Abstract
In this paper, we study a new type of flow shop and open shop models, which handle so-called “pliable” jobs: their total processing times are given, but individual processing times of operations which make up these jobs are flexible and need to be determined. Our analysis demonstrates that many versions of flow shop and open shop problems with pliable jobs appear to be computationally easier than their traditional counterparts, unless the jobs have job-dependent restrictions imposed on minimum and maximum operation lengths. In the latter case, most problems with pliability become NP-hard even in the case of two machines.
Highlights
In traditional flow shop and open shop models, n jobs of the set J = {1, 2, . . . , n} are processed by m machines Mi, 1 ≤ i ≤ m
We studied general properties of pliability models and performed a thorough complexity classification of flow shop and open shop problems with pliable jobs
For the unrestricted model with f = Lmax, we have a faster algorithm for the open shop problem; for the models with a common lower bound and f ∈ {Cmax, Lmax}, the flow shop problems are polynomially solvable, but the complexity status of the open shop counterparts is left as an open question
Summary
1 ≤ j ≤ n, consists of m operations Oi j , one operation on each machine Mi , with processing times pi j , 1 ≤ i ≤ m. The goal is to select an order of operations on Both models, flow shop and open shop, have a long history of study, see, e.g., Brucker (2007), Pinedo (2016). Pieces of the same job processed by two machines cannot overlap in time, while in the case of lot splitting or lot streaming overlapping may happen if the pieces belong to different sublots. Jobs typically consist of more than m operations, some of which are fixed and have to be processed by dedicated machines while others are flexible and need to be assigned to one of the appropriate machines. Operation parts can be moved to neighboring machines if those machines are equipped to handle them
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