Exact and heuristic methods for the discrete parallel machine scheduling location problem

TL;DR

This work tackles the discrete parallel machine makespan scheduling-location problem (DPMM ScheLoc), integrating machine-location decisions and job scheduling. It introduces a new arc-flow (AF) formulation, solves its linear relaxation via a column generation (CG) approach, and develops three heuristics (two MIP-based AF variants and an Iterated Local Search, ILS) embedded in a unified exact framework. Computational results show the AF model yields strong lower bounds, while the heuristics provide high-quality upper bounds quickly, enabling the framework to prove optimality for all published benchmarks and to produce near-optimal solutions for many new challenging instances. The study underscores the potential of tight AF formulations and framework-driven hybrid methods for integrated location–scheduling problems, with implications for scalable optimization in production and logistics settings.

Abstract

The discrete parallel machine makespan scheduling location (ScheLoc) problem is an integrated combinatorial optimization problem that combines facility location and job scheduling. The problem consists in choosing the locations of $p$ machines among a finite set of candidates and scheduling a set of jobs on these machines, aiming to minimize the makespan. Depending on the machine location, the jobs may have different release dates, and thus the location decisions have a direct impact on the scheduling decisions. To solve the problem, it is proposed a new arc-flow formulation, a column generation and three heuristic procedures that are evaluated through extensive computational experiments. By embedding the proposed procedures into a framework algorithm, we are able to find proven optimal solutions for all benchmark instances from the related literature and to obtain small percentage gaps for a new set of challenging instances.

Figures (5)

• Figure 1: Solution representation for an instance with $n=10$, $m=5$, and $p=3$.
• Figure 2: $AF$ solution for toy example of Figure \ref{['fig:example']}
• Figure 3: Example of a swap move.
• Figure 4: Concatenation cases when moving $j$ to a position before $l$.
• Figure 5: Concatenation cases when moving $j$ to a position after $l$.