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A heuristic for scheduling in a flowshop withthe bicriteria of makespan and maximumtardiness minimizationKarunakaran Chakravarthy & Chandrasekharan RajendranPublished online: 15 Nov 2010.

To cite this article: Karunakaran Chakravarthy & Chandrasekharan Rajendran (1999) A heuristic for schedulingin a flowshop with the bicriteria of makespan and maximum tardiness minimization, Production Planning &Control: The Management of Operations, 10:7, 707-714, DOI: 10.1080/095372899232777

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PRODUCTION PLANNING & CONTROL, 1999, VOL. 10, NO. 7, 707 –714

A heuristic for scheduling in a � owshop with thebicriteria of makespan and maximum tardinessminimization

KARUNAKARAN CHAKRAVARTHY and CHANDRASEKHARAN RAJENDRAN

Keywords � owshop scheduling, heuristics, makespan, tardi-ness

A bstract. This article deals with the development of a heuristicfor scheduling in a � owshop with the objective of minimizingthe makespan and maximum tardiness of a job. The heuristicmakes use of the simulated annealing technique. The proposedheuristic is relatively evaluated against the existing heuristic forscheduling to minimize the weighted sum of the makespan andmaximum tardiness of a job. The results of the computationalevaluation reveal that the proposed heuristic performs betterthan the existing one.

1. Introduction

Much research has been carried out on the problem ofscheduling in � owshops. Optimization algorithms for the

two- and three-machine � owshop problems for minimiz-ing the makespan have been developed by Johnson( 1954) , and Ignall and Schrage ( 1965) , respectively.Ignall and Schrage have also developed a branch-and-bound algorithm for minimizing the total � owtimeof jobs in a two-machine � owshop problem. In view ofthe NP-completeness of a vast majority of � owshopscheduling problems ( see Garey et al. 1976) , researchis also directed towards the development of heuristic ornear-exact procedures. Some of the noteworthy heuristicsare due to Campbell et al. ( 1970) , Dannenbring ( 1977) ,Gelders and Sambandam ( 1978) , King and Spachis( 1980) , Nawaz et al. ( 1983) , Widmer and Hertz ( 1989) ,Rajendran ( 1993) , I shibuchi et al. ( 1995) and Ho ( 1995) .While most exact and heuristic methods deal with the

A uthors: K. Chakravarthy and C. Rajendran, Industrial Engineering and Management Division ,Department of Humanities and Social Sciences, Indian Institute of Technology, Madras–600 036,India. e-mail: craaj@ hotmail.com

K A RUNA KA RA N CHA KRA VA R THY holds a BTech. degree in Mechanical Engineering from theIndian Institute of Technology Madras, and an MS degree in Industrial Engineering from theUniversity of Pittsburgh. His research interests are in sequencing and scheduling. He has publishedresearch articles in these areas. This paper is based on his BTech. thesis work. He is now working asa consultant in the USA.

CHA NDRA SE KHA RA N RA J ENDRA N is an Associate Professor of Operations Management at theIndian Institute of Technology, Madras. He has published a number of articles in journals, e.g.European J ournal of Operational Research, J ournal of Operational Research Society, International J ournal ofP roduction Economics, International J ournal of P roduction Research, and Production Planning & Control. Healso serves as a referee for these journals. He is a recipient of the Alexander von Humboldt ResearchFellowship of Germany.

Production P lanning & Control ISSN 0953–7287 print/ISSN 1366–5871 online # 1999 Taylor & Francis Ltdhttp://www.tandf.co.uk/JNLS/ppc.htm

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minimization of a single measure of performance, onlysome attempts have been made towards the developmentof optimization and heuristic algorithms with multipleobjectives for two- and multi-machine � owshop problems( see Ho and Chang 1991, Rajendran 1992, 1994, andNagar et al. 1995, 1996) . I t is often mentioned in theliterature ( see Baker 1974, French 1982, Pinedo 1995)that the study on the problem of scheduling with multipleobjectives deserves attention in view of its signi� cance inreal-life situations. Hence, in this article, the problem of� owshop scheduling with the twin objectives of minimiz-ing the makespan and maximum tardiness is considered,and a heuristic, based on the simulated annealing tech-nique, is proposed. A survey of literature has revealedthat Daniels and Chambers ( 1990) have developed aheuristic considering these bicriteria for the m-machine� owshop problem. For a given limiting value of themaximum tardiness, the heuristic by Daniels andChambers identi� es a subset of jobs that may occupythe � nal position of a sequence. Then, the eligible jobwith the largest completion time in the schedule thatminimizes the makespan is found out. In order to approx-imate this schedule, the heuristic by Nawaz et al. ( 1983) isused ( see Appendix for a description of the heuristic byNawaz et al. 1983) . This eligible job is placed in then-th position, where n refers to the number of jobs to bescheduled. Next, considering the remaining …n ¡ 1† jobs,the subset of eligible jobs and the actual eligible job foroccupying the …n ¡ 1†-th position are determined. Thisprocess is carried out until a complete sequenceis constructed. I f the � nal schedule violates the relevantmaximum tardiness limit, the procedure will backtrack tothe last job in the schedule with excess tardinessand replace it with a di� erent eligible job. Themakespan and maximum tardiness of jobs arecomputed for the sequence. The heuristic by Danielsand Chambers is referred to as DCH in this paper.In this article as well as in the article by Daniels andChambers, permutation schedules are considered.Note that in the case of � owshop scheduling, it is assumedthat all jobs have the same unidirectional routing,starting from the � rst machine, and that all jobs areindependent, i.e. no assembly operations are involved.Also, note that all jobs undergo processing on allmachines.

This article is organized as follows. First, we presentthe problem formulation, followed by a discussion onthe methods for generating a good seed sequence thatis given as an input to the simulated annealing heuristic ( SA) . The step-by-step procedure of the SA ispresented subsequently. The details of an extensive per-formance analysis of the proposed SA and DCH arediscussed.

2. Problem formulation

Let:n number of jobs to be scheduled in the � owshopm number of machines in the � owshopti j process time of job i on machine jd i due-date of job i¼ available partial schedule ( or an ordering of the

set of jobs already scheduled)q…¼; j† completion time of partial schedule ¼ on ma-

chine j

With the conditions of non-interference at machines,non-simultaneous processing of jobs and no job-passing,when job a is appended to ¼, the completion time ofpartial schedule ¼a on machine j , q…¼a; j†, can beobtained by the following recursive equation:

q…¼a; 1† ˆ q…¼; 1† ‡ ta1; and

q…¼a; j† ˆ max‰q…¼; j†; q…¼a; j ¡ 1†Š ‡ taj ; j ˆ 2; 3; . . . ; m …1†

Call the completion time of job a as C 0

a , i.e. we haveC 0

a ˆ q…¼a; m†. Let us call the resultant partial schedule ¼,where ¼ ¼a. The maximum tardiness of a job in acomplete schedule is given by

½ ˆ max‰maxfC 0

i ¡ d i; 0g; i ˆ 1; 2; . . . ; nŠ …2†

The makespan of is given by

M ˆ C 0

‰nŠ …3†

We use the following object ive ( or preference) function,also referred to by Daniels and Chambers:

Z ˆ ¶ M ‡ …1 ¡ ¶† ½ …4†

where ¶ denotes the relative importance ( or weights) ofthe individual components of the function. Note that theheuristic by Daniels and Chambers attempts to solve theproblem subject to ½ K , where K is a value speci� ed bythe scheduler. We have solved the present problem ofminimizing Z , subject to ½ K , where K is speci� edby us in the scheduling problem.

3. Initial heuristic sequence

Because the SA algorithm is a randomized search pro-cedure, we start with an initial seed sequence for thealgorithm to improve upon the seed sequence. The ratio-nale of starting with a good seed sequence is observed byJohnson et al. ( 1989) , especially to reduce the computa-tional time. In this paper, we use three methods for gen-erating three sequences and choose the best sequencewith respect to Z . These methods are as follows.

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3.1. Earliest due-date method

Using this method, we obtain a sequence by orderingthe jobs by ascending di values.

3.2. Least static slack method

Using this method, we obtain a sequence by orderingthe jobs by ascending SL i values, where

SL i ˆ di ¡Xm

jˆ 1

t i j …5†

3.3. N EH method

We make use of the heuristic by Nawaz et al. ( 1983) toobtain a sequence. See the Appendix for the mechanics ofthe NEH method. While generating partial sequences inthe NEH method, we evaluate them with respect to theirmakespan values and choose the best generated partialsequence.

I t is to be noted that while the � rst two methods seek tominimize the component of maximum tardiness, theNEH method seeks to minimize the component of themakespan in Z . Among the sequences obtained fromthese three methods, the sequence with the least valueof Z , subject to ½ K , is chosen. I f no sequence satis� esthe condition of ½ K , then the sequence with the leastvalue of ½ is chosen. The sequence, thus chosen, is givenas the input to the SA heuristic. Call this sequence S.

4. Improvement by the simulated annealingtechnique

The permutation sequencing problem has a largesolution space ( n factorial) . I t is well known that theSA approach yields near-optimal solutions to combinato-rially intractable problems. The SA algorithm, however,is generic, and has to be modi� ed in the context of thespeci� c problem under study. The generic SA algorithmwas � rst proposed by Kirkpatrick et al. ( 1983) . A genericprocedure of the SA technique for a minimizationproblem is given below.

Step 1. Get an initial solution, S :

Step 2. Set an initial temperature, T , …T > 0†.Step 3. While not frozen do the following:

Step 3.1. Do the follow ing L times:Step 3.1.1. Sample a neighbour S 0 from

S.

Step 3.1.2. Let = cost ( S 0 ) – cost ( S ) .Step 3.1.3. I f 0

then ( i.e. downhill move) setS ˆ S 0

else ( i.e. uphill move) setS ˆ S 0 , with a probabilityofexp …¡ =T †:

Step 3.2. Set T ˆ r T , where r is the reductionfactor.

Step 4. Return S.

The above algorithm is generic and is to be modi� edaccording to the problem under study. Note that theterm ‘temperature’ is essentially a control parameter todetermine the probability of acceptance of an inferiorsolution during the search process of SA.

We generate the neighbour S 0 by using the adjacentinterchange scheme ( AIS) . Using this scheme, we choosea job at random and then sample a uniform randomnumber. I f the uniform random number is less than orequal to 0.5, the chosen job is interchanged with theadjacent job to its left; otherwise, the chosen job is inter-changed with the adjacent job to its right. I f the chosenjob is at an extreme position, then it is interchanged withthe adjacent job to its right or left, depending upon theposition 1 or n. I t is to be noted that if ½ , the maximumtardiness of jobs in S 0 , is greater than K ( the limit onmaximum tardiness speci� ed in the problem) , the AISis repeated until ½ of S 0 is less than or equal to K .Thus, the feasibility of the generated sequence S 0 isensured.

We now present the details of mechanics and par-ameter settings of the SA algorithm used in this study.The initial temperature and the temperature reductionfactor are � xed such that a reasonable number of itera-tions can be be carried out before the algorithm ‘freezes’.We use a counter, called FR_CNT, which is used toterminate the SA algorithm. B refers to the sequencewith the least value of Z that is encountered at all tem-peratures. S 0 is the sequence which is obtained byemploying AIS on S . TOTAL and ACCEPT are twocounters that keep track of the total number of iterationsand total number of accepted moves, respectively, at aparticular temperature, and these counters are set to zeroat the start of every temperature. The iterations at aparticular temperature are terminated if ACCEPT isgreater than or equal to n, or when TOTAL is greaterthan or equal to k n, where k is an integer multiplier.The former check is used to infer whether the iterationsare successful in unearthing good solutions, and the lattercheck is for an upper limit on the number of iterations atany temperature. After the iterations are over, we deter-mine the percentage of accepted moves denoted by PER.

Scheduling in a � owshop 709

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I f PER is less than a pre-speci� ed value, FR_CNT isincremented by 1. This is done because the chances offurther obtaining good solutions are diminishing, andhence we move towards the terminal condition. The tem-perature reduction factor is used to reduce the tempera-ture, and the above-mentioned steps are repeated for thenew temperature until the algorithm ‘freezes’ or termi-nates. The termination of the SA algorithm is achievedby two conditions:

( 1) when FR_CNT reaches a prespeci� ed value; or( 2) when the temperature T falls to a prespeci� ed

value at which the algorithm is ‘frozen’. This con-dition is introduced to end the computation ( i.e.when FR_CNT is repeatedly reset to zero, therebyleading to a very large computational e� ort) .

I t is a customary practice to de� ne as equal to…Z S 0 ¡ Z S†. I n this paper, we de� ne as

ˆ …Z S 0 ¡ Z S† 100=Z S …6†

The advantage of this de� nition is that we have as adimensionless quantity, and indicates the relative per-centage deviation of the quality of the perturbed solutionfrom that of the original solution. Therefore, is notdependent upon the problem speci� cations, and hencewe can � x the initial and � nal temperature values morereasonably and accurately to minimize the computa-tional e� ort without sacri� cing the quality of � nal sol-ution. An initial temperature of 475 is chosen. Choosing ahigher value of temperature would result in the accep-tance of very inferior solutions. We have set the limits ( i.e.the associated probability of acceptance) for the inferior-ity of the perturbed solution S 0 . Using this setting, aninferior solution, inferior by 50% relative to the originalsolution S , is accepted with an associated probability of0.9. Therefore, we arrive at the initial temperature set-ting as given below:

P ˆ exp…¡ =T †; i:e: 0:9 ˆ exp …¡ 50=T †; or T 475

…7†

I t is now evident that this value of initial temperaturewill lead to the acceptance of an inferior solution ( inferiorby 50% relative to the original solution) with a probabil-ity of 0.9, when the SA algorithm starts. The temperaturereduction factor is � xed at 0.9. The � nal literature isbased on the value of initial termperature 475 and thereduction factor 0.9, and by allowing the algorithm torun for 30 ‘temperature stages’. We have therefore � xedthe value of the � nal temperature at 20, i.e. we have the� nal temperature to be 475 …0:9†30 , which is set to 20.A sample of 30 is considered to be a good sample statis-tically , and hence we have taken 30 ‘temperature stages’.The temperature-terminating conditions involve the lim-

its on ACCEPT and TOTAL. The minimum value ofACCEPT needed for the termination of iterations at atemperature is set to n, in order to restrict the computa-tional e� ort. We choose 25% as a good indicator of thesuccess in unearthing good sequences at a temperature,and hence the upper limit works out to be 4 n. A valueof 15% for the percentage of accepted moves is chosen asan overall indicator for the e� cacy of the SA algorithm,and consequently, FR_CNT is kept at 5, once again torestrict the computational e� ort. Moreover, pilot runshave also indicated that these parameter settings workquite well in yielding solutions of good quality, andthat any increased run time of the SA algorithm hasnot resulted in a substantial increase in the quality of� nal solutions.

5. Simulated annealing algorithm

I n this section we present the SA algorithm ( based onSridhar and Rajendran 1993) which we have adapted tothe scheduling problem in the � owshop under study. Theterminologies and notations used in the algorithm are� rst presented. The logical � ow chart of the SA algorithmis presented next.

5.1. N otations and terminolog y

FR_CNT freeze counter used to check whether thealgorithm could be frozen or not; whenthe counter reaches 5, the algorithm isdeemed to be frozen; the counter is resetto zero whenever we � nd that Z S 0 ZB

ACCEPT counter to keep track of the number ofaccepted moves at a particular tempera-ture

TOTAL counter to keep track of the total numberof moves at a particular temperature

PER stores the percentage of accepted movesat a particular temperature

B the best sequence obtained so far; when-ever B is updated by a better solution,FR_CNT is reset to zero, with therenewed hope of obtaining a still bettersolution later

S sequence on which AIS is employedS 0 sequence that is obtained by perturbing S

( i.e. by employing AIS on S )ZB ; Z S ; Z S 0 Z values of sequences B , S and S 0 , re-

spectivelyAIS…S ; S 0

; Z S 0 † the routine using AIS for which sequenceS is the input and S 0 is the output with itsobjective function value given by Z S 0

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DEL …B ; S 0 † the routine for computing the relativeimprovement ( or deterioration) of S 0

with respect to B . We have this measuregiven by

DELTA_B ˆ …Z S 0 ¡ Z B † 100=Z B ( 8)DEL …S ; S 0 † the routine for computing the relative

improvement ( or deterioration) of S 0

with respect to S. We have this measuregiven by

ˆ …Z S 0 ¡ Z S † 100=Z S …9†

P acceptance probability given byexp…¡ =T †

U uniform random number which is used tocheck for the acceptance of an inferiorsolution

5.2. SA algorithm: the log ical � ow chart

The SA algorithm, as a � ow chart, is shown in � gure 1.

6. Com putational experience w ith theperformance of the SA heuristic and the existingheuristic

An extensive relative evaluation of the SA heuristic( SA) and the heuristic by Daniels and Chambers( DCH) has been undertaken. Integer process times ofjobs have been sampled from a discrete uniform distri-bution in the range {1, 99}. The due-date of a job hasbeen randomly set in the range ‰

Pmjˆ 1 ti j ;

Pmjˆ 1

ti j ‡ 0:5 …n ¡ 1† mean process time of a job onmachine m}. This due-date setting would result in

Scheduling in a � owshop 711

Figure 1. F low chart of the proposed SA algorithm.

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approximately 50% of the jobs being tardy, and it is tightenough to distinguish between the performances of heur-istics. Such a due-date setting methodology was � rst pro-posed by Gelders and Sambandam ( 1978) . To specify K ,the limit on the maximum tardiness, for a given problem,we adopt the following procedure.

Note that a weak lower bound on makespan, LBM,can be obtained by the following:

LBM ˆ maxi

Xm

jˆ 1

t ij ‡ …n ¡ 1†

mean process time of a job on machine m …10†

The value of K for a problem is set as given below:

K ˆ maxi

fLBM ¡ D ig: …11†

Note that while DCH ensures the feasibility conditionthrough the backtracking procedure, the proposed SAensures feasibility through the AIS. In the AIS, the per-turbed sequence S 0 is considered only if the condition of ½

being less than or equal to K is satis� ed.The number of jobs has been varied from 10 to 50 in

steps of 10, and the number of machines has been variedfrom 5 to 25 in steps of 5. Three values of ¶ have been set,viz. 0.25, 0.5 and 0.75. In all, we have 75 di� erent experi-mental settings ( or problem sets) . In each problem set,we have generated 30 problems. In total, we have gen-erated 2250 problems. Let Z 0 and Z 0 0 be the values ofobjective functions given by the schedules of SA andDCH, respectively, for a given problem. The relativepercentage deviation ( RPD) of the solution by SA isgiven by

RPD…Z 0 † ˆ …Z 0¡ minfZ 0

; Z 0 0 g† 100= minfZ 0; Z 0 0 g

…12†

Similarly, the RPD…Z 0 0 † is calculated. The mean RPDvalues are computed over 30 such problems generated fora given problem set de� ned by n, m and ¶. We have alsoobserved the maximum RPD values in these 30 prob-lems. The results are tabulated in tables 1–3. I t is evidentfrom the tables that the SA is better than the DCH in allaspects.

7. Summary

This article has addressed the problem of scheduling ina � owshop with the bicriteria of minimizing the weightedsum of the makespan and maximum tardiness of a job. Aheuristic, based on the simulated annealing technique,has been proposed. When evaluated using a large num-ber of problems of various sizes, the proposed heuristic isfound to perform better than the existing heuristic.

712 K . Chakravarthy and C. Rajendran

Table 1. Relative performance of the procedures with ¶ ˆ 0:25.

Relative percentage increase in Z

SA DCHn m Mean Maximum Mean Maximum

10 5 0.553 1.067 4.597 10.85310 0.224 0.985 0.482 1.96815 0.074 0.436 0.641 2.85320 0.063 0.232 0.084 0.58725 0.026 0.130 0.167 0.644

20 5 0.579 1.045 2.324 7.87210 0.638 0.936 1.976 9.23415 0.173 0.647 0.751 2.96720 0.083 0.420 0.385 0.81025 0.069 0.256 0.236 0.756

30 5 0.238 0.936 1.976 9.23410 0.047 0.620 0.855 2.04815 0.379 0.835 0.799 2.70120 0.133 0.436 0.266 0.68925 0.104 0.380 0.245 0.921

40 5 0.143 0.358 0.883 3.99210 0.268 0.837 0.828 3.15815 0.336 0.973 0.715 2.55420 0.142 0.518 0.315 1.07825 0.127 0.570 0.274 1.389

50 5 0.299 0.530 1.697 6.06310 0.354 1.279 0.627 1.89615 0.153 0.567 0.885 2.51320 0.246 0.911 0.470 1.29825 0.167 0.849 0.321 1.786

Note: the sample size in every problem set is 30 and the total number ofgenerated problems

Table 2. Relative performance of the procedures with ¶ ˆ 0:50.

Relative percentage increase in Z

SA DCHn m Mean Maximum Mean Maximum

10 5 0.288 0.968 1.871 8.43410 0.161 0.973 0.447 4.37815 0.216 0.798 0.317 1.52420 0.086 0.683 0.289 1.66725 0.091 0.426 0.156 0.790

20 5 0.140 1.118 1.422 7.26610 0.079 0.804 0.873 1.85515 0.186 1.275 0.416 1.59120 0.072 0.571 0.127 0.87225 0.048 0.402 0.159 0.643

30 5 0.163 0.854 1.415 6.56110 0.232 1.231 0.269 1.48115 0.350 1.347 0.468 1.67120 0.144 0.588 0.279 0.98925 0.054 0.374 0.076 0.526

40 5 0.089 0.637 0.736 5.09410 0.204 1.215 0.417 2.06615 0.236 0.926 0.417 1.64620 0.082 0.490 0.286 0.70525 0.082 0.378 0.099 0.933

50 5 0.118 0.654 0.936 3.77010 0.206 0.944 0.193 0.92815 0.268 1.273 0.842 2.00720 0.096 0.344 0.221 0.90725 0.126 0.355 0.209 0.961

Note: the sample size in every problem set is 30 and the total number ofgenerated problems is 750.

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A cknowledgements

The authors thank the referee for the suggestions toimprove the earlier version of the paper.

References

BA KE R , K . R., 1974, Introduction to Sequencing and Scheduling ( NewYork: Wiley) .

CA MP B EL L , H. A ., DUDEK, R. A., and SMITH , M . L., 1970, Aheuristic algorithm for the n-job, m-machine sequencingproblem. M anag ement Science, 16, B630–B637.

DA NIE L S, R . L., and CHA MBE RS, R . J ., 1990, Multi-objective� ow shop scheduling. N aval Research Log istics, 37, 981–995.

DA NNE NB RING, D. G., 1977, An evaluation of � ow-shop sequen-cing heuristics. M anagement Science, 23, 1174–1182.

FRENCH , S., 1982, Sequencing and Scheduling : An Introduction to theM athematics of the J ob-shop ( Chichester: Ellis Horwood) .

GA REY , M . R ., J OHNSON , D. S., and SE THI, R., 1976, The com-plexity of � ow-shop and job-shop scheduling. M athematics ofOperations Research, 1, 117–129.

GEL DERS, L . F ., and SA MBA NDA M, N., 1978, Four simple heur-istics for scheduling a � owshop. International J ournal ofP roduction Research, 16, 221–231.

HO, J . C., 1995, Flowshop sequencing with mean � owtimeobjective. European J ournal of Operational Research, 81, 571–578.

HO, J . C., and CHA NG, Y .-L ., 1991, A new heuristic for the n-job, m-machine � owshop problem. European J ournal ofOperational Research, 52, 194–202.

I GNA L L , E., and SCHRA GE , L ., 1965, Application of the branch-and-bound technique to some � owshop scheduling problems.Operations Research, 13, 400–412.

I SHIB UCHI, H., M ISA KI, S., and TA NA KA , H., 1995, Modi� edsimulated annealing algorithms for the � ow shop sequencingproblems. European J ournal of Operational Research, 81, 388–398.

J OHNSON , D. S., A RA GON, C. R ., M CGEOCH, L . A., andSCHEVON , C., 1989, Optimisation by simulated annealing:an experimental evaluation: Part 1. Graph partitioning.Operations Research, 37, 865–891.

J OHNSON , S. M., 1954, Optimal two- and three-stage productionschedules. N aval Research Log istics Quarterly, 1, 61–68.

K ING, J . R ., and SP A CHIS, A. S., 1980, Heuristics for � ow-shopscheduling. International J ournal of P roduction Research, 18, 345–357.

K IRKP A TRICK, S., GEL A TT, C. D., J R , and V ECCHI, M . P ., 1983,Optimization by simulated annealing. Science, 220, 671–680.

NA GA R , A., HE RA GU, S. S., and HA DDOCK, J ., 1995, A branchand bound approach for a two-machine � owshop schedulingproblem. J ournal of the Operational Research Society, 46, 721–734.

NA GA R , A ., HERA GU, S. S., and HA DDOCK, J . , 1996, A combinedbranch and bound and genetic algorithm based approach fora � owshop scheduling problem. Annals of Operations Research,63, 397–414.

NA W A Z , M ., ENSCOR E , E. E., J R , and HA M, I ., 1983, A heuristicalgorithm for the m-machine, n-job � owshop sequencingproblem. OM EGA , 11, 91–95.

P INE DO, M ., 1995, SCHED ULIN G T heory , A lgorithms and Systems( Englewood Cli� s: Prentice Hall) .

RA J ENDRA N, C., 1992, Two-machine � owshop scheduling prob-lem with bicriteria. J ournal of the Operational Research Society,43, 871–884.

RA J ENDRA N, C., 1993, Heuristic algorithm for scheduling in a� owshop to minimize total � owtime. International J ournal ofP roduction Economics, 29, 65–73.

RA J ENDRA N, C., 1994, A heuristic for scheduling in � owshopand � owline-based manufacturing cell with multi-criteria.International J ournal of Production Research, 32, 2541–2558.

SRIDHA R , J ., and R A J E NDRA N, C., 1993, Scheduling in a cellularmanufacturing system Ð a simulated annealing approach.International J ournal of Production Research, 31, 2927–2945.

W IDME R , M ., and HER TZ , A., 1989, A new heuristic method forthe � owshop sequencing problem. European J ournal ofOperational Research, 41, 186–193.

A ppendix

The mechanics of the NEH method are explained witha numerical illustration. Suppose we have four jobs tobe scheduled. Find the sum of process times of job i, i.e.Pm

jˆ 1 ti j , and denote this sum by T i . Compute this sum fori ˆ 1; 2; . . . ; 4. Arrange the jobs in the descending orderof T i . Let the sequence thus obtained be [4–3–1–2] .Call this the seed sequence. Take out the � rst job fromthe seed sequence to form a partial sequence. We havethe partial sequence [4] , and the updated seedsequence is [3–1–2] . Take out the � rst job from the

Scheduling in a � owshop 713

Table 3. Relative performance of the procedures with ¶ ˆ 0:75:

Relative percentage increase in Z

SA DCHn m Mean Maximum Mean M aximum

10 5 0.298 1.038 0.907 7.19810 0.175 1.123 1.020 4.77615 0.086 0.948 0.246 1.58820 0.252 0.927 0.569 2.45525 0.192 1.119 0.581 2.761

20 5 0.086 1.094 0.832 7.36410 0.157 0.949 0.913 3.11415 0.129 0.858 0.217 1.41120 0.135 0.917 0.279 1.81625 0.263 1.058 0.421 1.848

30 5 0.173 0.987 0.439 6.16010 0.147 1.032 0.512 2.62015 0.145 0.947 0.218 1.50320 0.129 0.936 0.618 1.52325 0.120 0.945 0.258 1.517

40 5 0.089 1.079 0.763 5.41910 0.154 0.917 0.308 1.89715 0.134 0.827 0.451 1.40220 0.129 0.860 0.352 1.29725 0.097 0.774 0.255 1.085

50 5 0.076 0.936 1.217 4.50610 0.256 1.047 0.418 1.98315 0.275 0.855 0.394 1.29020 0.229 0.851 0.278 0.94625 0.094 0.693 0.108 0.943

Note: the sample size in every problem set is 30 and the total number ofgenerated problems is 750.

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updated seed sequence and insert it in all possiblepositions of the current partial sequence. We thus obtain[3–4] and [ 4–3] as the resultant partial sequences. Thebetter of these two resultant sequences, with respect tothe makespan, is chosen. Let us say that [3–4] yieldsa less makespan than [4–3] , and hence, [3–4] ischosen as the partial sequence. Now, take out the � rst

job from the updated seed sequence ( i.e. job 1) , andinsert it in all possible positions of [ 3–4] . We obtain[1–3–4] , [3–1–4] and [3–4–1] as the resultant partialsequences. Again, choose the best partial sequence. Thisprocess of inserting a job and choosing the best partialsequence is continued until a complete sequence is builtup.

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