Bicriterion scheduling problem in a job shop with parallel processors

International Journal of Production Economics, 25 ( 1991 ) 95-101 95 Elsevier

Bicriterion scheduling problem in a job shop with parallel processors

Ahmed Abu Cenna and Mario T. Tabucanon Industrial Engineering and Managemen! Division, Asian Institute of Technology, Bangkok, Thailand

(Received February 1, 1991; accepted August 2, 1991 )

Abstract

This paper deals with the problem of bicriterion scheduling with parallel identical machines. An algorithm is proposed taking into consideration the following criteria: minimizing total flowtime (measure of inventory) and minimizing the maximum tardiness (measure of customer service). Five different dispatching rules are compared, namely, Shortest Pro- cessing Time (SPT), Earliest Due Date (EDD), Minimum Slack Time (MST), Largest Processing Time (LPT) and the sequence given by the algorithm of Wassenhove and Gelders. The Wassenhove and Gelders algorithm gives the best result among the dispatching rules. Finally, the proposed algorithm is applied to a real-life situation to illustrate and justify its validity.

1. Introduction

Production scheduling is the allocation of resources over time to perform a collection of tasks. The resources are machines. Pure sequencing is a specialized scheduling problem in which the ordering of jobs completely determines a schedule. The effectiveness of a specific sequence may be measured in terms of average completion times, due date performance, inventory of jobs in process, labour, capacity, etc. Two commonly used measures of performance are mean flow time and maximum tardiness.

A manufacturing firm not engaged in mass production of a single item is usually faced with scheduling problems. Objectives in scheduling vary from firm to firm and often from time to time. Traditionally the aim would be either to achieve certain contractual target dates or simply to finish all the works as early as possible. But in the real world, managers perform the scheduling function in the light of multiple objectives, al- though research on the subject has continued to deal predominantly with the single-objective case.

In multiple-criteria decision making situations "satisficing" (a contraction of "to satisfy" and

"to sacrifice") is the name of the game, not op- timizing in the traditional sense. This concept, as applied to job shop scheduling, was utilized by Emmons [ 1 ] and Wassenhove and Gelders [2 ] for scheduling of jobs on a single machine. It ap- pears that hitherto this concept has not been applied for the case of multiple machines.

Baker [ 3 ] had shown that scheduling on a single machine is completely determined by the sequence of the jobs. But in the case of multiple- machine scheduling, a resource allocation problem is inherently superimposed and there ap- pears no universally acceptable single rule or heuristic for it. Thus, before using any heuristic there should be enough evidence to support its suitability.

Normal practice of dealing with a multiple- machine scheduling problem involves, among others, ascertaining the existence of a constructive algorithm. A constructive algorithm is one which, by following a simple rule which exactly determines the processing order, builds up an optimal solution for the problem. However, the body of literature suggests that only very few small-size problems are amenable to such analysis.

0925-5273/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

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Cheng and Sin [ 4 ], in their comprehensive re- views of literatures, had shown that there are many available algorithms for scheduling of jobs on parallel machines with a single criterion. They have reported that extensive treatment of scheduling problems can be found in the works of Con- way, Maxwell and Miller [5], Baker [3], Rin- nooy Kan [ 6 ], Lenstra and Rinnooy Kan [7 ], and French [ 8 ]. Panwalkar [ 9 ] presented a summary of over hundred priority dispatching rules analyzed through simulation technique. Gere [ 10 ] also discussed the definitions of priority rules, heuristics and dispatching rules and a number of conjectures about scheduling algorithms.

Smith [ 11 ] dealt with the problem of minimizing total flowtime while meeting due dates of jobs. Wassenhove and Gelders [2] extended Smith's idea towards solving a bicriterion scheduling problem. They developed an algorithm considering two criteria, namely, minimizing total flowtime and minimizing maximum tardiness. The concept of bicriterion scheduling, as a development of Smith's algorithm, was further extended by Chand and Schneeberger [12] for the problem of minimizing the weighted completion time and the maximum allowable tardiness.

In this paper we have developed an algorithm for two or more parallel machines in single-stage production. The algorithm deals with two criteria, total flowtime and maximum tardiness.

2. Notations and assumptions

The following notations will be used throughout the extent of this paper: n = Number of jobs available in the shop. m = Number of machines working in each

family. Pi = Processing time of job i. di = Due date of job i. C~ = Completion time of job i. /~i =Flowtime of job i (total processing

time of jobs completed earlier plus the processing time of its own).

ff = Average flowtime. c = Cost function of a schedule. S = A schedule. n = A permutation schedule. H = The set of permutation schedules.

T(n) H ( n )

U J A k Zmax

= Tardiness of schedule n. = Total flowtime of schedule n. = Total processing time of all available

jobs. = Set of all jobs. = A job. = Allowable maximum tardiness. = Position of a job in a set of jobs. = Maximum tardiness.

In this paper the following assumptions are valid unless otherwise specified: 1. Each job is an entity. 2. No pre-emption. Each job, once started, must

be completed before another job can be started on the same machine.

3. No cancellation. Each job must be processed to completion.

4. The processing times are independent of schedule. In particular, two things are settled here. Firstly, each setup time is sequence independent, i.e. the time taken to adjust a machine for a job is independent of the job last processed. Secondly, the machines are identical.

5. In-process inventory is allowed. 6. No machine can handle more than one job at a

time. 7. All machines are available throughout the

scheduling horizon. Breakdown of machines is not considered.

8. The technological constraints are known in ad- vance and are immutable.

9, The number of jobs, the number of machines, the processing times and the times at which the jobs are available for assignment are known and fixed.

3. Concept of optimal and efficient solutions

According to Tabucanon [ 13 ], an optimal solution in the classical sense is one in which the maximum values of all the objectives are at- tained simultaneously. The solution x* is optimal to the problem defined if and only if x*~S a n d f ( x * ) >~fz(x) for all land for x~S, where Sis the feasible region. In general, there is no optimal solution to a multiobjective problem. Optimality is not an illusion only when the objectives are nonconflicting. Therefore, one must be satisfied with obtained efficient solutions.

An efficient solution is one in which no in-

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crease can be obtained in any of the objectives without causing a simultaneous decrease in at least one of the objectives, e.g. the solution x* is efficient with bi-objectivef~ andS , if and only if there does not exist any x~X such that f(x) >~f(x*) for all l, and f(x) > f ( x * ) for at least one l. This solution is obviously not unique. In this paper "objectives" and "criteria" are used interchangeably.

4. Concept of efficiency in bicriterion scheduling

The choice of a single performance measure only represents part of the whole picture. For in- stance, if we seek to minimize the maximum tardiness of a schedule, then we will be reducing in a general sense the penalty cost incurred by the late completion of the jobs, but ignoring any inventory or utilization cost that are certainly incurred. The concept of efficiency, as described earlier, is fundamentally essential in bicriterion scheduling.

French [8] considers a single-machine problem in which Tmax is a suitable indicator of the penalty cost arising from the late completion of jobs and that F is an equally important indicator of in-process inventory cost. It is assumed that any other cost component can be neglected. Thus it is confirmed that the total cost is a function of Tma x and F alone, say c( Tmax,F ). Now it is reasonable to suppose that if either Tmax of F in- creases, then the total cost does, too. Therefore, c(Tmax,F) is an increasing function of both its arguments.

Now, suppose that S and S' are two schedules for which Tmax< T'max and F<F', where F' and T' m a x refer to S'.

In this case it is clear that S is more acceptable than S' because we have

c(TmaxF) <c(T'ma~,F')

Indeed, a little thought shows that we can always say that S is better than S' if

Tma x ~< T'ma x and F<~F'

with strict inequality holding in at least one case. We shall say that a schedule S' is efficient if

there does not exist a schedule S which dominates it.

5. Interpretation of efficiency

The concept of efficiency can be interpreted in conjunction with a problem ofbicriterion scheduling of n jobs on a single processor. All jobs are simultaneously available and are characterized by their processing time p, and due date d,. Two objectives to be considered are (i) to minimize flowtime (a measure for average in-process inventory) and (ii) to minimize maximum tardiness (a measure of customer service). This problem was expressed by Wassenhove and Gelders [2] as follows:

Let H = the set of permutation schedules, 7~ = a permutation schedule, C, =the completion time of the job i (given

a permutation schedule), T(g) =the maximum tardiness of schedule 7r, H(g ) = the total flowtime of schedule re.

Then, the problem (P) can be written as: ?/

min ~ C, = minH(~r) ( 1 ) ~ H t = 1 7c~H

min max {max(O,Ci-di)}=minT(~) (2) r e . H i = l.....,n ~ H

We know that objective ( 1 ) can be reached by ordering the jobs according to nondecreasing processing times (SPT) and objective (2) can be achieved by ordering the jobs according to nondecreasing due dates (EDD). But if we have to consider the objectives then the rules work rather poorly. For this we want to have a sequence that "does well" on both the objectives. To define such sequences the concept of efficiency was used by Wassenhove and Gelders [ 2 ].

A sequence zc*~//is efficient in problem (P) if there exists no 7re H such that

H(rc) ~<H(~z*) and T(~z) ~< T(Tr*)

where at least one relation holds with strict inequality. Similarly, a sequence 7rl dominates a sequence g2 when

H ( g I ) ~<H(7~2) and T(7~ 1 ) ~< T(7~2)

where at least one relation is a strict inequality. Clearly, the decision is to choose an efficient sequence among a number of efficient sequences.

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6. Algorithm for the sequencing of jobs with two performance measures

Smith [ 11 ] considered the single-machine sequencing problem with Tmax = 0, that is all the due dates can be met if the jobs are sequenced in EDD rule. He developed an algorithm which gives other sequences that minimize flowtime subject to the condition that Tmax--0. Wassenhove and Gelders [2] modified the Smith's algorithm to minimize F subject to Tmax ~< A, i.e. subject to no job being finished more than A after its due date. Wassenhove and Gelders developed an algorithm to generate all the possible efficient sequences. This algorithm can be described as follows:

Step 1: Put A= ~Tp,.

Step 2: Let Di=d i+A for all i. Step 3: Solve the new problem using the modi-

fied Smith's algorithm. If a solution exists, then it is efficient. Else go to Step 5.

Step 4: Compute T(g*) = max,=~ ...... {max (0,C,-d~)}. Put A = T(rc*)-1 . Go to step 2.

Step 5: Stop.

7. Algorithm for parallel processors with bicriterion performance measures

The problem addressed in this paper is to sequence N independent jobs on m parallel processors with two criteria, all jobs being available at an arbitrary time zero. The problem can be considered in two stages, namely (a) allocation of n jobs on m processors, and (b) sequencing of jobs to achieve both the criteria.

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The proposed algorithm which can deal with parallel processors and yet generates the efficient sequences meeting the two criteria, namely minimizing the maximum tardiness and minimizing the total flowtime, is as follows:

Step 1: Start with the given information of the jobs.

Step 2: Generate the efficient sequence of the jobs using Wassenhove and Gelders' algorithm.

Step 3: Distribute the jobs among the machines following one-at-a-time (H1) heuristics.

Step 4: Generate all the efficient sequences at each machine level as suggested by Was- senhove and Gelders.

Step 5: Determine the efficient sequences at the product family level.

Step 6: Go to Step 2 for next sequence. If no such sequence is possible go to Step 7.

Step 7: Compare all the efficient sequences at product family level to get the overall efficient sequences.

Step 8: Stop.

Using Wassenhove and Gelders' algorithm there may be several sequences created in a single queue before the allocation of jobs among the machines. For each of these sequences efficient sequences are created at the machine level. Fi- nally, the overall efficient sequences are taken. This technique is illustrated in Fig. 1.

In this paper some simple dispatching rules are also compared to those given by Wassenhove and Gelders. The algorithm, in the case of using simple dispatching rules, is a bit simpler and can be expressed as follows:

Step 1: Start with the given information of the jobs.

Step 2: Sequence the jobs with any of the dispatching rules.

Step 3: Distribute the jobs among the machines following one-at-a-time (H~) heuristics.

Step 4: Generate all the efficient sequences at each machine level as suggested by Was- senhove and Gelders.

Step 5: Determine the efficient sequences at the product family level.

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Step 6: Stop.

Four dispatching rules, namely Earliest Due Date (EDD), Shortest Processing Time (SPT), Minimum Slack Time (MST) and Largest Pro- cessing Time (LPT) were compared to that of using Wassenhove and Gelders' algorithm. For this comparison twenty sets of data were randomly generated.

9. Simulation of the model

In this paper a model is developed for the bicriterion sequencing problems where there are two or more parallel processors. As pointed out by Baker [3], there is no single heuristic or any single dispatching rule for parallel processors which always gives the optimum result. For this to be applicable with reasonable acceptance of the model, five different dispatching rules have been tested. As shown by Baker [3], for two parallel processors, heuristic H1 shows the best performance (in the single-criterion case), and thus is the only one used in this model.

For the purpose of simulation the jobs and their corresponding due dates are randomly generated. The processing time for each job is generated from pseudorandom numbers having single digit. Corresponding due dates are generated in the same way but run from 1 to 30 (taking the planning period as one month and one shift per day). The simulation also ensured that the due date can not be less than the corresponding processing time. As there are two machines available for each family as used in the simulation, the total processing time for one set of jobs was kept to around 60 but not less than 60 shifts.

Twenty sets of random data resembling one family each have been generated. The model is run for each of the data sets and as well for each dispatching rule. The efficient points of each data set and the corresponding dispatching rules are plotted to select the most efficient algorithm.

Finally, from all the twenty plotted graphs the overall efficient points have been taken. All the efficient points are grouped into the dispatching rules that they coincide with in the graphs. Then the dispatching rules are ranked according to their performance. From this experiment it can be concluded that using Wassenhove and Gelders'

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sequence as the priority rule gives the best performance in case of bicriterion scheduling with parallel machines. This is shown in Fig. 2.

With a careful observation of the results that have been drawn from the model, the following remarks can be made: 1 ) The efficient sequences for Wassenhove and

Gelders' algorithm superimpose with that of EDD for minimum tardiness.

2) In only few cases where MST and LPT rules give better result than that of EDD and of Wassenhove and Gelders'.

3 ) Among the simple dispatching rules, MST and LPT give better results than SPT and EDD rules, and Wassenhove and Gelders' sequence gives the best result among the dispatching rules tested.

Finally, from the simulation result, the following ranking of the dispatching rules can be made: Wassenhove and Gelders' sequence (W&G), Minimum Slack Time (MST), Largest Process-

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ing Time (LPT), Earliest Due Date (EDD), and Shortest Processing Time (SPT).

10. The case study

In order to illustrate the practicality of the developed model, a real-life case study was con- ducted in an electrical and electronic manufac- turer. As they are dealing with smaller components parts, they have to deal with a large variety of products. The specification of the products changes with different order and thus to keep raw material inventory is not logical. So they have to produce on order with a large lead time. Their scheduling time horizon is one month.

The production department comprises of a number of machines. They have to handle a large number of orders with a large variety of product specifications. The concept of family has been introduced to take advantage of product classification according to the technological constraints. When the products are divided into families it is revealed that each family of products is handled by two or more machines and the machines are operating in parallel. Most of the products only require single-stage operation. Machines are mostly semiautomatic. So for each family of products there is a single queue which is served by two or more parallel machines. For this study only the two-machine case has been considered.

The model was run with the actual data set. The model gives the possible efficient sequences with respect to total flowtime and maximum tardiness. The program also calculates the total tardiness for each of the efficient sequences. Actually the selection of a sequence depends on the weigh- tage given to the parameters maximum tardiness and total flowtime. A third criterion, total tardiness, helps the decision maker to take a decision.

The model helps the decision maker to take an important decision. The algorithm generates the efficient sequences with respect to the two parameters and the decision depends on how much importance the decision maker is giving to each of the parameters.

Another important point to be noted here is that total tardiness is not a regular measure with respect to the two criteria namely, total flowtime and maximum tardiness. But the parameter total

tardiness may also be considered while taking a decision. Thus this algorithms helps the top management taking a keen decision and this can be an introduction of scheduling considering three criteria.

11. Concluding remarks

Much effort has been given to develop a parallel-machines scheduling algorithm. There are algorithms available considering bicriterion scheduling with a single machine. This paper in- troduces bicriterion scheduling with parallel machines.

Wassenhove and Gelders' generated sequence is considered to be superior to those generated by other dispatching rules in the case of a single machine. The algorithm presented in this paper works in the case of parallel machines and generates efficient sequences with respect to total flowtime and maximum tardiness.

A program for this algorithm is written in FORTRAN77 for use in micro-computers. The program also computes total tardiness as the third factor for consideration in taking the decision. Hopefully the idea presented will further en- hance the application of multiple-criteria decision making in parallel-machines scheduling.

References

1 Emmons, H. and Pinedo, M., 1990. Scheduling stochas- tic jobs with due dates on parallel machines - Theory and methodology. Eur. J. Oper. Res., 47( 1 ): 49-55.

2 Wassenhove, V.L.N. and Gelders, L.F., 1980. Solving

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bicriterion scheduling problem. Eur. J. Op. Res., 4: 42- 48.

3 Baker, K.R., 1974. Introduction to Sequencing and Scheduling. John Wiley, New York.

4 Cheng, T.C.E. and Sin, C.C.C., 1990. A state-of-the-art review of parallel-machine scheduling research - In- vited Review. Eur. J. Oper. Res., 47(4): 271-292.

5 Conway, R.W., Maxwell, W.L. and Miller, L.W., 1967. Theory of Scheduling. John Wiley, New York.

6 Rinnooy Kan, A.H.G., 1976. Machine Scheduling Prob- lems, Classification, Complexity and Computations. Martinus Nijhoff, The Hague.

7 Lenstra, J.K. and Rinnooy Kan, A.H.G., 1977. Com- plexity of scheduling under precedence constrains. Op. Res., 26: 22-35.

8 French, S., 1982. Sequencing and Scheduling, An Intro- duction to the Mathematics of the Job Shop. John Wiley, New York.

9 Panwalkar, S.S. and Wafik, I., 1977. A survey of scheduling rules. Op. Res., 25( 1 ): 45-61.

10 Gere, W.S., 1966. Heuristics in job shop scheduling. Manage. Sci., 13(3): 167-190.

11 Smith, W.E., 1956. Various optimizers for single stage production. Nay. Res. Logist. Q., 3: 59-66.

12 Chand, S. and Schneeberger, 1986. A note on the single machine scheduling problem with minimum weighted completion time and maximum allowable tardiness. Nav. Res. Logist. Q., 33 (3): 551-557.

13 Tabucanon, M.T., 1988. Multiple Criteria Decision Making in Industry. Elsevier, Amsterdam.

14 Baker, K.R., 1984. Sequencing rules and due date as- signments in a job shop. Manage. Sci., 30(9): 1093- 1104.

15 Baker, K.R. and Merten, A.G., 1973. Scheduling with parallel processors and linear delay costs. Nav. Res. Log- ist Q., 20(4): 793-804.

16 Dogramaci, A., 1984. Production scheduling of independent jobs on parallel identical processors. Int. J. Prod. Res., 22(4): 535-548.

17 Dogramaci, A. and Surkis, J., 1979. Evaluation of a heuristic for scheduling independent jobs on parallel identical processors. Manage. Sci., 25 ( 12 ): 1208-1216.

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Bicriterion scheduling problem in a job shop with parallel processors