SINGLE MACHINE FAMILY SCHEDULING WITH TWO COMPETING AGENTS TO MINIMIZE MAKESPAN

We consider two-agent scheduling on a single machine, where there are job families and setup requirements exist between these families. Each agent's objective function is to minimize his own makespan. One of our goals is to find the optimal solution for one agent with a constraint on the other agent's makespan (constrained optimization). This problem is equivalent to the caudate Knapsack problem that we define in the paper. The other goal is to find single nondominated schedules (i.e., such that a better schedule for one of the two agents necessarily result in a worse schedule of the other agent), and to enumerate all nondominated schedules. Finally, two special cases, one with equal job processing times and the other with equal family setups are studied. We prove that the constrained optimization problems in both cases can be solved in polynomial time and that the cases have a polynomial number of nondominated schedules.