Logic-based Benders decomposition for bi-objective parallel machine selection and job scheduling with release dates and resource consumption

Abstract

This work addresses a new bi-objective parallel machine selection and job scheduling problem with release dates and resource consumption. It consists in optimally selecting subcontractors (machines) from a set of geographically dispersed locations and scheduling the orders (jobs) to the selected subcontractors for processing while meeting the order release dates and resource consumption restrictions, so as to simultaneously minimize the maximum completion time, i.e., the makespan, and the total cost including machine usage cost and resource consumption cost. The problem is first formulated into a bi-objective mixed-integer linear program based on linear ordering (LO-MILP), and then valid inequalities are explored based on property analysis. To solve it, an ɛ-constraint method based on LO-MILP (ɛ-LO-MILP) is first proposed. To more efficiently solve it, we also develop a tailored logic-based Benders decomposition combined with ɛ-constraint method (ɛ-LBBD) where a novel method to obtain a tight lower bound of the identical parallel machine with machine-dependent release dates and some problem-specific cuts are proposed. Numerical experiments on an illustrative example are conducted to show the applicability of the model and algorithm and intuitively reveal the trade-off between production efficiency and cost. Experimental results on 200 instances with up to 100 orders demonstrate that ɛ-LBBD can reduce the computation time by about 28.92% compared to ɛ-LO-MILP and yields better Pareto solutions than ɛ-LO-MILP and the well-known non-dominated sorted genetic algorithm II do.