Optimal Operating Policy for a Bottleneck with Random Rework

This paper presents a model of a bottleneck facility that performs two distinct types of operations: "regular" and "rework." Each job is subjected to a test after completing the regular operation at the bottleneck. If the job passes the test, then it continues its process downstream. Otherwise, the job will cycle back to the bottleneck stage for rework operation. Upon the completion of a batch of regular jobs, the decision maker observes the amount of rework and decides on whether to switch over to process the reworks or continue to process another batch of regular jobs. It is assumed that both switch-over time and cost are incurred when the facility switches from performing one type of operation to a different type. The goal of the analysis is to characterize the optimal operating policy for the bottleneck so that the average operating cost is minimized. In order to characterize the optimal operating policy, we first formulate the problem as a semi-Markov decision process. Then we show that there exists an optimal "threshold" operating policy that can be described as follows: upon completion of a batch of regular jobs, switch over to process the reworks only if the number of reworks exceeds a critical value. In addition, we develop a simple procedure to compute the critical value that specifies the optimal threshold policy. Moreover, we evaluate the impact of batch sizes, yield, and switch-over time on the optimal threshold policy.