Quantum-Inspired Differential Evolutionary Algorithm for Permutative Scheduling Problems

In the world of combinatorial optimization, a wide range of combinatorial problems exist for which the solution is permutative, such as knapsack problem, traveling salesman problem (TSP), vehicle routine problem (VRP), quadratic assignment problem (QAP), dynamic pickand-place (DPP) model of placement sequence and magazine assignment in robots and various types of scheduling problems. The objectives for these problems are usually to find the best sequence to realize the optimal, for example, to minimize the maximum completion time of jobs on machines or to find the shortest routing between several cities. These problems are very different from the continuous space problem because we are interested in sequence such as [1 2 3 4 5] which is permutative in nature. The solutions include a permutation-based sequence as well as the fitness. It is not feasible to solve this kind of problem using the approaches that solve only continuous problems. As an important part of the permutation-based combinatorial optimization problems, the permutative scheduling problems (PSP) account for a large proportion of the production scheduling which is the core content of advanced manufacturing system. Among them, the flow shop scheduling problem (FSP) and job shop scheduling problem (JSP) may be the best known permutation-based scheduling problems. Both of FSP and JSP have earned a reputation for being a typical strongly NP-complete combinatorial optimization problem (Garey, et al., 1976) and have been studied by many workers due to their importance both in academic and engineering fields. By now, the meta-heuristic algorithms achieve global or sub-optimal optima within acceptable time range are most popular for dealing with the permutation-based scheduling optimization problem like flow shop and job shop scheduling. These approaches are initiated from a set of solutions and try to improve these solutions by using some strategies or rules. The metaheuristics for PSP include genetic algorithm (GA) (Reeves & Yamada, 1998; Goncalves, et al., 2005), immune algorithm (IA) (Doyen, et al., 2003; Xu & Li, 2007), tabu search (TS) (Nowicki & Smutnicki, 1996; Pezzella & Merelli, 2000), simulated annealing (SA) (Hisao, et al., 1995), ant colony optimization (ACO) (Ying & Liao, 2004; Zhang, et al., 2006), particle swarm optimization (PSO) (Tasgetiren, et al., 2004; Liao, et al., 2007; Xia & Wu, 2006), local search (Stützle, 1998), iterated greedy algorithm (Rubén & Stützle, 2007), differential evolution (Pan, et al., 2008), and other hybrid approaches (Zheng & Wang, 2003; Qian, et al., 2008; Hasan, et al., 2009). We should notice that these meta-heuristics can obtain satisfactory solutions, while