 1 , J 3 } 225

F 2 {J 5 ,J 4 } 251

F 3 {J 4 } 216

T k c

C 216

}T T k C C c 1,max 225 Decision 225


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References

Baker, K. R., (1974), Introduction to sequencingand scheduling . John Wiley & Sons, NewYork.

Brah, S. A.; Hunsucker, J. L.; Shah, J., (1991),Mathematical modeling of schedulingproblems. Journal of Information &Optimization Sciences, 12(1), 113-137.

Framinan, J. M.; Leisten, R.; Ruiz-Usano, R.,(2005), Comparison of heuristics for flowtime minimisation in permutation flowshops.Computers & Operations Research , 32(5),1237-1254.

Gupta, J. N. D.; Stafford Jr., E. F., (2006),Flowshop scheduling research after fivedecades. European Journal of Operational

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Gupta, J. N., (1971), A functional heuristicalgorithm for the flowshop schedulingproblem. Operational Research Quarterly ,22(1), 39-47.

Hejazi, S. R.; Saghafian, S., (2005), Flowshop-scheduling problems with makespancriterion: A review. International Journal ofProduction Research , 43(14), 2895-2929.

Jungwattanakit, J.; Reodecha, M.; Chaovali-

twongse, P.; Werner, F., (2009), Acomparison of scheduling algorithms forflexible flowshop problems with unrelatedparallel machines, setup times and dualcriteria. Computers & Operations Research ,36(2), 358-378.

Kis, T.; Pesch, E., (2005), A review of exactsolution methods for the non-preemptivemultiprocessor flowshop problem. European

Journal of Operational Research , 164(3),592-608.

Koulamas, C.; Kyparisis, G. J., (2007), A note onthe two-stage assembly flowshop schedulingproblem with uniform parallel machines.

European Journal of Operational Research, 182(2), 945-951.

Kyparisis, G. J.; Koulamas, C., (2006), A note onmakespan minimization in two-stage flexibleflowshops with uniform machines. European

Journal of Operational Research, 175(2),1321-1327.

Kyparisis, G. J.; Koulamas, C., (2006), Flexibleflowshop scheduling with uniform parallelmachines. European Journal of Operational

Research, 168(2), 985-997.

Linn, R.; Zhang, W., (1999), Hybrid flowshopscheduling: A survey. Computers &

Industrial Engineering, 37(1-2), 57-61.

Low, C., (2005), Simulated annealing heuristic forflowshop scheduling problems with unrelatedparallel machines. Computers & Operations

Research, 32(8), 2013-2025.

Low, C.; Hsu, C.-J.; Su, C.-T., (2008), A two-stage hybrid flowshop scheduling problemwith a function constraint and unrelatedalternative machines. Computers & Opera-tions Research, 35(3), 845-853.

Roa, I.; Santos, D. L., (2000), Flowshops with Non-Identical Multiple Processors: A Studyon Makespans . Proceedings of The 5thAnnual International Conference onIndustrial Engineering -Theory, Applicationsand Practice, Hsinchu, Taiwan.

Ruiz, R.; Maroto, C., (2006), A genetic algorithmfor hybrid flowshops with sequencedependent setup times and machineeligibility. European Journal of Operational

Research , 169(3), 781-800.

Ruiz, R.; Serifoglu, F. S.; Urlings, T., (2008),Modeling realistic hybrid flexible flowshopscheduling problems. Computers &Operations Research , 35(4), 1151-1175.

Soewandi, H.; Elmaghraby, S. E., (2003),Sequencing on two-stage hybrid flowshopswith uniform machines to minimizemakespan. IIE Transactions , 35(5), 467-477.

Sule, D. R., (1997), Industrial scheduling . 1 st edition, PWS Publishing, Boston, 93-96.

Wang, H., (2005), Flexible flowshop scheduling:optimum, heuristics, and artificialintelligence solutions. Expert Systems , 22(2),78-85.

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