Single machine scheduling with a learning effect and discounted costs

In this paper, we consider a single machine scheduling problem with a learning effect and discounted costs. The learning effect of a job is assumed to be a function of its position. We show that discounted total completion time is minimized by the classical shortest processing time first (SPT) rule. For the following objective function, discounted total weighted completion time, we show by an example that the optimal schedule of the classical discounted weighted shortest processing time first (WDSPT) rule is not optimal in the presence of a learning effect. But for some special cases, we prove that the WDSPT rule can construct the optimal sequence. We give the worst-case error bound for the WDSPT rule in the general case. Some extensions of the problem are also given.