A Flexible Branch and Bound Method for the Job Shop Scheduling Problem

This paper deals with the makespan minimization problem of job shops. The problem is known as one of hard problems to optimize, and therefore, many heuristic methods have been proposed by many researchers. The aim of this study is also to propose a heuristic scheduling method for the problem. However, the difference between the proposed method and many other heuristics is that the proposed method is based on depth-first branch and bound, and thus it is possible to find an optimal solution at least in principle. To accelerate the search, when a node is judged hopeless in the search tree, the proposed flexible branch and bound method can indicate a higher backtracking node. The unexplored nodes are stored and may be explored later to realize the strict optimization. Two methods are proposed to generate the backtracking point based on the critical path of the current best feasible schedule, and the minimum lower bound for the makespan in the unexplored sub-problems. Schedules are generated based on Giffler and Thompson's active schedule generation algorithm. Acceleration of the search by the flexible branch and bound is confirmed by numerical experiment.