Boglárka G.-Tóth

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.

Sequential versus simultaneous approach in the location and design of two new facilities using planar Huff-like models

Companies frequently decide on the location and design for new facilities in a sequential way. However, for a fixed number of new facilities, the company might be able to improve its profit by taking its decisions for all the facilities simultaneously. In this paper we compare three different strategies: simultaneous location and independent design of two facilities in the plane, the same with equal designs, and the sequential approach of determining each facility in turn. The basic model is profit maximization for the chain, taking market share, location costs and design costs into account. The market share captured by each facility depends on the distance to the customers (location) and its quality (design), through a probabilistic Huff-like model. Recent research on this type of models was aimed at finding global optima for a single new facility, holding quality fixed or variable, but no exact algorithm has been proposed to find optimal solutions for more than one facility. We develop such an exact interval branch-and-bound algorithm to solve both simultaneous location and design two-facility problems. Then, we present computational results and exhibit the differences in locations and qualities of the optimal solutions one may obtain by the sequential and simultaneous approaches.

Obtaining an outer approximation of the efficient set of nonlinear biobjective problems

A new method for obtaining an outer approximation of the efficient set of nonlinear biobjective optimization problems is presented. It is based on the well known ‘constraint method’, and obtains a superset of the efficient set by computing the regions of δ-optimality of a finite number of single objective constraint problems. An actual implementation, which makes use of interval tools, shows the applicability of the method and the computational studies on a set of competitive location problems demonstrate its efficiency.

Empirical Investigation of the Convergence Speed of Inclusion Functions in a Global Optimization Context

This paper deals with the empirical convergence speed of inclusion functions applied in interval methods for global optimization. According to our experience the natural interval extension of a given function can be as good as a usual quadratically convergent inclusion function, and although centered forms are in general only of second-order, they can perform as one of larger convergence order. These facts indicate that the theoretical convergence order should not be the only indicator of the quality of an inclusion function, it would be better to know which inclusion function can be used most efficiently in concrete instances. For this reason we have investigated the empirical convergence speed of the usual inclusion functions on some test functions.

Location equilibria for a continuous competitive facility location problem under delivered pricing

The problem of finding location equilibria of a location-price game where firms first select their locations and then set delivered prices in order to maximise their profits is investigated. Assuming that firms set the equilibrium prices in the second stage, the game can be reduced to a location game for which a global minimiser of the social cost is a location equilibrium, provided that the demand is completely inelastic and the marginal production cost is constant. When the set of feasible locations is a region of the plane the minimisation of the social cost becomes a hard-to-solve global optimisation problem. We propose an exact interval branch-and-bound algorithm suitable for small and medium size problems and an alternating Weiszfeld-like heuristic for larger instances. The latter approach is based on a new iterative formula for which the validity of the descent property is proved. The proposed heuristic performs extremely well against the exact method when tested on small to medium size instances while requiring a tiny fraction of its computational time.

On determining the cover of a simplex by spheres centered at its vertices

The aim of this work is to study the Simplex Cover (SC) problem, which is to determine whether a given simplex is covered by spheres centered at its vertices. We show that the SC problem is equivalent to a global optimization problem. We investigate its characteristics.

AMIGO: Advanced Multidimensional Interval analysis Global Optimization algorithm

This paper analyzes and evaluates an efficient n-dimensional global optimization algorithm. It is an extended version of the algorithm of Casado et al. [1]. This algorithm takes advantage of all available information to estimate better bounds of the function. Numerical comparison made on a class of multiextremal test functions has shown that on average the new algorithm works faster than a traditional gradient based interval analysis global optimization method.

On the impact of spatial pattern, aggregation, and model parameters in planar Huff-type competitive location and design problems

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 maximize the profit obtained for the chain. Several types of constraints and costs are considered. Applying an interval analysis based global optimization method on several spatial patterns in a quasi-real-world environment, the behaviour of optimal solutions is investigated when changes are made in the basic model parameters. The study yields valuable insight for modellers into the impact of spatial pattern and various model parameters of the model on the resulting location and design decision. Spatial patterns differ in distribution of demand, of own and/or competing facilities, and of facility qualities. Studied model parameters include push force effects, investment restrictions and aggregation of demand.

Reconciling Franchisor and Franchisee: A Planar Biobjective Competitive Location and Design Model

This paper deals with a hard nonlinear biobjective optimization problem: finding the optimal location and design for a new franchised facility within a region where facilities (both of the franchise and not) already exist and compete for the market. The franchisor and the new franchisee both want to maximise their own profit in the market, but these two objectives are in conflict. Customers patronize all the facilities, old and new, proportionally to their attraction to them. Both resulting objective functions are neither convex nor concave. An interval branch and bound method is proposed to obtain an outer approximation of the whole set of efficient solutions. Computational experiments highlight the different kinds of information provided by this method and by a variation of the lexicographic method.

On Computational Aspects of a Regular n-Simplex Bisection

Branch-and-Bound (B&B) algorithms to solve Global Optimization (GO) may use n-simplicial partition sets. The n-simplex represents an n-dimensional body with n+1 vertices in (n+1)-dimensional space. The aim of this article is to investigate the properties of the binary tree generated by iterative bisection of the longest edge (LE) of the regular n-simplex as search space. This way of splitting an n-simplex reduces the appearance of bad shaped simplices which facilitates the convergence of the algorithm. It also helps to have a more uniform sampling of the search space since the function is evaluated at vertices of simplices. A motivation for this research is the estimation of the pending computational work load during the B&B GO algorithm. Such estimation may be helpful to steer parallel versions of the algorithms. In this paper we will show that the way the longest edge is selected affects on the number of sub-problems, the number of similar shapes of those sub-problems and their roundness factor. The computational cost to obtain those metrics increases with the dimension n. Here we show the results for n leq 3, where for n=3, we have a 3-dimensional body in a 4-dimensional space. Due to the exponential growth of the binary tree, high performance computing is useful in order to reach a high precision or when n eq 3. We make use of parallel computing under MATLAB software.