Single machine batch scheduling with two competing agents to minimize total flowtime

We study a single machine scheduling problem, where two agents compete on the use of a single processor. Each of the agents needs to process a set of jobs in order to optimize his objective function. We focus on a two‐agent problem in the context of batch scheduling. We assume identical jobs and identical (agent‐dependent) setup times. The objective function is minimizing the flowtime of one agent subject to an upper bound on the flowtime of the second agent. As in many real‐life applications, we restrict ourselves to settings where the batches of the second agent must be processed continuously. Thus, the batch sizes are partitioned into three parts, starting with a sequence of the first agent, followed by a sequence of the second agent, and ending by another sequence of the first agent. In an optimal schedule, all three are shown to be decreasing arithmetic sequences. We introduce an efficient $O(n^{\frac{3}{2}})$ solution algorithm (where n is the total number of jobs).