Minimizing the weighted sum of maximum earliness and maximum tardiness in a single-agent and two-agent form of a two-machine flow shop scheduling problem

In the past decade, multi-agent scheduling studies have become more widespread. However, the evaluation of these issues in the flow shop scheduling environment has received almost no attention. In this article, we investigate two problems. One problem is a common due date problem of constrained two-agent scheduling of jobs in a two-machine flow shop environment to minimize the weighted sum of maximum earliness and maximum tardiness of first-agent jobs and restrictions of non-eligibility on the tardiness of second-agent jobs. Another problem is a single-agent form of the two-agent problem when the number of second-agent jobs is zero. So, an optimal algorithm with polynomial time complexity is presented for the single-agent problem. For the two-agent problem, after it was shown to have minimum complexity of ordinary NP-hardness, a branch and bound algorithm, based on efficient lower and upper bounds and dominance rules, was developed. The computational results show that the algorithm can solve the large-size instances optimally.