Reliability evaluation of a multistate flexible flow shop with stochastic capacity for multiple types of jobs

A flexible flow shop (FFS) is a general manufacturing system that has been studied by numerous researchers. Owing to maintenance, partial failure, the possibility of failure, unexpected situations, etc., the number of functioning machines in a stage should be represented by multiple levels. It is appropriate to regard the capacity in each stage (i.e., the number of machines in a stage) as stochastic. Unlike the previous research, which dealt with the FFS problems under the assumption of the fixed capacity, this paper extends the deterministic capacity to the stochastic case in every stage. The FFS with stochastic capacity is modeled as a multistate flexible flow shop network (MFFSN), where each edge denotes a stage with stochastic capacity and each node denotes a buffer. The addressed problem is to evaluate network reliability, the probability that the MFFSN can complete a customer's order composed of multiple types of jobs within a time threshold. An efficient algorithm integrating a systematic branch-and-bound approach is proposed to obtain the lower boundary vectors, in terms of a pair of capacity vectors generated from two estimated demand vectors. Two practical cases, a tile production system and an apparel manufacturing system, are presented to demonstrate the proposed algorithm and to discuss the changes in network reliability within different time thresholds, respectively.