Pareto‐scheduling with double‐weighted jobs to minimize the weighted number of tardy jobs and total weighted late work

Abstract

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.

Bicriterion Pareto‐scheduling of equal‐length jobs on a single machine related to the total weighted late work

“…Problem $$$ {1}^{\ast }{\left|{p}_j=p,\left({d}_j{w}_j\right)\right|}^{\#}\left(\sum {U}_j,\sum {w}_j{Y}_j\right) $$$ is solvable in $$$ O\left(n\log n\right) $$$ time. Then problem $$$ 1{\left|{p}_j=p\right|}^{\#}\left(\sum {U}_j,\sum {Y}_j\right) $$$ is also solvable in $$$ O\left(n\log n\right) $$$ time, which improves the $$$ O\left({n}^2\right) $$$‐time solution in Guo et al (2022). Problem $$$ 1{\left|\mathrm{CO},{p}_j^{(B)}={p}^{(B)}\right|}^{\#}\left({f}_{\mathrm{max}}^{(A)},\sum {w}_j^{(B)}{Y}_j^{(B)}\right) $$$ is solvable in $$$ O\left({n}_A{n}_B\left({q}_A+{n}_B^2\right)\right) $$$ time, which improves the …”

“…Inspired by the two open problems $$$ 1{\left|{p}_j=p\right|}^{\#}\left(\sum {w}_j{U}_j,\sum {Y}_j\right) $$$ and $$$ 1{\left|{p}_j=p\right|}^{\#}\left(\sum {w}_j{U}_j,\sum {w}_j{Y}_j\right) $$$ in Guo et al (2022), we present some $$$ NP $$$‐hardness results for two families of problems in this section.…”

“…Problem $$$ {1}^{\ast }{\left|{p}_j=p\right|}^{\#}\left(\sum {U}_j,\sum {w}_j{Y}_j\right) $$$ is solvable in $$$ O\left({n}^3\right) $$$ time. Then problem $$$ 1{\left|{p}_j=p\right|}^{\#}\left(\sum {U}_j,\sum {w}_j{Y}_j\right) $$$ is also solvable in $$$ O\left({n}^3\right) $$$ time, which improves the $$$ O\left({n}^4\right) $$$‐time solution in Guo et al (2022). Problem $$$ {1}^{\ast }{\left|{p}_j=p,\left({d}_j{w}_j\right)\right|}^{\#}\left(\sum {U}_j,\sum {w}_j{Y}_j\right) $$$ is solvable in $$$ O\left(n\log n\right) $$$ time.…”

“…The authors presented a pseudo‐polynomial‐time algorithm for the general case and polynomial‐time algorithms for some special cases. Guo et al (2023) further extended the model in Guo et al (2022) to the case where each job $$$ {J}_j $$$ has $$$ m $$$ unrelated weights $$$ {w}_j^{(1)},{w}_j^{(2)},\dots, {w}_j^{(m)} $$$.…”

“…Two important review articles can be found in Sterna (2011Sterna ( , 2021. Guo et al (2022), we present some NP-hardness results for two families of problems in this section.…”