A comparative study of dispatching rules in dynamic flowshops and jobshops

European Journal of Operational Research 116 (1999) 156±170

Theory and Methodology

A comparative study of dispatching rules in dynamic ¯owshops and jobshops Chandrasekharan Rajendran a, Oliver Holthaus

b,*

a b

Industrial Engineering and Management Division, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India Faculty of Business Administration and Economics, Department of Production and Operations Management, University of Passau, Dr-Hans-Kap®nger-Str. 30, 94032 Passau, Germany Received 1 December 1996; accepted 1 December 1997

Abstract This paper presents a comparative study on the performance of dispatching rules in the following sets of dynamic manufacturing systems: ¯owshops and jobshops, and ¯owshops with missing operations and jobshops. Three new dispatching rules are proposed. We consider a total of 13 dispatching rules for the analysis of the relative performance with respect to the objectives of minimizing mean ¯owtime, maximum ¯owtime, variance of ¯owtime, proportion of tardy jobs, mean tardiness, maximum tardiness and variance of tardiness. First, we carry out the simulation study in ¯owshops with jobs undergoing processing on all machines sequentially and in jobshops with random routeing of jobs. The results of the study reveal some interesting observations on the relative performance of the dispatching rules in these two types of manufacturing systems. Next, we consider ¯owshops with missing operations on jobs and jobshops with random routeing of jobs. We observe some interesting results in the sense that the performance of dispatching rules is being in¯uenced by the routeing of jobs and shop¯oor con®gurations. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Flowshop; Jobshop; Dispatching rules; Simulation study

1. Introduction The problem of scheduling in dynamic jobshops has been extensively studied for many years and it attracts the attention of the researchers and practitioners equally. The problem is usually

*

Corresponding author. Tel.: 49 851 5 09 2454; fax: 49 851 5 09 2452; e-mail: [email protected].

characterized as one in which a set of jobs, each consisting of one or more operations to be performed in a speci®ed sequence on speci®ed machines and requiring some process times, is to be processed over a period of time. The objective is to determine the job schedules that minimize a measure (or multiple measures) of performance. The problem of scheduling in ¯owshops is usually investigated with a static arrival of jobs and hence, research is mostly directed towards the

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 0 2 3 - X

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

development of exact and approximate solution techniques (Baker, 1974; French, 1982; Pinedo, 1995). In many real-life situations, it is quite common to encounter a dynamic and stochastic arrival of jobs in manufacturing systems and hence the scheduler makes use of dispatching rules to derive the job schedules. While many studies have concentrated on the problem of scheduling in dynamic jobshops (see the survey articles of Blackstone et al., 1982; Haupt, 1989; Ramasesh, 1990 for a detailed discussion), there have been relatively few studies on dynamic ¯owshops (e.g. Scudder and Ho€mann, 1987; Hunsucker and Shah, 1992; Hunsucker and Shah, 1994). We also ®nd that there appears to be no earlier study presenting a comparative analysis of dispatching rules in jobshops and ¯owshops. These observations have been the motivation for the present work on dynamic ¯owshops and jobshops. This paper ®rst presents a literature review of earlier research work on dynamic ¯owshops and jobshops, followed by the identi®cation of the existing rules and the development of new dispatching rules for a performance analysis. The jobshop and ¯owshop models are subsequently presented with details of experimental set-ups. We consider two types of ¯owshops: one in which all jobs undergo processing on all machines and the other one in which jobs have missing operations on some machines. The results of the simulation experiments are discussed for the two cases separately: ¯owshops and jobshops, and ¯owshops with missing operations and jobshops. The interesting aspects of the ®ndings are also brought out. 2. Literature review A dispatching rule is used to select the next job to be processed from a set of jobs awaiting service at a facility that becomes free. The diculty of the choice of a dispatching rule arises from the fact that there are n! ways of sequencing n jobs waiting in the queue at a particular facility and the shop¯oor conditions elsewhere in the shop may in¯uence the optimal sequence of jobs at the present facility. Dispatching rules are normally intended to minimize the inventory and/or tardiness costs. It is

157

a customary practice to minimize the ¯owtimerelated and tardiness-related measures of performance since the associated inventory and tardiness costs are assumed to be directly proportional to the time periods of ¯owtime and tardiness of jobs, respectively, for the sake of theoretical research (Blackstone et al., 1982). The dispatching rules can be classi®ed into ®ve categories: (1) rules involving process times, (2) rules involving due-dates, (3) simple rules involving neither process times nor due-dates, (4) rules involving shop¯oor conditions and (5) rules involving two or more of the ®rst four classes. It has been observed that no single rule performs well for all important criteria related to ¯owtime and tardiness of jobs. In general, it has been noted that process-time based rules fare better under tight load conditions, while due-date based rules perform better under light load conditions (Conway, 1965; Rochette and Sadowski, 1976; Blackstone et al., 1982). Of course, the choice of a dispatching rule depends upon which criterion is to be met with, viz. the minimization of mean ¯owtime or mean tardiness or variance of ¯owtime or tardiness. A typical process-time based rule is the famous SPT (shortest process time) rule being used as a bench-mark quite often since this rule is found to be very e€ective in minimizing mean ¯owtime and also minimizing mean tardiness under highly loaded shop¯oor conditions (Conway, 1965; Rochette and Sadowski, 1976; Blackstone et al., 1982; Haupt, 1989). As for the due-date based rules, the earliest due-date (EDD) rule is perhaps the most popular rule. The rules such as the (®rst in, ®rst out) FIFO rule do not make use of any information on the process time or due-date of a job, and is often used as a benchmark since the FIFO rule is quite e€ective in minimizing the maximum ¯owtime and variance of ¯owtime in many cases. The more complex rules make use of both process time and due-date information, e.g. Least Slack rule, Critical Ratio, etc. (see Blackstone et al., 1982; Haupt (1989), for a detailed presentation of several rules). There are some rules which load the jobs depending on shop¯oor conditions rather than on the characteristics of jobs. An example of this type of rule is the WINQ rule (total work-content of jobs in the queue of the next operation of a

158

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

job) (Haupt, 1989). Day and Hottenstein (1970), Blackstone et al. (1982), Haupt (1989) and Ramasesh (1990) present excellent reports on some of the widely used and popular dispatching rules for jobshop scheduling. It is a customary practice for researchers to assume in the case of ¯owshop scheduling that all jobs are available at the beginning of the scheduling period (Baker, 1974; French, 1982; Pinedo, 1995). It is therefore quite natural that many optimizing and heuristic algorithms have been developed for minimizing makespan or total ¯owtime or both (e.g. Ignall and Schrage, 1965; Campbell et al., 1970; Dannenbring, 1977; Nawaz et al., 1983; Rajendran, 1994; Ho, 1995; Ishibuchi et al., 1995). However, when job arrivals are dynamic and stochastic, and job process times are not deterministic, the use of dispatching rules is resorted to in the case of ¯owshops (e.g. Hunsucker and Shah, 1992, 1994). These studies have considered some simple dispatching rules such as the SPT, FIFO and LPT (longest process time) rules, and have not considered the more ecient rules such as the COVERT rule (see Russell et al., 1987 for a detailed discussion of this rule) or the rules by Anderson and Nyirenda (1990), or Raghu and Rajendran (1993). Therefore, we observe that the existing studies on dynamic ¯owshops are not quite exhaustive. For the case of special operating conditions such as the pro®ts associated with jobs or lot splitting or di€erential job speeds relative to due-dates, some problem-speci®c dispatching rules have been developed (e.g. Scudder and Ho€mann, 1987; Smunt et al., 1996). It is evident that the problem of scheduling in dynamic ¯owshops has not received the attention as much as the problem of scheduling in dynamic jobshops. It is also clear that there seems to be no prior research on the comparative analysis of dispatching rules in ¯owshops and jobshops. Such a study assumes signi®cance since the nature of routeings is di€erent in ¯owshops and jobshops, viz. unidirectional routeing and random routeing of jobs, respectively, and this di€erence may in¯uence the relative performance of various dispatching rules. These observations have been the motivation for the present study. In Section 3, we identify the existing rules that are quite e€ective with respect to several measures of performance and subsequently present the

development of new dispatching rules for minimizing the ¯owtime-related performance measures. 3. Identi®cation of the best existing rules and development of new dispatching rules Before we present the dispatching rules considered in this study, we introduce the terminology used in this paper. Let s tij

o(i) Di Wi Ti Zi

time at which the dispatching decision is made; process time for operation j of job i; operation j of job i is performed on the machine that becomes free at the current instant s and this machine requires a job to be o€-loaded from the queue; total number of operations on job i; due-date for job i; total work-content of jobs in the queue of the next operation of job i. If operation j is the last operation for job i, then Wi is zero; time of arrival of job i; priority value assigned to job i at the time of decision of dispatching.

A study of the existing literature on jobshop scheduling reveals that the following rules are quite e€ective for di€erent measures of performance. (1) FIFO (®rst in, ®rst out): This rule is often used as a bench-mark. The job that has entered the queue at the earliest is chosen for loading. The FIFO rule is an e€ective rule for minimizing the maximum ¯owtime and variance of ¯owtime. (2) AT (arrival time): The job with the earliest arrival time in the shop is chosen for loading. The priority index of job i is given by Zi ˆ Ti :

…1†

This rule seeks to minimize the maximum ¯owtime and variance of ¯owtime. (3) EDD (earliest due-date): This rule is often used in industries for its simplicity of implementation in the shop¯oor. Since this rule performs well with respect to minimizing maximum tardi-

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

ness and variance of tardiness in the single-machine scheduling problem (Baker, 1974), we have chosen to include it in the present study. The priority index is given as follows: Zi ˆ Di :

…2†

(4) S/OPN (slack per remaining operation): The job with the least slack per remaining number of operations is chosen for loading. This rule is often used as bench-mark for evaluating the rules with respect to the tardiness-related measures of performance. The priority of job i is given as follows:  if si P 0; si =…o…i† ÿ j ‡ 1† …3† Zi ˆ si  …o…i† ÿ j ‡ 1† if si < 0; where the slack, si , is given by ! o…i† X tiq : si ˆ Di ÿ s ÿ

…4†

qˆj

(5) COVERT (cost over time): This rule has been extensively investigated in the literature (e.g. Russell et al., 1987) and is found to be e€ective when the tardiness-related measures of performance are of importance in jobshop scheduling. The COVERT rule computes a penalty function, ci , depending upon the slack of job i, si , and the sum of expected waiting times for the job's uncompleted operations, WTi , and hence determines the priority index, Zi , of the job. Mathematically, the priority index, Zi , given in the rule is expressed as follows: Zi ˆ ci =tij ; where 8 > < …WTi ÿ si †=WTi ci ˆ 0 > : 1

if 0 6 si < WTi ; if si P WTi ;

…5†

if si < 0:

The job with the largest Zi is chosen for loading and ties are broken by the smallest tij . The method of dynamic average waiting time (DAWT) has been used in the present study since this method is found to be quite e€ective (see Russell et al., 1987). (6) RR (rule by Raghu and Rajendran, 1993): This rule seeks to minimize both mean ¯owtime and mean tardiness of jobs. The simulation study of this rule has revealed that the rule outperforms

159

the SPT rule in minimizing mean ¯owtime in many cases, and that it outperforms the ATC rule by Vepsalainen and Morton (1987), and the CR + SPT and S/RPT + SPT rules by Anderson and Nyirenda (1990) in minimizing mean tardiness. The RR rule is based on the premise that in the computation of the priority index of a job, if proper weights are given to the components of process time and due-date of a job, depending on the utilization level of the machine, we could expect a good performance under a variety of shop¯oor conditions such as di€erent due-date settings and utilization levels. The RR rule also reckons the probable waiting time of the job at the machine of job's next operation. The priority index, Zi , of job i is given as follows: ÿ
Zi ˆ si  exp…ÿg†  tij =RPTi ‡ exp…g†  tij ‡ Wnxt ;

…6†

where g refers to the utilization level of the machine on which the job is to be loaded, RPTi denotes the sum of process times of uncompleted operations, including the current operation, on job i, and Wnxt indicates the probable waiting time of job i at the machine of its next operation. This waiting time is computed by taking into account the relative priority of job i, when the job enters the queue at the machine of its next operation. (7) SPT (shortest process time): This rule is perhaps the most commonly used rule for jobshop scheduling and is found to be very e€ective in minimizing mean ¯owtime and also in minimizing mean tardiness under highly loaded shop¯oor conditions (Conway, 1965; Rochette and Sadowski, 1976; Blackstone et al., 1982; Haupt, 1989). Moreover, this rule is the most e€ective in minimizing the proportion of tardy jobs in jobshops if the due-date setting is not too loose. (8) PT + WINQ (process time plus work-innext-queue): This rule has been recently proposed by Holthaus and Rajendran (1997) and is found to be the most e€ective in minimizing mean ¯owtime. The priority index by this rule is as given below: Zi ˆ tij ‡ Wi :

…7†

(9) PT + WINQ + AT (process time plus work-in-next-queue plus arrival time): This rule is

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C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

also proposed by Holthaus and Rajendran (1997), and is found to be quite e€ective in minimizing the maximum ¯owtime and variance of ¯owtime. The priority index is de®ned as follows: Zi ˆ tij ‡ Wi ‡ Ti :

…8†

(10) PT + WINQ + SL (process time plus work-in-next-queue plus negative slack): Again, developed by Holthaus and Rajendran (1997), this rule minimizes the maximum tardiness and variance of tardiness of jobs. The priority index is expressed as follows: Zi ˆ tij ‡ Wi ‡ minfsi ; 0g:

…9†

In all the above cases, except for the COVERT rule, the job with the least Zi is chosen for loading. We now present three new rules that seek to minimize the ¯owtime-related measures of performance. Proposed rule 1 ((PT + WINQ)/TIS rule): The motivation for this rule is the observation that the SPT and PT + WINQ rules tend to delay the completion times of jobs that have relatively large process times and hence these rules result in large values of maximum ¯owtime and variance of ¯owtime. The study by Holthaus and Rajendran (1997) introduces the PT + WINQ rule, that is found to be the most ecient rule for minimizing mean ¯owtime of jobs. Combining these two observations, and introducing a term corresponding to the resident time (or Time-In-Shop) of job i upto the current time instant, we present the new rule formally: ÿ
…10† Zi ˆ tij ‡ Wi =…s ÿ Ti †: The job with the least Zi is chosen for loading. This rule seeks to minimize not only the mean ¯owtime, but also the maximum ¯owtime and variance of ¯owtime because a job that has spent a longer resident time in the shop would be preferred for loading. Proposed rule 2 (PT/TIS rule): This rule is a simpli®ed version of rule 1. It makes use of the information only on the process time and resident time of job i. It is given as follows: Zi ˆ tij =…s ÿ Ti †:

…11†

The job with the least Zi is chosen for loading. Proposed rule 3 (AT-RPT rule): This rule makes use of the information on the resident time (or Time-In-Shop) and the total remaining process time of a job. It seeks to minimize the maximum ¯owtime and the variance of ¯owtime of jobs. The priority index Zi is given by Zi ˆ ÿ … s ÿ T i † ÿ

o…i† X

tiq :

…12†

qˆj

Since the term s is common to two jobs when their priority values are compared, the priority index gets reduced to the following: Zi ˆ T i ÿ

o…i† X tiq :

…13†

qˆj

The job with the least Zi is chosen for loading. We have now identi®ed a total of 10 existing rules, apart from the three new dispatching rules, for a performance analysis in ¯owshops and jobshops. 4. Experimental design for the simulation study A jobshop could be classi®ed into an open shop and a closed shop, depending upon the way in which jobs are routed in the shop. In a closed shop, the number of routeings available to a job is ®xed and an arriving job can follow one of the available routeings. In an open shop, there is no limitation on the routeing of a job and each job could have a di€erent routeing. In this paper we consider the open shop con®guration. The typical standard assumptions such as the processing of only one operation on a given machine at a given instant, no job preemption, an operation of any job to be performed after the completion of all its previous operations, machines being the only limiting resources, no machine breakdowns, no assembly of jobs, and no parallel machines (Haupt, 1989; Ramasesh, 1990) are also made in this study. As for the ®rst experimental evaluation in ¯owshops and jobshops, we assume the presence of 10 machines. In case of the ¯owshop, all jobs undergo processing on all 10 machines sequentially

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

starting from the ®rst machine. In case of the jobshop, a random permutation of 10 machines is chosen and the entering job undergoes processing in the chosen permutation order. It means that no machine will be revisited and all machines are visited. The reason for generating a random permutation of 10 machines and setting this permutation as the sequence of operations for a job in the jobshop is that we would like to have the same experimental settings in the jobshop and ¯owshop (where we have a unidirectional routeing of jobs over 10 machines). As for the second experimental evaluation in ¯owshops with missing operations and jobshops, we assume the presence of 10 machines. In case of the jobshop, the number of operations for an entering job is randomly sampled in the set f2; 3; . . . ; 9; 10g and the corresponding machine visitations are randomly generated with no machine being revisited. For example, if we have the number of operations for an entering job as 6, we sample six di€erent machines to be visited by the job, say, {6-1-7-10-8-3}. In case of the ¯owshop with missing operations, the machine visitation order will be {1-3-6-7-8-10}, in order to maintain the unidirectional routeing of jobs. In our opinion, such an approach of generating the machine visitations in ¯owshops and jobshops helps us to have the same experimental settings in both jobshops and ¯owshops, so that we could draw a meaningful comparison of the relative performance of dispatching rules in these two types of manufacturing systems. In all experimental setups, the process times are drawn from a uniform discrete distribution ranging from 1 to 49. The total work-content (TWK) method of duedate setting (Blackstone et al., 1982) is used in all experiments with the allowance factor, c ˆ 4, 6 and 8. The job arrivals are generated using an exponential distribution for inter-arrival times. Four machine-utilization levels Ug are tested in the experiments, viz. 80%, 85%, 90% and 95%. Thus, in all, there are three di€erent due-date settings and four di€erent utilization levels (i.e. four di€erent mean inter-arrival times), making a total number of 12 simulation experiment sets for every dispatching rule for ¯owshops and jobshops separately. It is a customary practice for researchers to conduct simulation experiments with di€erent pa-

161

rameter settings. We ®nd in the literature that the allowance factors in the range 3±8, and utilization levels in the range 80±95% are commonly considered (see Blackstone et al., 1982; and Haupt, 1989). While the number of machines in a jobshop could be theoretically anything, it is usually set in the range 6±12. This setting follows from the observations of Baker and Dzielinski (1960), and Nanot (1963) that the shop size is not a signi®cant factor in a€ecting the relative performance of rules and that a shop with about nine machines should adequately represent the complexity involved in large dynamic jobshop operations. In our study, each simulation experiment consists of 20 di€erent runs (or replications). In each run, the shop is continuously loaded with job-orders that are numbered on arrival. In order to ascertain when the system reaches a steady state, we have observed the shop parameters, such as utilization level of machines, mean ¯owtime of jobs, etc. It has been found that the shop reaches a steady state after the arrival of about 500 job-orders. Typically, the total sample size in simulation studies of jobshop scheduling is of the order of thousands of job completions (Conway et al., 1960; Blackstone et al., 1982). For a given total sample size, it is preferable to have a smaller number of replications and a larger run length, and the recommended number of replications is about 10 (Law and Kelton, 1984). Following these guidelines, we have ®xed the number of replications as 20, with the run length for every replication as 2000 completed job-orders. The statistical analysis of the experimental data with single-factor ANOVA with block design (Common Random Numbers for one block) and Duncan's Multiple Range Test (Montgomery, 1991; Lorenzen and Anderson, 1993) has shown that this sample size yields a variance which results in a Type I error of at most 1%. As for the computation of statistics from a given replication, we have collected data from orders numbering from 501 to 2500, and the shop is further loaded with jobs, until the completion of these 2000 numbered job-orders. This helps in overcoming the problem of `censored data' (Conway, 1965). The simulation program has been written in C++ and implemented on a Pentium PC.

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C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

5. Results and discussion The performances of the 13 rules under study are evaluated with respect to mean ¯owtime of ¯owtime …Fmax † and variance jobs …F †, maximum ÿ
of ¯owtime r2F , percentage of tardy jobs (%T), tardiness …Tmax † and mean tardiness …T †, maximum ÿ
variance of tardiness r2T . The results of the simulation study for some sets of parameter values are presented in Tables 1±4. These results are obtained by taking the mean over 20 replications. The complete set of results (corresponding to four utilization levels and three allowance factor values) could not be presented for want of space and also due to the reason that the other results are not substantially di€erent from what are currently given. We now discuss some typical results of the experimental analysis for the conventional ¯owshops and jobshops, followed by the analysis for ¯owshops with missing operations and jobshops. 5.1. Flowshops and jobshops Tables 1 and 2 present the values of F ; Fmax ; r2F ; %T, T ; Tmax and r2T yielded by the 13 dispatching rules in conventional ¯owshops (Table 1) and in jobshops (Table 2) using the utilization levels Ug ˆ 80% and Ug ˆ 95% and the allowance factors c ˆ 4 and c ˆ 6. For each combination of utilization level, allowance factor and performance measure, those mean values which are signi®cantly smaller than the other mean values are marked with an asterisk. 5.1.1. Mean ¯owtime The rules SPT and PT + WINQ reduce to be the same in the case of ¯owshops since the WINQ component is the same for all jobs. For ¯owshops, the SPT and PT + WINQ rules emerge to be the best, while the RR rule performs very well with utilization levels up to 90%. When the ¯owshop is heavily loaded at 95% utilization level, the performance of the RR rule improves as the due-date setting becomes loose. At tight due-date settings, the slack component of the RR rule serves to hasten the tardy jobs and in the process, the performance of the RR rule is very good with respect

to minimizing the mean tardiness at the expense of an enhanced mean ¯owtime. Moreover, the lookahead component, Wnxt , of the RR rule does not seem to be quite e€ective in ¯owshops since all jobs have the same operation as the next one. As for jobshops, the performance of the RR rule is better at high utilization levels than at low utilization levels due to the relative e€ectiveness of its look-ahead component, Wnxt , whereas the performances of the PT + WINQ and SPT rules are consistent and good in the same order. The difference in the performance of the RR rule in ¯owshops and jobshops is due to the presence of the look-ahead component, Wnxt , in it and this component is more meaningful in the case of jobshops (due to non-unidirectional routeing of jobs) than in ¯owshops. It is also known that the rules making use of the information on process and waiting times of jobs perform quite well in jobshops, and therefore we observe the good performances of the PT + WINQ, SPT and RR rules. 5.1.2. Maximum ¯owtime and variance of ¯owtime For minimizing maximum ¯owtime the performances of the FIFO, AT, AT-RPT, PT/TIS, (PT + WINQ)/TIS and PT + WINQ + AT rules are almost comparable in the case of ¯owshops. The reason for the similarity between the performances of the PT + WINQ + AT and AT-RPT rules on the one hand, and those of the FIFO and AT rules on the other hand is that the WINQ and RPT components in the former two rules do not play a dominant role in the priority index computations in ¯owshops since all jobs have the same unidirectional routeing and the expected values of the total remaining process times of di€erent jobs will not substantially di€er. However, for jobshops, the AT-RPT and PT + WINQ + AT rules emerge to be the best for most of the cases. It is noteworthy that the AT rule emerges to be better than the FIFO rule. The reason is that the AT rule hastens the job that has a larger resident time in the entire shop¯oor rather than myopically considering resident time in the current queue. With respect to the minimization of variance of ¯owtime the rules PT/TIS and (PT + WINQ)/TIS are very e€ective in the case of ¯owshops, whereas for jobshops, the AT-RPT and PT + WINQ + AT

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

163

Table 1 Performance of rules in conventional ¯owshops Ug (%)

c

Rule

F

Fmax

rF 2

%T

80

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

636.5 636.2 645.4 607.5 613.8 528.3 524.2  520.3  520.3 540.6 553.4 606.1 524.3



1237.7 1237.7  1256.4 1535.9 1533.6 1549.0 1525.6 3665.0 3665.0  1275.2  1236.4  1199.8 1612.3

39344.2 39260.9 40115.0 59307.2 58221.3 49988.6 41903.0 87233.1 87233.1  30107.3  28624.5 35243.0 54542.6

9.1 9.0 10.2 4.2 4.7 2.6  1.6 4.4 4.4 2.5 3.0 6.8 6.7

13.7 13.6 15.8 4.5 5.3  1.0  1.4 21.1 21.1 2.8 3.2 9.6 6.4

562.7 560.8 583.9  190.6  204.9  198.4  227.3 2610.4 2610.4 355.1 374.9 492.8  243.4

3207.4 3184.8 3780.4 742.2 920.4  161.7  229.1 23070.7 23070.7 568.6 589.8 2184.4 774.4

80

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

636.5 636.2 645.4 602.1 604.8  520.9 524.2  520.3  520.3 540.6 553.4 606.1  521.3



39344.2 39260.9 40115.0 76003.7 73666.9 66332.9 47365.7 87233.1 87233.1  30107.3  28624.5 35243.0 69707.4

0.7 0.7 0.8  0.0  0.0  0.2  0.0 1.3 1.3  0.1  0.1 0.4 1.7

0.8 0.8 1.1  0.0  0.0  0.1  0.0 7.8 7.8  0.1  0.1 0.5 1.6

295.6 294.6 324.4  0.0  0.0 26.5  0.0 2102.3 2102.3 50.5 90.9 237.2 204.2

210.5 208.6 278.9  0.0  0.0  1.1  0.0 9892.0 9892.0  8.1  15.1 124.5 198.8

95

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

1998.8 1998.8 2006.1 1977.8 1986.6 1425.6 1628.3  1280.7  1280.7 1401.3 1503.3 1926.4 1715.2



2974.4 2975.1  2976.5 3292.2 3293.4 21425.7 4482.8 35988.8 35988.8  2954.0  2880.8  2912.9  3024.0

243347.6 243161.7  242573.2 280154.9 280116.3 2512371.2 442143.8 6803343.8 6803343.8  229385.5  196695.7  243068.8 293585.2

94.2 94.2 94.5 94.7 94.8 76.2 76.2  25.5  25.5 74.3 80.7 92.6 87.5

1008.3 1008.2 1014.6 984.5 993.2  457.3 687.5 573.4 573.4  459.9 544.8 939.5 759.1

2309.3 2310.8 2332.2 1950.8 1969.8 20311.0 3365.9 34922.6 34922.6 1923.2 1883.0 2217.1  1671.0

220889.9 220788.4 224693.9 200524.4 205411.4 2341636.3 270522.1 6029416.4 6029416.4  127902.2  124081.1 208227.4  155500.9

95

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

1998.8 1998.8 2006.1 1953.6 1964.7 1471.3 1370.0  1280.7  1280.7 1401.3 1503.3 1926.4 1472.4





75.0 75.0 75.2 72.9 73.5 44.5 38.2  15.3  15.3 40.3 48.8 71.4 59.5

586.9 586.7 592.3 536.4 545.6  186.8  168.9 469.9 469.9  179.9 227.6 532.1 264.4

2034.5 2036.0 2055.9 1430.3 1443.8 12133.4 1326.5 34389.4 34389.4 1522.1 1513.4 1945.0  934.5

155609.3 155530.6 157960.6 117890.6 119734.8 694286.3  59862.6 5636553.9 5636553.9  61857.6  65108.1 142990.2  58037.4



1237.7 1237.7  1256.4 1741.3 1718.8 2126.3 1647.5 3665.0 3665.0  1275.2  1236.4  1199.8 2220.2 



2974.4 2975.1  2976.5 3467.9 3466.4 13711.1 3340.1 35988.8 35988.8  2954.0  2880.8  2912.9  3005.0 

 

243347.6 243161.7  242573.2 321699.1 318035.8 974967.2 316480.8 6803343.8 6803343.8  229385.5  196695.7  243068.8 386009.1 

T

Tmax

rT 2

164

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

Table 2 Performance of rules in jobshops Ug (%)

c

Rule

F

Fmax

rF 2

%T

80

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

839.1 811.3 809.9 785.3 780.0 651.4 642.6  633.6  630.2 684.1 670.9 761.3  634.1

1962.0 1536.8  1422.7 1825.0 1606.8 1681.0 1592.3 3777.6 2703.2 1546.5 1762.0 1479.2 1669.4

73655.1 46482.4  38070.2 71070.5 62078.1 65733.0 54204.1 122044.2 74648.3 42018.0 43833.2 41168.7 58716.2

27.1 22.9 22.9 15.0 5.7  3.2  2.2 8.6 8.8 7.8 9.0 16.7 11.2

62.4 39.4 38.9 21.3 6.1  1.9  1.8 35.1 23.7 9.9 14.8 25.3 13.4

1068.2 731.0 733.6 485.2  305.1  318.4  272.8 2626.0 1604.5 510.4 801.9 655.9  373.4

17985.2 8533.8 8263.7 4065.5 1229.1  449.0  332.7 31329.9 12837.2 2091.4 4205.1 5385.7 1927.2

80

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

839.1 811.3 809.9 770.3 812.0  635.5 684.7  633.6  630.2 684.1 670.9 761.3  632.2

1962.0 1536.8  1422.7 1949.2 2039.2 2162.0 1930.2 3777.6 2703.2 1546.5 1762.0 1479.2 2052.1

73655.1 46482.4  38070.2 91904.5 121691.2 94553.8 88172.3 122044.2 74648.3 42018.0 43833.2 41168.7 70490.1

4.1 2.0 2.0  0.1  0.0  0.1  0.0 2.3 1.6 0.3 0.7 1.2 1.9

7.7 2.7 2.8  0.1  0.0  0.1  0.0 10.9 4.6 0.2 1.1 1.3 2.1

698.6 452.5 458.8 29.2  0.4  4.8  2.4 2086.1 1104.4 150.9 435.0 365.9 262.4

2580.5 642.8 684.9  4.2  0.0  0.0  0.1 11901.9 2952.0 47.3 333.9 300.8 339.6

95

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

2418.2 1958.0 1956.8 1936.0 2079.3 1530.5  1347.7 1466.8 1428.8 1556.8 1513.3 1785.8 1493.2

4452.4 2978.4 2853.4 3248.3 5420.6 21714.8 3580.2 25875.4 12306.1 3091.0 3937.2 2862.1  2714.4

389561.2 159891.9  144409.3 196001.2 690911.9 2216269.4 239339.2 4817213.2 1320991.3 196891.1 261927.1  146081.1 161474.1

98.1 98.0 98.1 98.3 97.3 80.5 74.7  38.6 57.8 88.8 81.5 95.7 88.5

1419.4 958.6 957.3 936.1 1079.0 541.0  386.0 660.4 543.8 572.3 547.2 790.1 522.1

3528.4 2187.7 2190.1 1928.0 4340.8 20645.7 2274.2 24704.8 11183.8 2015.7 2955.4 2049.3  1485.4

383376.4 168511.5 162342.9 146660.8 641894.2 2034123.3 123806.7 4079225.6 1006503.7 134509.2 199304.8 144365.1  83276.2

95

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

2418.2 1958.0 1956.8 1925.9 1829.8 1514.5  1296.7 1466.8 1428.8 1556.8 1513.3 1785.8 1416.2

4452.4 2978.4  2853.4 3496.1 4420.8 11302.6 3063.2 25875.4 12306.1 3091.0 3937.2  2862.1  2862.6

389561.2 159891.9  144409.3 243087.0 335557.8 643845.8 194157.2 4817213.2 1320991.3 196891.1 261927.1  146081.1 239849.4

88.0 78.4 78.5 78.8 61.9 36.3  24.2  22.4 30.4 50.9 45.1 68.9 54.3

951.1 512.2 512.1 470.1 376.8 158.4  62.7 504.8 333.5 223.6 241.4 376.9 175.2

3123.9 1908.8 1933.2 1462.7 2676.1 9629.0  1018.1 24124.1 10623.6 1570.1 2538.2 1751.3  889.6

314105.9 121015.5 118370.9 90536.6 195127.7 430994.7  19207.9 3705205.9 803002.8 69300.0 119471.9 92919.0  25100.3

T

Tmax

rT 2

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

rules have the best performance. With respect to all ¯owtime-related performance measures the PT/ TIS and (PT + WINQ)/TIS rules seem to o€er good compromise solutions between the SPT, PT + WINQ and AT rules. The reason is that we reckon the process time as well as the resident time of jobs in the PT/TIS and (PT + WINQ)/TIS rules. Of course, the ®nal choice between the PT/ TIS, (PT + WINQ)/TIS, SPT, PT + WINQ, PT + WINQ + AT and AT rules depends on the relative importance that the decision maker assigns to the measures of performance such as mean ¯owtime, maximum ¯owtime and variance of ¯owtime. 5.1.3. Percentage of tardy jobs Up to 85% utilization levels, the RR rule fares better than the SPT rule. At 90% and 95% utilization levels, the SPT rule emerges better than the RR rule, while the RR rule fares well under loose due-date settings. The reason is that at highly loaded conditions with tight due-date settings, the slack component of the RR rule is dominant and it loses its contribution as the allowance factor increases. At this stage, the look-ahead component of the RR rule serves to enhance the throughput of jobs and consequently renders a less number of jobs tardy. It is also interesting to observe that the RR rule is not e€ective in ¯owshops as much as in jobshops. The reason is due to the relative e€ectiveness of the look-ahead component, Wnxt , in jobshops (due to random routeing of jobs) as against the case in ¯owshops. 5.1.4. Mean tardiness The RR rule continues to be the best (or not signi®cantly worse than the best rule) for this objective except for one case where we encounter the ¯owshop with a high utilization level of 95% and a tight due-date setting of c ˆ 4. As discussed earlier, the look-ahead component, Wnxt , in the RR rule is not so e€ective in ¯owshops since all jobs have the same unidirectional routeing. The relative performance of the COVERT rule seems to be better in ¯owshops than in jobshops. The reason is due to a better estimation of waiting time of jobs on machines, when there is a unidirectional routeing of jobs in ¯owshops.

165

5.1.5. Maximum tardiness and variance of tardiness The RR rule seems to be quite e€ective for many cases of jobshops and ¯owshops at utilization levels of 80% and 85%. However, at highly loaded conditions (90% and 95% utilization levels) and tight due-date settings (c ˆ 4), we ®nd that the PT + WINQ + SL rule emerges to be better than the RR rule for both jobshops and ¯owshops. Also at 95% utilization level and c ˆ 6, the PT + WINQ + SL rule is a very e€ective rule for minimizing maximum tardiness and variance of tardiness. The reason is that many jobs are likely to be more tardy under high utilization levels than under low utilization levels and hence, the component of negative slack, SL, becomes dominant and it serves to enhance the performance of the PT + WINQ + SL rule under highly loaded conditions. In the case of the RR rule, we reckon the slack, irrespective of its positive or negative value, and hence, the rule performs well under di€erent conditions.

5.2. Flowshops with missing operations and jobshops Tables 3 and 4 present the values of F ; Fmax ; r2F ; %T, T ; Tmax and r2T yielded by the 13 dispatching rules in ¯owshops with missing operations on jobs (Table 3) and in jobshops (Table 4) using the utilization levels Ug ˆ 80% and Ug ˆ 95% and the allowance factors c ˆ 4 and c ˆ 6. For each combination of utilization level, allowance factor and objective, those mean values which are signi®cantly better than the other mean values are marked again with an asterisk. 5.2.1. Mean ¯owtime For both ¯owshops with missing operations on jobs and jobshops, the PT + WINQ rule emerges to the best (or not signi®cantly worse than the best performing rule) in all cases except for one case where we encounter the jobshop with a high utilization level of 95% and a due-date setting of c ˆ 6. It is also interesting to note that at higher utilization levels (85%, 90%, 95%) the PT + WINQ rule performs signi®cantly better than the SPT rule.

166

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

Table 3 Performance of rules in ¯owshops with missing operations on jobs Ug (%)

c

Rule

F

80

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

471.5 470.2 479.3 460.3 459.4 390.5 368.3  362.3  361.4 393.2 391.8 433.0 369.2

80

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

471.5 470.2 479.3 460.3 450.9 367.8 367.0  362.3  361.4 393.2 391.8 433.0  363.7

95

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

95

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

Fmax

rF 2

%T

T

Tmax

rT 2

1480.3 1058.5  1034.2 1633.5 1559.2 1735.7 1568.9 3409.2 3039.8 1146.0 1162.3  1015.5 1526.0

72195.4 40843.9  37203.4 107960.4 87709.4 69805.2 62956.3 92662.4 83326.4  36325.2  39426.8  37437.6 65343.7

26.2 30.9 34.0 17.0 15.8 7.9  5.2 8.7 9.2 17.9 16.3 24.9 13.9

38.7 55.3 65.0 21.1 19.2  5.4  4.2 29.5 25.7 21.6 20.3 38.2 12.2

751.5 757.4 781.8  386.4 503.1 721.6  336.1 2530.3 2070.2 654.6 704.4 693.1  303.4

7788.5 11221.0 13117.0 3574.2 3427.7 1848.3  651.8 24075.2 17638.4 4053.3 4134.8 7427.5 1362.1

1480.3 1058.5  1034.2 1983.9 1860.8 1961.3 1685.1 3409.2 3039.8 1146.0 1162.3  1015.5 2051.2

72195.4 40843.9  37203.4 145356.2 107613.3 75223.7 70812.0 92662.4 83326.4  36325.2  39426.8  37437.6 74165.5

6.9 13.3 15.6  0.9  0.7  0.5  0.2 3.2 3.2 5.7 4.9 9.6 4.0

8.1 22.1 27.8  0.6  0.4  0.1  0.1 13.1 10.3 5.9 5.8 13.9 3.2

526.8 692.7 726.5  100.1  69.4  45.3  41.0 2148.4 1671.0 513.1 567.4 613.6 261.7

1664.4 5318.0 6829.6  100.0  58.1  2.7  3.3 11940.2 7616.1 1167.3 1310.8 3132.1 373.7

1353.1 1358.2 1358.8 1328.6 1339.7 1002.7 1070.5 928.0  857.2 1011.2 992.0 1204.9 1091.4

3327.8 2237.9 2219.7 2877.1 3889.0 23227.5 5782.9 29870.2 18450.7 2545.8 3310.6  2142.6 2603.2

535203.7 192813.3  181969.7 343178.6 313302.6 3001687.6 561990.1 4795080.9 2039522.5 216605.3 264667.8 199307.3 313607.7

92.1 93.7 93.5 93.1 94.1 70.4 75.0  28.4 35.8 83.5 78.7 89.4 84.2

763.0 768.8 770.5 737.1 745.7  419.6 502.6 485.8  396.0  438.4  424.5 622.2 518.9

2515.4 1971.4 1993.0 1627.6 3498.5 22313.2 4842.6 29057.6 17594.2 2025.0 2835.4 1814.3  1380.7

272115.0 153425.6 153907.6 139014.9 224101.2 2693328.9 259677.4 4208952.6 1635856.6 116667.8 142564.1 132322.2  102675.8

1353.1 1358.2 1358.8 1311.9 1316.0 1034.4 955.8 928.0  857.2 1011.2 992.0 1204.9 996.6

3327.8 2237.9 2219.7 3199.8 3473.8 16793.2 3550.2 29870.2 18450.7 2545.8 3310.6  2142.6 2848.8

535203.7 192813.3  181969.7 459776.6 381495.2 1656189.5 425115.3 4795080.9 2039522.5 216605.3 264667.8 199307.3 394469.1

76.5 78.5 78.4 76.4 76.3 45.5 47.1  17.6 21.7 57.8 52.6 70.4 60.7

506.6 524.0 527.7 457.0 450.1 221.6  196.4 409.5 308.6 245.1 240.4 397.5 244.0

2189.4 1910.3 1931.6 1280.2 2530.7 15417.6 1922.2 28651.3 17190.5 1850.8 2684.2 1742.8  981.2

175121.7 136154.3 138920.9 87282.1 120969.0 1266981.0 63602.1 3964078.8 1470441.4 79426.1 103412.0 107777.3  45103.1









C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

167

Table 4 Performance of rules in jobshops with missing operations on jobs Ug (%)

c

Rule

F

Fmax

rF 2

%T

80

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

516.2 501.7 502.1 488.3 474.0 411.0 390.9  386.9  386.7 426.3 415.2 463.5 391.6

1712.3 1168.3  1042.1 1702.2 1485.8 1676.8 1514.9 3570.3 2396.2 1243.2 1457.4 1184.2 1510.3

86475.9 45856.2  35690.3 115346.1 85788.5 72698.8 68808.7 106870.8 77373.2 42144.1 48029.0 44190.3 66546.7

32.4 33.8 35.9 20.4 10.6 6.1  4.8 9.9 11.8 20.3 17.6 26.9 15.7

53.4 54.3 64.5 24.2 8.1  3.3  3.3 32.8 25.3 23.3 23.3 38.1 15.4

896.2 665.4 733.9 497.4  329.9 560.5  342.4 2694.4 1606.3 621.1 786.6 621.2  383.2

11121.4 9343.8 11770.1 3886.2  1106.5  828.1  487.0 28315.4 12187.3 3953.7 5267.5 6668.2 1878.7

80

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

516.2 501.7 502.1 482.9 484.9 391.3 408.4  386.9  386.7 426.3 415.2 463.5  387.5

1712.3 1168.3  1042.1 1994.1 1927.9 2023.6 1850.5 3570.3 2396.2 1243.2 1457.4 1184.2 1954.4

86475.9 45856.2  35690.3 149206.5 124193.8 83858.1 91365.4 106870.8 77373.2 42144.1 48029.0 44190.3 73236.6

9.2 12.8 15.6 0.7  0.1  0.3  0.1 3.4 3.6 6.0 5.2 9.4 4.1

11.9 19.0 26.1 0.3  0.0  0.0  0.0 14.6 8.3 5.7 6.6 12.4 3.6

616.2 595.0 691.5 139.1  17.8  25.0  23.4 2309.8 1324.7 483.7 646.0 540.7 283.4

2620.8 4046.2 5988.6 38.3  0.3  0.6  0.6 14520.8 4531.8 1007.7 1639.8 2527.3 478.9

95

4

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

1434.2 1246.2 1250.8 1214.3 1297.8 1003.7 917.4 953.3  886.1 1015.0 954.9 1113.0 982.9

3994.3 2251.8  2105.1 2860.2 3901.0 20655.2 4364.4 25105.7 12559.3 2587.2 3731.0 2366.7 2557.3

629934.0 171821.8  145608.1 329951.6 400364.7 2387068.7 413092.3 3765178.7 1140129.0 211853.5 336688.3 188748.5 278247.4

94.0 94.7 94.4 93.5 95.0 69.5 76.1  32.5 48.4 87.4 75.4 90.3 83.9

841.3 653.9 659.9 621.1 701.5 419.0  342.7 490.1 381.9 432.7 391.3 526.7 408.7

3108.9 1738.1 1804.7 1606.0 2929.8 19734.2 3246.2 24244.4 11758.0 1908.2 3082.4 1792.3  1441.2

342886.4 118542.4 120134.5 117838.4 274367.4 2068493.3 160095.9 3194334.2 791364.9 109687.1 188180.0 109092.3  78475.8

95

6

FIFO AT AT-RPT EDD S/OPN COVERT RR SPT PT+WINQ PT/TIS (PT+WINQ)/TIS PT+WINQ+AT PT+WINQ+SL

1434.2 1246.2 1250.8 1201.2 1183.1 1003.4  856.0 953.3 886.1 1015.0 954.9 1113.0 924.1

3994.3 2251.8  2105.1 3202.0 3582.1 13819.1 3296.0 25105.7 12559.3 2587.2 3731.0 2366.7 2815.8

629934.0 171821.8  145608.1 450885.7 389178.7 1118868.8 339012.0 3765178.7 1140129.0 211853.5 336688.3 188748.5 363602.4

81.2 76.7 76.2 74.9 71.0 39.7 36.7  19.6 28.5 59.0 47.3 66.7 55.8

573.9 407.6 419.4 345.8 316.7 177.6  94.5 402.3 261.7 229.7 221.1 304.4 175.8

2730.9 1673.5 1758.8 1295.8 2090.7 12405.3 1423.5 23829.0 11368.5 1706.2 2851.1 1682.7  1065.2

229657.9 98775.1 105183.9 72094.6 117733.2 760191.1  33234.3 2959005.8 662724.9 72438.7 133587.1 81594.3  33268.6

T

Tmax

rT 2

168

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

5.2.2. Maximum ¯owtime and variance of ¯owtime In the case of ¯owshops, for minimizing maximum ¯owtime the PT + WINQ + AT rule has the best performance, whereas for minimizing variance of ¯owtime the AT-RPT rule emerges to the best (or not signi®cantly worse than the best performing rule) in all cases. For jobshops, the AT-RPT rule has the best performance with respect to both objectives. Not only in jobshops, but also in ¯owshops with missing operations the AT rule emerges to be better than the FIFO rule. Once again, as in the earlier study of conventional ¯owshops and jobshops, the PT/TIS and (PT + WINQ)/TIS rules perform quite well with respect to all ¯owtime-related measures of performance, and the choice between the PT/TIS, (PT + WINQ)/TIS, SPT, PT + WINQ, PT + WINQ + AT, AT and ATRPT rules depends on the preference structure used by the decision maker. 5.2.3. Percentage of tardy jobs Just as in the comparative study of jobshops and conventional ¯owshops, up to 85% utilization levels, the RR rule fares better (or not signi®cantly worse) than the SPT rule. For 90% and 95% utilization levels with tight due-date settings, the SPT rule emerges better than the RR rule. These observations are similar to those in the case of the comparative analysis of conventional ¯owshops and jobshops, and the reasons for this trend are also the same as discussed earlier. 5.2.4. Mean tardiness Similar to the results observed earlier in the study on jobshops and conventional ¯owshops, the RR rule continues to be the best (or not signi®cantly worse than the best performing rule) for this objective except for one case where we encounter the ¯owshop with a high utilization level of 95% and a tight due-date setting of c ˆ 4. As discussed earlier, the look-ahead component in the RR rule seems to be not so e€ective in ¯owshops with missing operations as much as in jobshops. The performance of the COVERT rule seems to be better in ¯owshops than in jobshops. The reason could be again attributed to a better estimation of waiting time of jobs on machines, when there is a unidirectional routeing of jobs in ¯owshops.

5.2.5. Maximum tardiness and variance of tardiness Once again, the RR rule seems to be quite effective for many cases of jobshops and ¯owshops under low utilization levels. However, at highly loaded conditions (90% and 95% utilization levels) and tight due-date settings (c ˆ 4), we ®nd that the PT + WINQ + SL rule emerges to be better than the RR rule (or not signi®cantly worse than the RR rule) for both jobshops and ¯owshops. Also at 95% utilization level and c ˆ 6, the PT + WINQ + SL rule continues to fare better. These observations are the same as in the case of the comparative analysis of conventional ¯owshops and jobshops, and the reasons for this trend are also the same as discussed earlier. 5.3. Some general observations on dispatching rules We wish to make some observations at this stage. The objective of minimizing mean ¯owtime leads to the minimization of mean waiting time of jobs, and hence to the minimization of mean inprocess inventory. The objective of minimizing mean tardiness leads to the minimization of mean customer dissatisfaction level. Similarly, other measures of performance are related to in-process inventory and customer dissatisfaction. While computing the mean or maximum or variance of ¯owtime/tardiness, we implicitly assume that all jobs are equal in importance or that they have the same holding/tardiness costs. In other words, we assume that the holding/tardiness costs are directly proportional to the ¯owtime/tardiness, respectively. It can be seen that the chosen measures of performance are surrogate measures of the minimization of cost-based parameters. As for minimizing a complex cost function consisting of inprocess inventory and tardiness (or penalty) costs, we propose that a decision maker/manager could make use of an additive function with appropriate relative weights to the costs, and use the cost function as the basis for choosing an appropriate dispatching rule. It is to be noted that the relative weights for di€erent costs would depend on the utility function/preference structure that the decision maker employs, and hence the choice of a dispatching rule would also vary accordingly. We

C. Rajendran, O. Holthaus / European Journal of Operational Research 116 (1999) 156±170

would also like to mention here that Holthaus and Rajendran (1997) have addressed the development of new dispatching rules that seek to minimize the mean and maximum ¯owtime of jobs, and maximum tardiness of jobs. These new rules and the best existing rules have been evaluated through a simulation study of a jobshop. The present study addresses the development of a new dispatching rule that seeks to minimize the maximum ¯owtime of jobs, apart from the development of two new dispatching rules that seek to simultaneously minimize the mean and maximum ¯owtime of jobs. In addition, we have now addressed the evaluation of dispatching rules in dynamic ¯owshops, which is perhaps the ®rst of its kind to be reported in the literature, with a variety of measures of performance and experimental conditions. A comparative analysis of the relative performance of a number of dispatching rules in dynamic ¯owshops and jobshops has also been reported, which is once again the ®rst of its kind. We have also considered the ¯owshop with missing operations, and carried out the performance analysis of dispatching rules in this type of ¯owshop. The present study is therefore unique in view of these considerations and it contributes to the existing body of knowledge in the ®eld of jobshop and ¯owshop scheduling. 6. Conclusion While a lot of research work has been carried out on the study of dispatching rules in jobshop scheduling, there have been relatively few attempts to study the relative performance of dispatching rules in dynamic ¯owshops. Furthermore, a comparative study on the performance of dispatching rules in dynamic ¯owshops and jobshops is interesting and revealing because the job routeings in¯uence the relative performance of dispatching rules. This study has been one such attempt to compare the relative performances of dispatching rules in two sets of manufacturing systems: dynamic ¯owshops and jobshops, and dynamic ¯owshops with missing operations and jobshops. Three new dispatching rules have been proposed and a total of 13 dispatching rules have been

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considered for a performance analysis with respect to the objectives of minimizing mean ¯owtime, maximum ¯owtime, variance of ¯owtime, proportion of tardy jobs, mean tardiness, maximum tardiness and variance of tardiness. The ®rst simulation study has been carried out in ¯owshops with jobs undergoing processing on all machines sequentially and in jobshops with random routeing of jobs. The results of the study have revealed some interesting observations on the performance of the dispatching rules in these two types of manufacturing systems. The second simulation study has considered the ¯owshops with missing operations on jobs and jobshops with random routeing of jobs. It has been observed that the relative performance of dispatching rules is being in¯uenced by the routeing of jobs and shop¯oor con®gurations. The performance of various rules with respect to every measure of performance has been discussed in detail. Overall, a proposed rule, AT-RPT rule, emerges to be very e€ective in minimizing the maximum ¯owtime and variance of ¯owtime of jobs. While the PT + WINQ rule performs very well in minimizing mean ¯owtime of jobs, the PT + WINQ + SL rule performs well in minimizing the maximum tardiness and variance of tardiness of jobs. The RR rule appears to be a good rule with respect to the tardiness-related performance measures in many cases of manufacturing systems under study. This rule reckons the process time, slack and waiting time at the next operation of the job, and also the shop load conditions. Hence the rule adapts itself e€ectively to a number of di€erent shop¯oor conditions. Likewise, we ®nd that two proposed rules, viz. the (PT + WINQ)/TIS and PT/TIS rules, seem to o€er good compromise solutions with respect to the ¯owtime-related performance measures in most cases of manufacturing systems under study. These rules consider not only the process time, but also the resident time of a job, thereby enhancing the performance with respect to the mean and maximum ¯owtime. As mentioned earlier, the choice of a dispatching rule ®nally depends on the relative importance that the decision maker assigns to each measure of performance. The results also indicate that the rules that include information about process time, total

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work-content of jobs in the queue of next operation of a job, arrival time and due-date fare very well in simultaneously minimizing many measures of performance in jobshops as well as in ¯owshops.

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±1997. The authors are thankful to the three referees for their constructive comments and suggestions to improve the earlier version of the paper.

References Anderson, E.J., Nyirenda, J.C., 1990. Two new rules to minimize tardiness in a jobshop. International Journal of Production Research 28, 2277±2292. Baker, K.R., 1974. Introduction to Sequencing and Scheduling. Wiley, New York. Baker, C.T., Dzielinski, B.P., 1960. Simulation of a simpli®ed job shop. Management Science 6, 311±323. Blackstone, J.H., Phillips, D.T., Hogg, G.L., 1982. A state-ofthe-art survey of dispatching rules for manufacturing job shop operations. International Journal of Production Research 20, 27±45. Campbell, H.G., Dudek, R.A., Smith, M.L., 1970. A heuristic algorithm for the n-job, m-machine sequencing problem. Management Science 16, B630±B637. Conway, R.W., Johnson, B.M., Maxwell, W.L., 1960. An experimental investigation of priority dispatching. Journal of Industrial Engineering 11, 221±230. Conway, R.W., 1965. Priority dispatching and job lateness in a job shop. Journal of Industrial Engineering 16, 228± 237. Dannenbring, D.G., 1977. An evaluation of ¯ow-shop sequencing heuristics. Management Science 23, 1174±1182. Day, J.E., Hottenstein, M.P., 1970. Review of sequencing research. Naval Research Logistics Quarterly 17, 11±39. French, S., 1982. Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop. Ellis Horwood, Chichester, UK. Haupt, R., 1989. A survey of priority rule-based scheduling. OR Spektrum 11, 3±16. Ho, J.C., 1995. Flowshop sequencing with mean ¯owtime objective. European Journal of Operational Research 81, 571±578.

Holthaus, O., Rajendran, C., 1997. Ecient dispatching rules for scheduling in a job shop. International Journal of Production Economics 48, 87±105. Hunsucker, J.L., Shah, J.R., 1992. Performance of priority rules in a due date ¯ow shop. OMEGA 20, 73±89. Hunsucker, J.L., Shah, J.R., 1994. Comparative performance analysis of priority rules in a constrained ¯ow shop with multiple processors environment. European Journal of Operational Research 72, 102±114. Ignall, E., Schrage. L., 1965. Application of the branch-andbound technique to some ¯owshop scheduling problems. Operations Research 13, 400±412. Ishibuchi, H., Misaki, S., Tanaka, H., 1995. Modi®ed simulated annealing algorithms for the ¯ow shop sequencing problems. European Journal of Operational Research 81, 388± 398. Law, A.M., Kelton, W.D., 1984. Con®dence intervals for steady-state simulations: I. A survey of ®xed sample size procedures. Operations Research 32, 1221±1239. Lorenzen, T.J., Anderson, V.L., 1993. Design of Experiments: A No-Name Approach, Marcel Dekker, New York. Montgomery, D.C., 1991. Design and Analysis of Experiments, 3rd ed. Wiley, New York. Nawaz, M., Enscore, Jr., E.E., Ham, I., 1983. A heuristic algorithm for the m-machine, n-job ¯owshop sequencing problem. OMEGA 11, 91±95. Nanot, Y.R., 1963. An experimental investigation and comparative evaluation of priority disciplines in job-shop-like queueing networks. Research Report No. 87. Management Sciences Research Project, UCLA. Pinedo, M., 1995. Scheduling Theory, Algorithms and Systems, Prentice Hall, Englewood Cli€s, NJ. Raghu, T.S., Rajendran, C., 1993. An ecient dynamic dispatching rule for scheduling in a job shop. International Journal of Production Economics 32, 301±313. Rajendran, C., 1994. A heuristic for scheduling in ¯owshop and ¯owline-based manufacturing cell with multi-criteria. International Journal of Production Research 32, 2541±2558. Ramasesh, R., 1990. Dynamic job shop scheduling: A survey of simulation research. OMEGA 18, 43±57. Rochette, R., Sadowski, R.P., 1976. A statistical comparison of the performance of simple dispatching rules for a particular set of job shops. International Journal of Production Research 14, 63±75. Russell, R.S., Dar-el, E.M., Taylor III, B.W., 1987. A comparative analysis of the COVERT job sequencing rule using various shop performance measures. International Journal of Production Research 25, 1523±1540. Scudder, G.D., Ho€mann, T.R., 1987. The use of cost-based priorities in random and ¯ow shops. Journal of Operations Management 7, 217±232. Smunt, T.L., Buss, A.H., Kropp, D.H., 1996. Lot splitting in stochastic ¯ow shop and job shop environments. Decision Sciences 27, 215±238. Vepsalainen, A.P.J., Morton, T.E., 1987. Priority rules for job shops with weighted tardiness costs. Management Science 33, 1035±1047.