Lagrangian Relaxation Algorithms for Re-entrant Hybrid Flowshop Scheduling

This paper focuses on a re-entrant hybrid flowshop scheduling (RHFS) problem with the objective of minimizing the sum of weighted completion time of jobs. In the re-entrant hybrid flowshop considered here, there are several stages, each with identical parallel machines. This problem is strongly NP-hard since it is more complicated than general hybrid flowshop which is already proven to be NP-hard. We present the first implementation of the Lagrangian Relaxation (LR) for the problem. The complication and time-consumption of solving all the subproblems at each iteration in subgradient optimization motivate the development of the surrogate subgradient method where only one subproblem is minimized at each iteration and an adaptive multiplier update scheme of Lagrangian multipliers is designed. The computational experiments are performed on randomly generated test problems and the results demonstrate that the proposed method can solve the problem effectively in a reasonable amount of the computation time.