ON EQUIVALENT-JOB FOR JOB - BLOCK IN 2X n SEQUENCING PROBLEM WITH TRANSPORTATION-TIMES

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1 Journal of the Operations Research Society of Japan Vol. 24, No. 2, June The Operations Research Society of Japan ON EQUIVALENT-JOB FOR JOB - BLOCK IN 2X n SEQUENCING PROBLEM WITH TRANSPORTATION-TIMES Park ash Lal Maggu M. s. College Ghanshiam Das M. S. College Ravinder Kumar M. S. College (Recei\'ed January 2, 980; Final July, 980) Abstract This paper has the objective to obtain optimal schedule principle for 2 x n sequencing problem wherein transportation times of jobs and equivalent-jobs for job-blocks are assumed to occur. The optimal schedule rule is based upon a theorem as established in the paper.. Introducti on Beman [] and Johnson [2] considered a problem involving the scheduling of n jobs on two machines. Mitten [6, 7] and Johnson [] treated a scheduling problem with arbitrary time lags. Maggu and Das [4] established "equivalent -job for job-blocks theorem" for 2xn sequencing problem. In the '2-machine, n-job' makespan problem the concept of equivalent jobs for blocks in job sequencing was introduced by Maggu and Das as follows: Consider the jobsequence S = (a, a, a,. a ) of n jobs with the condition that jobs a and 2 n k a + must occur in the sequences as a block, Le., if a is the i-th job then k k a + must be the (i+)-th job. Now it is possible to define a job S (say) with k processing times tsa and t on two machine A and B respectively which can lb replace the jobs a and ak-~l for the purpose of finding the minimum schedule k time. When S replaces a and a + to produce a new sequence S', the completion times on both machines is changed by a value which is independent of the k k particular sequence S. Hence the substitution does not change the relative merit of different sequences. 6

2 Equivalent-Job with Transportation-Times Sequencing 7 Further, Maggu and Das [5J considered 2xn flowshop problem wherein transportation times of jobs from one machine to another are assumed to occur. In all the preceding papers, except [5J, the transportation times of jobs from one machine to another are neglected. Owing to theoretical and as well practical interest this paper has the object of providing a decomposition algorithm to obtain optimal schedule of jobs for the case of 2-machine n-job flow-shop problem wherein a block of ordered jobs is assumed to be an equivalent to a single job and, furthermore, the transportation times of jobs from one machine to another subsequent machine are given. Now to mee,t the obj ect of this paper to develop an algorithm producing an optimal schedule, for the two-machine n-job flow-shop problem, minimizing the total elapsed time we prepare the basis for the same in the form of a theorem as follows. 2. Description of Problem Theorem: Consider a flow-shop problem consisting of two machines A and B, and a set of no-jobs to be processed on these machines. We are given processing time tix for each job i, on machine X = A, B. Each machine can handle at most one job at: a time and the processing of each job must be finished on machine A before it can be processed on machine B. the order of treatments in the process A and B are the same. the transportal:ion time of job i from machine A to B. It has been assumed that Let t i denote! In the transportation process, several jobs can be handled simultaneously. Let be an equivalent job for a given ordered set of jobs (a k, a k + l ). Then processing times tla and tsb on machines A and B are given by tsa (t A + ; ) + (t + t ) min(t A + t,t +t) a k a k ak+ia ak+ a k + a k + akb a k tsb = (t B +,; ) + (t + t ) - min(t B + t, t ak + A + t ) a k a k ak+ib a k + a k a k a k + l and the transportation time of S from machine A to B is given by ts=o. Proof: Consider the two sequences 8 and S' of jobs as: S (ai' a 2, a,... a k _ l, a k, o'k+l' a k + 2,... a n ) Let T ix d'~note completion time of job i on machine X, when jobs are processed according to S. Let T~X denote! completion time of job i on mach:lne

3 8 P.L. Maggu. G. Das and R. Kumar X, when jobs are done according to S' Now it is clear to see that T = max (T A + t a. ' T ) + t o.kb o.k. k o.k_lb o.kb max (T A + t + to. B' T + t B) a./( o. k k o.k_lb o. k T = max (T + o.k+lb o.k+la t, T ) + t o.k+l o.kb o.k+lb max (T + t, T + t + t o.kb o.k+la o.k+l o.ka a k Again, we have T o.k + 2 B max (T o.k+2 A + t, T ) + t o.k+2 o.k+lb o. k + 2 B max (T + t, T + t + t o.k+lb ' o.k + 2 A a k +2 o.k+la o. k + l max Now,

4 Equivalent-job with Transportation-Times Sequencing 9 Hence max (T A + t uk+ 2 uk+ 2 () Similarly for sequence S' we have T'SB = max (T'SA + ts' T'U k _ B) + tsb max (T'SA + ts + tsb l"u k _ B + t SB ) max max (T' A + t T'OA + to + tab u k + 2 u k + 2 I-' I-' I-' (2) Now without loss of generality one can assume easily: tsa (tu A + t ) + (t + t ) - C u k k u k + A u k + ts o tsb (t B + t ) +. (t B + t ) - C u k u k u k + uk+ with C min (t + t t E' uk+l + Uk+ u t ) k ' U k Now T uk _ A T B U k _ T uk _ B T' uk _ A + tsa + t uk+l T + (t A + t U ) + (t + t ) - C u k _ A U k k u k + A u k + + t uk+za T + t + t uk + A + t + (t + t - C) u k _ A uka uk+l u k u k + T uk + 2 A + (t uk + t uk + - C) ()

5 40 P.L. Maggu, G. Das and R. Kumar Hence (t B + t ) + (t B + t ) - C u k u k u k + u k + Now min max Hence T' u _ B + tab fj k Substituting in (2), we have

6 Equiva/ent-Job with Transportation-Times Sequencing 4 max max (4) From () and (4), we have (5) Let D =, t + t - C a k a k +l Then from () and (5), we have T' C!k+2 B T' C!k+2 A T T a k + 2 B a k + 2 A + D + D (6) (7) From (6) and (7), it is clear that replacement of job-block (a k, a k + l ) in S by job f increases the completion times of later job a k + by a constant D in 2 S' as compared for that job (Le. a + ) in S. Let T and T' be the completion k 2 times of sequences Sand S' respectively. Then from the above discussion., it is easily observed that T' = T + D. Hence we can replace a single job f as equivalent to the job-block (a k, a k + ) of the given ordered job-pair a l k, ak+l in S.

7 42 P.L. Maggu, G. Das and R. Kumar Note : Extension of the equivalent - job concept of a block consisting of an ordered set of k?2 jobs is obvious. Note 2: Following as in [4] it can be indicated that equivalency is associative but not co~nutative i.e.. Decomposition Algorithm The above theorem gives a numerical method to obtain an optimal sequence for the 2 x n sequencing problem wherein concepts of Equivalent-job for jobblock and transportation times of jobs are involved. Let the given problera in the tableau form be described as follows: job Machine A (i) (A.) "lt A t. "l- Bl Machine B 2 t,) A, t'l A B2 B n ---- t A n n B n (B.) "lwhere t. denotes the transportation time for job i from machine A to B, and "l- A., B. (as usual) denote the processing times of job i on machines A and B "l- "lrespectively. Let S = (r, s,..., u) be an equivalent-job of an ordered block of jobs r, s,..., u. Then the algorithm to determine optimal schedule minimizing total elapsed time is decomposed into following steps: Step (): Step (2): Obtain processtng-times on machines A and B for each equivalentjob of a job-block by using above theorem. Also find the transportation time of each equivalent job. Now form a reduced problem replacing the given job-block in the original proble!m by their equivalent-jobs. Step (): Let G and H be fictitious machines. Then form a new reduced problem from step 2, where G i and Hi are the processing times of job

8 Equivalent-lob with Transportation-Times Sequencing 4 i on G and H defined by G. -z, H. -z, A. + t. -z, -z, B. + t. -z, -z, Step (4): Find optimal sequence for the, reduced problem in step. Step (5): In the optimal sequence of step (4) replace back each equivaentjob by their ordered job-block. Now this sequence gives us the! optimaity for the original problem. Justification of the Algorithm: Step () is justified from theorem. FroDl step (2) we obtain a reduced two machine: problem with intermediate transportation process equivalent to our origina. problem. When intermediate transportation exists under the assumption that several jobs can be handled simultaneously in the transportation process, the problem can be reduced to a two process problem with fictitious ma.chines (which justifies the step () in the algorithm), a fact clearly stated in the references [] (p.2), [2] (p.5l) and [';] (p.4). Using [2] we can. obtain optimal sequence of reduce!d problem as per step. Again step (5) is justified from the theorem. Numerical Example: Consider the following tableau form of a 2X5 flow-shop problem where symbols ti' Ai' Bi have the usual meanings as defined above. job Machine A t. Machine -z, (i) (A.) -z, (B.) -z, 6 B Let S = (,, 5). Then optimal sequence is through 5 of the Decomposition Aogrithm. Let 0 = (, ) Then S = (0, 5). obtained by following steps. Now as per step (.):

9 44 P.L. Maggu. G. Das and R. Kumar E (2 + ) + (5 + 6) - min (5 + 6, 6 + ) (6 + ) + (7 + 6) - min (5 + 6, 6 + ) Again (7 + 0) + (9 + 2) min ( + 0, 9 + 2) ( + 0) + (8 + 2) min ( + 0, 9 + 2) By step (2) replacing job-block (,, 5) by equivalent job S, the reduced problem is: job (i) Machine A (A.) '.. t.,_--,'.. Machine (B) (B.) '.. 7 o By step (), let G and H be fictitious machines then the new reduced problem is

10 Equivalent-Job with Transportation-Times Sequencing 45 job Machine C Machine H (i) (C.).- (H.) where C. = A. + t., H. B. + t By step (4), the optimal sequence of newly reduced problem due to Johnson's procedure is (, 4, 2) or (4, S, 2). By step (5), replacing back (,, 5, 4, 2) or (4,,, 5, 2). = (,. 5), we have optimal sequence The total elapsed time T for each optimal sequence is calculated in the following tableau forms: job t. Machine A ---=.- Machine B (i) in - out in - out o S _ o T) 4 :r : _ T). 4. Observation It may be observed that the problem dealt in this paper is thus basically the two process problem under the transportation process as studied in theorem (c.l []) wherein the concept of Block-job has been equipped with. It may be noted that here the job-block is formed equivalent to a given set of

11 46 P.L. Maggu. G. Das and R. Kumar preordered jobs. 5. Acknowledgement The authors are thankful to the refrees for their critical and useful comments to revise the paper. References [] Bellman, R.: Mathematical Aspects of Scheduling Theory. J. Soc. Ind. and Appl. Math., Vo.4, 956. [2] Johnson. S. M.: Optimal Two-and three-stage production schedules with set up times included. Nov. Res. Log. Quart., Vo., 954. [] Johnson, S. M.: Discussion: Sequencing n jobs on two machines wi.th arbitrary time lags. Management Science, Vo.5, 959. [4] Maggu, P. L. and Das, G.: Equivalent jobs for job-blocks in job sequencing. OPSEARCH, Vo.4, 977. [5] Maggu, P. L. and Das, G.: On 2xn Sequencing Problem wi.th Transportation Times of jobs. (Accepted) Pure Appl. Math. Sci., Vo.XII, No.-2, (980) [6] Mitten, L. G.: Sequencing n jobs on two machines with arbitrary time lags. Management Science, Vo.5, 959. [7] Mitten, L. G.: A scheduling problem. J. Ind. Eng., Vo.0, 959. Parkash La MAGGU: Post Box 65, Saharanpur 24700, India