Discrete Optimization
The bicriterion semi-obnoxious location (BSL) problem solved by an ϵ-approximation

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Abstract

Locating an obnoxious (undesirable) facility is often modeled by the maximin or maxisum problem. But the obnoxious facility is often placed unrealistically far away from the demand points (nodes), resulting in prohibitively high transportation cost/time. One solution is to model the problem as a semi-obnoxious location problem.

Here we model the problem as a bicriterion problem, not in advance determining the importance of the obnoxious objective compared to the cost/time objective.

We consider this model for both the planar and the network case. The two problems are solved by an approximation algorithm, and the models are briefly compared by means of a real-life example.

Introduction

In the two traditional single facility location problems, a new facility is located (placed) so as to minimize transportation costs (minisum), or as to minimize the distance to the farthest customer (minimax). In the minisum problem, we sum all the distances between the new facility and the customers, multiplied by a weight depending on the individual customer. In the minimax problem, we minimize the largest weighted distance. The minisum model can be relevant when locating a warehouse and the minimax model can be used to locate a fire station. These models are presented in Love et al. [14] and Francis et al. [9], both including many references. The obnoxious location problem is a more recent class of problems, where the two most common are the maxisum and maximin models. When locating an obnoxious (undesirable) facility the goal is to place it as far from the existing facilities (demand points, customers) as possible. See [8] or [6] for a review.

There is little literature combining the desirable and the obnoxious facility location models. In this paper, we model the combined problem as a Bicriterion Semi-obnoxious Location (BSL) problem. One objective function is obnoxious and one is desirable. We also consider both the network case and the planar case of the problem. In biobjective optimization our goal is to find the set of efficient solutions. These solutions are such that there does not exist another solution that has a better value in one objective without having a worse value in the other objective. The concept of efficient solutions is the same as Pareto optimal solutions. In the network case, where the demand points are nodes in a network and we try to locate the new facility in a node or on an edge, we have found no references, but ongoing research is presented in Hamacher et al. [10]. In the planar case, where the feasible locations are in R2, we have found only three references, namely, two papers by Brimberg and Juel [2], [3], and a paper by Carrizosa et al. [5].

In the bicriterion model, developed in the first paper by Brimberg and Juel [2], the first objective is the minisum objective and the second objective (the obnoxious criterion) is the minisum objective, where the Euclidean distance is raised to a negative power. It is proposed to solve the problem (finding the efficient solutions) in two steps. First a convex combination with parameter λ∈[0,1] of the two objectives (weighting method [17]) is formed. The resulting objective is neither convex nor concave. By varying λ a trajectory of efficient solutions may be determined. In the paper an algorithm based on this is outlined. A numerical example is presented.

In the second paper by Brimberg and Juel [3], a different bicriterion model is considered. In this model, the first objective is again the minisum objective, but the second objective (obnoxious) is now the maximin objective. They present two different solution methods for this model, but only one of them is guaranteed to find the complete set of efficient solutions.

In the bicriterion model developed in the third paper by Carrizosa et al. [5], the first objective (the obnoxious criterion) is modeled as the maxisum, and the second objective is modeled as the minisum problem. A solution procedure based on the Big Square Small Square (BSSS) approach is suggested. The procedure finds an approximation of the set of efficient solutions but no computational experience is reported. It should also be mentioned, that the approximation is in value space, and not in decision space.

The theory of the planar and network models is quite different, and the two models are not often compared, even though they try to describe the same real-life problem. We apply the two models on a real-life example in Section 4.

Next, we present the basic model for the BSL problem. We assume that there are n existing facilities (demand points). In the planar case they are denoted aj=(aj1,aj2), j=1,…,n. In the network case, they are denoted v1,v2,…,vn. We want to place a new facility at location x in order to minimize both the (transportation) costs and the obnoxiousness. Let S denote the set of feasible solutions, f(x) the obnoxious objective function and g(x) the cost objective function. The general model looks as follows:minf(x)ming(x)s.t.x∈S.We assume f depends negatively on the distance function and g depends positively on the distance function. This means, when we increase the distance between the new facility and an existing facility, this will have a decreasing effect on f and an increasing effect on g, e.g., less obnoxiousness but higher transportation costs.

Definition 1

A feasible solution x to (1) is efficient iff there does not exist another feasible solution x̄ to (1) such that f(x̄)⩽f(x), g(x̄)⩽g(x) and (f(x̄),g(x̄))≠(f(x),g(x)). Otherwise x is inefficient.

Efficiency is defined in the decision space. There is a natural counterpart in the criterion space. The feasible region in criterion space is denoted by Z and is given by Z={z(x)∈R2|z(x)=(f(x),g(x)), x is feasible in (1)}.

Definition 2

z(x)∈Z is a nondominated criterion vector iff x is an efficient solution to (1). Otherwise z(x) is a dominated criterion vector.

For a textbook introduction to multicriteria analysis see Steuer [17] or more recently Ehrgott [7]. We note that several efficient solutions may correspond to the same nondominated criterion vector.

As mentioned we consider two cases of the problem, namely, the planar case, denoted the BSPL problem, where the feasible solutions form a region in the plane, and the network case, denoted the BSNL problem, where the set of demand points are vertices in a network.

The BSPL problem is solved using a variant the BSSS method described by Hansen et al. [11]. The BSSS method is similar to a Branch and Bound method and inherits some of its weaknesses, namely, that the convergence of it may be slow unless good (in this case lower) bounds are used. In this paper we are able to find exact lower bounds in most situations. Actually, if the level sets of one of the objectives are convex, an exact lower bound with respect to this objective may be found. In the numerical real-life example presented in Section 4 the adjusted method works very well. We also adopt the BSSS method to solve the BSNL problem as well (in which we also use exact lower bounds). The real-life example is also solved for the network case, and interestingly, similar results as in the planar case are obtained.

The remaining part of the paper is organized as follows. In Section 2, we describe the BSPL problem and the solution approximation algorithm, and in Section 3 the BSNL problem and its solution method is described. In Section 4 a real-life application of the two models is presented. Section 5 contains the conclusions.

Section snippets

The planar case: the BSPL problem

The BSPL problem is formulated in the following way. There are n facilities (demand points) located at points a1,a2,…,an, and the objective is to locate a semi-obnoxious facility at x so as to minimize a weighted sum of the distances raised to a negative power, and to minimize the weighted sum of the distances between the existing facilities and the new facility. The first criterion f(x) may be thought of as a pollution effect and the second criterion g(x) as transportation costs. This model

The network case: the BSNL problem

In this section, we adapt the BSSS method to the network case. However, instead of dividing big squares into smaller squares, we divide edges into sub-edges. The idea of using a BSSS like method to solve a problem on a network has been explored before, namely, in a paper by Karkazis and Boffey [13]. In their paper, they provide a BSSS like algorithm for solving a problem on a line segment (network) with one objective function using lower bounds. The main difference between our approach and the

A real-life application

To illustrate the usefulness of the two models, we present an application. Currently, there is a debate in Denmark as to the location of a new international airport in the mainland Jutland in order to replace an existing one. The existing airport is located near a small city called Tirstrup approximately 45 km to the north-east of Århus, the largest city in Jutland (with about 215.000 inhabitants). The existing airport is located in an area where not many people are living and where not many

Concluding remarks

In this paper, we have set up two bicriterion location models for locating one obnoxious facility, namely, one for the planar case and one for the network case. Efficient (well-working) solution algorithms based on the well-known BSSS algorithm has been proposed. Both models are easily extended to multiple criteria. All that needs to be changed is the DCR operation.

Even though the planar and the network model may seem distinct in structure, they are designed to solve the same real-life problem.

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