Single-facility location problems in two regions with ℓ₁- and ℓ_q-norms separated by a straight line

Highlights

• Single facility location problem is studied, whose demand points are in two regions.

• Different norms are used in each region that are separated by a straight line.

• Gate(1,q) algorithm solves the initial single-facility location problems.

• A comparative study with other well-known algorithms is carried out.

Abstract

In this paper, Fermat–Weber location problems with demand points located in two regions Ω₁ and Ω_q separated by a straight line, are addressed. Ω₁ and Ω_q are, respectively, endowed by different norms ℓ₁ and ℓ_q, with q > 1. In order to compute distances between points in different regions, the concept of Gate point is applied.

With the aim of solving the problem, a new algorithm, called Gate(1, q), is designed. This algorithm uses the characterization of the solutions in the regions Ω₁ and Ω_q, and the straight line.

A comparative study with other well-known algorithms is carried out in order to test the performance of the proposed algorithm. The results are promising since they show that the Gate(1, q) algorithm leads to a more accurate solution than those of the other algorithms in a relatively short computing time.

Introduction

Modelling actual distances is a crucial topic in several spatial problems, and has been the subject of research since the early 70s (Brimberg, Walker, Love, 2007, Fernández, Fernández, Pelegrin, 2002, Love, Morris, 1972, Love, Morris, 1979, Love, Walker, 1994). The optima l location of facility problems depends greatly on the distances used, and therefore location problems have been studied using a variety of distance measures. However, the environment space where a facility is to be located is seldom homogeneous but instead is composed of several parts with different distances (Brimberg, Kakhki, Wesolosowsky, 2003, Parlar, 1994). For example, this non-homogeneity occurs when the region under consideration contains both urban and rural zones. Even within cities, there are areas where the distance fits better to a rectangular or to a Manhattan distance and other areas where a more complex model of distance is needed that involves some of those derived from the ℓ_p-norm with p > 1 (Fernández, Fernández, Pelegrin, 2002, Love, Walker, 1994).

Single-facility location problems with mixed distance measures are studied in Planchart and Hurter (1975), where a distance function is obtained by adding the Manhattan and the Euclidean distances, and where the problem for mixed ℓ_p norms is also generalized. An algorithm where different gauges can be mixed is presented in Michelot and Lefebvre (1987). Furthermore, Carrizosa and Rodríguez-Chía (1997) addressed a p-facility minisum problem with a metric induced by a gauge and a finite set of rapid transit lines, where the location problem is reduced in order to solve a finite number of (multi-)Weber problems. Location of transfer facilities with mixed distances has also been dealt with in López-de-los Mozos, Mesa, and Schöbel (2017).

In this paper, it is assumed that a straight line, $r\equiv y=mx$ with m > 0, splits the space IR² into two regions Ω_p and Ω_q endowed with ℓ_p- and ℓ_q-norms, respectively, where 1 ≤ p < q < ∞, and r ⊂ Ω_q. Let us denote this construction as ${\Omega }_{pq}^m$. Let us indicate that the particular case m = 0 is studied in Brimberg et al. (2003).

The location problem studied in ${\Omega }_{pq}^m,$ which is a generalisation of the well-known single-facility minisum, or the Weber–Fermat problem (Francis, McGinnis, White, 1992, Love, Morris, Wesolowsky, 1988), can be stated as follows:

$$\underset{X\in I{R}^2}{\min }\{\sum _{i\in S_p}{w}_{i}d(P_i,X)+\sum _{j\in S_q}{w}_{j}d(Q_j,X)\},$$

where ${\left\{P_i\right\}}_{i\in S_p}\subset {\Omega }_p$ and ${\left\{Q_j\right\}}_{j\in S_q}\subset {\Omega }_q$ are two sets of points, S_p and S_q are the corresponding index sets, and $w_i,w_j\in {IR}^{+}$ with i ∈ S_p, j ∈ S_q, and the distance is given by the length of the shortest path (geodesic path) connecting them (see Mitchell & Papadimitriou, 1991).

With respect to the literature on this topic, the optimization problem (1) was studied for the ℓ₁- and ℓ₂-norms in Parlar (1994), where it was proved that the objective function is a non-convex function. This problem was then re-formulated as a mixed integer program and so a modified Weiszfeld-type procedure was proposed for its solution and its results were also compared with an adaptive random search procedure for three example problems.

Problem (1) was solved in Brimberg et al. (2003) for the case where a vertical straight line separates the regions endowed with ℓ₁- and ℓ₂-norms, respectively. For the case of a slanted straight line and these norms, the problem is transformed into a convex multi-facility location for which a Weiszfeld procedure with a hyperbolic approximation is proposed. A characterization of the crossing points on the slanted straight line is given in Zaferanieh, Kakhki, Brimber, and Wesolosowsky (2009), where, based on the assumption that the optimal solution lies on the rectangular hull of the existing facilities, they propose an efficient Big Square Small Square procedure. Furthermore, a bounded region in IR² with a norm inside the region and another norm in the rest of the space is considered in Brimberg, Kakhki, and Wesolosowsky (2005), where this new problem is solved by a finite number of convex programs searching for certain points and for related corner and interior demarcation curves.

The main contribution of this paper is the design of an algorithm, called the Gate(1, q), which is specifically designed to solve the optimization problem (1) for $p=1$ and q > 1, by taking into account the properties of the Gate points (Franco, Velasco, & Gonzalez-Abril, 2012) and the shortest path between points in Ω₁.

The remainder of the paper is arranged as follows. The geodesic path between two points in ${\Omega }_{pq}^m$ is studied in Section 2. By analysing the possible solutions in Ω₁ and Ω_q, with q > 1, to problem (1), a global solution is found in IR², in Section 3. The Gate(1, q) algorithm is presented in Section 4, and several experiments involving the Gate(1, q), the Multi-facility location procedure, and the Modified Weiszfeld Procedure, are carried out in Section 5. Finally, conclusions are drawn.