Optimizing batch operations with batch-position-dependent learning effect and aging effect

Motivated by applications in the metal pickling process, we study the optimization scheduling problems of batch operations with batch-position-dependent learning effect and aging effect. In production, a set of metal workpieces are pickled on a fixed capacity single batch facility. If the pickling solution is regarded as the machine, the processing time of the metal work-pieces will be longer and increase in a power function because of the aging effect. At the same time, workers need to heat the solution. A worker can finish the task more and more quickly because of the learning effect, resulting in shorter processing time which decreases in a power function. Then we consider three models of batch operations. In the first model, the work-pieces have identical sizes and an optimal algorithm is proposed with time complexity of O(nlogn). In the second model, the work-pieces have identical processing time and it is shown that the model is NP-hard in the strong sense. Then, we propose an approximation algorithm. The absolute and asymptotic worst-case ratios of the algorithm are 2 and 1.494. In the third model, the work-pieces have arbitrary processing time which is proportional to their sizes and the model is also NP-hard in the strong sense. Finally, an approximation algorithm is proposed with an absolute and asymptotic worst-case ratio is less than