Dispatching in flowshops with bottleneck machines

Computers & Industrial Engineering 52 (2007) 89–106 www.elsevier.com/locate/dsw

Dispatching in flowshops with bottleneck machines Chandrasekharan Rajendran

a,*

, Knut Alicke

b

a

b

Department of Management Studies, Indian Institute of Technology Madras, Chennai 600 036, India Institute for Material Flow and Logistics, Department of Mechanical Engineering, University of Karlsruhe, D-76128 Karlsruhe, Germany Received 21 June 2003; received in revised form 29 September 2006; accepted 10 October 2006 Available online 14 December 2006

Abstract This paper addresses the problem of dispatching in flowshops with bottleneck machines. The presence of bottleneck machines results in the restricted throughput in flowshops. The objective is to develop dispatching rules for scheduling by taking into account the presence of bottleneck machines. The measures of performance are the minimization of total flowtime of jobs, the minimization of the sum of earliness and tardiness of jobs, and the minimization of total tardiness of jobs, considered separately. Many existing conventional dispatching rules and the proposed dispatching rules have been extensively investigated for their performance by generating a large number of problems of various sizes and bottleneck conditions. The results of the experimental investigation show that the proposed dispatching rules emerge to be superior to the conventional dispatching rules.  2006 Elsevier Ltd. All rights reserved. Keywords: Flowshop; Bottleneck machines; Dispatching rules; Flowtime; Earliness; Tardiness

1. Introduction Many attempts have been made over the decades to address the problem of scheduling in flowshops. The most common objective of scheduling is the minimization of the makespan. Some important studies on flowshop scheduling with the makespan objective are due to Johnson (1954), Ignall and Schrage (1965), Campbell, Dudek, and Smith (1970), Dannenbring (1977), Nawaz, Enscore, and Ham (1983), Widmer and Hertz (1989), Taillard (1990), Ogbu and Smith (1990), Ishibuchi, Misaki, and Tanaka (1995), Ben-Daya and Al-Fawzan (1998), and Framinan, Leisten, and Rajendran (2003). It is to be noted that in view of a vast majority of flowshop scheduling problems being NP-complete in nature, many researchers focussed their efforts on the development of heuristics that yield optimal or near-optimal solutions for large-sized problems. Another important objective of scheduling is the minimization of total flowtime (or the sum of completion times of jobs) that results in minimum in-process inventory, and heuristics with this objective have been developed by Miyazaki, Nishiyama, and Hashimoto (1978), Gelders and Sambandam (1978), Rajendran (1993), Ho (1995), Rajendran and *

Corresponding author. Tel.: +91 44 2257 4559; fax: +91 44 2257 0509. E-mail address: [email protected] (C. Rajendran).

0360-8352/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2006.10.006

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Ziegler (1997), Liu and Reeves (2001), and Rajendran and Ziegler (2004). Another objective of significance is the minimization of total tardiness of jobs, and some attempts have been made to develop efficient heuristics (e.g., Gelders & Sambandam, 1978; Hasija & Rajendran, 2004; Kim, 1993; Parthasarathy & Rajendran, 1998). Most of the studies on flowshop scheduling have assumed the presence of machines which are near-bottleneck with respect to each other. In other words, no machine is assumed (or expected) to be the bottleneck in the manufacturing system. However, in reality, it is possible that one bottleneck machine (or a couple of machines which are almost equally bottleneck) can be present in the system. In such a case, it is quite natural that we should develop solution methodologies with specific consideration to bottleneck machines. An important work in the related area has been the ‘Theory of Constraints’ (TOC), proposed by Goldratt (1990). According to Goldratt, there can be constraints within a production system emerging out of either of two conditions. The first condition arises when the demand on a resource exceeds its capacity. Such a resource is called a bottleneck (Umble & Srikanth, 1990). The second condition comes into picture when a non-bottleneck machine may become a bottleneck, especially when jobs do not arrive at this resource at the required or planned time to perform the operation. Such resources are temporary bottlenecks or wandering bottlenecks. In the TOC literature, a capacity-constrained resource (CCR) is defined to be any resource that restricts the throughput in the system. It is interesting to note that a CCR is not necessarily a bottleneck, nor vice versa (Umble & Srikanth, 1990). The TOC technique used to accomplish the exploitation of system’s constraint is drum–buffer–rope. The drum is a system constraint or other critical resource that sets the pace of production. The ropes are signalling methods that tie the release of raw materials to the production at the drum. The buffers are extra material on the shop floor resulting from work being released earlier than the absolute minimum time required to complete all production steps. A survey of literature on the TOC and the related idea of Drum–Buffer–Rope (DBR) production systems reveals that much of the research does not consider the aspects related to operating policies and issues such as establishing priorities between competing jobs (e.g., Lambrecht & Segaert, 1990; Luebbe & Finch, 1992; Plenert, 1993; Russell & Fry, 1997; Spencer, 2000). A related aspect to the current work is the Order Review and Release system (ORR). The concept of regulating the flow of incoming job-orders was first introduced by Wight (1970) in order to control in-process inventories. The process of ORR involves the following steps. As an order arrives at the shop, it is placed in the pre-shop buffer. The ORR policy controls the release of the order into the shop. Upon entering the shop, the order becomes work-in-process inventory. Primarily, two ORR approaches are being used to control the release of work into the shop. Philipoom and Fry (1992) defined them as real-time and a-priori methodologies. In a real-time methodology, jobs are released based on the current shop congestion. Additional orders are released when the amount of work in the shop falls below a pre-defined level. a-priori methodologies release jobs based on pre-determined release time. These release times are determined through backward or forward loading, based on shop conditions at the time the release date is set. A number of attempts have been made to explore these two strategies (e.g., Ahmed & Fisher, 1992; Bergamaschi, Cigolini, Perona, & Portioli, 1997; Cigolini, Perona, & Portioli, 1998; Hendry & Kingsman, 1991; Hendry & Wong, 1994; Melnyk & Ragatz, 1989; Philipoom & Fry, 1999; Ragatz & Mabert, 1988). It is evident from the literature on ORR that there exists no best way for the ORR management, and that ORR performance is highly dependent on shop and job conditions such as shop utilization, due-date tightness, and dispatching rules. As a matter of fact, many research works have been devoted to the investigation of the impact of such conditions on the ORR performance. Yet another significant and related work was reported by Baker (1984a). Baker observed that ORR would reduce the effectiveness of shop scheduling and that the choice of dispatching rules would be more important to shop performance than the controlled release of jobs. According to Baker, ORR limits the number of jobs available to the scheduling system, thereby limiting the number of scheduling alternatives available. Hence, it is evident that the development and investigation of dispatching rules which take into account specifically the presence of bottleneck machines in shops will prove very crucial to the performance of any shop, apart from influencing the performance of ORR. In all algorithms and heuristics discussed so far with respect to the problem of scheduling in flowshops, one of the basic assumptions is that only permutation schedules are considered, and hence heuristics have been developed for that special case. A permutation schedule has the property that the order of processing jobs is identical on all machines. In other words, once a sequence is established for the first machine, the order of processing jobs on the subsequent machines is governed by the ‘first-in-first-out’ rule. One of the reasons

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for focussing on permutation schedules in flowshop scheduling is the difficulty (‘computational hardness’) in finding out an optimal schedule which will comprise different sequences on different machines. As for the problem of scheduling in the real-life flowshops where there is no necessity that only permutation schedules be implemented, the use of dispatching rules is mostly resorted to. In fact, the use of dispatching rules is the easiest, simplest and the most popular approach to scheduling of jobs in real-life situations. The primary reasons for the use of dispatching rules in many studies on scheduling are the associated ease in generating feasible schedules, and implementing and controlling them at the shopfloor level in real-life situations, as opposed to sophisticated optimization techniques (see Pinedo, 1995). Moreover, it is also a customary practice in industries to accumulate job-orders and release them into the shop so that capacity-planning problem can be solved better than in the case of dynamic release of job-orders into the shop. Additionally, in view of dynamic changes with respect to schedules taking place in the shopfloor in real-life situations, it is also a customary practice to go in for dispatching rules. For these reasons, we have chosen to study the current static flowshop scheduling problem with the stated assumptions, and hence, we consider the development of dispatching rules in flowshops with bottleneck machines. Three measures of performance are considered in the current study: the minimization of total flowtime of jobs; the minimization of the sum of earliness and tardiness of jobs; and the sum of tardiness of jobs. Hence, the present work can be divided into two phases: the first phase dealing with the flowtime-objective and the second phase dealing with the due-date-based objectives. 2. Development of dispatching rules with flowtime-objective by considering the presence of a single bottleneck machine Dispatching decisions are made when a machine becomes free after processing a job. The problem is to identify the job to be taken up for processing on the machine from the waiting line of jobs. The dispatching decision is made by the use of dispatching rules. For the objective of minimizing mean flowtime of jobs, the most commonly used dispatching rule is the SPT (shortest processing time) rule (see Blackstone, Phillips, & Hogg, 1982; Haupt, 1989; Ramasesh, 1990), and is found to be effective in flowshop scheduling as well (Rajendran & Holthaus, 1999). Apart from this rule, we also consider the rule where the priority of a job is given by its total or sum of remaining processing times (see Blackstone et al., 1982; Haupt, 1989). We call this rule as the TPT rule. Let us first consider the case where there is only one bottleneck machine (BM) in the flowshop consisting of m machines in series. We now propose two dispatching rules. According to the first rule, when the dispatching decision is to be made with respect to jobs waiting at a machine ahead of the BM, then the job that can reach the BM at the earliest is chosen for loading. However, if the machine is present after the BM, the rule identifies the job that can leave the flowshop at the earliest. Basically, the rule seeks to minimize the overall waiting time of jobs and also enhance the throughput in the flowline, by taking into account the relative position of the BM. The second rule makes the dispatching decision on the basis of processing times of jobs on the BM, if the jobs are waiting at a machine ahead of the BM; however, if the jobs are waiting at a machine after the BM, then the imminent processing times of jobs are considered for the dispatching decision. Once again, the dispatching decision is made depending upon the position of the machine (where jobs are waiting) relative to the BM. 2.1. Proposed rule 1 Let the bottleneck machine be machine q. Let the processing time of job i on machine k be tik. Let s be the current time at which the dispatching decision is to be made with respect to machine j. Suppose job i is taken up for processing on machine j.For P the case where j 6 q, we can define a lower bound on the completion time q of job i on machine q, given by s þ k¼j tik . Based on this lower bound, we can define the priority for job i as follows. ! q X tik ; or simply Zi ¼ s þ k¼j

Zi ¼

q X k¼j

tik :

ð1Þ

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However, if j > q, we define the priority index for job i as follows. ! m X tik ; or simply Zi ¼ s þ k¼j

Zi ¼

m X

ð2Þ

tik :

k¼j

The job with the minimum Zi is chosen for loading. We call this proposed rule as PR-1. 2.2. Proposed rule 2 According to this rule, the priority of a job is again governed by the position of machine j relative to the bottleneck machine. For the case where j 6 q, we define the priority for job i as follows. Z i ¼ tiq

ð3Þ

However, if j > q, we define the priority index for job i as follows. Z i ¼ tij :

ð4Þ

Note that the priority of a job is based on the processing time on the bottleneck machine, when j is 6q. When j is >q, the conventional SPT rule is operative. The job with the minimum Zi is chosen for loading. We call this rule as the BSPT (bottleneck-based shortest processing time) rule. 3. Performance evaluation of the dispatching rules The four dispatching rules included in the relative evaluation are the TPT, SPT, BSPT and PR-1 rules. The number of jobs to be scheduled in the flowshop, denoted by n, is varied from 20 to 60 in steps of 20, and the number of machines, denoted by m, is varied from 5 to 25 in steps of 5. Integer processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. The reason for choosing this range of uniform distribution of processing times is that such a distribution generates unbiased and difficult flowshop scheduling problems to be solved, and hence most research works on flowshop scheduling considered this distribution to generate flowshop problems (e.g., Campbell et al., 1970; Dannenbring, 1977; Liu & Reeves, 2001; Nawaz et al., 1983; Taillard, 1993). As for the BM, integer processing times are sampled from a discrete uniform distribution in the range [51, 99]. Note that while the mean processing time of a job on the BM is more than the mean processing time on any non-BM, the variation in processing times of jobs on the BM is less than the variation in processing times of jobs on any non-BM, in order to ensure that the effect of presence of the bottleneck machine on the performance of dispatching rules is felt reasonably strongly. As for the position of the BM in the flowshop, we have chosen it as one of [m · 0.25], [m · 0.50] and [m · 0.75]. In all, we have 45 different problem sets. For every problem set, we have randomly generated thirty problems (or replications). All the 1350 flowshop problems are solved by using each of the four dispatching rules under consideration. The solutions with respect to total flowtime (or sum of flowtime of jobs) are noted. The relative percentage increase (RPI) in the total flowtime of jobs given by the dispatching rule k for a given problem p (denoted by Fp(k)) is computed as follows: RPIp ðkÞ ¼ ðF p ðkÞ  minfF p ðk 0 Þ;

k 0 ¼ 1; 2; 3; 4gÞ  100= minfF p ðk 0 Þ;

k 0 ¼ 1; 2; 3; 4g:

Here, the dispatching rules are indicated by the respective k values as follows: k = 1: k = 2: k = 3: k = 4:

the the the the

TPT rule, SPT rule, BSPT rule, and PR-1 rule.

ð5Þ

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P  30 The mean RPI for a given rule k (i.e., Mean-RPIðkÞ ¼ p¼1 RPIp ðkÞ =30) and the maximum RPI for rule k (i.e., Max-RPI(k) = max{RPIp(k), p = 1, 2, . . ., 30}) are calculated for every problem set. A problem set is defined in terms of n, m, and the position of BM. In addition, in all performance evaluations of dispatching rules in the current study, we have carried out Duncan’s multiple range test by using the absolute values of the measure of performance (namely, in this part of the study by using total flowtime of jobs Fp(k) yielded by rule k for problem p in a given problem set) corresponding to various rules for a given problem set, at the significance level of 0.05. The set of best rules is indicated by 1, the set of next best rules is indicated by 2, and so on. Some results of relative evaluation of dispatching rules with respect to the performance measure of minimization of total flowtime of jobs are presented in Table 1; the complete set of results for different positions of the BM could not be presented for want of space. It is interesting to observe that the performance of the rules with respect to the mean RPI is similar to that with respect to the maximum RPI, thereby indicating that these performances of the rules with respect to these two measures are related. Therefore, we conduct the statistical analysis and show the results under the column-set ‘Mean RPI’ only. The following observations can be made from the results. The performance of the TPT rule is not good. The performance of the BSPT rule is also not good. In fact, the performance of the BSPT rule mostly worsens as the number of machines increases and as the position of the BM shifts towards the end of flowshops. Likewise, the performance of the SPT rule deteriorates. The reason is that the SPT rule makes use of local information, i.e., the imminent processing times of jobs, and hence, as the size of flowshop increases, the use of local information leads to worse solutions. For the same reason, the proposed BSPT rule behaves badly. However, the performance of the TPT rule improves as the number of machines increases because the TPT rule can be viewed as a global rule, as opposed to the SPT and BSPT rules. The proposed PR-1 rule emerges to be the best (or significantly not worse than any other rule). As discussed earlier, the superior performance of the PR-1 rule can be ascribed to its effective consideration of the relative position of the BM, when the dispatching decision is made. As an additional experimental work for performance evaluation of various dispatching rules, we have looked into the effect of relative magnitude of processing times of jobs on non-BMs and the BM on the performance of dispatching rules. Towards this work, we have sampled the integer processing times of jobs on

Table 1 Performance evaluation of four dispatching rules in flowshops with the machine [m · 0.50] as BM n

m

Relative percentage increase in total flowtime of jobs Mean TPT 2

Maximum SPT

BSPT

PR-1

2

3

1

TPT

SPT

BSPT

PR-1

20

5 10 15 20 25

4.61 3.812 3.091,2 2.421 2.461,2

3.63 5.562 5.512 5.912 4.892

11.30 10.343 10.833 9.643 8.743

0.11 0.141 0.341 0.511 0.351

12.49 9.43 9.68 9.15 8.90

9.27 12.16 16.02 11.39 11.88

21.03 20.42 19.64 19.27 15.11

1.44 1.84 3.04 3.84 3.83

40

5 10 15 20 25

5.792,3 3.732 3.002 2.851,2 2.391,2

3.112 4.302 4.612 5.092 4.802

7.873 9.493 9.573 9.243 7.953

01 0.011 0.071 0.061 0.191

11.54 11.66 8.26 7.27 7.50

6.24 8.90 11.12 11.73 9.95

14.89 14.72 15.50 18.16 13.69

0 0.49 1.57 1.27 2.37

60

5 10 15 20 25

6.193 5.662,3 4.332 3.812 2.381,2

3.092 3.482 4.232 5.062 4.452

5.993 7.053 8.613 8.863 8.263

01 0.021 01 0.021 0.001

9.64 11.72 9.25 7.67 6.40

5.43 8.15 7.78 8.86 8.56

9.68 10.31 13.47 13.49 14.04

0 0.56 0 0.70 0.28

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BM are sampled from a discrete uniform distribution in the range [51, 99].

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non-BMs from a discrete uniform distribution in the range [21, 99], and we have sampled the integer processing times of jobs on the BM from a discrete uniform distribution in the range [51, 99]. Note that while the mean processing time of a job on a non-BM is 0.8 times the mean processing time on the BM, the variation in processing times of jobs on the BM is less than the variation in processing times of jobs on any non-BM. This is done so in order to ensure that the effect of presence of the bottleneck machine on the performance of dispatching rules is felt reasonably well. But for this change, all other experimental settings in Table 1 have been retained in this part of experimental work, and the performance of dispatching rules has been noted. Results are tabulated in Table 2. It is evident from the table that the pattern of performance of dispatching rules is the almost the same as in Table 1, indicating that the performance of the rules is not much sensitive to the relative magnitude of processing times of jobs on non-BMs and the BM. 4. Development of another dispatching rule and its relative evaluation with respect to the PR-1 rule for flowtimebased measures Having been encouraged by the superior performance of the PR-1 rule, we have attempted to improve it. This attempt has yielded a modified version of the PR-1 rule, called as the PR-2 rule. This rule assigns weights to processing times of jobs, depending upon the position of the machine (where jobs are waiting) relative to the BM. Also, the processing times closer to the machine are assigned with larger weights than the processing times farther away from the machine. The PR-2 rule operates as follows. For the case when j is 6 q, define the priority index as follows: Zi ¼ s þ

q X

ðq  k þ 1Þtik ;

or simply

k¼j

Zi ¼

q X ðq  k þ 1Þtik :

ð6Þ

k¼j

However, if j > q, we define the priority index for job i as follows: Table 2 Further performance evaluation of four dispatching rules in flowshops with the machine [m · 0.50] as BM n

m

Relative percentage increase in total flowtime of jobs Mean TPT 1

Maximum SPT 1

BSPT

PR-1

2

1

TPT

SPT

BSPT

PR-1

20

5 10 15 20 25

2.43 2.561,2 1.761,2 1.631 1.401,2

2.21 3.612 4.092 4.952 3.892

7.75 8.433 7.773 7.002 6.713

0.86 0.431 0.461 0.441 0.621

7.84 7.89 7.43 6.82 8.00

5.10 9.30 13.93 9.97 9.76

16.04 18.98 16.33 13.02 11.32

5.43 4.17 3.48 4.95 2.82

40

5 10 15 20 25

2.921 2.051,2 2.521,2 2.171,2 1.511,2

1.591 3.832 4.372 4.152 3.622,3

6.612 9.583 8.253 7.583 6.083

0.541 0.271 0.311 0.201 0.381

10.42 8.68 6.65 6.76 6.95

3.95 7.30 8.84 8.19 10.16

17.11 14.63 12.14 13.00 10.57

3.77 2.90 5.92 3.07 3.43

60

5 10 15 20 25

2.861 2.341,2 2.561,2 2.041,2 1.081,2

1.661 3.122 3.312 3.592 2.972

6.402 8.643 7.973 6.883 6.183

0.361 0.111 0.151 0.121 0.121

7.40 5.59 7.30 6.24 3.32

3.57 6.63 6.84 7.13 7.25

13.02 12.65 13.79 11.91 12.71

2.39 2.32 3.00 2.29 1.83

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [21, 99]. c. Processing times of jobs on the BM are sampled from a discrete uniform distribution in the range [51, 99].

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Zi ¼ s þ

m X ðm  k þ 1Þtik ;

95

or simply

k¼j

Zi ¼

m X

ðm  k þ 1Þtik :

ð7Þ

k¼j

We call this rule as PR-2. The rationale for assigning weights to processing times of jobs is based on the observations made by Rajendran (1993), and Rajendran and Ziegler (1997). Basically, the processing times close to machine j are assigned greater weights than those away from machine j, because the former set of processing times has greater impact on the flowtime or completion time of jobs than the latter set of processing times. We can also observe that if the size of the given flowshop is large, this impact is more pronounced than in a small-sized flowshop. Also, note that the priority of a job is governed by the position of machine j relative to the BM. In order to evaluate the relative performance of the PR-1 and PR-2 rules, the same set of problems, presented in Table 1, is again considered for different positions of the bottleneck machine in the flowline. The mode of evaluation of the two rules is similar to the mode of evaluation of rules, given in Section 3. The mean and the maximum relative percentage increases in the values of total flowtime yielded by the two rules for some problem sets are presented in Table 3. It is interesting to note that the proposed PR-2 rule performs better when the number of machines is greater than 15 and the position of the BM is towards the end of flowshops. The reason for this behavior of the PR-2 rule is that the impact of assignment of weights to processing times is realised only when the number of machines is large, and also when more number of machines are present before the BM in the flowline. However, when the size of a flowshop is small or medium and the BM is not present towards the end of the flowline, it appears that the proposed PR-1 rule performs effectively. 5. Development of dispatching rules for flowtime-based measures by considering the presence of two bottleneck machines Our next investigation pertains to the situation in which two BMs are present in the flowshop. We propose two dispatching rules for scheduling in such cases. Basically, the first rule in this case (called PR-3 rule) is based on the PR-1 rule in the sense that the priority indices for jobs are computed by taking into the account Table 3 Performance evaluation of the proposed two dispatching rules in flowshops with the machine [m · 0.75] as BM n

m

Relative percentage increase in total flowtime Mean

Maximum

PR-1

PR-2

PR-1

1

PR-2

1

20

5 10 15 20 25

0.35 0.781 1.571 1.871 1.651

0.57 0.681 0.621 0.311 0.231

3.26 5.66 6.95 5.57 4.83

1.88 3.43 3.16 2.64 2.23

40

5 10 15 20 25

0.161 0.391 1.511 1.601 1.531

0.371 0.711 0.161 0.421 0.371

1.61 2.39 5.33 7.07 5.01

1.22 2.35 1.84 3.92 3.98

60

5 10 15 20 25

0.051 0.311 0.341 1.561 1.191

0.431 0.541 0.731 0.431 0.351

0.88 2.69 1.89 6.86 5.18

1.11 2.43 2.80 2.78 3.14

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BM are sampled from a discrete uniform distribution in the range [51, 99].

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the total processing times either upto the imminent BM or upto the end of the flowline. The second rule (called PR-4 rule) differs from the first one by considering the ‘overall’ or ‘global’ BM, instead of considering the imminent or ‘local’ BM, for computing the total processing times of jobs. 5.1. Proposed dispatching rule PR-3 Suppose we have two bottleneck machines present in the flowshop. Let their positions be q and q 0 , and let q be
q X

tik ;

if j 6 q

tik ;

if q < j 6 q0

tik ;

otherwise:

k¼j

¼

q0 X k¼j

¼

m X

ð8Þ

k¼j

It is evident that the rule attempts to schedule the jobs with minimum sum of processing times upto the imminent BM, thereby minimizing the waiting time of jobs at the imminent BM. Basically, the focus of scheduling is on the imminent BM. For the case where two BMs need not necessarily have the same relative influence or impact on the throughput of jobs in the flowline, we still follow the PR-3 rule without any modification. 5.2. Proposed dispatching rule PR-4 The second proposed dispatching rule works, depending upon the relative influence of the two BMs. Let us assume that these two BMs are present in positions q and q 0 in the flowshop, and let q be
q0 X

tik ;

if j 6 q0

tik ;

otherwise:

k¼j

¼

m X

ð9Þ

k¼j

It is evident that the rule attempts to schedule the jobs with minimum sum of processing times upto the last BM, thereby attempting to minimize the waiting time of jobs at the last BM. Basically, the focus of scheduling is on the last BM in the flowline, and it does not consider any other BM in the flowline. However, when one BM has more influence or impact on the throughput relative to the other BM, we modify the PR-4 rule. Let us assume that the position of the greater BM is Q in the flowline. Note that Q can be equal to either q or q 0 , depending upon where the greater BM is present. The PR-4 rule operates as follows. Zi ¼

Q X

tik ;

if j 6 Q

tik ;

otherwise:

k¼j

¼

m X

ð10Þ

k¼j

We call the BM with greater influence on throughput as the ‘global BM’, and the BM with less impact on throughput as the ‘local BM’. The focus of the rule PR-4 is on the global BM.

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6. Performance analysis of dispatching rules for flowtime-based measures in the presence of two bottleneck machines In order to evaluate the relative performance of the PR-3 and PR-4 rules, the same set of problems, presented in Table 1, is again considered, except for the positions of two BMs and the sampled processing times on the two BMs. We have also considered the SPT rule as a benchmark rule. As for the presence of the two bottleneck machines in the flowline, we have adopted the following three settings: 1. Positions of two BMs are [m · 0.333] and [m · 0.667], and discrete processing times of jobs on both BMs are sampled from the uniform distribution in the range [51, 99]. 2. Positions of two BMs are [m · 0.333] and [m · 0.667], and discrete processing times of jobs on the former BM are sampled from the uniform distribution in the range [76, 124], and the discrete processing times of jobs on the latter BM are sampled from a uniform distribution in the range [51, 99]. 3. Positions of two BMs are [m · 0.333] and [m · 0.667], and discrete processing times of jobs on the former BM are sampled from the uniform distribution in the range [51, 99], and the discrete processing times of jobs on the latter BM are sampled from a uniform distribution in the range [76, 124]. Note that processing times of jobs on other machines, i.e., non-BMs, have been sampled from a uniform distribution in the range [1, 99]. The mode of evaluation of the three rules under consideration is similar to the mode of evaluation of rules, given in the earlier sections. The mean and the maximum relative percentage increases in the values of total flowtime yielded by the three rules for different problem sets are presented, and some typical results are presented in Tables 4 and 5. The following observations are evident from the results. Table 4 Performance evaluation of three dispatching rules in flowshops with the machines [m · 0.333] and [m · 0.667] as BM n

m

Relative percentage increase in total flowtime of jobs Mean

Maximum

SPT

PR-3

PR-4

SPT

PR-3

PR-4

20

10 15 20 25 30

2.361 3.382 4.402 4.792 5.632

0.031 0.121 0.061 0.121 0.201

0.031 0.081 0.071 0.041 0.201

5.46 8.96 13.09 10.37 12.18

0.28 0.71 0.80 0.82 3.06

0.28 1.03 0.80 0.58 3.06

40

10 15 20 25 30

2.131 3.702 4.132 4.632 4.702

0.011 0.021 0.041 0.081 0.041

0.001 0.041 0.031 0.031 0.091

4.35 8.38 8.33 8.74 8.37

0.24 0.22 0.47 0.45 0.46

0.14 0.23 0.25 0.29 0.47

60

10 15 20 25 30

1.481 2.851 3.242 3.472 8.112

0.001 0.011 0.021 0.031 0.031

0.001 0.011 0.021 0.041 0.081

2.85 5.39 5.60 5.63 8.11

0.03 0.11 0.15 0.24 0.24

0.11 0.08 0.14 0.43 0.45

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BM [m · 0.333] are sampled from a discrete uniform distribution in the range [76, 124], and processing times of jobs on the BM [m · 0.667] are sampled from a discrete uniform distribution in the range [51, 99].

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Table 5 Performance evaluation of three dispatching rules in flowshops with the machines [m · 0.333] and [m · 0.667] as BM n

m

Relative percentage increase in total flowtime of jobs Mean

Maximum

SPT

PR-3

PR-4

SPT

PR-3

PR-4

20

10 15 20 25 30

4.472 5.512 6.152 6.382 6.752

2.531,2 2.691 1.861 1.951 1.461

0.071 0.471 0.421 0.261 0.641

11.00 12.95 14.25 15.11 12.94

7.15 7.43 7.32 7.62 6.75

2.11 4.22 3.85 2.71 4.25

40

10 15 20 25 30

4.012 6.072 5.722 5.712 6.212

2.121,2 2.341 1.691 1.411 1.451

0.061 0.071 0.101 0.251 0.281

7.03 11.89 11.82 10.93 13.19

5.24 6.93 6.47 5.65 5.92

1.01 1.53 1.05 2.08 2.04

60

10 15 20 25 30

3.382 5.092 4.792 5.172 5.232

1.951,2 2.311,2 1.521 1.581 1.631

0.011 0.041 0.181 0.111 0.221

5.44 8.97 7.78 9.01 8.78

3.68 5.00 4.14 3.96 4.49

0.02 1.05 1.48 2.13 2.05

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BM [m · 0.333] are sampled from a discrete uniform distribution in the range [51, 99], and processing times of jobs on the BM [m · 0.667] are sampled from a discrete uniform distribution in the range [76, 124].

1. For the case where the given two BMs have the same influence or impact on the throughput in the flowline, the proposed PR-3 rule performs better than (though not significantly better than) the PR-4 rule. The reason is that both BMs in this case can be termed as ‘global’, and the focus of the PR-3 rule is on the imminent global BM, whereas the PR-4 rule ignores the presence of the first global BM. The performance of the SPT rule is found to be consistently and significantly bad. 2. For the case where the given two BMs have different relative impact on the throughput, with the global BM being present ahead of the local BM, there is not much of a difference between the performances of the PR3 and PR-4 rules (see Table 4). The reason is that the relative impact of the local BM is not felt much, when the global BM is present ahead of the local BM in the flowline. Again, the performance of the SPT rule is significantly bad. 3. For the case where the given two BMs have different relative impact on the throughput, with the global BM being present after the local BM, the PR-4 rule performs better than (though not significantly better than) the PR-3 rule (see Table 5). The reason is that the focus of scheduling in the PR-4 rule is on the global BM, whereas the focus of scheduling in the PR-3 rule is on the local BM. In addition, the global BM is present after the local BM in the flowline. Once again, the performance of the SPT rule is significantly bad. Overall, we make the following recommendations. 1. When the BMs have an equal influence or impact on the throughput in a flowline, it is advantageous to schedule with a focus on the imminent BM. This helps us to minimize the waiting time of jobs upto each BM. 2. When one BM has a greater impact on the throughput in a flowline than the other BMs, it appears advantageous to schedule with a focus on the greater BM.

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7. Development of dispatching rules considering the presence of a single bottleneck machine for job-due-datebased measures The most commonly used dispatching rules with the objective of minimizing the job-earliness and job-tardiness are the SPT (shortest processing time) rule, the EDD (earliest due-date) rule and the slack rule (see Blackstone et al., 1982; Haupt, 1989; Ramasesh, 1990). Apart from these rules, we also consider the ODD (operation due-date) rule that is found to be a good rule with repect to the due-date related measures such as the minimization of tardiness of jobs (see Baker, 1984b; Jayamohan & Rajendran, 2000; Kanet & Hayya, 1982). Let us consider the case where there is only one bottleneck machine (BM) in the flowshop consisting of m machines in series. We now propose two dispatching rules. According to the first rule, when the dispatching decision is to be made with respect to jobs waiting at a machine ahead of the BM, then the job with the earliest operation due-date with respect to the BM is chosen for loading. However, if the machine is present after the BM, the rule identifies the job with the earliest due-date. Basically, the rule seeks to minimize the earliness and tardiness of jobs by having a tight monitoring of progress of jobs in accordance with their operation due-dates and also by taking into account the relative position of the BM. Apart from considering the operation duedate or the earliest due-date (as in the first proposed rule), the second rule reckons with the sum of processing times of jobs upto the BM, if the jobs are waiting at a machine ahead of the BM; however, if the jobs are waiting at a machine after the BM, then the sum of processing times of jobs upto the end of the flowline is considered for the dispatching decision. Once again, the dispatching decision is made depending upon the position of the machine (where jobs are waiting) relative to the BM. The mathematical formulation of the two rules is now presented. 7.1. Proposed rule 1: DD/BJ rule Let the flowshop consist of m machines in series, and let the BM be machine q. Let the processing time of job i on machine k be tik, and let the due-date of job i be Di. In the case of flowshops with all jobs being available at time zero, the operation due-date of job i for machine j, dij, is given as follows: !, ! j m X X d ij ¼ Di  tik tik ð11Þ k¼1

k¼1

Let s be the current time at which the dispatching decision is to be made with respect to machine j. For the case where j 6 q, we define the priority for job i as follows. Z i ¼ d iq :

ð12Þ

However, if j > q, we define the priority index for job i as follows. Z i ¼ d im ; Z i ¼ Di :

or simply ð13Þ

The job with the minimum Zi is chosen for loading. We call this proposed rule as DD/BJ, indicating that the job priority is determined with respect to either the due-date based on the BM or the job due-date. The objective of this rule is to render the jobs adhere to their operation due-dates focused on either the BM or the end of the flowline, depending upon the position of the current machine relative to the BM. 7.2. Proposed rule 2: (TPT+DD)/BJ rule According to this rule, the priority of a job is again governed by the position of machine j relative to the BM. In addition to the factors considered in the DD/BJ rule, we consider the sum of processing times. For the case where j 6 q, we define the priority for job i as follows.

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Z i ¼ d iq þ

q X

tik

ð14Þ

k¼j

However, if j > q, we define the priority index for job i as follows. Z i ¼ Di þ

m X

tik

ð15Þ

k¼j

The job with the minimum Zi is chosen for loading. We call this rule as the (TPT+DD)/BJ rule, indicating that the total processing time is considered in addition to the operation due-date. The objective of adding the component of the sum of processing times of a job to the computation of job priority is to enable an enhanced throughput of jobs in the flowshop, and hence an enhanced performance with respect to tardiness of jobs. 8. Performance evaluation of the dispatching rules for due-date-based measures We have identified from a survey of literature (Baker, 1984b; Blackstone et al., 1982; Haupt, 1989; Jayamohan & Rajendran, 2000; Kanet & Hayya, 1982; Ramasesh, 1990) the following four rules as benchmark rules in our study. These rules are used not only in almost all theoretical studies, but also in real-life situations as scheduling rules (also see Pinedo, 1995). Hence, we have chosen to use these rules in our study as well. 1. The ODD rule: According to this rule, the priority index of job i at machine j is given as follows: Z i ¼ d ij

ð16Þ

2. The EDD rule: According to this rule, the priority index of job i at machine j is given below: Z i ¼ Di

ð17Þ

3. The SPT rule: According to this rule, the priority index of job i at machine j is given as follows: Z i ¼ tij

ð18Þ

4. The SLACK rule: According to this rule, the priority index of job i at machine j is given as follows: Z i ¼ Di 

m X

tik  s

ð19Þ

k¼j

In all the above rules, the job with the least Zi is chosen for loading. The four benchmark dispatching rules and the proposed two dispatching rules are evaluated by generating a large number of flowshops problems of varying sizes. The number of jobs to be scheduled in the flowshop is varied from 20 to 60 in steps of 20, and the number of machines is varied from 5 to 25 in steps of 5. Integer processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. As for the BM, integer processing times are sampled from a discrete uniform distribution in the range [75, 125]. As for the position of the BM in the flowshop, we have chosen it as one of [m · 0.25], [m · 0.50] and [m · 0.75]. In all, we have 45 different problem sets. For every problem set, we have randomly generated thirty problems (or replications). Towards the purpose of setting due-dates for jobs, we have assigned the due-date of job i, denoted by Di, as follows. Di ¼

m X

tij þ u  ðn  1Þ  50;

ð20Þ

j¼1

where u is a uniform random number sampled in the range [0, 1]. The rationale behind this setting of due-dates of jobs is that if job i is the first job in a permutation sequence, a weak lower bound on makespan is given by P m j¼1 t ij þ ðn  1Þ  50. By the due-date setting used in the present work, we can expect about 50% of jobs to be tardy because u varies uniformly in the range [0, 1]. Gelders and Sambandam (1978) have also followed the similar approach for due-date setting in their work. In fact, observations from some pilot runs have confirmed

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our expectation with respect to the values of tardiness and earliness of jobs. All the 1350 flowshop problems thus generated are solved by using each of the six dispatching rules under consideration. The solutions yielded by each of the dispatching rules with respect to the sum of earliness and tardiness of jobs, and also the sum of tardiness of jobs are noted. The relative percentage increase (RPI) in the sum of earliness and tardiness of jobs (ET) given by the dispatching rule k (denoted by ET(k)) for a given problem is computed as follows. RPIðkÞ ¼ ðET ðkÞ  minfET ðk 0 Þ; k 0 ¼ 1; 2; 3; 4; 5; 6gÞ  100= minfET ðk 0 Þ; k 0 ¼ 1; 2; 3; 4; 5; 6g

ð21Þ

Here, the dispatching rules are indicated by the respective k values as follows: k = 1: k = 2: k = 3: k = 4: k = 5: k = 6:

the the the the the the

ODD rule, EDD rule, SPT rule, SLACK rule. DD/BJ rule, and (TPT+DD)/BJ rule,

The mean RPI in the sum of earliness and tardiness of jobs yielded by a given rule over 30 problems for a given problem set is computed. The problem set is defined in terms of n, m, and the position of BM. Some typical results of relative evaluation of dispatching rules with respect to the measure of minimization of the sum of earliness and tardiness of jobs are presented in Table 6. The relative evaluation of dispatching rules with respect to the sum of tardiness of jobs is carried out in the way similar to the computation presented in Exp. 21, and the results are presented in Table 7. In addition, the statistical test has been conducted to determine if there exists a significant difference between the performance of the rules under investigation (similar to that discussed in Section 3). The following observations can be made from the results of this investigation. 1. The (TPT+DD)/BJ rule performs significantly the best (or significantly not worse than any other rule) with respect to the minimization of the sum of earliness and tardiness of jobs, as well as with respect to the minimization of the sum of tardiness of jobs. Table 6 Performance evaluation of six dispatching rules in flowshops with the machine [m · 0.50] as BM n

m

Mean relative percentage increase in the sum of earliness and tardiness of jobs ODD 2,3

20

5 10 15 20 25

8.06 16.213 22.843 28.473 28.654

40

5 10 15 20 25

3.372,3 9.102,3 11.083 15.774 17.625

60

5 10 15 20 25

2.151,2 6.312,3 7.812.3 8.752,3 9.993

EDD 2

SPT

SLACK 3

DD/BJ 1

(TPT+DD)/BJ

33.28 28.164 31.744 33.394 32.695

9.42 17.203 18.853 25.953 30.744,5

2.38 2.291 4.241 3.761 7.722

1.351 1.971 2.241 2.161 3.531

4.192,3 6.402 7.052 8.553 8.823

29.624 29.744 28.364 30.685 29.296

5.923 11.533 11.213 14.394 13.964

2.161,2 2.381 3.171 3.832 4.402

0.441 0.161 0.421 0.561 0.771

3.241,2 4.642 5.412 5.572 6.682

29.623 29.634 27.894 25.714 23.914

4.192 8.533 9.293 10.543 10.683

1.821,2 1.421 2.691 2.091 2.391

0.581 0.191 0.121 0.381 0.361

5.12 9.052 8.482 12.822 13.543

4

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BM are sampled from a discrete uniform distribution in the range [75, 125].

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Table 7 Performance evaluation of six dispatching rules in flowshops with machine [m · 0.50] as the BM n

m

Mean relative percentage increase in the tardiness of jobs ODD 2,3

EDD 3

SPT

SLACK 4

4

DD/BJ 1,2

(TPT+DD)/BJ

20

5 10 15 20 25

5.33 13.392 19.454 25.854 26.274

7.01 12.582 13.693 17.243 18.173

10.99 14.842 19.434 24.924 24.794

12.37 21.663 25.385 32.525 37.005

3.42 3.841 7.662 5.492 9.662

1.501 0.741 1.011 1.001 2.341

40

5 10 15 20 25

2.191 7.373 8.423 13.043,4 14.814

5.492,3 7.853 8.343 10.463 10.443

8.954 10.994 11.903,4 16.044,5 15.904

7.323,4 13.284 12.664 16.775 15.864

3.041,2 3.082 3.992 5.012 5.442

0.881 0.161 0.441 0.401 0.591

60

5 10 15 20 25

1.471,2 5.112 6.233 7.032 8.202,3

3.812,3 5.142 6.023 6.362 7.552

8.154 9.763 10.444 10.063 10.993,4

4.823 9.153 10.054 11.483 11.704

2.251.2.3 1.561 3.112 2.561 2.931

0.831 0.101 0.151 0.411 0.371

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BM are sampled from a discrete uniform distribution in the range [75, 125].

2. The DD/BJ performs the next best (or significantly not worse than the ODD and EDD rules). 3. The SPT and SLACK rules are the worst performing rules. 4. The performance of the proposed rules improves as the position of the bottleneck BM shifts to the right. All these observations prove the point that in the presence of the BM, we need to develop scheduling rules that explicitly consider the presence of the bottleneck machine. Also, the performance of the EDD rule appears to be the best among the existing rules because this rule considers the job information beyond the BM and hence it performs better than the rules that consider the local information (e.g., the SPT rule). In fact, as the size of flowshop increases, the performance of the rules using local information deteriorates. As discussed earlier, the superior performance of the proposed rules can be attributed to their effective consideration of the relative position of the BM, when the dispatching decision is made. These observations validate our stand that we need to develop scheduling rules explicitly to take into account the presence of the BM in flowshops. 9. Extension of the proposed dispatching rules for due-date-based measures in the presence of multiple bottleneck machines Our next investigation pertains to the situation in which more than one BM is present in the flowshop. Basically, we make use of the proposed rules in the same form except with the following modifications. 9.1. Modification for the proposed rule DD/BJ Let us denote machine q as the next imminent BM with respect to machine j (machine j being the machine where the dispatching decision needs to be made). Note that there can be more BMs after machine q in the flowshop. However, the priority of job i is derived by considering machines j and q only. In this case, we define Zi in the same way as defined in Eq. (12). This definition of Zi holds as long as there exists a BM after machine j. When there exists no BM after machine j, we define Zi as in Eq. (13).

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9.2. Modification for the proposed rule (TPT+DD)/BJ Let us denote machine q as the next imminent BM with respect to machine j (machine j being the machine where the dispatching decision needs to be made). Note that there can be more BMs after machine q in the flowshop. However, the priority of job i is derived by considering machines j and q only. In this case, we define Zi in the same way as defined in Eq. (14). This definition of Zi holds as long as there exists a BM after machine j. When there exists no BM after machine j, we define Zi as in Eq. (15).

10. Performance analysis of dispatching rules for due-date-based measures in the presence of multiple bottleneck machines In order to evaluate the relative performance of the dispatching rules under study, the same set of problems with the number of jobs varying from 20 to 60 and the number of machines varying from 5 to 25, presented in Table 6, is again considered, except for the positions of multiple BMs. As for the presence of multiple bottleneck machines in the flowline, we have adopted the following two settings: 1. Positions of two BMs are [m · 0.333] and [m · 0.667], and discrete processing times of jobs on both BMs are sampled from the uniform distribution in the range [75, 125]. 2. Positions of three BMs are [m · 0.25], [m · 0.50] and [m · 0.75], and discrete processing times of jobs on both BMs are sampled from the uniform distribution in the range [75, 125]. Note that processing times of jobs on other machines, i.e., non-BMs, have been sampled from a uniform distribution in the range [1, 99]. The mode of evaluation of the six rules under consideration is similar to the mode of evaluation of rules, given in Exp. 21. The mean relative percentage increases in the values of the sum of earliness and tardiness of jobs, and the sum of tardiness of jobs yielded by the six rules for different problem sets are computed, and some typical results are presented in Tables 8 and 9. The following observations are evident. Table 8 Performance evaluation of six dispatching rules in flowshops with machines [m · 0.25], [m · 0.50] and [m · 0.75] as BMs n

m

Mean relative percentage increase in the sum of earliness and tardiness of jobs ODD 1

EDD 1

SPT

SLACK 2

1

DD/BJ 1

(TPT+DD)/BJ

20

5 10 15 20 25

2.18 10.103 14.592 15.413 16.883

2.64 6.332 6.021 7.612 5.281

27.19 24.294 23.913 23.974 19.643

3.63 11.503 12.732 15.703 11.912

2.18 2.781 3.441 3.021 5.441

2.571 1.941 4.791 2.701 4.771

40

5 10 15 20 25

1.241 5.073 7.153 8.132,3 12.303

2.131 4.172,3 4.862,3 5.312 6.292

25.992 27.174 26.894 22.714 24.274

2.311 7.153 6.883 8.773 9.603

1.241 1.971,2 2.641,2 2.051 1.481

0.761 0.901 1.661 0.831 2.401

60

5 10 15 20 25

0.541 3.872 6.512 6.874 7.902

1.521 2.391,2 3.431,2 3.712,3 5.322

27.472 27.633 26.783 24.605 22.253

1.911 4.212 5.082 6.343,4 7.822

0.541 1.341,2 1.291 1.621,2 1.641

0.301 0.721 0.701 0.851 0.411

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BMs are sampled from a discrete uniform distribution in the range [75, 125].

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Table 9 Performance evaluation of six dispatching rules in flowshops with machines [m · 0.25], [m · 0.50] and [m · 0.75] as BMs n

m

Mean relative percentage increase in the tardiness of jobs ODD 1,2

EDD 1,2

SPT

SLACK 3

2

DD/BJ 1,2

(TPT+DD)/BJ

20

5 10 15 20 25

2.10 8.102 12.122 13.192,3 15.343,4

3.40 9.362 9.282 11.412 9.292

11.48 13.543 14.303 16.373 13.203

4.48 15.173 16.833 20.434 16.874

2.10 3.611 3.181 3.651 6.702

1.321 1.301 2.821 1.291 3.541

40

5 10 15 20 25

1.241 3.692,3 5.662,3 7.072 10.462,3

2.371 5.283 6.303 7.002 8.082

10.902 11.915 13.934 12.353 14.584

2.611 8.384 8.493 10.733 11.613,4

1.241 2.011,2 2.801,2 2.631 1.611

0.531 0.571 1.181 0.571 1.701

60

5 10 15 20 25

0.581 2.821,2 5.273 4.872 6.782

1.651 2.821,2 4.062,3 4.512 6.242

11.342 12.903 13.164 11.583 11.943

2.061 4.682 5.823 7.242 8.892,3

0.581 1.431 1.301,2 1.721 1.991

0.191 0.571 0.421 0.551 0.301

Notes: a. Sample size in every problem set is 30 and the total number of generated problems is 450. b. Processing times of jobs on non-BMs are sampled from a discrete uniform distribution in the range [1, 99]. c. Processing times of jobs on the BMs are sampled from a discrete uniform distribution in the range [75, 125].

1. The (TPT+DD)/BJ rule continues to perform significantly the best (or significantly not worse than any other rule) with respect to the minimization of the sum of earliness and tardiness of jobs, as well as with respect to the minimization of the sum of tardiness of jobs. 2. The DD/BJ again performs the next best (or significantly not worse than the ODD and EDD rules). 3. The SPT and SLACK rules are the worst performing rules. 4. The performance of the proposed rules improves as the position of the bottleneck BM shifts to the right. 5. As the number of BMs increases, the difference in the performance of the EDD and DD/BJ rules decreases. This is not unexpected because the presence of more BMs leads to a more balanced flowline, and hence the DD/BJ rule tends to behave in the same way as the EDD rule. The same observation holds for the ODD rule (behaving similar to the DD/BJ rule) which is quite evident from its performance in the small-sized flowshops with m = 5.

11. Summary A survey of literature on flowshop scheduling has revealed that no work appears to have been done on issues related to dispatching jobs in the presence of bottleneck machines (BMs). Based on this finding, the present work has been undertaken to develop dispatching rules for static flowshops with BMs. The minimization of total flowtime of jobs, the minimization of the sum of earliness and tardiness of jobs, and the minimization of total tardiness of jobs have been considered as the measures of performance. New dispatching rules, with a focus on BMs, have been developed. A number of flowshop problems, with varying number of machines, jobs and positions of BMs, have been generated for evaluating the proposed dispatching rules and some benchmark rules often used in the literature on scheduling. The results reveal that the proposed dispatching rules perform much better than the benchmark rules with respect to minimization of total flowtime, sum of earliness and tardiness, and total tardiness of jobs, considered separately.

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