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.     

MARKET CAPTURE BY TWO COMPETITORS: THE PREEMPTIVE LOCATION PROBLEM*

ABSTRACT. We consider a location and allocation game for two competitor firms, A and B, that each seek to locate p facilities in a network. A market is captured by a particular firm if that market's closest facility belongs to that firm rather than a competitor. The question is as follows: Firm A wants to locate its p facilities so that B, which enters also with p facilities after Firm A has located its facilities, will capture the minimum market value possible. That is, Firm A wishes to preempt Firm B in its bid to capture market share to the maximum extent possible. A model is presented that addresses this issue, together with solution methods and computing times.     

LOCATING A SINGLE NEW FACILITY AMONG EXISTING,UNEQUALLY ATTRACTIVE FACILITIES*

ABSTRACT. The location of a new facility in a competitive environment is investigated. It is assumed that customers patronize the facility with the highest utility value rather than the closest facility. Simulations show that the new approach yields a superior location in terms of market size captured. Sensitivity analyses of the problem parameters illustrate the sensitivity of the location and the sensitivity of the market size captured to the facility's quality.     

SPATIAL ASYMMETRIC DUOPOLY WITH AN APPLICATION TO BRUSSELS' AIRPORTS*

ABSTRACT In this paper, the problem of a city with access to two firms or facilities (shopping malls, airports, commercial districts) selling a differentiated product (shopping, flights) and/or offering a differentiated workplace is studied. Transport connections to one facility are congested. A model is presented for this asymmetric duopoly game that can be solved for a Nash equilibrium in prices and wages. A comparative statics analysis is used to illustrate the properties of the equilibrium. A numerical model is then applied to the two Brussels airports. Three stylized policies are implemented to address the congestion problem: expansion of transport capacity, congestion pricing, and a direct subsidy to the uncongested facility. Our results indicate that the degree of intrinsic differentiation between the two firms is crucial in determining the difference in profit and market share. Price and wage differences also depend on trip frequency and consumer preferences for diversity. Congestion pricing is the most effective policy tool but all three options are shown to have attractive attributes.     

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.     

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.     

THE OPTIMAL LOCATIONS OF BRANCH FACILITIES AND MAIN FACILITIES WITH CONSUMER SEARCH*

ABSTRACT. This paper examines the socially optimal locations of branch facilities (or small stores) and main facilities (or large stores) on a finite linear market that is uniformly populated from position 0 to position 1. Each consumer has a probability w of finding the desired service (or product) at a branch facility, and a probability 1 of finding the desired service (or product) at a main facility. Two types of consumer search are considered: phone search and visit search. Different assumptions are made about the numbers of branch facilities and main facilities (each involving one or two facilities of each type). Under visit search, the socially optimal locations of branch facilities tend to be closer to main facilities than under phone search, and this tendency is more pronounced for smaller values of w.     

A Family of Location Models for Multiple-Type Discrete Dispersion

One of the defining objectives in location science is to maximize dispersion. Facilities can be dispersed for a wide variety of purposes, including attempts to optimize competitive market advantage, disperse negative impacts, and optimize security. With one exception, all of the extant dispersion models consider only one type of facility, and ignore problems where multiple types of facilities must be located. We provide examples where multiple-type dispersion is appropriate and based on this develop a general class of facility location problems that optimize multiple-type dispersion. This family of models expands on the previously formulated definitions of dispersion for single types of facilities, by allowing the interactions among different types of facilities to determine the extent to which they will be spatially dispersed. We provide a set of integer-linear programming formulations for the principal models of this class and suggest a methodology for intelligent constraint elimination. We also present results of solving a range of multiple-type dispersion problems optimally and demonstrate that only the smallest versions of such problems can be solved in a reasonable amount of computer time using general-purpose optimization software. We conclude that the family of multiple-type dispersion models provides a more comprehensive, flexible, and realistic framework for locating facilities where weighted distances should be maximized, when compared with the special case of locating only a single type of facility.     

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.     

Externalities of Nuclear Power Plants: Further Evidence

Several prior studies found no detrimental external effects of nuclear power plants when estimating the distance gradient for housing prices within a hedonic model. Other papers found significant negative effects of nuclear power when studying real asset prices in cross sections of broad market areas.We suggest a resolution and verify that an installation effect occurs after controlling for the tendency of facility builders to seek out cheap land. The study assembles a panel of all commercial market areas, including indicators for nuclear facilities, in the contiguous United States observed 11 times over roughly equal intervals covering the span from 1945 to 1992.     

The Probabilistic Minisum Flow Interception Problem: Minimizing the Expected Travel Distance until Intercept under Probabilistic Interception

We develop a variant of the flow interception problem (FIP) in which it is more desirable for travelers to be intercepted as early as possible in their trips. In addition, we consider flows being intercepted probabilistically instead of the deterministic view of coverage assumed in the FIP literature. We call the proposed model the probabilistic minisum FIP (PMFIP); it involves minimizing the sum of the expected distance that each flow travels until intercepted at a facility among placed facilities. This extension allows us to evaluate the effect of facility location under any given value of the interception probability and to apply the model to a variety of situations. We apply the proposed model to an example network by assuming a hypothetical situation in which people gather at a stadium from various nodes on the network, and receive some goods or services on the way to the stadium. We analyze optimal solutions obtained by varying the number of facilities and interception probability. It is shown that the expected travel distance until intercept is greatly reduced by means of a few optimally located facilities under a moderate interception probability.     

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.     

Geographic information system‐based evaluation of spatial accessibility to maternal health facilities in Siaya County,Kenya

Maternal mortality is a major problem in middle‐income and low‐income countries, and the availability and accessibility of healthcare facilities offering safe delivery is important in averting maternal deaths. Siaya County, in Kenya, has one of the highest maternal mortality rates in the country—far more than the national average. This study aimed to evaluate geographic access to health facilities offering delivery services in Siaya County. A mixed‐methods approach incorporating geographic information system analysis and individual data from semi‐structured interviews was used to derive travel time maps to facilities using different travel scenarios: AccessMod5 and ArcGIS were used for these tasks. The derived maps were then linked to georeferenced household survey data in a multilevel logistic regression model in R to predict the probability of expectant women delivering in a health facility. Based on the derived travel times, 26 per cent (13,140) and 67 per cent (32,074) of the estimated 46,332 pregnant women could reach any facility within one and two hours, respectively, while walking with the percentage falling to seven per cent (3,415) and 20 per cent (8,845) when considering referral facilities. Motorised transport significantly increased coverage. The findings revealed that the predicted probability of a pregnant woman delivering in a health facility ranged between 0.14 and 0.86. Significant differences existed in access levels with transportation‐based interventions significantly increasing coverage. The derived maps can help health policy planners identify underserved areas and monitor future reductions in inequalities. This work has theoretical implications for conceptualising healthcare accessibility besides advancing the literature on mixed methodologies.     

Multi‐Type,Multi‐Zone Facility Location

The placement of facilities according to spatial and/or geographic requirements is a popular problem within the domain of location science. Objectives that are typically considered in this class of problems include dispersion, median, center, and covering objectives—and are generally defined in terms of distance or service‐related criteria. With few exceptions, the existing models in the literature for these problems only accommodate one type of facility. Furthermore, the literature on these problems does not allow for the possibility of multiple placement zones within which facilities may be placed. Due to the unique placement requirements of different facility types—such as suitable terrain that may be considered for placement and specific placement objectives for each facility type—it is expected that different suitable placement zones for each facility type, or groups of facility types, may differ. In this article, we introduce a novel mathematical treatment for multi‐type, multi‐zone facility location problems. We derive multi‐type, multi‐zone extensions to the classical integer‐linear programming formulations involving dispersion, centering and maximal covering. The complexity of these formulations leads us to follow a heuristic solution approach, for which a novel multi‐type, multi‐zone variation of the non‐dominated sorting genetic algorithm‐II algorithm is proposed and employed to solve practical examples of multi‐type, multi‐zone facility location problems.     

Activity Levels at Hub Facilities in Interacting Networks

Hubs are a special type of central facility which are designed to act as switching points for intemodal flows. For instance, a set of ten interacting cities might all be connected to one of two major hubs. All flows between the cities would then be routed via the hubs. There is an obvious saving in the number of routes necessary to interconnect the cities when hubs are utilized, with a concomitant high level of activity at the facilities. This paper takes a heuristic approach to the evaluation of networks and hub locations to find locally optimal designs. It is shown that minimization of transportation costs may require assignment of nodes to a facility other than the nearest. A discount on the interhub transportation costs promotes a wider spacing of facilities. In a system with several hubs, minimization of total hub usage tends to concentrate demand very heavily into one central facility.     

基于弱势群体需求的北京服务设施可达性集成研究

关注人群属性和需求,探讨服务设施供给的社会公平是新时代服务设施研究的重要议题,对于人口总量巨大、社会构成多元、社会空间分异凸显的大城市尤其如此。尽管研究某一弱势群体对单项服务设施需求的文献不断涌现,但是结合居民主观的服务设施需求偏好的可达性集成研究尚不多见。本文以北京作为案例城市,首先采用居民主观调查数据分析四类弱势群体对于公共服务设施的需求结构,然后基于POI数据,借助GIS分析这四类弱势群体服务设施需求偏好下的北京市公共服务设施综合可达性情况,且对综合可达性较差的区域进行了空间识别。本研究综合定性与定量方法,探讨主观与客观数据相匹配的可达性集成方法,可以为今后的相关研究提供借鉴,并能够对北京建设国际一流和谐宜居之都提供理论支撑。     

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.     

A Cost-Benefit Location-Allocation Model for Public Facilities: An Econometric Approach

This research develops and operationalizes a facility location-allocation model based on cost-benefit principles derived from welfare economics. Despite the theoretical advantages of cost-benefit location-allocation models, the difficulties associated with estimating household preferences for public facilities have heretofore prevented their application. This research demonstrates that the hedonic-pricing methodology can be effectively used to estimate preferences for public facilities. Specifically, household preferences for Baltimore public middle schools were estimated from the spatial variation in housing prices using the random bidding model. To provide an example of the methodology, the cost-benefit location-allocation objective function was maximized to simultaneously determine the optimal number, quality, and locations of Baltimore middle schools. The cost-benefit approach to facility location constitutes a major improvement over existing methods because it directly incorporates user preferences into the objective function and because the number and quality of facilities can be determined endogenously rather than being specified as a constraint a priori.     

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.     

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.