確率制約を伴う機械スケジューリング問題の解法

Abstract

We consider an n-job one machine scheduling problem in which the processing time of each job i is a random variable subject to a normal distribution N(mi' v() and the object is to maximize the weighted number of early jobs subject to the constraint that some specified jobs must be early. It is assumed that mp < mq implies v~ ~ vJ, where mi and vl are, respectively, a known mean value and a known variance associated with each job i. If such constraint is relaxed, the problem has been shown to be NP-complete, suggesting strongly that there exists no efficient exact algorithm whatever for the problem. Moreover, it is assumed that if mp < mq or v~ < vJ, then wp ~ W q' where wi is a known weight associated with each job i. It is well known for the problem with arbitrary weights to be NP-complete even in the deterministic case (Le., vl = 0). We show that the problem with the above assumptions can be solved in O(n2) time and that it has a practical application. ' ~ ~ a nownmean