Scheduling of Jobs with Multiple Weights on a Single Machine for Minimizing the Total Weighted Number of Tardy Jobs

We consider the scheduling of jobs with multiple weights on a single machine for minimizing the total weighted number of tardy jobs. In this setting, each job has m weights (or equivalently, the jobs have m weighting vectors), and thus we have m criteria, each of which is to minimize the total weighted number of tardy jobs under a corresponding weighting vector of the jobs. For this scheduling model, the feasibility problem aims to find a feasible schedule such that each criterion is upper bounded by its threshold value, and the Pareto scheduling problem aims to find all the Pareto-optimal points and for each one a corresponding Pareto-optimal schedule. Although the two problems have not been studied before, it is implied in the literature that both of them are unary NP-hard when m is an arbitrary number. We show in this paper that, in the case where m is a fixed number, the two problems are solvable in pseudo-polynomial time, the feasibility problem admits a dual-fully polynomial-time approximation scheme, and the Pareto-scheduling problem admits a fully polynomial-time approximation scheme.