COMPETITIVE LOCATION UNDER UNCERTAINTY OF COSTS

ABSTRACT. In this paper, we study the centroid problem from competitive location theory for a linear market with uniform demand, assuming that the leader has imperfect information about the follower's fixed and marginal costs. It is shown that the general version of this problem can be formulated as a nonlinear programming problem and the exact solution can be obtained analytically in a special case. A simple strategy is also given for the general problem, and it is proven that this strategy has a guaranteed error bound. It is demonstrated that uncertainty of costs might lead to market failure in the centroid problem, but this disappears if the game is repeated and the firms learn from observing each other's moves. It is also shown that it is possible for the leader to obtain optimal expected profit at a low perceived risk, with only sufficient, and not necessarily perfect, information. These two observations lead to our primary conclusion from the study that although cost uncertainty is a realistic feature of most competitive location models, there are very effective ways of dealing with it.     

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

The vector assignment p‐median problem (VAPMP) is one of the first discrete location problems to account for the service of a demand by multiple facilities, and has been used to model a variety of location problems in addressing issues such as system vulnerability and reliability. Specifically, it involves the location of a fixed number of facilities when the assumption is that each demand point is served a certain fraction of the time by its closest facility, a certain fraction of the time by its second closest facility, and so on. The assignment vector represents the fraction of the time a facility of a given closeness order serves a specific demand point. Weaver and Church showed that when the fractions of assignment to closer facilities are greater than more distant facilities, an optimal all‐node solution always exists. However, the general form of the VAPMP does not have this property. Hooker and Garfinkel provided a counterexample of this property for the nonmonotonic VAPMP. However, they do not conjecture as to what a finite set may be in general. The question of whether there exists a finite set of locations that contains an optimal solution has remained open to conjecture. In this article, we prove that a finite optimality set for the VAPMP consisting of “equidistant points” does exist. We also show a stronger result when the underlying network is a tree graph.     

Multiple Facilities Location in the Plane Using the Gravity Model

Two problems are considered in this article. Both problems seek the location of p facilities. The first problem is the p median where the total distance traveled by customers is minimized. The second problem focuses on equalizing demand across facilities by minimizing the variance of total demand attracted to each facility. These models are unique in that the gravity rule is used for the allocation of demand among facilities rather than assuming that each customer selects the closest facility. In addition, we also consider a multiobjective approach, which combines the two objectives. We propose heuristic solution procedures for the problem in the plane. Extensive computational results are presented.     

The Spatial Structuring of Central Places

The spatial structuring of “classical” central place systems is the primary concern of this research. Emphasizing the development of such systems from basic spatial relations between centers and hinterland locations (rather than as a result of an underlying geometry of regions), this paper takes an optimization approach to the siting of central places in a single-good system. The objective function is shown to be one that maximizes both market coverage of demand and market overlap. Access to this objective is given through the natural slack covering model.     

On the Logit Approach to Competitive Facility Location

The random utility model in competitive facility location is one approach for estimating the market share captured by a retail facility in a competitive environment. However, it requires extensive computational effort for finding the optimal location for a new facility because its objective function is based on a k -dimensional integral. In this paper we show that the random utility model can be approximated by a logit model. The proportion of the buying power at a demand point that is attracted to the new facility can be approximated by a logit function of the distance to it. This approximation demonstrates that using the logit function of the distance for estimating the market share is theoretically founded in the random utility model. A simplified random utility model is defined and approximated by a logit function. An iterative Weiszfeld-type algorithm is designed to find the best location for a new facility using the logit model. Computational experiments show that the logit approximation yields a good location solution to the random utility model.     

Results of a New Approach to Solving the p-Median Problem with Maximum Distance Constraints

Considerable interest has been directed in the past to developing approaches for solving the p-median problem with maximum distance constraints. All current solution techniques consider potential facilities to be located only at nodes of the network. This paper deals with the solution of this problem under the condition where facility placement is not restricted to nodes. The examples given show that improvement in weighted distance can be obtained by solving the unrestricted site problem. In addition, feasible solutions can be obtained for smaller numbers of facilities than possible by all nodal facility placement.     

A Synthesis of Aggregation Methods for Multifacility Location Problems: Strategies for Containing Error

When solving a location problem using aggregated units to represent demand, it is well known that the process of aggregation introduces error. Research has focussed on individual components of error, with little work on identifying and controlling total error. We provide a focussed review of some of this literature and suggest a strategy for controlling total error. Consideration of alternative criteria for evaluating aggregation schemes shows that the method selected should be compatible with the objectives of the analyses in which it is used. Experiments are described that show that two different measures of error are related in a nonlinear way to the number of aggregate demand points (q), for any value of the number of facilities (p). We focus on the parameter q/p and show that it is critical for determining the expected severity of the error. Many practical implementations of location algorithms operate within the range of q/p where the rate of change of error with respect to q/p is highest.     

A Bi‐objective Optimization Model for Designing Safe Walking Routes for School Children

This study uses a mathematical optimization approach to design safe walking routes from school to home for children. Children are thought to be safer when walking together in groups rather than alone. Thus, we assume that the risk of walking along a given road segment in a group is smaller than that of walking the same segment alone. At the same time, the walking route between school and home for each child should not deviate substantially from the shortest route. We propose a bi‐objective model that minimizes both the total risk (particularly, the total distance walked alone) and the total walking distance for all children. We present an integer programming formulation of the proposed problem and apply this formulation to two instances based on actual road networks. We obtain Pareto optimal solutions using a mathematical programming solver and analyze the characteristics of the solutions and their potential applicability to real situations. The results show that the proposed model produces much better solutions compared with the solution where each child walks along the shortest path from school to home. In some optimal solutions, only a small deviation from the shortest path results in a dramatic reduction of the risk objective.     

Surviving in a Competitive Spatial Market: The Threshold Capture Model

Many facility location decision models ignore the fact that for a facility to survive it needs a minimum demand level to cover costs. In this paper we present a decision model for a firm that wishes to enter a spatial market where there are several competitors already located. This market is such that for each outlet there is a demand threshold level that has to be achieved in order to survive. The firm wishes to know where to locate its outlets so as to maximize its market share taking into account the threshold level. It may happen that due to this new entrance, some competitors will not be able to meet the threshold and therefore will disappear. A formulation is presented together with a heuristic solution method and computational experience.     

Numerical Analysis,Computers and Problem-solving

Abstract

The number of problem which can be solved by the method of classical mathematics is limited, especially those which have application to practical problems. Numerical analysis offers a way out, but demand the co-operation of scientist, applied mathematician, and expertise in computer software and numerical methods. The following article discusses examples amenable to numerical analysis and describes the Oxford University-Industry Study Group which has successfully contributed to the solution of applied problems. It also argues for the establishment of computer program libraries which have recently become a valuable adjunct to problem solving.     

The Optimal Nodal Location of Public Facilities With Price-Sensitive Demand

The purpose of this paper is to present some models for the location of public facilities in nodal networks that explicitly maximize social welfare by accounting for price-elastic demand functions. The models presented here are general; yet they are mathematically equivalent to the plant location problem and are therefore amenable to solution procedures developed for the plant location problem. The models presented here distinguish between two institutional environments that reflect the degree of power of the consumer to choose which facility to patronize. If consumers can be assigned arbitrarily to facilities and can be denied service, then the environment is one of public fiat. If consumers must be served at the facility of their choice, then a “serve-allcomers” environment exists. Separate models for each environment are specified, and the relationship between optimal assignments and pricing policies is developed.     

Determining Market Areas Captured by Competitive Facilities: A Continuous Equilibrium Modeling Approach

Many existing models concerning locations and market areas of competitive facilities assume that customers patronize a facility based on distance to that facility, or perhaps on a function of distances between the customer and the different facilities available. Customers are generally assumed to be located at certain discrete demand points in a two-dimensional space, or continuously distributed over a one-dimensional line segment. In this paper these assumptions are relaxed by employment of a continuum optimization model to characterize the equilibrium choice behavior of customers for a given set of competitive facilities over a heterogeneous two-dimensional space. Customers are assumed to be scattered continuously over the space and each customer is assumed to choose a facility based on both congested travel time to the facility and on the attributes of the facility. The model is formulated as a calculus of variations problem and its optimality conditions are shown to be equivalent to the spatial customer-choice equilibrium conditions. An efficient numerical method using finite element technique is proposed and illustrated with a numerical example.     

On Covering Problems and the Central Facilities Location Problem

A number of variations of facilities location problems have appeared in the research literature in the past decade. Among these are problems involving the location of multiple new facilities in a discrete solution space, with the new facilities located relative to a set of existing facilities having known locations. In this paper a number of discrete solution space location problems are treated. Specifically, the covering problem and the central facilities location problem are shown to be related. The covering problem involves the location of the minimum number of new facilities among a finite number of sites such that all existing facilities (customers) are covered by at least one new facility. The central facilities location problem consists of the location of a given number of new facilities among a finite number of sites such that the sum of the weighted distances between existing facilities and new facilities is minimized. Computational experience in using the same heuristic solution procedure to solve both problems is provided and compared with other existing solution procedures.     

Modeling Geographic Ferrous Scrap Markets: Regional Prices and Interregional Transactions in the United States*

Abstract. The U.S. Regional Ferrous Scrap Model analyzes spatial variations in prices for two grades of ferrous scrap using a logistic model of choice under differentiated products. The model uses a computer‐generated equilibrium framework to solve for prices that support the observed spatial distribution of supply and demand quantities. This paper presents the model's formal structure and its solution algorithm. The model specification is highly disaggregated with 1,212 supply and 240 demand regions. Characteristics of the equilibrium solution are described for prices and interregional flows. Sensitivity of equilibrium values to changes in model parameters is reported.     

YIELDING OSCILLATOR UNDER TRIANGULAR GROUND ACCELERATION PULSE

An analytical solution is presented for the response of a bilinear inelastic simple oscillator to a symmetric triangular ground acceleration pulse. This type of motion is typical of near-fault recordings generated by source-directivity effects that may generate severe damage. Explicit closed-form expressions are derived for: (i) the inelastic response of the oscillator during the rising and decaying phases of the excitation as well as the ensuing free oscillations; (ii) the time of structural yielding; (iii) the time of peak response; (iv) the associated ductility demand. It is shown that when the duration of the pulse is long relative to the elastic period of the structure and its amplitude is of the same order as the yielding seismic coefficient, serious damage may occur if significant ductility cannot be supplied. The effect of post-yielding structural stiffness on ductility demand is also examined. Contrary to presently-used numerical algorithms, the proposed analytical solution allows many key response parameters to be evaluated in closed-form expressions and insight to be gained on the'response of inelastic structures to such motions. The model is evaluated against numerical results from actual near-field recorded motions. Illustrative examples are also presented.     

WEBER'S PROBLEM WITH ATTRACTION AND REPULSION*

ABSTRACT. Weber's problem consists of locating a facility in the plane in such a way that the weighted sum of Euclidean distances to n given points be minimum. The case where some weights are positive and some are negative is shown to be a d.-c. program (i.e., a global optimization problem with both the objective function and constraint functions written as differences of convex functions), reducible to a problem of concave minimization over a convex set. The reduced problem is then solved by outer-approximation and vertex enumeration. Moreover, locational constraints can be taken into account by combining the previous algorithm with an enumerative procedure on the set of feasible regions. Finally, the algorithm is extended to solve the case where obnoxiousness of the facility is assumed to have exponential decay. Computational experience with n up to 1000 is described.     

CENTER POINTS, EQUILIBRIUM POSITIONS, AND THE OBNOXIOUS LOCATION PROBLEM

ABSTRACT. Real variable analysis has een used to great benefit in a variety of classical problems in location theory. In this paper we explore basic complex variable techniques in one formulation of the obnoxious location problem. A general definition of center points is first given and used to formulate several alternate versions of the obnoxious location problem. A logarithmic transformation is then used to demonstrate some equivalences between these families of distinct location problems (defined via center points). A prototype logarithmic potential function which results from this formulation is then investigated, and it is demonstrated that the extremal solutions with this objective reside on the boundary of its domain of definition. An application using zero- and one-dimensional centers is discussed, and a generalization to the spatial obnoxious problem is also briefly examined. We define a zero-dimensional center as a critical point of the logarithmic potential function, and it is shown that these centers are equivalent to the solutions of the Complex Moment Problem.     

Designing Reliable Center Systems: A Vector Assignment Center Location Problem

The p‐center problem is one of the most important models in location theory. Its objective is to place a fixed number of facilities so that the maximum service distance for all customers is as small as possible. This article develops a reliable p‐center problem that can account for system vulnerability and facility failure. A basic assumption is that located centers can fail with a given probability and a customer will fall back to the closest nonfailing center for service. The proposed model seeks to minimize the expected value of the maximum service distance for a service system. In addition, the proposed model is general and can be used to solve other fault‐tolerant center location problems such as the (p, q)‐center problem using appropriate assignment vectors. I present an integer programming formulation of the model and computational experiments, and then conclude with a summary of findings and point out possible future work.     

THE MINIMAX-MIN LOCATION PROBLEM*

We consider a new objective function for the placement of a public facility with reference to variations in accessibility : the minimization of the range between the maximal and the minimal distances to users. Some properties of the solution are given; algorithms for the Euclidean and rectilinear distance cases are presented.     

城市居民休闲度假旅游需求实证研究——以南京为例

我国旅游业正在从观光游览型向休闲度假型转变。运用数理统计分析方法,以南京为例探析对城市居民休闲度假旅游需求进行研究。研究发现偏爱休闲度假的居民在旅游资源喜好、旅游活动偏好、购买行为、信息来源渠道、出游方式等方面与一般居民有着明显差异。据此探讨休闲度假产品的开发,分析未来的休闲度假旅游发展趋势,并提出相应的休闲度假旅游产品开发对策与建议。