A Combination of Flow Shop Scheduling and the Shortest Path Problem

TL;DR

This paper introduces a complex combinatorial optimization problem combining flow shop scheduling with the shortest path problem, proving its NP-hardness and proposing approximation algorithms for specific cases.

Contribution

It formally defines the combined problem, proves its NP-hardness, and offers approximation algorithms for cases with fixed or variable number of machines.

Findings

The problem is NP-hard even with two machines.

The problem is strongly NP-hard in the general case.

Proposed approximation algorithms improve solution quality for specific scenarios.

Abstract

This paper studies a combinatorial optimization problem which is obtained by combining the flow shop scheduling problem and the shortest path problem. The objective of the obtained problem is to select a subset of jobs that constitutes a feasible solution to the shortest path problem, and to execute the selected jobs on the flow shop machines to minimize the makespan. We argue that this problem is NP-hard even if the number of machines is two, and is NP-hard in the strong sense for the general case. We propose an intuitive approximation algorithm for the case where the number of machines is an input, and an improved approximation algorithm for fixed number of machines.