Approximation algorithms for the three-machine proportionate mixed shop scheduling

TL;DR

This paper improves approximation algorithms for the three-machine proportionate mixed shop scheduling problem, providing a tighter 4/3 ratio, an FPTAS for specific cases, and revealing NP-hardness when adding certain open-shop jobs.

Contribution

It introduces a 4/3-approximation algorithm for the problem, proves its asymptotic tightness, and develops an FPTAS for cases with largest flow-shop jobs, also analyzing the problem's complexity.

Findings

Improved 4/3-approximation algorithm for the problem.

Established asymptotic tightness of the 4/3 ratio.

Developed an FPTAS when the largest job is a flow-shop job.

Abstract

A mixed shop is a manufacturing infrastructure designed to process a mixture of a set of flow-shop jobs and a set of open-shop jobs. Mixed shops are in general much more complex to schedule than flow-shops and open-shops, and have been studied since the 1980's. We consider the three machine proportionate mixed shop problem denoted as $M3 \mid prpt \mid C_{\max}$, in which each job has equal processing times on all three machines. Koulamas and Kyparisis [{\it European Journal of Operational Research}, 243:70--74,2015] showed that the problem is solvable in polynomial time in some very special cases; for the non-solvable case, they proposed a $5/3$-approximation algorithm. In this paper, we present an improved $4/3$-approximation algorithm and show that this ratio of $4/3$ is asymptotically tight; when the largest job is a flow-shop job, we present a fully polynomial-time approximation scheme (FPTAS). On the negative side, while the $F3 \mid prpt \mid C_{\max}$ problem is polynomial-time solvable, we show an interesting hardness result that adding one open-shop job to the job set makes the problem NP-hard if this open-shop job is larger than any flow-shop job. We are able to design an FPTAS for this special case too.