Heuristics for scheduling in flowshops and flowline-based manufacturing cells to minimize the sum of weighted flowtime and weighted tardiness of jobs

Computers & Industrial Engineering 37 (1999) 671±690

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Heuristics for scheduling in ¯owshops and ¯owline-based manufacturing cells to minimize the sum of weighted ¯owtime and weighted tardiness of jobs Chandrasekharan Rajendran a,*, Hans Ziegler b a

Industrial Engineering and Management Division, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India b Faculty of Business Administration and Economics, Department of Production Management, University of Passau, Dr.-Hans-Kap®nger-Street 30, 94032 Passau, Germany Accepted 7 December 1999

Abstract The problem of scheduling in two di€erent types of ¯owshops (all jobs available at time zero, di€erent job availability times known a priori) and in ¯owline-based manufacturing cells is considered with the objective of minimizing the sum of weighted ¯owtime and weighted tardiness of jobs. First, heuristic preference relations are developed by the consideration of lower bounds on the completion times, operation due-dates, and weights for holding and tardiness of jobs. A heuristic algorithm for scheduling is then proposed by making use of the heuristic preference relations. Two more heuristic algorithms are developed by implementing an improvement scheme to enhance the quality of the solution given by the ®rst heuristic algorithm. The proposed and the existing heuristics are evaluated with respect to the three problem classes under consideration by solving a large number of randomly generated problems. The results of an extensive computational investigation for various problem sizes are presented. It has been observed that all three proposed heuristics perform better than the existing heuristics in giving a solution of superior quality and that the ®rst proposed heuristic yields a good solution by requiring a negligible CPU time. In addition, an experimental investigation is carried out to evaluate the e€ectiveness of the improvement scheme when implemented in the existing heuristics, and also the e€ectiveness of heuristics based on simulated annealing. The results are discussed in detail. 7 2000 Elsevier Science Ltd. All rights reserved.

* Corresponding author. 0360-8352/99/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 8 3 5 2 ( 0 0 ) 0 0 0 0 3 - 6

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1. Introduction A ¯owshop is a conventional manufacturing system where all machines are arranged in the order of performing operations on jobs. A Cellular Manufacturing System (CMS) is a layout of cells consisting of machines such that the parts requiring operations within a cell are grouped into a family with no or minimum intercell movement. All parts (or jobs) in a part-family need not be processed on all machines in a cell i.e., a job can have missing operations on some machines. The con®guration of a cell can be of two types, namely a ¯owline layout and a jobshop layout. The formation of a ¯owline layout has de®nite advantages over the jobshop layout such as a simpli®ed ¯ow of material with no back-tracking, less material handling and a greater control of production activities [6]. In this paper, the term `cell' refers to the ¯owline-based cell in a cellular manufacturing system. The ¯owshop scheduling problem, which consists in the determination of an optimal schedule for the jobs on the machines, has been a keen area of research for many years. Optimization algorithms for the two- and three-machine ¯owshop problems with respect to di€erent objectives have been developed by Johnson [14] and Ignall and Schrage [12]. The NPcompleteness of various scheduling problems has been discussed widely in the literature [8,22]. As the vast majority of ¯owshop scheduling problems is NP-complete, research is mostly directed towards the development of heuristic or near-optimal methods. Some of the heuristic procedures developed so far are due to Campbell et al. [4], Dannenbring [5], King and Spachis [15], Nawaz et al. [19], Widmer and Hertz [29], Osman and Potts [21], Taillord [28], Ogbu and Smith [20], Ishibuchi et al. [13], and Ben-Daya and Al-Fawzan [3]. While all these heuristics seek to minimize makespan, heuristics have also been proposed to minimize total or mean ¯owtime (e.g. [10,17,23]). For the problem of scheduling in a ¯owline-based manufacturing cell with missing operations for jobs in a part-family, Logendran and Nudtasomboon [16] and Sridhar and Rajendran [26] have proposed heuristics to minimize makespan and total ¯owtime, respectively. The objective of scheduling to minimize total ¯owtime is a signi®cant objective in many real-life situations, especially with respect to the minimization of inventory or holding costs, and is a more important objective than that of minimizing makespan [2,7]. Likewise, the objective of scheduling to minimize total tardiness of jobs is an important objective in practice too, because tardiness may result in the contractual penalty of belated delivery and the loss of customer goodwill. Therefore, it is evident that the objective of scheduling to minimize the sum of ¯owtime and tardiness of jobs is one of the most signi®cant and important objectives with respect to real-life considerations. While studying the ¯owshop scheduling problem, it is generally assumed that all jobs are equal in importance. In reality, such an assumption need not necessarily hold. For example, jobs have di€erent unit costs, holding costs and contractual penalties for belated delivery, and hence they cannot be treated equivalently with respect to the objective of minimizing the costs associated with ¯owtime and tardiness. Therefore, the objective of minimizing the sum of ¯owtime and tardiness of jobs has to be modi®ed to that of minimizing the sum of weighted ¯owtime and weighted tardiness of jobs. The resulting problem of scheduling to minimize the sum of weighted ¯owtime and weighted tardiness of jobs is still NP-complete, and heuristic algorithms have therefore been developed [9].

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In this paper, we study three problem classes: static ¯owshops where all jobs are available at the beginning of the scheduling period; ¯owshops where jobs are available at di€erent times known a priori; and ¯owline-based manufacturing cells. We propose three ecient heuristic algorithms for scheduling to minimize the sum of weighted ¯owtime and weighted tardiness of jobs. First, heuristic preference relations are developed by the consideration of lower bounds on the completion times, operation due-dates, and weights for holding and tardiness of jobs. A heuristic algorithm is then proposed by making use of the heuristic preference relations. Two more heuristic algorithms are developed by implementing an improvement scheme to enhance the quality of the solution given by the ®rst heuristic algorithm. The proposed heuristics and the existing heuristics by Gelders and Sambandam [9] are evaluated by solving a large number of randomly generated problems. The results of an extensive computational investigation for various problem sizes are presented. In addition, an experimental investigation is carried out to evaluate the e€ectiveness of the improvement scheme when implemented in the existing heuristics, and the e€ectiveness of heuristics based on simulated annealing. The performances of simulated annealing heuristics that make use of a randomly generated sequence and the sequence given by the ®rst proposed heuristic as the seed sequences are also evaluated. 2. Problem formulation Let tij hi ti Ai Di n m s p q…s, j† dP ij

processing time of job i on machine j cost (or relative weight) of holding for job i per unit time cost (or relative weight) of tardiness for job i per unit time availability time of job i due-date of job i total number of jobs to be scheduled total number of machines in the ¯owshop or cell ordered set of jobs already scheduled, out of n jobs; partial sequence set of unscheduled jobs completion time of partial sequence s on machine j (i.e. the release time of machine j after processing all jobs in partial sequence s) due-date of job i for operation j complete sequence of n jobs (or a complete permutation of n jobs)

It is well known that for minimizing the total ¯owtime of jobs in a ¯owshop comprising two machines, there always exists an optimal schedule having an identical sequence for all jobs on each of the machines [12]. However, this is not necessarily the case for m > 2: Although using di€erent sequences of the jobs on some or all of the machines might be necessary to obtain an optimal schedule with respect to minimizing total ¯owtime or minimizing total tardiness, this is no common practice in industry. Therefore, in studying ¯owshops it is usually assumed that the job sequence is to be chosen to be identical on each of the machines. A schedule of this

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type is called a permutation schedule. It is de®ned completely by the sequence of all the jobs. In this paper, permutation schedules are considered. In the problem of static ¯owshop scheduling it is generally assumed that all jobs are available at the beginning of a scheduling period, i.e. Ai ˆ 0 for all i. This may not necessarily be the case in all real-life situations. Di€erent jobs can become available at di€erent times during a scheduling period. When the availability times of jobs are deterministic and are known a priori, the formulation of the problem is as follows. Consider a partial sequence si built by appending job i to s: The completion time of si on machine j can be found by the following recursive equation:  q…si, 1† ˆ max q…s, 1†; Ai ‡ ti1 ,  q…si, j† ˆ max q…s, j†; q…si, j ÿ 1† ‡ tij ,

for j ˆ 2, 3, . . . , m

…1†

where q…f, j† ˆ 0 for all j. The weighted ¯owtime of job i or holding cost for job i is given by Ci ˆ …q…si, m† ÿ Ai †hi The weighted tardiness of job i is given by  Ti ˆ max q…si, m† ÿ Di ; 0 ti :

…2†

…3†

The objective of minimizing the sum of weighted ¯owtime and weighted tardiness of jobs for a P complete schedule can be stated as follows: minimize Z ˆ

n X

…Ci ‡ Ti †

…4†

iˆ1

The recursive equations for a permutation sequence in the context of cell scheduling are di€erent from those for ¯owshop scheduling due to the presence of missing operations for jobs [24,26]. The missing operations for jobs are associated with zero processing times of jobs in the process±time matrix. The readers may refer to the works of Sridhar and Rajendran [26] and Rajendran [24] for details of recursive equations for cell scheduling.

3. Development of the proposed heuristic algorithms The development of the proposed heuristic algorithms is ®rst presented, followed by their formal descriptions. The proposed heuristic algorithm 1 is developed for scheduling jobs in a static ¯owshop and later extended to address the problem of scheduling in a ¯owshop where jobs are available at di€erent times and their availability times are known a priori. The proposed heuristic algorithm 1 for static ¯owshop scheduling holds for cell scheduling too. Since the proposed heuristic algorithms 2 and 3 make use of an improvement scheme to enhance the quality of the seed sequence given by the proposed heuristic algorithm 1, they remain the same for all cases.

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3.1. Development of the proposed heuristic algorithm 1 The proposed heuristic 1 consists of generating 2m sequences and choosing the best sequence as the heuristic sequence. Before we present the theoretical foundations for the development of the heuristic sequence, we discuss some observations on the static ¯owshop scheduling problem with respect to the objective of minimizing the sum of weighted ¯owtime and weighted tardiness of jobs. Considering the problem of scheduling on a single machine, it is known that the Shortest Processing Time (SPT) rule minimizes the total ¯owtime of jobs and that when jobs are associated with weights for holding and tardiness, the job with larger weights for holding and tardiness is favored for sequencing earlier than the job with smaller weights for holding and tardiness [2,7]. Suppose we extend the principle of scheduling the jobs according to the SPT rule for the one-machine problem to the m-machine ¯owshop problem by considering the total processing times of jobs on m machines. This extension may not suce because such an extension ignores the in¯uence of individual processing times on ¯owtimes of jobs. For example, a job with relatively large processing times on initial machines in the ¯owshop will result in large completion times of the job on all machines and hence in a large ¯owtime of the job. Consequently, the succeeding jobs too will have large completion times on the machines and hence have large ¯owtimes. Therefore, we recommend that the completion times of jobs on the machines should be reckoned, apart from the holding and tardiness costs of jobs, to minimize the sum of weighted ¯owtime and weighted tardiness of jobs. This approach takes into the account the in¯uence of individual processing times on the completion times of jobs and is, therefore, expected to be quite e€ective. This principle is made use of in the development of the following heuristic preference relations. Consider Eq. (1). Suppose it is assumed that job i waits at the ®rst machine only and that it does not wait at any other machine in the shop. A lower bound on the completion time of job i on machine j is given by LB…si, j† ˆ q…s, 1† ‡

j X

tiq :

…5†

qˆ1

The sum of lower bounds on the completion times of job i on machines 1±m is given by m m m X X X LB…si, j† ˆ q…s, 1† ‡ …m ÿ j ‡ 1†tij jˆ1

jˆ1

…6†

jˆ1

We propose a heuristic preference relation that can be stated as follows. Partial sequence sa is preferred to partial sequence sb if m X

m X

LB…sa, j†

jˆ1

ha ‡ ta

<

LB…sb, j†

jˆ1

hb ‡ tb

,

where jobs a and b belong to set p:

…7†

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It is evident from expression (7) that if we start withPa null schedule, we can build up a sequence by ordering the jobs in the ascending order of ‰ m jˆ1 …m ÿ j ‡ 1†tij Š=…hi ‡ ti †: Any tie in ordering is broken by choosing the job with the minimum value of …1=…hi ‡ ti †). Extending the discussion for the general case where job i waits at machine k and does not wait at any other subsequent machine, a lower bound on the completion time of job i on machine j, where jrk, is given by LB…si, j† ˆ q…s, k† ‡

j X

tiq :

…8†

qˆk

For this case the heuristic preference relation can be stated as follows. Partial sequence sa is preferred to partial sequence sb if m X

m X

LB…sa, j†

jˆk

<

ha ‡ ta

LB…sb, j†

jˆk

hb ‡ tb

:

…9†

By letting k ˆ 2, 3,::, m in expression (9),Pwe can generate …m ÿ 1† additional sequences by ordering the jobs in ascending order of ‰ m jˆk …m ÿ j ‡ 1†tij Š=…hi ‡ ti †: Once again, any tie in ordering is broken by choosing the job with the minimum value of …1=…hi ‡ ti †). Therefore, totally m sequences are generated and the best sequence, with respect to the sum of weighted ¯owtime and weighted tardiness of jobs, is chosen as the seed sequence. Let us now discuss the problem from a di€erent perspective, i.e. by the consideration of duedates of jobs. It is well known that the Earliest Due-Date (EDD) rule minimizes maximum lateness and is also quite an e€ective heuristic rule to minimize the total tardiness of jobs for the problem of scheduling on a single machine. An extension of EDD rule, called Earliest Weighted Operation Due-Date (EWODD), for scheduling in the m-machine ¯owshop to simultaneously minimize the sum of weighted ¯owtime and weighted tardiness of jobs is presented in this study. The rule can be stated as follows. Step 1: Compute for every job i Di  d ij0

ˆ

j X kˆ1

m X

tik ,

j ˆ 1, 2, 3, . . . , m

…10†

tik

kˆ1

Step 2: For every j, from 1 to m, obtain a sequence by arranging the jobs in the ascending order of …dij0 =…hi ‡ ti †). Any tie in ordering is broken by choosing the job with the minimum …1=…hi ‡ ti †† value. Evaluate the sequence with respect to Z. Step 3: Choose the sequence with the least value of Z. The basic idea of this rule is in the apportionment of the due-date of a job for every operation (or machine) in proportion to the job's processing times cumulated upto the operation considered. Such an approach leads to a tight monitoring of the progress of jobs with respect

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to their due-dates for every operation and hence it leads to a better adherence to jobs' completions by their due-dates. By using the proposed EWODD rule, we also take into account the minimization of the sum of weighted ¯owtime and weighted tardiness of jobs since the weights for holding and tardiness of jobs are also reckoned. The proposed heuristic algorithm 1 combines the principles discussed above and seeks to minimize the sum of weighted ¯owtime and weighted tardiness of jobs. The steps of the heuristic algorithm 1, termed as HA-1, can be fully described as follows. Step 1: Set k ˆ 1: Step 2: For each job i calculate m X …m ÿ j ‡ 1†tij

zi ˆ

jˆk

…11†

hi ‡ ti

tie in ordering is broken by choosing Step 3: Order the jobs according to ascending zi : Any P k be the sequence thus obtained. the job with the minimum …1=…hi ‡ ti †† value. Let Compute the sum of weighted ¯owtime and weighted tardiness of jobs for this sequence. Let it be denoted by Z k : Step 4: Set k ˆ k ‡ 1: Return to Step 2 if kRm; otherwise proceed to Step 5. Step 5: Compute for every job i Di  d ij0

ˆ

j X qˆ1

m X

tiq ,

j ˆ 1, 2, 3, . . . , m

…12†

tiq

qˆ1

Step 6: Set k ˆ 1: in ordering is broken by Step 7: Order the jobs in the ascending …dik0 =…hi ‡ ti †). Any tie Pm‡k be the sequence thus choosing the job with the minimum …1=…hi ‡ ti †† value. Let obtained. Compute the sum of weighted ¯owtime and weighted tardiness of jobs for this sequence. Let it be denoted by Z m‡k : Step 8: Set k ˆ k ‡ 1: Return toP Step 7 if kRm; otherwise proceed to Step 9. Step 9: Determine the sequence S such that  Z S ˆ min Z k , k ˆ 1, 2, . . . , 2m : This sequence with Z S constitutes the heuristic solution 1. STOP. For the problem of scheduling jobs in a cell, the proposed heuristic algorithm 1 remains the same. As for the problem of scheduling in a ¯owshop where jobs are available at di€erent times and their availability times are known a priori, we present a modi®cation in HA-1 to take into account the presence of the availability time of a job. The modi®cations pertain to Steps 2 and 5 of the proposed heuristic, while the other steps remain the same. Step 2: For each job i calculate

678

C. Rajendran, H. Ziegler / Computers & Industrial Engineering 37 (1999) 671±690 m X …m ÿ j ‡ 1†tij0

zi ˆ

jˆk

, where ti10 ˆ ti1 ‡ Ai , and tij0 ˆ tij ,

hi ‡ ti

for j ˆ 2, 3, . . . , m

…13†

Step 5: Compute for every job i Di  d ij0 ˆ

j X qˆ1

m X qˆ1

tiq0 ,

j ˆ 1, 2, 3, . . . , m

…14†

tiq0

where tiq0 is computed as in Step 2. The rationale for the modi®cations in the computation of zi and dij0 is that we replace the availability time of job i by an extended and ®ctitious processing time for the ®rst operation. However, when the schedule times, weighted ¯owtime and weighted tardiness of jobs in a given sequence are computed, we make use of the recursive equations presented earlier.

3.2. Development of the proposed heuristic algorithms 2 and 3 Heuristic algorithms 2 and 3 attempt to improve the quality of solution yielded by the proposed heuristic algorithm 1 by employing an improvement scheme on the heuristic solution 1. The improvement scheme used in this paper is based on the sequential insertion of each job in the seed sequence at each possible di€erent position of a generated sequence and has been used by Rajendran and Ziegler [25] to enhance the quality of a heuristic solution with respect to the problem of scheduling a ¯owshop, where setup and removal times are separated from processing times, with the objective of minimizing the makespan. The proposed heuristic algorithm 2, called HA-2, makes use of this improvement scheme to enhance the quality of the solution given by HA-1. APformal description of HA-2, where ‰xŠS denotes the job found in the S PS b , being xth position of sequence Pbthe sequence given by HA-1, and ‰yŠ denotes the job , is given as follows. found in the yth position of sequence Pb PS Step 1: Set x ˆ 1 and ˆ : Step 2: Set y ˆ 1: Step 3: If ‰xŠS 6ˆ‰yŠb Py P then generate a sequence which di€ers from b only by having moved job ‰xŠS to the y yth position, P y and calculate the0 sum of weighted ¯owtime and weighted tardiness, i.e. Z , of ; otherwise set y ˆ y: sequence Step 4: Set y ˆ y ‡ 1: Return to P Step 3 if yRn; otherwise proceed to Step 5. Step 5: Determine the sequence i such that  Z i ˆ min Z y j y ˆ 1, 2, . . . , n, and y6ˆy 0 : P P If Z i < Z b , set b ˆ i :

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Step 6: Set P x ˆ x ‡ 1: Return to Step 2 if xRn; otherwise proceed to Step 7. Step 7: b is the heuristic sequence 2. STOP. The improvement scheme is basically a local search method. It is based on a neighborhood search structure that is often called a shift, and similar approaches have been used by researchers to improve a heuristic seed sequence, especially in the use of meta-heuristics such as Simulated Annealing and Tabu search (e.g. [13,28,29]). In the proposed heuristic 2, the improvement scheme identi®es the best position of insertion for a given job and the resulting sequence is used to update the seed or current sequence if there is an improvement in the quality of solution. We propose one more heuristic algorithm, HA-3, that attempts to yield a sequence with an enhanced quality of solution. This heuristic employs the improvement scheme on the sequence yielded by the proposed heuristic HA-2 to explore a possible improvement with respect to the sum of weighted ¯owtime and weighted tardiness of jobs. The motivation for the development of HA-3 has come from the pilot computational experiments with HA-1, HA-2 and HA-3 for many problems out of the problem sets given in Tables 2, 7 and 8 with di€erent number of jobs and machines. It has been observed that HA-3 improves the solutions given by HA-2 by 1±2%, on an average. However, a further application of the improvement scheme on the solution given by HA-3 has resulted in an improvement of not more than 0.5%, on an average, in the quality of heuristic solution of HA-3. Hence, it is concluded that HA-3 yields a solution of good quality and no more application of the improvement scheme is necessary since the increase in the computational e€ort does not result in an associated substantial improvement in the quality of the ®nal solution.

4. Performance analysis The performance analysis of the proposed and existing heuristics is undertaken for the static ¯owshop, the ¯owshop with di€erent job availability times and for the manufacturing cell. The details of the analysis are now presented for each case separately. 4.1. Performance analysis of heuristics for ¯owshop scheduling For minimizing the sum of weighted ¯owtime and weighted tardiness of jobs in the mmachine static ¯owshop, four heuristics are proposed by Gelders and Sambandam [9]. These heuristics build up a schedule by appending a job to an available partial schedule on the basis of idle time of machines, lower bound on makespan, and the due-date, holding and tardiness costs of the appended job. The heuristics later improve the schedules by using the adjacent pairwise interchange scheme. Out of the four heuristics, the ®rst two heuristics have been found to be more e€ective than the next two heuristics. Hence, we have chosen to solve the ¯owshop problem by using the ®rst two heuristics of Gelders and Sambandam, and then take the schedule that yields the best value of the sum of weighted ¯owtime and weighted tardiness of jobs. This best schedule is termed as GS in this paper. We compare the performance of the three proposed heuristics with that of GS. The three proposed heuristic algorithms are termed

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as HA-1, HA-2 and HA-3. All heuristics have been coded in FORTRAN and implemented on an 80486-PC. 360 small-sized problems and 540 large-sized problems have been randomly generated. For the small-sized problems, the number of jobs has been 6 and 7; for the largesized problems, the number of jobs has been varied from 20 to 60, in steps of 20. In all cases, the number of machines has been varied from 5 to 30 in steps of 5, and the number of problems generated in each problem set, de®ned by n  m, has been equal to 30. The processing times of the jobs on the machines have been sampled from a rectangular distribution ranging from 1 to 99. Most researchers have used this distribution in their experimentations (e.g. [10,11,13,17±19,23,26±29]) because evidence exists that such a distribution allows for more dicult problems to be solved [4,5]. The holding and tardiness costs (or weights) for jobs have been sampled from a rectangular distribution ranging from 1 to Pmdue-date of a job is sampled from a rectangular distribution ranging from Pm10 [9]. The t and … ij jˆ1 jˆ1 tij ‡ 50…n ÿ 1†† [9]. This due-date setting is rather tight and such a tight duedate setting will bring out the better or poorer relative performance of a heuristic with respect to the other heuristics under evaluation for minimizing the weighted tardiness of jobs. In fact, the results of performance analysis in this study also indicate that the total weighted tardiness of jobs given by the heuristic sequences has been large enough to distinguish between the relative performances of the heuristics under evaluation. For this reason, we have followed the approach used by Gelders and Sambandam [9]. For evaluating the heuristics for small-sized problems, the absolute percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs in a heuristic sequence from that of an optimal sequence has been used, i.e. Zk ÿ Z  100 Z

…15†

where Zk is the objective function value given by the kth heuristic and Z  is the optimal objective function value. The absolute percentage increases in the sum of weighted ¯owtime and weighted tardiness of jobs in the sequence given by the kth heuristic are calculated for all 30 problems of a given problem set. The mean absolute percentage increase (MAPI) for the 30 problems is then computed. The results of the analysis are presented in Table 1 for n ˆ 6 and 7. It is evident from the table that the proposed heuristics perform better than GS in reducing the mean absolute percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs. As far as large-sized problems are concerned, the mode of evaluation is the measurement of the relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs in a heuristic solution from the best value of sum of weighted ¯owtime and weighted tardiness of jobs given by all heuristic solutions, i.e. Zk ÿ Z # 100, Z#

…16†

where Z # is the minimal objective function value yielded by all heuristic solutions under consideration. It is evident that for each of the problems solved at least one of the heuristics will have a relative percentage increase in the sum of weighted ¯owtime and weighted tardiness

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equal to zero. Such relative percentage increases in the sum of weighted ¯owtime and weighted tardiness are calculated for each of the 30 problems for a given problem set. The mean relative percentage increase (MRPI) for the 30 problems is then calculated. The results of the performance analysis are presented in Table 2 for n ˆ 20, 40 and 60 jobs. It is evident from the table that the proposed heuristics perform much better than GS in reducing the mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs and that HA-3 emerges to be by far the best, yielding always the best solution. As far as the computational e€ort is concerned, the proposed heuristic 1 is the fastest, followed by the proposed heuristics 2 and 3, and then by the heuristics of Gelders and Sambandam. The details of the computation e€ort for solving various problem-sizes are given in Table 3. It is noteworthy that the mean CPU time required by HA-1 for solving most problems is negligible and that HA-1 also emerges to be better than the existing heuristics of Gelders and Sambandam with respect to the quality of solutions. These details also show that computation time does not restrict the practical application of any of the proposed heuristics. We have also investigated the e€ectiveness of the heuristics in minimizing the two components of Z separately, i.e. minimizing the weighted ¯owtime and minimizing the weighted tardiness of jobs respectively to analyze whether the improvement of Z by applying the proposed heuristics is realized by a better performance with respect to one of the components and a poorer performance with respect to the other component. It has been found that the proposed heuristics also perform better than GS with respect to individually minimizing the components of weighted ¯owtime and weighted tardiness of jobs. Hence the proposed heuristics improve the weighted ¯owtime and weighted tardiness of jobs simultaneously. The details of results are not presented in the paper for want of space. Table 1 Performance of the four heuristics for the small-sized ¯owshop problems (mean absolute percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs)a,b n

m

HA-1c

HA-2c

HA-3c

GSc

6

5 10 15 20 25 30 5 10 15 20 25 30

4.8579 2.7275 3.1671 2.0456 2.7990 2.3686 4.8854 3.0319 3.5061 3.1908 2.3651 2.1863

0.8964 0.4616 0.4932 0.2978 0.2207 0.0513 1.0526 0.3628 0.5470 0.2679 0.2241 0.2907

0.8553 0.2726 0.3046 0.2438 0.1230 0.0355 0.7948 0.2336 0.3749 0.1463 0.0772 0.1691

4.4063 2.1171 2.7397 1.9900 3.1106 2.3710 4.7642 2.5146 3.5191 2.9379 2.5937 2.6179

7

a

Sample size in every problem set is 30 and the total number of generated problems is 360. An entry close to zero indicates the good performance of the corresponding heuristic. c HA-1, HA-2 and HA-3 refer to the proposed heuristic algorithms 1, 2 and 3, respectively, and GS refers to the better of the two heuristics of Gelders and Sambandam. b

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C. Rajendran, H. Ziegler / Computers & Industrial Engineering 37 (1999) 671±690

The next phase of the performance analysis deals with the possible implementation of improvement techniques such as the metaheuristics of Simulated Annealing, Genetic Algorithms and Threshold Accepting. We have chosen to consider only those metaheuristics which yield solutions of good quality within the CPU times comparable to those of the proposed heuristics. A study by Aarts et al. [1] on the problem of static job shop scheduling has revealed that the simulated annealing heuristic performs best in the sense that it ®nds better solutions than the other local search algorithms within the same CPU time. In a study on ¯owshop scheduling with the objective of minimizing makespan, Ishibuchi et al. [13] have proposed a neighborhood generation mechanism based on best-move strategy and used it in simulated annealing. They have made use of the following acceptance probability function. Replace the current solution x by the solution y with the probability   DZ…x4y † , p…x4y † ˆ min 1, exp c where DZ…x4y † ˆ Z…x† ÿ Z…y †:

…17†

Ishibuchi et al. have made use of a constant control parameter, c, as against the conventional decreasing function to determine the value of c. In addition, they have randomly and

Table 2 Relative performance of the four heuristics for the large-sized ¯owshop problems (mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs)a n

m

HA-1

HA-2

HA-3

GS

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

10.5972 11.3017 9.3817 7.8024 8.1713 7.2436 14.5516 15.0631 13.1149 12.3012 10.9972 10.4910 15.7886 15.9788 14.8897 13.5040 12.9614 11.0672

0.8882 1.1152 1.0197 0.9798 1.0403 0.6544 1.2979 1.5855 1.3487 1.3843 1.1180 1.0503 1.2485 1.3919 1.5775 1.4448 1.3388 0.8326

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14.6510 13.4616 11.2454 11.0186 9.4809 8.5516 22.1519 20.4015 18.4077 18.2801 16.8887 16.1999 23.7886 24.6446 23.0187 22.1286 20.8841 18.5657

40

60

a

Sample size in every problem set is 30 and the total number of generated problems is 540.

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independently generated K solutions from the neighborhood of the current solution x and chosen the best solution among the generated K solutions as the solution y. They have observed that the performance of their simulated annealing algorithm is less sensitive to the choice of the cooling schedule than that of the standard simulated annealing algorithms. In addition, their simulated annealing heuristic is more generic than the simulated annealing heuristics by Osman and Potts [21] and Ogbu and Smith [20]. The reason is that the parameter settings by Osman and Potts, and Ogbu and Smith are speci®c to the ¯owshop scheduling problem with the makespan objective, whereas the simulated annealing heuristic by Ishibuchi et al. has only one parameter setting in the acceptance probabilty function, i.e. c, and the setting is robust and not speci®c to the makespan objective. Moreover, Ishibuchi et al. have also observed that their simulated annealing heuristic performs better than the Tabu search algorithm with the restriction of comparable CPU times. For these reasons, we have chosen to use the simulated annealing heuristic by Ishibuchi et al. in our study. There are some variants suggested by Ishibuchi et al. in their study and hence an initial investigation was taken up to identify the best variant for the present study. A number of pilot runs for many problems out of the problem sets given in Table 2 with di€erent number of jobs and machines has indicated that the best-move strategy performs, on an average, better than the ®rst-move strategy. In addition, the two settings of c ˆ 5 and 10 [13] have not yielded any substantial di€erence in the quality of solutions generated and hence we have decided to use the setting of c ˆ 5 in the present study. As for the number of neighborhood solutions generated, we have found that there is a di€erence in the quality of solutions when the settings Table 3 Mean CPU time (in seconds) required for di€erent procedures for the large-sized ¯owshop problemsa n

m

HA-1

HA-2

HA-3

GS

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

R 0.1 R 0.1 R 0.1 R 0.1 R 0.1 0.13 R 0.1 R 0.1 R 0.1 0.20 0.32 0.45 R 0.1 0.11 0.22 0.35 0.49 0.65

0.78 0.91 1.05 1.20 1.38 1.58 2.24 3.38 4.53 5.60 6.71 7.86 5.40 7.51 9.78 12.41 15.11 17.94

1.54 1.79 2.08 2.36 2.71 3.10 4.42 6.70 8.92 11.02 13.11 15.34 10.70 14.89 19.37 24.53 29.87 35.35

0.24 0.39 0.74 1.28 1.96 2.80 1.05 2.57 5.69 9.23 13.68 18.93 4.82 9.75 19.85 36.87 59.10 82.59

40

60

a

Sample size in every problem set is 30 and the mean CPU time is computed for these 30 generated problems.

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C. Rajendran, H. Ziegler / Computers & Industrial Engineering 37 (1999) 671±690

of K ˆ n and 2n [13] are used. In other words, the increased number of generated neighborhood sequences has resulted in a solution of better quality. It has to be noted that Ishibuchi et al. have used a randomly generated sequence as the seed sequence in their simulated annealing heuristic. Since the heuristic algorithm 1 proposed in this paper yields a solution within a negligible CPU time, we have ®rst taken up the investigation of determining the in¯uence of the randomly generated sequence and the heuristic sequence 1, as the seed sequences, on the quality of ®nal solutions with respect to the simulated annealing heuristic by Ishibuchi et al. The simulated annealing heuristics with the randomly generated seed sequence and parameter settings of K ˆ n and 2n are termed as SA1-R and SA2-R respectively. The simulated annealing heuristics with the seed sequence given by the proposed heuristic 1 and parameter settings of K ˆ n and 2n are termed as SA1-H and SA2-H respectively. The same set of problems indicated in Table 2 has been used. The method of evaluation of SA1-R, SA2-R, SA1-H and SA2-H is similar to that in Table 2 in the sense that the mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs given by the kth simulated annealing heuristic is computed. The results are presented in Table 4. It is evident from the table that the simulated heuristics with the heuristic sequence 1 as the seed sequence perform better than the simulated annealing heuristics with a randomly generated seed Table 4 Relative performance of the four simulated annealing heuristics for the large-sized ¯owshop problems (mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs)a n

m

SA1-Rb

SA2-Rb

SA1-Hb

SA2-Hb

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

6.2982 5.1818 3.9387 3.9168 3.6815 3.4557 5.6926 4.8094 4.8415 4.7928 3.6951 3.7671 5.0820 5.1033 4.9187 4.0967 3.8708 3.6124

2.9147 2.5611 1.8958 2.2620 2.3572 1.6518 4.2170 3.0128 2.9448 2.7684 2.3028 1.9824 3.5603 2.7016 3.3266 2.4353 2.6333 2.1762

0.8914 1.3569 1.3847 0.9449 1.7050 1.0448 1.4852 1.2528 1.2247 1.4468 1.2737 1.1988 0.8725 1.2701 1.0701 1.2574 1.1981 0.9792

0.4810 0.3271 0.4706 0.3864 0.2701 0.5290 0.1276 0.2436 0.1560 0.1632 0.3392 0.2435 0.1521 0.1635 0.1804 0.2565 0.1010 0.2101

40

60

a

Sample size in every problem set is 30 and the total number of generated problems is 540. SA1-R and SA2-R refer to the Simulated Annealing heuristic of Ishibuchi et al. [13] with a randomly generated seed sequence and the setting of K ˆ n and 2n, respectively, and SA1-H and SA2-H refer to the Simulated Annealing heuristic of Ishibuchi et al. with the seed sequence given by the heuristic algorithm HA-1 and the setting of K ˆ n and 2n, respectively. b

C. Rajendran, H. Ziegler / Computers & Industrial Engineering 37 (1999) 671±690

685

sequence. This result seems to apparently contradict the general observation on the performance of metaheuristics with respect to the choice of starting solutions. The reasons for our ®nding are that we have chosen to restrict the computation times of the simulated annealing heuristics comparable to those of the proposed heuristics and that the metaheuristics show their real strength when computation times are of no concern [1]. It is therefore evident that the choice of seed sequence will not substantially in¯uence the quality of the ®nal solution only if there is no restriction on CPU times. Since we have chosen to evaluate the performance of simulated annealing heuristics with restrictions on the CPU times to be comparable to those required by the proposed heuristics, the simulated annealing heuristics with the seed sequence given by the proposed heuristic 1 and the parameter settings of K ˆ n and 2n are used for further analysis of performance in the current study. We have also chosen to evaluate the performance of GS when the improvement scheme used in this paper (see Section 3.2) is employed on the heuristic sequence given by GS. This approach is termed as GS + IS. We now have the proposed heuristics HA-2 and HA-3, GS + IS, SA1-H and SA2-H for a relative evaluation. The same set of problems given in Table 2 is used and the method of relative evaluation of the heuristics is also the same. The results are presented in Table 5. The computation times required by these heuristics are given in Table 6. Table 5 Relative performance of the ®ve procedures for the large-sized ¯owshop problems (mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs)a n

m

HA-2b

HA-3b

GS + ISb

SA1-Hb

SA2-Hb

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

1.3429 1.8421 1.6557 1.4237 1.7818 1.0197 1.7479 1.9071 1.7761 1.9003 1.3053 1.4742 1.5904 1.7521 1.8235 1.7794 1.6405 1.0758

0.4505 0.7189 0.6328 0.4398 0.7321 0.3644 0.4439 0.3189 0.4215 0.5111 0.1850 0.4196 0.3382 0.3573 0.2432 0.3308 0.2993 0.2416

2.7475 2.5190 2.4327 2.1110 1.9102 1.4007 4.4639 3.6734 3.7571 3.1359 2.8999 2.4948 5.5479 5.2713 4.5232 3.5022 3.8530 3.7102

1.4353 2.1086 2.1496 1.4999 2.4509 1.3473 1.9294 1.9780 1.9630 1.9603 2.1039 1.7117 1.5168 1.7782 1.6982 1.9022 2.0872 1.4960

1.0209 1.0750 1.2243 0.9414 1.0083 0.8319 0.5694 0.9649 0.8891 0.6741 1.1637 0.7545 0.7913 0.6686 0.8014 0.8990 0.9861 0.7215

40

60

a

Sample size in every problem set is 30 and the total number of generated problems is 540. HA-2 and HA-3 refer to the proposed heuristic algorithms 1 and 3, respectively. GS + IS refers to the better of two heuristics of Gelders and Sambandam supplemented by the insertion based improvement scheme, and SA1-H and SA2-H refer to the Simulated Annealing heuristics of Ishibuchi et al. with the seed sequence given by the proposed heuristic algorithm HA-1 and the setting of K ˆ n and 2n, respectively. b

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It is evident from the tables that the proposed heuristics continue to perform better than the other heuristics with respect to the quality of solutions and the computation times. It is also interesting to observe that the implementation of improvement scheme greatly enhances the quality of solution given by GS. However, this process is associated with a large computation time. In addition, we have also conducted an experimental investigation to evaluate the performance of the heuristics HA-2, HA-3, GS + IS, SA1-H and SA2-H with respect to minimizing the components of weighted ¯owtime and weighted tardiness of jobs. We have observed that the proposed heuristics continue to perform better than the other heuristics with respect to minimizing the individual components of the objective function Z. We have also conducted an experimental investigation to evaluate the performance of the heuristics when the weights for holding and tardiness of jobs are sampled from the rectangular distribution ranging from 1 to 20. The better performance of the proposed heuristics is also observed in this case. Furthermore, we have attempted to improve the quality of the solution given by the heuristic sequence 3 by giving this sequence to the simulated annealing heuristic by Ishibuchi et al. with the setting of K ˆ 2n: The rationale is that while the proposed heuristics make use of a local search based on a conventional approach, the simulated annealing heuristics can help in getting out of a local optimum. The results of such an investigation has revealed that there is no substantial improvement in the quality of the solution yielded by HA-3 in that the average improvement is in the order of only about 0.7%. Therefore, we believe that the proposed heuristics are indeed very e€ective in minimizing the sum of weighted ¯owtime and weighted Table 6 Mean CPU time (in seconds) required for di€erent procedures for the large-sized ¯owshop problemsa n

m

HA-2

HA-3

GS + IS

SA1-H

SA2-H

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

0.78 0.91 1.05 1.20 1.38 1.58 2.24 3.38 4.53 5.60 6.71 7.86 5.40 7.51 9.78 12.41 15.11 17.94

1.54 1.79 2.08 2.36 2.71 3.10 4.42 6.70 8.92 11.02 13.11 15.34 10.70 14.89 19.37 24.53 29.87 35.35

1.00 1.28 1.74 2.42 3.31 4.30 3.21 5.88 10.15 14.75 20.19 26.45 10.15 17.21 29.41 48.99 73.85 99.95

0.40 0.57 0.78 0.99 1.24 1.44 3.05 4.73 6.37 7.82 9.45 11.00 7.50 10.48 13.76 17.37 21.13 25.08

0.77 1.10 1.51 1.93 2.41 2.80 6.04 9.51 12.67 15.43 18.65 21.70 14.89 20.80 27.33 34.40 41.85 49.63

40

60

a

Sample size in every problem set is 30 and the mean CPU time is computed for these 30 generated problems.

C. Rajendran, H. Ziegler / Computers & Industrial Engineering 37 (1999) 671±690

687

tardiness of jobs. The details of all these experiments are not reported in the paper for want of space. 4.2. Performance analysis of heuristics for ¯owshop scheduling when jobs are available at di€erent times We observe that the heuristics by Gelders and Sambandam can still be used for scheduling of jobs in the ¯owshop problem where jobs are available at di€erent times and their availability times are known a priori. This is so because the heuristics by Gelders and Sambandam construct sequences by appending jobs, one by one, to the available partial sequence and in such a case we only need to use the recursive equations and computations as presented in Eqs. (1)±(3). As presented earlier, we make use of the proposed heuristic algorithm HA-1, with modi®cations in Steps 2 and 5, for scheduling in the ¯owshop with di€erent availability times of jobs. Heuristics HA-2 and HA-3 make use of the sequence given by this modi®ed heuristic HA-1. The relative performance evaluation of the proposed heuristics time of job i and GS is carried out for the same set of problems as in Table 2. The availability P t ‡ 50…n ÿ 1††]. is randomly sampled from the rectangular distribution between 0 and ‰0:5… m ij jˆ1 This setting of availability times of jobs ensures a wide variation in the availability times of di€erent jobs and hence the relative performances of the heuristics under evaluation can be distinguished. The results of performance analysis of heuristics for scheduling in a ¯owshop Table 7 Relative performance of the four heuristics for the large-sized ¯owshop problems with jobs available at di€erent times (mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs)a,b n

m

HA-1

HA-2

HA-3

GS

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

19.0904 14.7834 12.0672 10.2438 10.4882 9.5742 22.7799 20.6209 16.7182 14.5923 14.8094 13.5666 25.9818 20.5377 18.4373 17.5029 16.0984 15.0086

1.2882 1.3936 1.0243 1.3350 1.1181 1.2695 2.0378 1.8509 1.9789 1.5550 1.3193 1.5445 2.0803 1.9674 1.8407 1.7675 2.0177 1.9067

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18.1789 13.2703 14.0882 12.0879 11.2361 12.0708 23.7864 22.1894 20.8333 20.8350 21.0483 19.9666 24.0092 26.0939 23.2444 25.0421 24.1417 24.1445

40

60

a b

Sample size in every problem set is 30 and the total number of generated problems is 540. Jobs are available at di€erent times and their availability times are known a priori.

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Table 8 Relative performance of the four heuristics for the large-sized cell problems (mean relative percentage increase in the sum of weighted ¯owtime and weighted tardiness of jobs)a,b n

m

HA-1

HA-2

HA-3

GS

20

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

15.9127 15.7252 13.2938 12.0266 11.6030 9.3787 20.3574 19.6945 17.8819 17.6775 15.7738 13.7244 20.9293 21.2467 21.0241 18.8756 18.0364 16.0578

1.3750 1.4192 1.5021 1.5489 1.6031 0.9477 1.5734 1.9861 1.6114 1.9728 1.9066 1.4250 1.7658 1.8752 1.9948 1.8682 1.4845 1.5902

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19.7043 21.7173 18.1710 18.0149 14.2747 13.0817 33.2595 28.4326 24.7368 24.8443 23.1754 20.4693 37.0112 31.9897 31.8236 30.5252 27.4275 26.3308

40

60

a b

Sample size in every problem set is 30 and the total number of generated problems is 540. The percentage of missing operations on jobs is 20%.

with di€erent availability times of jobs are presented in Table 7. The results indicate that the proposed heuristics perform better than the existing heuristics in minimizing the sum of weighted ¯owtime and weighted tardiness of jobs. Especially, HA-3 generated always the best solution. A further performance analysis has been carried out to evaluate the relative performance of HA-2, HA-3, GS + IS, SA1-H and SA2-H in the context of di€erent availability times of jobs in a ¯owshop. The method of evaluation of these heuristics is same as in Table 5. The results indicate that the proposed heuristics perform better than the other heuristics. We have also observed the relative e€ectiveness of the proposed heuristics with respect to minimizing the components of weighted ¯owtime and weighted tardiness. It is found that the performances of the proposed heuristics continue to be superior in minimizing both components of the objective function Z separately. The details of these results are not included in the paper to make it concise. 4.3. Performance analysis of heuristics for scheduling jobs in a part-family in cells We have used the recursive equations for scheduling jobs in a part-family in the manufacturing cell and evaluated the performance of the proposed heuristics HA-1, HA-2 and HA-3, and GS. We have assumed the percentage of missing operations in a cell to be 20% [16,26]. This is achieved by considering the same set of problems used in Table 2 and assigning the processing time with zero value if a sampled uniform random number is less than 0.2. The

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results of performance analysis of the heuristics are presented in Table 8. We have also evaluated the relative performances of the heuristics HA-2, HA-3, GS + IS, SA1-H and SA2H for scheduling jobs in a cell. The results indicate that the proposed heuristics are also quite e€ective for scheduling in a manufacturing cell. The details are not included for want of space.

5. Conclusion The problem of scheduling in ¯owshops without and with di€erent job availability times, and ¯owline-based manufacturing cells is considered with the objective of minimizing the sum of weighted ¯owtime and weighted tardiness of jobs. First, heuristic preference relations are developed by the consideration of lower bounds on the completion times, operation due-dates, and weights for holding and tardiness of jobs. A heuristic algorithm is then proposed by making use of the heuristic preference relations. Two more heuristic algorithms are developed by implementing an improvement scheme to enhance the quality of the solution given by the ®rst heuristic algorithm. The proposed and the existing heuristics are evaluated by a large number of randomly generated scheduling problems. The results of an extensive computational investigation for various problem sizes are presented. For all problem classes under consideration, it has been observed that all three proposed heuristics perform better than the existing heuristics in giving a solution of superior quality and that the ®rst proposed heuristic yields a good solution by requiring a negligible CPU time. In addition, an experimental investigation is carried out to evaluate the e€ectiveness of the improvement scheme when implemented in the existing heuristics, and also the e€ectiveness of heuristics based on simulated annealing is studied. The results are discussed in detail. It has been observed that the e€ectiveness of the simulated annealing heuristics can be improved by giving the sequence yielded by one of the proposed heuristic algorithms as the seed sequence, without increasing the computation e€ort, instead of using a randomly generated seed sequence.

Acknowledgements This research work was carried out when the ®rst author had been at the University of Passau and was supported by the Alexander von Humboldt Research Fellowship during 1996± 97. The authors thank the referee for the suggestions and comments on the earlier version of the paper.

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