Minimizing makespan on parallel batch processing machines with two-dimensional rectangular jobs

Abstract

This paper addresses the parallel batch processing machines scheduling problem with two-dimensional bin packing constraints. One typical application of this problem is last-mile truck delivery. A set of rectangular items (or jobs) that have delivery times are processed by a set of homogeneous vehicles (or machines). Each vehicle can deliver a batch of items at a trip following a fixed route as long as these items in the batch can be placed onto the vehicle without overlapping. The total delivery time of a batch is determined by the longest delivery time of the items in the batch. The problem is to allocate the items, which may be rotated, into batches and then assign the batches to the vehicles so as to minimize the makespan. A mixed integer programming model of the problem is formulated. A number of heuristic algorithms are proposed and compared. Some heuristics for two-dimensional packing problems are adapted to the problem. The performance of the algorithms is evaluated based on computational experiments. The experimental results show that the best-fit maximal rectangle (BFMR) heuristic is the best, in which the batch with the smallest remaining capacity is selected for an item, the empty maximal rectangle with the least space in the batch is chosen to place the item, and finally the generated batches are allocated to the vehicles according to the longest batch delivery time algorithm.