A fixed-parameter algorithm for a routing open shop problem: unit processing times, few machines and locations

TL;DR

This paper presents a fixed-parameter algorithm for a routing open shop problem with unit processing times, few machines, and locations, optimizing scheduling with travel times between jobs.

Contribution

The authors develop a fixed-parameter algorithm for a routing open shop problem considering machine travel times, with complexity depending on the number of vertices and machines.

Findings

Algorithm solves the problem in exponential time relative to vertices and machines

Efficient for small numbers of machines and locations

Provides a theoretical foundation for routing and scheduling integration

Abstract

The open shop problem is to find a minimum makespan schedule to process each job $J_i$ on each machine $M_q$ for $p_{iq}$ time such that, at any time, each machine processes at most one job and each job is processed by at most one machine. We study a problem variant in which the jobs are located in the vertices of an edge-weighted graph. The weights determine the time needed for the machines to travel between jobs in different vertices. We show that the problem with $m$ machines and $n$ unit-time jobs in $g$ vertices is solvable in $2^{O(gm^2\log gm)}+O(mn\log n)$ time.