Pareto-optimization of three-agent scheduling to minimize the total weighted completion time, weighted number of tardy jobs, and total weighted late work

We consider three-agent scheduling on a single machine in which the criteria of the three agents are to minimize the total weighted completion time, the weighted number of tardy jobs, and the total weighted late work, respectively. The problem is to find the set of all the Pareto-optimal points, that is, the Pareto frontier, and their corresponding Pareto-optimal schedules. Since the above problem is unary NP-hard, we study the problem under the restriction that the jobs of the first agent have inversely agreeable processing times and weights, that is, the smaller the processing time of a job is, the greater its weight is. For this restricted problem, which is NP-hard, we present a pseudo-polynomial-time algorithm to find the Pareto frontier. We also show that, for various special versions, the time complexity of solving the problem can be further reduced.