A bi-objective approach to discrete cost-bottleneck location problems

Abstract

This paper considers a family of bi-objective discrete facility location problems with a cost objective and a bottleneck objective. A special case is, for instance, a bi-objective version of the (vertex) p-centdian problem. We show that bi-objective facility location problems of this type can be solved efficiently by means of an $\varepsilon$-constraint method that solves at most $(n-1)\cdot m$ minisum problems, where n is the number of customer points and m the number of potential facility sites. Additionally, we compare the approach to a lexicographic $\varepsilon$-constrained method that only returns efficient solutions and to a two-phase method relying on the perpendicular search method. We report extensive computational results obtained from several classes of facility location problems. The proposed algorithm compares very favorably to both the lexicographic $\varepsilon$-constrained method and to the two phase method.

Appendix: A link to the weighted p-centdian problem

The p-centdian problem is a combination of the p-median and the p-center problems where the objective function is a convex combination of the median/cost and the center/bottleneck objectives. Given a graph $G=(V,E)$, let P(G) be the set of all points on G and d(l, h) be the distance of a shortest path between points l and h on G (note that the points l and h might be interior points on the edges of G). Furthermore, given a set of points $S\subseteq {P(G)}$ we define . In addition, let $w_j\ge 0$ and $v_j \ge 0$ be weights of the node $j\in V$ representing, for example, the number of customers or some other measure of attractiveness of the node j. For $0\le \lambda \le 1$ the p-centdian problem may then be stated as

$$\begin{aligned} \begin{aligned} \min&\ \lambda \sum _{j\in V} w_jd(S,j) + (1-\lambda ) \max _{j\in V} v_jd(S,j)\\ s.t.:&\ S\subseteq {P(G)}\\&\ \vert S\vert =p \end{aligned} \end{aligned}$$

(8)

It has been shown in Pérez-Brito et al. (1997) that there exists a finite set of points on G containing an optimal solution to the p-centdian. In what follows, the problem

is denoted the bi-objective p-centdian problem. It turns out that there can be infinitely many Pareto optimal solutions to this problem.

Theorem 2

The set of Pareto optimal solutions to the bi-objective p-centdian problem on a network can be uncountably infinite even for $p=1$ when $S\subseteq P(G)$.

Fig. 2

Illustrations of the example used in the proof of Theorem 2, a illustration of the graph used in the proof of Theorem 2, b solutions in outcome space for the graph used in the proof of Theorem 2

Proof

We show the result by giving an example having this property. Consider the graph given in Fig. 2a, where the edge lengths are given by $d(1,2)=d(2,1)=1$ and $d(2,3)=d(3,2)=2$. Furthermore, suppose $p=1$, that is, we want to place one facility on G. The intersection points on G are denoted by $n_4$, $n_5$, and $n_6$. It is easily verified that locating a facility at node 2 is an optimal solution to the 1-median problem with outcome vector (3, 2) whereas placing a facility at $n_5$ is an optimal solution for the 1-center problem, resulting in the outcome vector (3.5, 1.5). As the solutions are unique optimal solutions they are efficient. In Fig. 2b the outcome vectors for all points on the graph G have been plotted. All points on the edge (1, 2) map into the dashed line, while all solutions on the edges $(n_5,n_6)$ and $(n_6,3)$ map into the dotted line. However, all points on the edge $(2,n_5)$ map into the solid line which is non-dominated. As there are an uncountable infinite number of points on this edge, the result follows. $\square$

Note that the bi-objective p-centdian is not covered by the general BO–CBLP as the set I of potential facility sites has to be finite in the definition of the BO–CBLP. However, letting the sets I and J equal the set V, the (vertex) p-centdian problem (8) can be stated as the following scalarized version of the BO–CBLP:

where , , $f_i=0$, $c_{ij}=w_jd(i,j)$, and $t_{ij}=v_jd(i,j)$ for all $i\in I$ and $j\in J$. In many practical applications it will suffice to consider placing facilities only at the nodes of the graph. For $0< \lambda < 1$ a solution to the vertex-p-centdian problem corresponds to a supported efficient solution of the BO–CBLP (3) with the sets I, J, $\mathcal {X} _i$, $\mathcal {Y}$ and the coefficients $f_i$, $c_{ij}$, and $t_{ij}$ defined as above. This means that solving the BO–CBLP yields a solution to the vertex-p-centdian problem for each value of $\lambda >0$. Most literature considers special structured graphs such as trees when solving the p-centdian problem. This approach offers a means to solve the vertex-p-centdian problem for all values of $0<\lambda <1$ on a general graph with no assumption other than that facilities should be located on nodes only.