Flowshop Sequencing Problem with Ordered Processing Time Matrices

Abstract

A flowshop sequencing problem having an ordered processing time matrix is defined. Job and machine characteristics resulting in processing time relationships that have logical and practical bases are discussed. An optimizing solution procedure for a special class of ordered matrix problem is presented along with proof of optimality.

A concise survey of efficiently solvable special cases of the permutation flow-shop problem

“…We conclude our survey by showing how the ideas presented thus far apply to so-called ordered flow-shop problems [35,36]. The jobs of a flow-shop are said to be ordered if there exists a permutation n of the jobs such that Pn(i)j^Pn (2)…”

“…We shall prove that a permutation n is optimal whenever the jobs are ordered from the largest to the smallest [35].…”

Min-Max Regret Version of an m-Machine Ordered Flow Shop with Uncertain Processing Times

“…Johnson (1954) presented a seminal algorithm, referred to as Johnson's rule, that efficiently solves the problem of minimizing the makespan in a two-machine flow shop. Since the problem of minimizing the makespan in a flow shop with more than two machines is strongly NP-hard (Garey and Johnson, 1976), however, researchers have considered a flow shop with special structures for processing times, such as the ordered flow shop (Smith et al, 1975(Smith et al, , 1976 and the proportionate flow shop (Allahverdi, 1999;Hou and Hoogeveen, 2003;Ow, 1985). Smith et al (1975Smith et al ( , 1976 considered an m-machine ordered flow shop.…”

“…Since the problem of minimizing the makespan in a flow shop with more than two machines is strongly NP-hard (Garey and Johnson, 1976), however, researchers have considered a flow shop with special structures for processing times, such as the ordered flow shop (Smith et al, 1975(Smith et al, , 1976 and the proportionate flow shop (Allahverdi, 1999;Hou and Hoogeveen, 2003;Ow, 1985). Smith et al (1975Smith et al ( , 1976 considered an m-machine ordered flow shop. Smith et al (1976) showed that an optimal schedule has an inverted V-shape, which is defined as a sequence in which all jobs scheduled before the job with the longest processing time are in shortest processing time (SPT) order and all jobs scheduled after the job with the longest processing time are in longest processing time (LPT) order.…”

“…Smith et al (1976) showed that an optimal schedule has an inverted V-shape, which is defined as a sequence in which all jobs scheduled before the job with the longest processing time are in shortest processing time (SPT) order and all jobs scheduled after the job with the longest processing time are in longest processing time (LPT) order. Smith et al (1975) considered the case where the job processing times on the first (last) machine are the longest and showed that the problem can then be solved in polynomial time. They showed that if the first (last) machine is the slowest, the LPT (SPT) order is an optimal permutation schedule.…”