A novel hybrid bee algorithm for two-machine flow-shop scheduling problems with batch delivery

Abstract

Because of high costs for the delivery, manufacturers are usually required to dispatch their products in a batch delivery system. However, using such a system leads to some negative effects, such as increasing the number of tardy jobs. The current paper aims to investigate the two-machine flow-shop scheduling problem where jobs are processed in series on two stages, and then to be dispatched to customers in batches. The objective is to minimize the batch delivery cost and tardiness cost, related to the number of tardy jobs. First, a mixed-integer linear programming (MILP) model is proposed to explain the problem. As this is an NP-hard problem, the MILP model cannot solve large-size problems in a reasonable CPU running time. To solve large-size instances, some metaheuristic algorithms are provided, including Bee Algorithm (BA), Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and a novel Hybrid Bees Algorithm (HBA). Using Friedman and Wilcoxon signed-ranks tests, then, these intelligent algorithms are compared and the results are analyzed. The results indicate that the HBA provides the best performance for large-size problems.