A Hybrid Bee Algorithm for Two-Machine Flow-Shop Scheduling Problems with Batch Delivery

Because of the high costs for the delivery, manufacturers are generally needed to dispatch their products in a batch delivery system. However, using such a system leads to some adverse effects, such as increasing the number of tardy jobs. The current paper investigates the two-machine flow-shop scheduling problem where jobs are processed in series on two stages and then 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 model (MILP) is proposed to explain this problem. Because the problem under consideration is NP-hard, the MILP model cannot solve large-size instances in a reasonable running time. Some metaheuristic algorithms are provided to solve the large-size instances, including BA, PSO, GA, and a novel Hybrid Bees Algorithm (HBA). Using Friedman and Wilcoxon signed-ranks tests, these intelligent algorithms are compared, and the results are analyzed. The results indicate that the HBA provides the best performance for large-size problems.