We consider the single‐machine Pareto‐scheduling problem to minimize the weighted number of tardy jobs and total weighted late work simultaneously. The problem is to find the set of all the Pareto‐optimal points, that is, the Pareto frontier, and their corresponding Pareto‐optimal schedules. We consider the corresponding weighted‐sum scheduling problem and primary‐secondary scheduling problems, being subproblems of the general Pareto‐scheduling problem. The NP‐hardness of the general problem follows directly from the NP‐hardness of the two constituent single‐criterion problems. We present a pseudo‐polynomial algorithm and a fully polynomial‐time approximation scheme (FPTAS) running in weakly polynomial time to deal with the general problem. When all the jobs have a common due date, we further provide an FPTAS running in strongly polynomial time. We also study some special cases of the general problem where the jobs have equal processing times, a common due date, or a common weight, and analyze their computational complexity status.

<p style='text-indent:20px;'>In this paper, we study the single-machine Pareto-scheduling of jobs with multiple weighting vectors for minimizing the total weighted late works. Each weighting vector has its corresponding weighted late work. The goal of the problem is to find the Pareto-frontier for the weighted late works of the multiple weighting vectors. When the number of weighting vectors is arbitrary, it is implied in the literature that the problem is unary NP-hard. Then we concentrate on our research under the assumption that the number of weighting vectors is a constant. For this problem, we present a dynamic programming algorithm running in pseudo-polynomial time and a fully polynomial-time approximation scheme (FPTAS).</p>

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.