A single-machine batch scheduling problem is investigated. Each job has a positive processing time and due-date. Setup times are assumed to be identical for all batches. All batch sizes cannot exceed a common upper bound. As in many practical situations, jobs have to be subject to flexible precedence constraints. The aim of this paper is to find an optimal batch sequence. The sequence is to minimize the maximal completion time and maximize the minimum value of desirability of the fuzzy precedence. However, there usually exists no batch sequence optimizing both objectives at a time. Therefore, we seek some non-dominated batch sequences after the definition of non-dominated batch sequence. Based on an iterative Procedure HL proposed by Cheng et al., an efficient algorithm is presented to find some non-dominated batch sequences.
This paper considers a variant of the bottleneck transportation problem. For each supply-demand point pair, the transportation time is an independent random variable. Preference of each route is attached. Our model has two criteria, namely: minimize the transportation time target subject to a chance constraint and maximize the minimal preference among the used routes. Since usually a transportation pattern optimizing two objectives simultaneously does not exist, we define non-domination in this setting and propose an efficient algorithm to find some non-dominated transportation patterns. We then show the time complexity of the proposed algorithm. Finally, a numerical example is presented to illustrate how our algorithm works.