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.     

Dispersion of Nodes Added to a Network

For location problems in which optimal locations can be at nodes or along arcs but no finite dominating set has been identified, researchers may desire a method for dispersing p additional discrete candidate sites along the m arcs of a network. This article develops and tests minimax and maximin models for solving this continuous network location problem, which we call the added-node dispersion problem (ANDP). Adding nodes to an arc subdivides it into subarcs. The minimax model minimizes the maximum subarc length, while the maximin model maximizes the minimum subarc length. Like most worst-case objectives, the minimax and maximin objectives are plagued by poorly behaved alternate optima. Therefore, a secondary MinSumMax objective is used to select the best-dispersed alternate optima. We prove that equal spacing of added nodes along arcs is optimal to the MinSumMax objective. Using this fact we develop greedy heuristic algorithms that are simple, optimal, and efficient (O( mp )). Empirical results show how the maximum subarc, minimum subarc, and sum of longest subarcs change as the number of added nodes increases. Further empirical results show how using the ANDP to locate additional nodes can improve the solutions of another location problem. Using the p-dispersion problem as a case study, we show how much adding ANDP sites to the network vertices improves the p-dispersion objective function compared with (a) network vertices only and (b) vertices plus randomly added nodes. The ANDP can also be used by itself to disperse facilities such as stores, refueling stations, cell phone towers, or relay facilities along the arcs of a network, assuming that such facilities already exist at all nodes of the network.     

Adaptive Cell Tower Location Using Geostatistics. 基于地统计学的适应性基站发射塔选址研究

In this article, we address the problem of allocating an additional cell tower (or a set of towers) to an existing cellular network, maximizing the call completion probability. Our approach is derived from the adaptive spatial sampling problem using kriging, capitalizing on spatial correlation between cell phone signal strength data points and accounting for terrain morphology. Cell phone demand is reflected by population counts in the form of weights. The objective function, which is the weighted call completion probability, is highly nonlinear and complex (nondifferentiable and discontinuous). Sequential and simultaneous discrete optimization techniques are presented, and heuristics such as simulated annealing and Nelder–Mead are suggested to solve our problem. The adaptive spatial sampling problem is defined and related to the additional facility location problem. The approach is illustrated using data on cell phone call completion probability in a rural region of Erie County in western New York, and accounts for terrain variation using a line‐of‐sight approach. Finally, the computational results of sequential and simultaneous approaches are compared. Our model is also applicable to other facility location problems that aim to minimize the uncertainty associated with a customer visiting a new facility that has been added to an existing set of facilities.     

Location Modeling Utilizing Maximum Service Distance Criteria

The location set-covering problem (LSCP) and the maximal covering location problem (MCLP) have been the subject of considerable interest. As originally defined, both problems allowed facility placement only at nodes. This paper deals with both problems for the case when facility placement is allowed anywhere on the network. Two theorems are presented that show that when facility placement is unrestricted, for either the LSCP or MCLP at least one optimal solution exists that is composed entirely of points belonging to a finite set of points called the network intersect point set (NIPS). Optimal solution approaches to the unrestricted site LSCP and MCLP problems that utilize the NIPS and previously developed solution methodologies are presented. Example solutions show that considerable improvement in the amount of coverage or the number of facilities needed to insure total coverage can be achieved by allowing facility placement along arcs of the network. In addition, extensions to the arc-covering model and the ambulance-hospital model of ReVelle, Toregas, and Falkson are developed and solved.     

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.     

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 Regionally Constrained p-Median Problem

In many applications involving the location of public facilities, the activity that is being located is added to the landscape of existing public facilities and services. Further, there also exists a political and behavioral landscape of differing jurisdictions, representations, and perceptions. This paper presents a new form of the p-median problem which addresses two types of “regional” constraints that can arise in public facilities planning. A formulation is given for this specially constrained median problem along with a solution approach. Several examples are presented with computational results which indicate that this type of constrained problem could lead to better facilities planning.     

Location-Allocation Models of Some Traffic Facilities

The objective of this paper is to formulate location-allocation models of linear facilities, such as bridges, railroad crossings, and staircases. The location-allocation models in this paper differ from existing location-allocation models in that the service provided by the facilities is consumed by trips, with associated origins and destinations. The first part of the paper shows the formulation of these location-allocation problems in one-dimensional space. It is demonstrated that these problems can also be formulated by the use of a rectilinear Voronoi diagram in two-dimensional space. The second part of the paper demonstrates fundamental relationships between the number of facilities and the total length of indirect trips and between the number of facilities and the total number of indirect trips for uniformly and independently distributed origins and destinations.     

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.     

The Multi‐level Location Set Covering Model

The classical Location Set Covering Problem involves finding the smallest number of facilities and their locations so that each demand is covered by at least one facility. It was first introduced by Toregas in 1970. This problem can represent several different application settings including the location of emergency services and the selection of conservation sites. The Location Set Covering Problem can be formulated as a 0–1 integer‐programming model. Roth (1969) and Toregas and ReVelle (1973) developed reduction approaches that can systematically eliminate redundant columns and rows as well as identify essential sites. Such approaches can often reduce a problem to a size that is considerably smaller and easily solved by linear programming using branch and bound. Extensions to the Location Set Covering Model have been proposed so that additional levels of coverage are either encouraged or required. This paper focuses on one of the extended model forms called the Multi‐level Location Set Covering Model. The reduction rules of Roth and of Toregas and ReVelle violate properties found in the multi‐level model. This paper proposes a new set of reduction rules that can be used for the multi‐level model as well as the classic single‐level model. A demonstration of these new reduction rules is presented which indicates that such problems may be subject to significant reductions in both the numbers of demands as well as sites.     

An Equity Model for Locating Environmentally Hazardous Facilities

This paper develops a multiobjective mathematical location model to identify possible locations for environmentally hazardous facilities. Risk and equity are recognized as the most important criteria in determining site selection. In contrast to earlier models, the equity objective explicitly considers the existing distribution of environmental burdens when siting new hazardous facilities. Proposed environmentally hazardous facilities are located so that the burdens associated with new and existing hazards are shared as equally as possible among all areas. The application of the model, in a case study of the Greenpoint/Williamsburg neighborhood in Brooklyn, New York, illustrates the trade‐offs associated with various risk and equity scenarios. Sensitivity analyses demonstrate how the existing distribution of environmental burdens may act as a constraint and limit the degree of equity that may be obtained when locating new facilities.     

Two New Location Covering Problems: The Partial P-Center Problem and the Partial Set Covering Problem

Two new covering problems are introduced. The partial covering P-center problem minimizes a coverage distance in such a way that a given fraction of the population is covered. The partial set covering problem seeks the minimum number of facilities needed to cover an exogenously specified fraction of the population within a given coverage distance. The problems are formulated as integer linear programming problems. Bisection search algorithms are outlined for the two problems. The search algorithm repeatedly solves a Lagrangian relaxation of the maximal covering problem. Computational results for the Lagrangian relaxation of the maximal covering problem and for the bisection search algorithms are presented on problems with up to 150 nodes.     

An Interactive Approach to the Central Facility Location Problem: Locating Pediatric Hospitals in Warsaw

Most of the conventional approaches to the central facility location problem neglect the interaction of analyst and decision-maker during the locational choice process. This paper presents a new interactive approach to the central facility location problem. It is assumed that the problem is formulated by an analyst as a multiobjective optimization problem. Then the decision-maker searches for a satisfactory solution working directly with the computer system. The interactive procedure was implemented on IBM-PC XT/AT as the decision support system DINAS (Dynamic Interactive Network Analysis System) which enables the solution of various multiobjective location-allocation problems. DINAS has been successfully used for solving a real-world planning problem, namely for finding locations for pediatric hospitals in the Warsaw region.     

Competitive Multifacility Location on Networks: the (r∣Xp)‐Medianoid Problem

Abstract.  We investigate the ( r ∣ X _p )‐medianoid problem for networks. This is a competitive location problem that consists of determining the locations of r facilities belonging to a firm in order to maximize its market share in a space where a competitor is already operating with p facilities. We consider six scenarios resulting from the combination of three customer choice rules (binary, partially binary, and proportional) with two types of services (essential and unessential).
Known discretization results about the existence of a solution for the set of nodes are extended. Some examples and computational experience using heuristic algorithms are presented.     

Two New Location Covering Problems: The Partial P-Center Problem and the Partial Set Covering Problem

Two new covering problems are introduced. The partial covering P-center problem minimizes a coverage distance in such a way that a given fraction of the population is covered. The partial set covering problem seeks the minimum number of facilities needed to cover an exogenously specified fraction of the population within a given coverage distance. The problems are formulated as integer linear programming problems. Bisection search algorithms are outlined for the two problems. The search algorithm repeatedly solves a Lagrangian relaxation of the maximal covering problem. Computational results for the Lagrangian relaxation of the maximal covering problem and for the bisection search algorithms are presented on problems with up to 150 nodes.     

Errors Caused by Rounded Data in Two Simple Facility Location Problems

As a rule, data to be used in locational analysis are either rounded up or rounded down. Therefore, error is incurred if such location data are used. The objective of this paper is to examine location error and cost error due to rounding in unweighted minisum and minimax problems in one-dimensional continuous space. Several conclusions on rounding effects are obtained by examining the respective mean-squared errors. First, rounding tends to exert more serious influence on the minisum problem than on the minimax problem. Second, in both location problems, the location error shows a pattern that is the inverse of that of the cost error.     

Asymmetric Entrant Locaiton with a Discrete Consumer Distribution: Spatial Price Discrimination Versus Mill Pricing

In this paper I examine the profit-maximizing locations of entrants. Suppose that firms practice spatial price discrimination and consumer locations are discrete, such as five equally spaced towns on a roadway. With completely inelastic consumer demand an entrant between two existing firms is often indifferent between the symmetric (central) location and a continuum of asymmetric (noncentral) locations. However, downward-sloping consumer demand often causes the entrant to strictly prefer either of two asymmetric locations to any other location. These results are very different from those found in mill-pricing (free-on-board or f.o.b.-pricing) models.     

Protecting Critical Assets: The r-Interdiction Median Problem with Fortification

Many systems contain bottlenecks, critical linkages, and key facilities. Such components, when lost due to a man-made or natural disaster, may imperil a system in performing its intended function. This article focuses on reducing the impact of an intentional strike against a supply system where supply facilities can be fortified in order to prevent such events. It is assumed that fortification resources are limited and must be used in the most efficient manner. In a recent article, Church, Scaparra, and Middleton (2004) introduced the r-interdiction median problem, which can be used to identify the most important facilities in a supply system. In this article, we extend that model to address the option of fortifying such sites against possible interdiction. We present a new integer-linear programming model that optimally allocates fortification resources in order to minimize the impact of interdiction. Computational results are presented in using this model for several hypothetical problems. We also discuss the general properties of fortification and demonstrate that the presence of fortification can impact which system elements are considered critical.     

Elimination of Source A and B Errors in p-Median Location Problems

The p-median problem is a powerful tool in analyzing facility location options when the goal of the location scheme is to minimize the average distance that demand must traverse to reach its nearest facility. It may be used to determine the number of facilities to site, as well as the actual facility locations. Demand data are frequently aggregated in p-median location problems to reduce the computational complexity of the problem. Demand data aggregation, however, results in the loss of locational information. This loss may lead to suboptimal facility location configurations (optimality errors) and inaccurate measures of the resulting travel distances (cost errors). Hillsman and Rhoda (1978) have identified three error components: Source A, B, and C errors, which may result from demand data aggregation. In this article, a method to measure weighted travel distances in p-median problems which eliminates Source A and B errors is proposed. Test problem results indicate that the proposed measurement scheme yields solutions with lower optimality and cost errors than does the traditional distance measurement scheme.     

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.