Optimization of batch operations with a truncated batch-position-based learning effect

Motivated by applications in art tile manufacturing and metal working industries, we study the optimization problem with a truncated batch-position-based learning effect. In production, a set of semi-products need to be processed on a single batch facility which has a fixed capacity. Several semi-products can be processed together in one batch if their total size does not exceed the facility capacity. We consider a truncated batch-position-based learning effect which is a typical behavior of workers. During the learning period, the worker can finish the task more and more quickly because of learning effects. After the learning period, the worker reaches the best ability and the ability keeps steady. Then we consider two models of manufacturing with batch operations. In the first model, semi-products have identical sizes and we propose an optimal algorithm with time complexity of O(nlog n). In the second model, semi-products have arbitrary sizes which are proportional to their processing times and the model is shown to be NP-hard in the strong sense. We propose two types of learning effects including fast and slow truncated batch-position-based learning effects. Then we propose an approximation algorithm with an absolute and asymptotic worst-case ratio less than 2. Finally, we conduct computational experiments and the results show the effectiveness of our algorithms. We also provide managerial insights and detailed suggestions for decision makers of manufacturing companies based on our results.