A perturbation-based heuristic for the capacitated multisource Weber problem

Abstract

This paper proposes a perturbation-based heuristic for the capacitated multisource Weber problem. This procedure is based on an effective use of borderline customers. Several implementations are considered and the two most appropriate are then computationally enhanced by using a reduced neighbourhood when solving the transportation problem. Computational results are presented using data sets from the literature, originally used for the uncapacitated case, with encouraging results.

Introduction

The continuous capacitated location–allocation problem with a fixed number of open facilities each with a constant capacity, which is also known as the capacitated multisource Weber problem, may be stated as follows: given the location of each fixed point (customer point), the demand at each fixed point, the transportation cost for the area of interest, the number of facilities to open, and the capacity of each of these facilities, the aim is to determine the location of each facility, and the allocation of customers to these open facilities (if more than one facilities are to be opened). Given

Parameters

n

the number of fixed points (or customer points)

w_j

demand or weight of customer j (j = 1, … , n)

$a_j=(a_j^1\text{,}{a}_j^2)$

location of customer j where $a_j\in {\mathfrak{R}}^2$, (j = 1, … , n)

M

the number of facilities to be located

b

fixed capacity of a facility where b ∈ N

Decision variables

$X_i=(X_i^1\text{,}{X}_i^2)$

coordinates of facility i where $X_i\in {\mathfrak{R}}^2$

x_ij

quantity assigned from facility i to customer j, i = 1, … , M, j = 1, … , n

The problem can be formulated as follows:

$$\text{Minimise}\sum _{i=1}^M\sum _{j=1}^n{x}_{\mathit{ij}}d({\mathit{X}}_i\text{,}{\mathit{a}}_j)$$

$$\begin{array}{cc}\hfill & \text{subject to}\hfill \\ \hfill & \sum _{j=1}^n{x}_{\mathit{ij}}⩽b\forall i=1\text{,}\dots \text{,}M\text{,}\hfill \end{array}$$

$$\sum _{i=1}^M{x}_{\mathit{ij}}=w_j\forall j=1\text{,}\dots \text{,}n\text{,}$$

$$x_{\mathit{ij}}⩾0\forall i=1\text{,}\dots \text{,}M\text{;}j=1\text{,}\dots \text{,}n\text{,}$$

where d(X_i, a_j) represents the Euclidean distance between facility i and customer j.

(1) denotes the objective function which is the total transportation cost, (2) ensures that capacity constraints of the facilities are not violated, (3) guarantees that the demand of every customer is satisfied and (4) refers to non-negativity of the decision variables x_ij.

It can be noted that once the set of open facilities has been decided upon (e.g., if we fix the open facilities in the formulation), the resulting problem reduces to the usual Transportation Problem (TP) which can be solved optimally in polynomial time. In short, the problem is to find the best facility configuration.

In this study the value of b is set to $⌈\frac{{\sum }_{j=1}^n{w}_j}{M}⌉$ where ⌈x⌉ is the smallest integer larger than or equal to x. Note that if $b>\frac{{\sum }_{j=1}^n{w}_j}{M}$ we introduce a dummy customer with a 0 transportation cost and a demand equals to the remaining demand, e.g., $b-\frac{{\sum }_{j=1}^n{w}_j}{M}$. This customer is used only when solving the TP, but not at the location and the allocation stages.

Most of the work in the literature on the capacitated facility location concentrates on the discrete problem and the methods mainly used include dual-ascent based [11], cross decomposition method [13], constructive-type heuristic [10], [7] and Lagrangian relaxation heuristics [2], [1].

Other related work on the continuous location problem include Eben-Chaine et al. [8] who studied the case of capacitated facility location on a line, and Brimberg and Mladenovic [4], Brimberg et al. [3] and recently by Salhi and Gamal [12] who investigated the multisource Weber problem. To our knowledge, it is only Cooper [6] in the 1970s who attempted the capacitated continuous case. He presented exact and approximate methods for solving the transportation-location problem. The heuristic method described in this work is a modification of the alternating transportation-location method introduced in [6]. Here, the location method and the usual TP are alternately applied until there is no epsilon improvement in cost. We shall describe Cooper’s method [6] as this will be used as the foundation for our perturbation-based heuristic.

Firstly, M facilities are randomly chosen from the fixed points. Then, the TP using these M open facilities is solved to find the allocation for the capacitated problem. For each of the M independent set of allocations, containing n_i fixed points where i = 1, … , M and ${\sum }_{i=1}^M{n}_i⩾n$, the new location of the facilities is found using the iterative procedure based on the Weiszfeld Equation which is given below:

$$X_i^{1^{(k)}}=\frac{{\sum }_{j_i=1}^{n_i}\frac{w_{j_i}{a}_{j_i}^1}{d\left(X_i^{(k-1)}\text{,}{a}_{j_i}\right)}}{{\sum }_{j_i=1}^{n_i}\frac{w_{j_i}}{d\left(X_i^{(k-1)}\text{,}{a}_{j_i}\right)}}\text{and}{X}_i^{2^{(k)}}=\frac{{\sum }_{j_i=1}^{n_i}\frac{w_{j_i}{a}_{j_i}^2}{d\left(X_i^{(k-1)}\text{,}{a}_{j_i}\right)}}{{\sum }_{j_i=1}^{n_i}\frac{w_{j_i}}{d\left(X_i^{(k-1)}\text{,}{a}_{j_i}\right)}}\text{,}$$

where the superscript k denotes the iteration number and $w_{j_i}$ represents all or a fraction of the jth customer demand that is assigned to facility i. Obviously $w_{j_i}⩽w_j$ as some customers may have their demand split because of the solution of the TP and hence some customers can be used more than once in Eq. (5) with their appropriate demand adjusted accordingly.

The location problem and the TP are alternately solved until there is no epsilon improvement in cost.

According to [6], ATL yields a convergent monotone nonincreasing sequence of values for the objective function. However, there is no guarantee that it will converge to the global minimum but the result, when not optimal, is found empirically to lie within ∼10%, and usually within 2–3%, of the optimal solution when tested on small instances.

The rest of the paper is structured as follows: in the next section, the modification on Cooper’s ATL is presented. Section 3 describes our perturbation-based heuristic and section Section 4 presents a neighbourhood reduction for solving the TP. Section 5 provides our computational results and our findings as well as some research issues are given in the last section.