Scheduling with general position-based learning curves

Learning effect in scheduling problems has received growing attention since Biskup introduced the position-based model, where the learning curve is expressed as a power function of a job position. Hurley pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, the learning rates show considerable variation within industries or firms. The variation extends not only across firms at a given time, but also within firms over time. For instance, the learning curves usually have an initial downward concavity, and no further improvements are made after some amount of production. Beside the standard power model, learning curve is seldom discussed in scheduling. In this paper, we offer a surprising simple yet realistic learning effect model which has the flexibility to describe different learning curves easily. For instance, the standard power model, the well-known S-shaped and the plateau functions are special cases of the proposed model. We then present the optimal solution for some scheduling problems.