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1 A New Heuristic Method for the Permutation Flow Shop Scheduling Problem Author(s): Joao Vitor Moccellin Source: The Journal of the Operational Research Society, Vol. 46, No. 7 (Jul., 1995), pp Published by: Palgrave Macmillan Journals on behalf of the Operational Research Society Stable URL: Accessed: 16/11/ :39 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Palgrave Macmillan Journals and Operational Research Society are collaborating with JSTOR to digitize, preserve and extend access to The Journal of the Operational Research Society.

2 Journal of the Operational Research Society (1995) 46, ? 1995 Operational Research Society Ltd. All rights reserved /95 $12.00 S A New Heuristic Method for the Permutation Flow Shop Scheduling Problem JOAO VITOR MOCCELLIN University of Sao Paulo, Brazil A new heuristic method for the permutation flow shop scheduling problem is presented and compared with two other heuristics named NEH and SPIRIT. The new heuristic method is based on a property of the scheduling problem that provides an upper bound on the idle time of the last machine between any two adjacent jobs regardless of their position in the sequence of jobs. The results from computational experience have shown that the new heuristic outperforms, in solution quality, all others for problems having up to 50 jobs and 30 machines. Key words: heuristics, sequencing, tabu search, travelling salesman problem INTRODUCTION The permutation flow shop scheduling problem is a production problem where a set of n jobs have to be processed in the same sequence by each of m machines. This is the case in manufacturing facilities where jobs (parts) are moved from machine to machine by material handling devices. Since there is no job passing, which means the job sequence is the same for all machines, the number of possible schedules for n jobs is n!. Usually the problem consists of finding a sequence for the jobs that minimizes the makespan; that is, the overall time to complete the schedule. The search for a solution to the problem of finding an optimal or near optimal sequence has yielded both exact and heuristic solution techniques. Over the past three decades many heuristics have been formulated for sequencing jobs in a flow shop. One class of such heuristics finds a good solution to the problem by improving an initial sequence through Tabu Search (TS) techniques. Recently, TS methods have been proposed for solving the permutation flow shop scheduling problem: see, for instance, Widmer and Hertz', Taillard2 and Reeves3. In the paper by Widmer and Hertz' a heuristic named SPIRIT is presented, whose basic steps are: (1) get an initial solution to the flow shop sequencing problem by using an analogy with the travelling salesman problem; (2) improve the initial solution by using Tabu Search techniques. Computational experience with a number of known heuristics has shown that SPIRIT and the constructive heuristic NEH of Nawaz et al.4 outperform other heuristics. For any n-job sequence S, the makespan M(S) can be expressed by: n n-1 M(S) = Epmj + E (1) j=1 j=0 where Pmj= the processing time on the last machine m of the job in the jth position of S; X[m [j+= the idle time on the last machine m between the end of the job in position j and the start of the job in position (j + 1) of S, and j= 0 means a 'dummy' job, with zero processing times, which is always before the first job in S. If we consider Xm[j+,] as the distance between the jobs in the jth and (j + 1)th positions of S, then the permutation flow shop scheduling problem becomes a directed travelling salesman problem. The sequence S* that minimizes the makespan M(S) is given by the shortest route Correspondence: J. V. Moccellin, School of Engineering of Sdo Carlos, University of Sdo Paulo, Brazil

3 884 Journal of the Operational Research Society Vol. 46, No. 7 from the dummy job (j= 0) to the last one (j = n). Of course, we do not know the idle times Xm[j?,] in advance unless we have considered a particular sequence. In the first basic step of the SPIRIT heuristic procedure the analogy with the travelling salesman problem uses the 'distance' from job u to job v defined by the following expression: m duv = Piu + - -(m k)lpku Pk-l,vl + Pmv. (2) k=2 Widmer and Hertz' have set as an open question the knowledge of whether a 'distance' other than that given by the above expression would give a better initial solution for the tabu search method. A primary contribution of this paper is to give a satisfactory answer to such a question. THE NEW HEURISTIC PROCEDURE The heuristic procedure we introduce in this paper, which is named FSHOPH (flow shop heuristic), is similar to SPIRIT. The essential difference from SPIRIT is in the basic step of finding an initial solution for the scheduling problem. An analogy with the travelling salesman problem is still used but the distance between two jobs is quite different from that used by Widmer and Hertz. In the FSHOPH heuristic the distance between two jobs is based on the following property of the permutation flow shop scheduling problem5. 'Let u and v be two arbitrary jobs from the set of n jobs. For any sequence S with jobs u and v respectively scheduled in positions j and (j + 1), j = 0, 1, 2,..., n - 1, one has that: UBXUV = max [O, UBX u + (Pkv - Pk+l,u)] (3) k=1, 2,..., m - 1, where UBX'v = O, and UBXhV = an upper bound on the idle time of machine h between the end of job u and start of job v, Phi = processing time of job i on machine h'. The above property is derived from a feasibility condition between machine and job idle times of the flow shop sequencing problem. This condition is also used by Selen and Hott6 in their mixed-integer goal-programming formulation of the problem. By using the recurrence relation (3) successively for k = 1, 2,..., m - 1, we obtain an upper bound UBX m on the idle time of the last machine m between any two adjacent jobs u and v regardless of the position of these two jobs. Thus, from expressions (1) and (3) we have the following. For any n-job sequence S an upper bound on the makespan M(S) is given by n n-1 UBM(S) = Epmj + EUBXm j=1 j=o If we consider UBXm][j+?] as the 'distance' between jobs in positions j and (j + 1), then the original permutation flow shop scheduling problem can be heuristically solved by an analogy with the directed travelling salesman problem. In the first basic step of the FSHOPH heuristic the distance from job u to job v is given by UBXmJ and we search for the route (sequence) that minimizes the upper bound UBM(S). In order to find a good solution of our travelling salesman problem we use the well-known Farthest Insertion Travelling Salesman Procedure (FITSP). The second basic step of the FSHOPH heuristic is the same as that of SPIRIT. Widmer and Hertz considered problems having up to 20 jobs and 20 machines. In order to deal with larger problems we propose the selection of the Stopping Condition (parameter Nbmax) according to Table 1. The tabu search procedure, which is used to improve the heuristic solution, is stopped when the current best solution is not improved during Nbmax successive iterations. The values in Table 1 have arisen from an experiment in which we have

4 J. V. Moccellin-Method for the Permutation Flow Shop Scheduling Problem 885 constrained the computation times not to exceed 10 minutes on a microcomputer 386 DX with a mathematical coprocessor, for problems having up to 50 jobs and 30 machines. TABLE 1. Selection of Nbmax Number of jobs Number of machines Nbmax n s 30 m 6 30 n 30 < n S 40 m < 15 n 40 < n < 50 m < 10 n 50 - n 60 m < 7 n 30 < n m < < n < m < n m < < n m s n m < < n '75 m < < n < m < n m < n 75 7 m < < n 100 m < < n < m S < n m < <n m 30 1 COMPUTATIONAL EXPERIENCE The NEH, SPIRIT and FSHOPH heuristics have been tested on a total of 1020 problems with m E {4, 7, 10, 15, 20, 30} and n E {6, 8, 10, 15, 20, 30, 40, 50, 75, 100}. Each of the m x n combinations with n - 50 was replicated 20 times. For n > 75 only five problems for each value of m were generated. The operation processing times were randomly generated integers uniformly distributed over the interval [0, 10]. In the computational tests the heuristics were coded in Turbo-Pascal and have been run on a microcomputer 386 DX with an mathematical coprocessor. The principal results from the computational experience are summarized in Tables 2 and 3. Table 2 shows the average percentage quality of the solutions for a sample of small problems (n = 6, 8, 10). The quality Q(S) of a solution (sequence S) is defined as Q(S) = [M(S) - M*]/M*, where M(S) is the makespan provided by sequence S, and M* is the optimal sequence makespan. Table 3 summarizes the comparison of the heuristics for small and medium-size problems (120 problems for each value of n, n = 6, 8, 10, 15, 20, 30, 40, 50) showing the percentage of success of each heuristic method. This percentage is defined as the total number of times the heuristic obtains the best makespan (alone or in conjunction with others) divided by the number of generated problems. TABLE 2. Average percentage quality of solutions on small problems Number NEH SPIRIT FSHOPH of jobs % % % b (120)a (0.015)c (0.105) (0.100) (76) (0.033) (0.397) (0.396) (21) (0.058) (1.085) (1.071) Total average (0.035) (0.529) (0.522) anumber of problems with known optimal solutions baverage percentage solution quality caverage computation time in seconds.

5 886 Journal of the Operational Research Society Vol. 46, No. 7 TABLE 3. Comparison NEH-SPIRIT-FSHOPH Number % % % of jobs NEH SPIRIT FSHOPH a (0.015)b (0.105) (0.100) (0.033) (0.397) (0.396) (0.058) (1.085) (1.071) (0.181) (6.437) (6.402) (0.419) (24.28) (22.89) (1.370) (129.9) (117.1) (3.199) (237.4) (240.1) (6.201) (350.9) (348.8) Total apercentage of success baverage computation time in seconds CONCLUSIONS For the set of small problems with known optimal solutions we observe in Table 2 that, on average, FSHOPH generates solutions whose deviations from the optima are around 0.5% (0.92% for SPIRIT, and 1.53% for NEH). For small and medium-size problems (n from 6 to 50 jobs) the simulation results from Table 3 have shown that FSHOPH has yielded the best solution in 76.5% of the test problems, followed by SPIRIT (46.9%) and NEH (35.0%). FSHOPH takes similar computation times to SPIRIT but produces significantly better solutions, whereas NEH takes much less computation time than either but is outperformed in solution quality by both other heuristics. For larger problems (n = 75, 100 jobs) the NEH heuristic was the best because the computation time required by both FSHOPH and SPIRIT to obtain good solutions is greatly increased. REFERENCES 1. M. WIDMER and A. HERTZ (1989) A new heuristic method for the flow shop sequencing problem. Eur. J. Opl Res. 41, E. TAILLARD (1990) Some efficient heuristic methods for the flow shop sequencing problem. Eur. J. Opl Res. 47, C. R. REEVES (1993) Improving the efficiency of tabu search for machine sequencing problems. J. Opl Res. Soc. 44, M. NAWAZ, E. E. ENSCORE JR and I. HAM (1983) A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega 11, J. V. MOCCELLIN (1993) New properties of the permutation flow shop scheduling problem. Paper presented at the 36th Joint National Meeting of ORSA/TIMS, Phoenix, AZ, USA, November W. J. SELEN and D. D. Horr (1986) A mixed-integer goal-programming formulation of the standard flow-shop scheduling problem. I. Opl Res. Soc. 37, Received January 1994; accepted November 1994 after three revisions