Customer Order Scheduling Problem on Parallel Machines with Identical Order Size

This paper considers a scheduling problem where a customer orders multiple products(jobs) from a production facility. The objective is to minimize the sum of the order(batch) completion times. While a machine can process only one job at a time, multiple machines can simultaneously process jobs in a batch. Although each job has a unique processing time, we consider the case where batch processing times are identical. This simplification allows us to develop heuristics with improved performance bounds. This problem was motivated by a real world problem encountered by foreign electronics manufacturers. We first establish the complexity of the problem. For the two parallel machine case, we introduce two simple but intuitive heuristics, and find their worst case relative error bounds. One bound is tight and the other bound goes to 1 as the number of orders goes to infinity. However, neither heuristic is superior for all instances. We extend one of the heuristics to an arbitrary number of parallel machines. For a fixed number of parallel machines, we find a worst case bound which goes to 1 as the number of orders goes to infinity. Then, a tighter bound is found for the three parallel machine case. Finally, the heuristics are empirically evaluated.