| Summary: | 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 <i>m</i> weights (or equivalently, the jobs have <i>m</i> weighting vectors), and thus we have <i>m</i> 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 <i>m</i> is an arbitrary number. We show in this paper that, in the case where <i>m</i> 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.
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