Improved Scheduling for the Three-Machine Proportionate Open Shop and Mixed Shop Minimum Makespan Problems

Guanqun Ni,Lei Chen

doi:10.1109/access.2020.3030694

Abstract

Open shop scheduling problems have many practical applications, in which the jobs can be in any order with the only restriction that their durations do not overlap with each other. In some cases, the order of a subset of jobs dictates the flow shop model while the remaining jobs can be processed in any order according to the open shop model, and the corresponding problem is called mixed shop scheduling. For every job, if its processing times on all machines are the same, then the shop is called proportionate shop. In this article we investigate the three-machine proportionate open shop and mixed shop minimum makespan problems proposed by Koulamas and Kyparisis. Our result improves the previous work in three ways. First, for both the open shop and the mixed shop scheduling problems, we derive additional sufficient conditions under which the corresponding problems are solvable in polynomial time. Second, for the non-solvable cases of the open shop scheduling problem we present an improved 13/12-approximation algorithm. Third, for the non-solvable cases of the mixed shop scheduling problem we present an improved approximation algorithm with the worst-case ratio bound of (7/6 + ϵ). All the algorithms proposed in this article run in polynomial time.

Highlights

Open shop scheduling problems have many interesting applications
For the non-solvable cases of the O3|prpt|Cmax problem, we present an improved approximation algorithm with the worst-case ratio bound of 13/12
EXACT ALGORITHMS FOR THE O3|prpt |Cmax PROBLEM we first investigate conditions under which the O3|prpt|Cmax problem is solvable in polynomial time, and present an improved 13/12-approximation algorithm for the non-solvable case of the O3|prpt|Cmax problem

Summary

INTRODUCTION

Open shop scheduling problems have many interesting applications. Two typical examples are the scheduling of medical tests in an outpatient health care facility and the scheduling of exams in an academic institution. Theorem 1: If p1 ≤ T3i∗−1, or p1 ≥ T3i∗ , or p1 = p2 + Tin∗ , the O3|prpt|Cmax problem is solvable in polynomial time and Cm∗ax = max{P, 3p1}. Theorem 3: If p2 < p1 < p2 + Tin∗ , T3i∗−1 < p1 < T3i∗ and pi∗ ≤ p1/2, there is a polynomial algorithm for the O3|prpt|Cmax problem with the worst-case ratio bound of 13/12. We focus on the non-solvable case of the M 3|prpt|Cmax problem and present a (7/6 + )-approximation algorithm for any given > 0, noting that the algorithms proposed in [1].

AN APPROXIMATION ALGORITHM FOR THE

CONCLUSION