A Genetic Algorithm to Solve the Hybrid Flow Shop Scheduling Problem with Subcontracting Options and Energy Cost Consideration

This paper analyses the hybrid flow shop scheduling problem (HFSSP) with subcontracting options and time depending energy costs. While the consideration of energy costs in scheduling has increased considerably in recent years, subcontracting is rarely analysed in scheduling literature. A mathematical MILP formulation is given to define the exact problem and to calculate optimal solutions for small instances. The objective is to minimise the total production costs for internal and external manufacturing including transportation and energy costs. Since, already the general HFSSP is NP-hard the considered problem is difficult to solve to optimality. Therefore, a genetic algorithm (GA) based on a detailed matrix encoding procedure is proposed. To the best of my knowledge this is the first time that a heuristic approach is presented for the considered problem. An algorithm for intelligent swaps to make use of waiting time and a right-shifting procedure to take advantage of time depending energy costs prove to be suitable to improve the performance of the GA significantly. It can be shown that the GA finds nearly optimal solutions in a very short time.