A hybrid two-phase encoding particle swarm optimization for total weighted completion time minimization in proportionate flexible flow shop scheduling

A proportionate flow shop (PFS) is a special case of the m machine flow shop problem. In a PFS, a fixed sequence of machines is arranged in s stages (s > 1) with only a single machine at each stage, and the processing time for each job is the same on all machines. Notably, PFS problems have garnered considerable attention recently. A proportionate flexible flow shop (PFFS) scheduling problem combines the properties of PFS problems and parallel-identical-machine scheduling problems. However, few studies have investigated the PFFS problem. This study presents a hybrid two-phase encoding particle swarm optimization (TPEPSO) algorithm to the PFFS problem with a total weighted completion time objective. In the first phase, a sequence position value representation is designed based on the smallest position value rule to convert continuous position values into job sequences in the discrete PFFS problem. During the second phase, an absolute position value representation combined with a tabu search (TS) is applied starting from the current position of particles that can markedly improve swarm diversity and avoid premature convergence. The hybrid TPEPSO algorithm combines the cooperative and competitive characteristics of TPEPSO and TS. Furthermore, a candidate list strategy is designed for the TS to examine the neighborhood and concentrate on promising moves during each iteration. Experimental results demonstrate the robustness of the proposed hybrid TPEPSO algorithm in terms of solution quality. Moreover, the proposed hybrid TPEPSO algorithm is considerably faster than existing approaches for the same benchmark problems in literature.