An efficient benders decomposition for the $p$-median problem

Highlights

• The Benders sub-problems of the p-median problem can be solved very quickly with a polynomial-time algorithm.

• The efficient implementation of a two-stage Benders decomposition algorithm performs better than state-of-the-art exact methods in solving instances of up to 238025 clients and sites, and also previously unsolved benchmark instances.

• The two-stage Benders decomposition algorithm can be adapted to other location problems such as the uncapacitated facility location problem and the optimal diversity management problem.

Abstract

The $p$-median problem is a classic discrete location problem with numerous applications. It aims to open $p$ sites while minimizing the sum of the distances of each client to its nearest open site. We study a Benders decomposition of the most efficient formulation in the literature. We show that the Benders cuts can be separated in linear time. The Benders reformulation leads to a compact formulation for the $p$-median problem. We implement a two-phase Benders decomposition algorithm that outperforms state-of-the-art methods on benchmark instances by an order of magnitude and allows to exactly solve for the first time several instances among which are large TSP instances and BIRCH instances. We also show that our implementation easily applies to the uncapacitated facility location problem.

Introduction

Discrete location problems aim at choosing a subset of locations from a finite set of candidates in which to establish facilities in order to allocate a finite set of clients. The most common objective for these problems consist in minimizing the sum of the fixed costs of the facilities and the allocation costs of supplying the clients. Within these problems, the $p$-median problem $\left(pMP\right)$ is one of the fundamental problems (Laporte, Nickel, & Saldanha-da Gama (2019)). In the $\left(pMP\right)$, we have to choose $p$ locations from the set of candidate sites, no fixed costs are considered and the allocation costs are equal to the distance between clients and sites. More formally, given a set of $N$ clients $\{C_1,\dots ,C_N\}$ and a set of $M$ potential sites to open $\{F_1,\dots ,F_M\}$, let $d_{ij}$ be the distance between client $C_i$ and site $F_j$ and $p\in \mathbb{N}$ the number of sites to open. The objective is to find a set $S$ of $p$ sites such that the sum of the distances between each client and its closest site in $S$ is minimized. The $\left(pMP\right)$ is an NP-hard problem (Kariv & Hakimi (1979)) and leads to applications where the sites correspond to warehouses, plants, shelters, etc. This includes the contexts of emergency logistics and humanitarian relief (An, Zeng, Zhang, & Zhao (2014); Mu & Tong (2020); Takedomi, Ishigaki, Hanatsuka, & Mori (2022)). Another important application is a particular clustering problem, usually called k-medoids problem in which the set of clients and sites are identical. In this problem, sub-groups of objects, variables, persons, etc. are identified according to defined criteria of proximity or similarity (Klastorin (1985); Marín & Pelegrín (2019); Park & Jun (2009); Ushakov & Vasilyev (2021); Voevodski (2021)).

A great interest in solving large location problems has led to the development of various heuristics and meta-heuristics in the literature. However, the exact solution of large-scale instances remains a challenge. Some location problems have recently been efficiently solved using the Benders decomposition method within a branch-and-cut approach (see e.g., Cordeau, Furini, & Ljubić (2019); Fischetti, Ljubic, & Sinnl (2017); Gaar & Sinnl (2022)). Among them, the uncapacitated facility location problem $\left(UFL\right)$ is probably the most studied location problem. In the $\left(UFL\right)$ the number of sites to be opened is not fixed, but an opening cost is associated with each site.

In this paper, we explore a Benders decomposition for the $\left(pMP\right)$. We propose a polynomial time algorithm for the separation of its Benders cuts. We implement an efficient two-phase Benders decomposition algorithm which provides better results than the best exact solution method in the literature Zebra (García, Labbé, & Marín (2011)). We present our results on about 230 benchmark $\left(pMP\right)$ instances of different sizes (up to 238,025 clients and sites) satisfying or not the triangle inequality. We finally extend our implementation to solve the $\left(UFL\right)$ and present some results.

The rest of the paper is organized as follows. Section 2 presents the literature review of the $\left(pMP\right)$. Section 3 describes our Benders decomposition method. Section 4 presents the computational results. In Section 5 we draw some conclusions together with research perspectives.