Parallel batch scheduling: Impact of increasing machine capacity

•Study the impact of increasing capacity on p-batch machines of minimizing makespan.•Define two performance ratios to quantify the impact of increasing capacity.•Deduce upper bounds of the two ratios to evaluate the impact of increasing capacity.•Design an optimal algorithm to find optimal capacity for preemptive case.•Develop an algorithm to decide capacity and total costs for non-preemptive case. Scheduling performance naturally improves with increased machine capacity, but the per-unit improvement typically decreases or even keeps unchanged with excessive capacity. We consider parallel batch scheduling on identical machines to minimize the makespan, with and without preemption. The machine capacity is the maximum number of jobs that a machine can process simultaneously. We quantitatively analyze the impact of capacity augmentation on the makespan in the form of increasing machine capacity. Considering machines costs, we need to determine the machine capacity that minimizes the weighted sum of the makespan and capacity costs. First, we obtain an upper bound of the ratio between the minimum makespans of two scheduling problems with different machine capacities. Second, noting the intractability of the scheduling problem without preemption, we analyze the upper bound of the ratio between the obtained makespans by heuristics of two scheduling problems with different machine capacities. Third, for the preemptive case, we develop a polynomial time algorithm to obtain the optimal machine capacity, and also present another polynomial time algorithm to obtain the optimal machine capacity as well as for bounding the approximation ratio of the non-preemptive case. Fourth, we design an approximation algorithm using machine capacity found in the preemptive case to yield a schedule for the non-preemptive case, and analyze the worst case performance ratio of the algorithm. This research provides new insights into performance improvement with increased machine capacity in parallel batch scheduling, and analyzes the trade-off between the makespan and capacity costs.