Computers ind. Engng Vol. 14, No. 3, pp. 361-364, 1988 Printed in Great Britain. All rights reserved
0360-8352/88 $3.00+0.00 Copyright 1988 Pergamon Press plc
MULTI -PRODUCT FLOWSHOP SCHEDULING WITH ORDERED PROCESSING T IMES
R. C. MISHRA, P. C. PANDEY and J. L. GAINDHAR Department of Mechanical and Industrial Engineering, University of Roorkee, Roorkee 247 667, India
(Received for publication 7 September 1987)
Abstract--The problem of scheduling n jobs on m machines has received considerable research attention over the last 30 years or so. As a result, several optimization and heuristic algorithms are now available. This paper presents a single-pass heuristic scheduling procedure, to yield optimum solution, when the processing times of the jobs are ordered.
A flowshop is characterized by more or less continuous and uninterrupted flow of work through a series of work stations. In such a case, the job flow is unidirectional and all the jobs observe an identical technological processing sequence. Based on several simplifying assumptions, the flowshop sequencing problem has been studied extensively in the past, and a number of algorithms have been proposed [1-3]. In general, these assumptions normally relate to jobs, machines or the operating policies in the shop.
The research effort of the past three decades concerning flowshop scheduling, has yielded three basic formulation and solution approaches; these being mathematical programming, curtailed enumeration and heuristic procedures.
Smith et al.  and Panwalker and Khan  have described a subcategory of the classical flowshop problem in which the processing times of different jobs follow a special order. An ordered flowshop problem possesses two unique characteristics. The first concerns a relationship among all the jobs of a particular problem, if a particular job has a smaller processing time on any machine than any other job, then the processing time of the former job will be less than or equal to the processing time of the latter on all corresponding machines. The second distinguishing characteristic of ordered problems concerns a relationship among all machines of a particular problem. The machine with minimum processing time for a given job will also have the minimum processing times for every other job.
In this paper the authors have proposed a single pass heuristic scheduling procedure when the processing times of the jobs are ordered. The characteristic feature of the proposed solution methodology is its simplicity and ease of computation.
Smith et al.  have proposed an algorithm for the ordered flowshop problem subject to the following processing time constraints (this was followed by another algorithm for the general case ):
(a) If PTit > PTk, for i, K ~ N; for some t e M, Then PT~j ~ PTkj for all J e M.
(b) If PTrj > PT, k for some r e N; for J, K ~ M, Then PTij >~ PTik for all i e N.
In both the cases, proofs of optimality for the minimal makespan sequence, generated by the respective algorithms, are also available. Panwalkar and Smith  have reported a simple procedure to find the optimal sequence that minimizes the mean completion times of the jobs.
362 R.C. MisnaA et al.
ORDERED FLOWSHOP IN PRACTICE
Smith et al.  have cited the following practical situations which lead to ordered problems. An ordering of job processing times can be caused by several factors operating either singly or jointly. One of these factors is the number of units of the jobs. If the nature of operations on each job is very similar but the jobs differ greatly in the number of units of production, then the processing time elements will reflect these differences in an ordered manner common to all jobs. Another factor affecting processing times of any job on all machines is job complexity. One job may require simple operations on each unit in the job while another job may require involved time-consuming operations. In such cases, job complexity can have a profound effect upon processing times of the jobs on all machines.
Ordering of processing times according to the machines can also result from the different nature of operations performed on the machines. If one machine operation is complex for all jobs and another operation is simple for all jobs the former machine will normally have larger unit processing times than will corresponding jobs on the latter machine. The operation complexity factor could affect all jobs and thus is a source of ordered processing time.
Smith et al.  are of the view that all sequencing problems, from a practical situation, will not meet the requirements for classification as ordered problems. However, such a situation does exist to an extent that special attention should be given to it.
ORDERED FLOWSHOP PROBLEM
The authors have considered a flowshop problem where the processing times satisfy the following conditions.
Let PT(n,m) be the processing time of the nth job on the ruth machine (n --- 1,2 . . . . . N; m = 1,2 . . . . , M); N denotes the set of n jobs and M denotes the set of m machines.
It is assumed that the job processing times satisfy the following relationships:
PT(n + 1, m)
Multi-product flowshop scheduling 363
SCHEDULING AN ORDERED FLOWSHOP WITH NATURAL SEQUENCE OF JOBS
It can be shown that for conditions (1)-(4), stated above, the natural sequence of the jobs (i.e. 1,2, . . . , N) gives an optimal solution of the problem to minimize the makespan (throughput time), provided the sequence thus obtained satisfies the condition . A proof of the optimal sequence algorithm is given in the Appendix. The generated sequence for the situation in Table 1 yields the makespan of 584 units, and satisfies condition (4). Thus the natural sequence of jobs, in the present case, results into minimum throughput time.
It has been shown that for the special class of ordered flowshop problems described in this paper. The natural processing of the jobs in the natural order on different machines leads to minimum throughput time, provided the sequence thus generated satisfies the idle time constraint.
1. R. A. Dudek and O. F. Tetuton Jr. Development of M-stage decision rule for scheduling n jobs through M machines. Opl Res. 12, 417-497 (1964).
2. J. N. D. Gupta. A search algorithm for generalized flowshop scheduling problems. Comput. Opns Res. 2, 83-90 (1975).
3. J. R. King and A. S. Spachis. Heuristics for flowshop scheduling. Int. J. Prod. Res. 18(3), 345-356 (1980). 4. M. L. Smith, S. S. Panwalkar and R. A. Dudek. Flowshop sequencing problem with ordered processing time
matrices. Mgmt Sci. 21(5), 544-549 (1975). 5. S. S. Panwalkar and A. W. Khan. An ordered flowshop sequencing problem with mean completion time criterion. Int.
J. Prod. Res. 14(5), 631-635 (1976). 6. M. L. Smith, S. S. Panwalkar and R. A. Dudek. FIowshop sequencing problem with ordered processing time
matrices, a general case. Nay. Res. Log. Q. 23, 481--486 (1976).
Let N represent the set of n jobs and M represent the set of m machines; and PT(n,m) is the processing time of the nth job on the mth machine. Let,
Tom = 0form E M.
Thus we have,
PT(n + 1, m)
364 R. C. MISHRA et al.
and N N
7"-~ E T~u + E IT'iM. (A.6) i~ l i=1
From equations (A.5) and (A.6) it can be seen that the first terms in these equations are the same and constant. However, the second terms are different and these decide the magnitude of the makespan. As per equation (A.4), ITnu