Single machine batch scheduling problem with fuzzy batch size

In a batch scheduling problem, jobs are grouped (group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. Though all batch sizes cannot exceed a common upper bound, the upper bound is flexible and satisfaction degree with respect to the upper limit to be maximized is given. Also the other two objectives, i.e., the maximum completion time and the flow-time are to be minimized. Usually there exists no solution optimizing three objectives at a time. Therefore we define non-dominated solutions consisting of batch size, batch number and allocation of jobs to batches. First we propose an efficient algorithm for a sub-problem with fixed upper limit of batch size, fixed batch number based on a lagrange relaxation procedure. Based on the properties of non-dominated solutions clarified in this paper, we propose an efficient algorithm to find some non-dominated solutions. Finally we summarize the results in this paper and discuss further research problems.