Frank Plastria

Static competitive facility location: An overview of optimisation approaches

Abstract We give an overview of the research, models and literature about optimisation approaches to the problem of optimally locating one or more new facilities in an environment where competing facilities are already established.

GBSSS: The generalized big square small square method for planar single-facility location

Abstract Big Square Small Square is a geometrical branch-and-bound algorithm, devised by P. Hansen et al. (OR 33, 1251–1265), for the solution of constrained planar minisum single-facility location problems with L p norms and continuous non-decreasing costfunctions. The method basically works by splitting the studied planar region into squares, and either rejecting or further processing these squares by the evaluation of a lower bound. We present a modified version of this algorithm aimed at correcting a small failure to converge, accelerating the calculations, minimising the information to be stored, and, most importantly, determining a region of near-optimality. Furthermore the method is applicable to any planar single-facility problem with distances measured by mixed norms and as an objective any continuous function of the distances. This includes nearly all the models which have been proposed in the literature.

Solving a Huff-like competitive location and design model for profit maximization in the plane

Abstract A chain wants to set up a single new facility in a planar market where similar facilities of competitors, and possibly of its own chain, are already present. Fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility, determined by the different perceived qualities of the facilities and the distances to them, through a gravitational or logit type model. Both the location and the quality (design) of the new facility are to be found so as to maximise the profit obtained for the chain. Several types of constraints and costs are considered. Two solution methods are developed and tested. The first is a repeated local optimisation heuristic, extending earlier proposals to the supplementary design question and the presence of locational constraints. The second is an exact global optimisation technique based on reliable computing using interval analysis, incorporating several novel features. An example and comparative computational results demonstrate that this difficult and very multi-modal problem can be solved by such techniques. The local optimisation method turns out not to be very robust in its results, even after numerous repetitions, whereas the global optimisation method yields very useful and complete information on guaranteed near to optimal solutions after an important but still quite acceptable computational effort.

Discrete models for competitive location with foresight

We adapt the competitive location model based on maximal covering to include the knowledge that a competitor will enter the market later with a single new facility. The objective is to locate facilities under a budget constraint in order to maximise the remaining market share after the competitors later entry. We develop mixed zero-one programming formulations for each of the following three strategies: the maxmin strategy where the worst possible competitor choice is considered, the minimum regret strategy, and the von Stackelberg strategy in which the competitor also optimises its market share. Some computational results show the feasibility and limits of these models.

Reformulation descent applied to circle packing problems

Several years ago classical Euclidean geometry problems of densest packing of circles in the plane have been formulated as nonconvex optimization problems, allowing to find heuristic solutions by using any available NLP solver. In this paper we try to improve this procedure. The faster NLP solvers use first order information only, so stop in a stationary point. A simple switch from Cartesian coordinates to polar or vice versa, may destroy this stationarity and allow the solver to descend further. Such formulation switches may of course be iterated. For densest packing of equal circles into a unit circle, this simple feature turns out to yield results close to the best known, while beating second order methods by a time-factor well over 100.This technique is formalized as a general reformulation descent (RD) heuristic, which iterates among several formulations of the same problem until local searches obtain no further improvement. We also briefly discuss how RD might be used within other metaheuristic schemes.

Lower subdifferentiable functions and their minimization by cutting planes

This paper introduces lower subgradients as a generalization of subgradients. The properties and characterization of boundedly lower subdifferentiable functions are explored. A cutting plane algorithm is introduced for the minimization of a boundedly lower subdifferentiable function subject to linear constraints. Its convergence is proven and the relation is discussed with the well-known Kelley method for convex programming problems. As an example of application, the minimization of the maximum of a finite number of concave-convex composite functions is outlined.

Optimal location and design of a competitive facility

Abstract.A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits.It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favour of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in problems in which optimal solutions might not exist.

Formulating logical implications in combinatorial optimisation

Abstract When practical problems are formulated as combinatorial optimisation models one must often include logical implications between decisions. It is useful to express these implications as linear constraints involving binary variables, since linear constraints offer the possibility of using linear programming and branch and bound as an initial solution method. Often this formulation step of the modelling process seems far from evident to many practitioners, students and researchers. Therefore it is of interest to make simple rules available to clarify and help in this process. In this educationally oriented tutorial paper we introduce such a rule, LIP, the logical implication principle. It offers an easy and automatic way to translate elementary logical implications involving 0–1 variables into linear constraints. Based on an extremely simple cutting plane, we demonstrate how the rule is extended using complementarity and implicit binary variables, and leads to a simple but powerful instrument. This is illustrated by several examples of applications in various fields of Operational Research. The paper culminates in the description of a novel set of constraints which fully eliminate all permutation-equivalent solutions to numbered clustering problems.

Undesirable facility location with minimal covering objectives

An undesirable facility is to be located within some feasible region of any shape in the plane or on a planar network. Population is supposed to be concentrated at a finite number n of points. Two criteria are taken into account: a radius of influence to be maximised, indicating within which distance from the facility population disturbance is taken into consideration, and the total covered population, i.e. lying within the influence radius from the facility, which should be minimised. Low complexity polynomial algorithms are derived to determine all nondominated solutions, of which there are only O(n3) for a fixed feasible region or O(n2) when locating on a planar network. Once obtained, this information allows almost instant answers and a trade-off sensitivity analysis to questions such as minimising the population within a given radius (minimal covering problem) or finding the largest circle not covering more than a given total population.

Gauge Distances and Median Hyperplanes

A median hyperplane in d-dimensional space minimizes the weighted sum of the distances from a finite set of points to it. When the distances from these points are measured by possibly different gauges, we prove the existence of a median hyperplane passing through at least one of the points. When all the gauges are equal, some median hyperplane will pass through at least d-1 points, this number being increased to d when the gauge is symmetric, i.e. the gauge is a norm.Whereas some of these results have been obtained previously by different methods, we show that they all derive from a simple formula for the distance of a point to a hyperplane as measured by an arbitrary gauge.