A multi-objective discrete differential evolution algorithm for energy-efficient two-stage flow shop scheduling under time-of-use electricity tariffs

In response to the appeal for environmental protection and the efficient utilization of energy, local governments in China actively promote time-of-use (TOU) electricity tariffs for manufacturing enterprises. Motivated by a real-life industrial scenario of large special-purpose pressure vessel production, this paper addresses an energy-efficient two-stage flow shop scheduling problem under TOU tariffs to minimize the total electricity cost and the mean tardiness. Since the problem is computationally intractable, we focus on developing a multi-objective discrete differential evolution (MDDE) algorithm. Specifically, based on the optimal properties of the problem, we tailor an encoding scheme that consists of two job sequences and an idle time vector. The novel mutation and crossover operators are designed to generate the trail individuals, and the hypervolume contribution indicator is incorporated into the bi-criteria selection operator to measure the quality of the solution. Furthermore, two neighborhood structures are designed to iteratively improve the non-dominated solutions in the external archive set. We evaluate the performance of the MDDE algorithm via extensive computational experiments. The experimental results indicate that the MDDE algorithm can obtain the good approximate Pareto front, and it outperforms the well-known NSGA-II, SPEA2 and MOEA/D algorithms in solution quality.