Scheduling multiprocessor tasks in a two-stage flow-shop environment

Computers ind. Engng Vol. 33, Nos 1-2, pp. 269-272, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352•97 $17.00 + 0.00 PII: S 0 3 6 0 - 8 3 5 2 ( 9 7 ) 0 0 0 9 0 - 9

Pergamon

SCHEDULING

MULTIPROCESSOR

TASKS

IN

A

TWO-STAGE FLOW-SHOP

ENVIRONMENT CEYDA OGUZ 1 and M. FIKRET ERCAN 2 1Department of Management and 2Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

ABSTRACT In this paper, we developed several efficient heuristic algorithms to schedule unit processing time multiprocessor tasks in a two-stage hybrid flow-shop for minimizing makespan. We also derived two effective lower bounds for the problem. Then, we analyzed the average performance of the heuristic algorithms by computing the average relative gap of each heuristic solution from the lower bound. The results of the computational experiment to test the average performance of the proposed heuristic algorithms on a set of randomly generated problems showed that three of the proposed heuristic algorithms perform well. © 1997 Elsevier Science Ltd

KEYWORDS Average performance; heuristic; hybrid flow-shop; makespan; multiprocessor task scheduling.

INTRODUCTION We consider the following problem: There is a set fl of n independent and simultaneously available jobs to be processed in a two-stage flow-shop where stage j has m j identical parallel processors, j = 1, 2. Each job Ji E ,7 has two multiprocessor tasks (MPTs), namely (i, 1) and (i, 2). MPT (i, j) should be processed without interruption by a certain number of identical parallel processors at stage j , hence, each MPT (i, j ) is characterized by its processing time, Pij, and its processor requirement, sizeij (i = 1, 2 , . . . , n and j = 1,2). All the processors are continuously available from time 0 onwards and each processor can handle no more than one MPT at a time. In this model, it is assumed that all MPTs have the same processing time, hence, Pij = 1 (i = 1 , 2 , . . . , n, j = 1, 2). Jobs flow through from stage 1 to stage 2 by utilizing any of the processors while satisfying the flow-shop and the MPT constraints. The objective is to find an optimal schedule for the jobs in this flow-shop environment so as to minimize the maximum completion time of all jobs (makespan), Cmax. Using the well-known three-field notation with extensions introduced later (see Drozdowski, 1996), we denote this scheduling problem as H F 2 1 s i z e i j , p q = l[Cmax. H F 2 denotes a two-stage flow-shop with several identical parallel processors in each stage which is usually referred to as a hybrid flow-shop. MPT scheduling and hybrid flow-shop scheduling (HFS) have traditionally been studied separately. In previous studies, HFS problems were based on the assumption that a task can be processed by at most one processor at a time whereas MPT scheduling problems on parallel processors (MPTS-PP) assumed a single stage setting. HFS problems with classical job definition have been studied by various researchers. In several studies, different heuristic algorithms were developed 269

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and their average performance was analyzed (see, for example, Gupta, 1988; Guinet et al., 1996). In other studies, the worst-case error bound of the proposed heuristic algorithms were analyzed (Shriskandarajah and Sethi, 1989; Lee and Vairaktarakis, 1994). Recently, a classification of the previously developed heuristics for HFS and a complete analysis of the worst-case performance of the classes was provided (Chen, 1995). In another study, global lower bounds on HFS were presented (Santos et al., 1995). The MPTS-PP problem for single stage shops has been addressed by several authors. These papers mainly focus on the complexity issues of MPTS-PP problems. That is, either problems are shown to be NP-hard or polynomial time algorithms are presented (see, for example, Lloyd, 1981; Blazewicz et al., 1986; Du and Leung, 1989). In a recent study, an extensive survey on scheduling MPTs is presented (Drozdowski, 1996). The motivation for this paper comes from a computer vision system which has three layers, each consisting of identical parallel processors. A job with non-preemptive MPTs should be processed in all three layers in order. The first and the second layers are deterministic. The third layer is non-deterministic and time consuming, but has a deadline since the system works in real-time. Hence, we concentrate on minimizing the maximum completion time of all tasks on the first and second layers to provide more time for third layer tasks.

HEURISTIC ALGORITHMS The HF21sizeij,pi j = llCm~: problem is NP-hard, because Plsizei,pi = llC,,~x is NP-hard (Lloyd, 1981). Due to this result and because a quick and a good solution is required from a practical point of view, we focus on developing efficient heuristic algorithms to find approximate solutions to the HF21sizeij,pi j = llCm~x problem. The basic structure of all heuristics that we propose for the HF21sizeij,Pi j = llCmax problem are as follows:

Basic Structure of Heuristic Algorithms (BSHA): Step 1. Sequence jobs. Step 2. Schedule jobs in Stage 1. Step 3. Schedule jobs in Stage 2. We use five rules to sequence jobs, two rules to schedule jobs in Stage 1 and two rules to schedule jobs in Stage 2 to obtain different heuristic algorithms. For sequencing jobs, we modify two wellknown priority sequencing rules, namely the Shortest Processing Time (SPT) and the Longest Processing Time (LPT), by using processor requirements to have Shortest Processor Requirement (SPR) and Longest Processor Requirement (LPR) rules, respectively. Another sequence is obtained by using Johnson's Algorithm (Johnson, 1954), in which processing times are replaced with processor requirements. We call this sequence a Modified Johnson Sequence (MJS).

Sequencing Rules ( S EQ ): 1. SPR[1]: Find a sequence S by applying SPR rule to Stage 1 requirements. 2. LPR[1]: Find a sequence S by applying LPR rule to Stage 1 requirements. 3. SPR[2]: Find a sequence S by applying SPR rule to Stage 2 requirements. 4. LPR[2]: Find a sequence S by applying LPR rule to Stage 2 requirements. 5. M J S : Let S1 = {ilsizeil <_sizei~} and $2 = {ilsizeix > sizei2}. Sort S1 and $2 according to SPR and LPR rules, respectively. Obtain S by concataneting S1 and $2. Scheduling Rules in Stage 1 (SCH1): 1. Modified List Scheduling (MLS): Given a sequence S of the jobs, construct a schedule in Stage 1 by assigning the first unassigned job Ji in S to the first available sizeil processors if the number of available processors is not less than sizeil in the current time slot. Otherwise, assign job Ji to the sizeil processors in the next time slot.

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2. Modified First Fit (MFF): Given a sequence S, construct a schedule in Stage 1 by assigning the first unassigned job Ji with sizeil to ml processors together with job Jk with sizekl, where size~l = ml - sizeil, i y~ k. If there is no such job Jk, assign job Ji to the end of the schedule.

Scheduling Rules in Stage 2 (SCH2): 1. LPR-Avallable (LPRA): As the jobs are processed and finished in Stage 1, assign the jobs in the L P R order wherever processors are available. 2. List Scheduling-Available (LSA): As the jobs are processed and finished in Stage 1 in order, assign the jobs in the same order wherever processors are available. To evaluate the performance of the heuristics, we derived two lower bounds on the value of an optimal schedule. Since each M P T is characterized by its processor requirement, an immediate lower bound is given by LB1 = [ ( ~ i sizeil)/ml] + 1, where Ix] denotes the smallest integer no smaller than x. The first term in LB1 is equal to the maximum processing time of MPTs in Stage 1, if we allow preemption. The second term is the processing time of the last task to be processed at Stage 2. Similarly, another immediate lower bound is given by LB2 = [ ( ~ i sizei2)/m2] + 1. Hence, the lower bound on the value of an optimal schedule for the HF21sizeij,pi j = llCmax problem will be LB = max{LB1, LB2}.

COMPUTATIONAL STUDY The performance of the proposed heuristics was tested with problems that were generated as follows. For every M P T ( i , j ) of job J~, i = 1, 2 , . . . , n, an integer processor requirement in stage j was generated from a uniform distribution over [1, mj], j = 1, 2. The number of jobs was selected so that n = 10,50,100,300. The number of processors was chosen such that ml = 2m2 = 2 k and ml = m2 = 2 k with k = 2,3,4,5. The first design corresponds to the actual design of the computer vision system that motivated us. In each design, for every combination of n and (ml;m2), 50 problems were generated. From the analyses of the performance of different heuristic algorithms for the combination of five SEQ rules, two SCH1 rules and two SCH2 rules, it is seen that three of them (H1, H2 and H3) outperformed all others. H1, H2, and H3 stand for H(SPR[1],MFF, LSA), H(SPR[1], MFF, LPRA), and H ( M J S , MFF, LSA) where H(A, B, C) denotes the heuristic algorithm that employs rules A, B, and C in Steps 1, 2, and 3 of BSHA, respectively. Hence, we present the computational results of H1, H2, and H3 only. In Table 1, we report the average relative error C(Hl) - L B ) / L B x 100 where C(HI) denotes the Cm~ obtained by heuristic Hl, l = 1,2,3. The results demonstrate that all three heuristic algorithms are very effective in solving the HF21sizeij,pij = llCma z problem, although H1 and H2 perform much better than H3 and neither H1 nor H2 outperforms the other. From Table 1, we observe that as the number of jobs increases, the average relative error decreases.

CONCLUSION In this paper, we have considered the HF21sizeij , Pij = llCm~ problem which is A f t - h a r d in the strong sense. In view of this complexity result, we presented several heuristic procedures for the solution of the problem. We also developed an effective lower bound on the value of an optimal schedule. Then, we analyzed the average performance of the proposed heuristic algorithms. The computational results show that two of the proposed heuristic algorithms are very effective and can be used for problems with a large number of jobs and processors. As an extension, the HF21sizeij,pijlCm~x problem should be analyzed and efficient heuristics should be developed.

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Table 1: Average relative error of each heuristic.

n 10 10 10 10 50 50 50 50 100 100 100 100 300 300 300 300

k 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

ml =2m2 H(1) H(2) 0 . 9 4 0.94 4.60 4.64 9.52 9.81 8.65 8.65 0.00 0.00 3.06 2.94 6.20 6.00 8.32 8.40 0.00 0.00 2.43 2.47 4.63 4.61 7.14 6.69 0.00 0.00 1.10 1.84 2.84 3.16 4.03 4.00

H(3) 1.77 5.60 12.47 11.54 0.00 3.99 8.03 10.76 0.00 3.79 6.18 8.36 0.00 2.79 4.44 5.82

n 10 10 10 10 50 50 50 50 100 100 100 100 300 300 300 300

k 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

ml = m2 H(1) H(2) H(3) 3 . 2 9 3 . 2 9 4.68 7.99 7.95 8.81 9 . 3 7 9 . 3 7 12.35 9 . 2 4 9 . 2 4 13.89 3.56 3 . 6 7 4.92 5 . 0 0 5 . 2 5 6.92 8.43 7.72 9.59 10.39 10.29 12.22 2 . 1 4 2 . 5 4 3.87 5 . 1 1 4 . 8 8 6.89 6 . 1 0 5.80 7.59 8.06 7.88 9.56 1.37 2 . 5 8 4.09 2 . 3 6 2 . 7 8 4.67 4 . 0 1 4.42 5.93 5 . 1 2 5 . 1 8 6.48

Another research direction is to construct a different model which will incorporate all three layers of the computer vision system and to develop on-line algorithms for this real-time problem.

Acknowledgement- This research was supported in part by The Hong Kong Polytechnic University under grant number 0351-086-A3-230.

REFERENCES Blazewicz, J., M. Drabowski and J. Weglarz (1986). Scheduling multiprocessor tasks to minimize schedule length. IEEE Trans. Comput., C-35/5,389-393. Chen, B. (1995). Analysis of classes of heuristics for scheduling a two-stage flow shop with parallel machines at one stage. J. Opl Res. Soc., 46,234-244. Drozdowski, M. (1996). Scheduling multiprocessor tasks - An overview. Eur. J. Opl Res., 94, 215-230. Du, J. and J.Y-T. Leung (1989). Complexity of scheduling parallel task systems. SIAM J. Discrete Math., 2,473-487. Guinet, A., M.M. Solomon, P.K. Kedia and A. Dussauchoy (1996). A computational study of heuristics for two-stage flexible flowshops. Int. J. Prod. Res., 34, 1399-1415. Gupta, J.N.D. (1988). Two stage hybrid flowshop scheduling problem. J. Opl Res. Soc., 39, 359-364. Johnson, S.M. (1954). Optimal two and three-stage production schedules with setup times included. Naval Res. Logist. Q., 1, 61-68. Lee, C-Y. and G.L. Vairaktarakis (1994). Minimizing makespan in hybrid flowshops. Opns Res. Lett., 16, 149-158. Lloyd, E.L (1981). Concurrent task systems. Opns Res., 29, 189-201. Santos, D.L., J.L. Hunsucker and D.E. Deal (1995). Global lower bounds for flow shops with multiple processors. Eur. J. Opl Res., 90, 112-120. Shriskandarajah, C. and S.P. Sethi (1989). Scheduling algorithms for flexible flowshops: worse and average case performance. Eur. J. Opl Res., 43, 143-160.