Optimal Number of Hierarchical Facilities with Failures

This article proposes a continuous approximation model for determining the number of hierarchical facilities when lower level facilities are subject to failures. The average distance from customers to the nearest open facility is derived for two types of customer behavior. The optimal number of facilities that minimizes the average distance is then obtained. The analytical expression for the optimal number of facilities demonstrates how the location of facilities, the failure probability, and the customer behavior affect the optimal hierarchy and the average distance. The result shows that introducing the hierarchy can reduce the average distance if the failure probability is small and the penalty for failing to use facilities is large. The model provides a fundamental understanding of the optimal hierarchy and is useful for designing hierarchical facility systems.     

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 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.     

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.     

A Covering Tour Model for Planning Mobile Health Care Facilities in SuhumDistrict, Ghama

The provision of adequate primary health care in developing countries is often troublesome. The problem is to provide a sufficient number of facilities to be geographically accessible, yet few enough to be properly stocked and staffed. In many less developed countries accessibility problems are exacerbated by extensive rainy seasons in which travel is only possible on paved roads. Using the covering tour model we investigate the use of mobile facilities to resolve this dilemma in Suhum District, Ghana. The model minimizes a mobile facility's travel while serving all population centers within range of a feasible stop. Computational results show that in the rainy season the model cannot provide full coverage; over six percent of the population is beyond a covering distance of eight kilometers. In the dry season, 99 percent of the population can be served by a tour at a covering distance of seven kilometers.Beyond a distance of four kilometers, the dry season problem becomes a trade-off between the distance traveled by healthcare patrons and mobile facilities. These results illustrate the importance of flexibility of mobile systems: if accessibility cannot be provided in all seasons it may still be provided at favorable times of the year.     

Order Distance in Regular Point Patterns

This article examines the k th nearest neighbor distance for three regular point patterns: square, triangular, and hexagonal lattices. The probability density functions of the k th nearest distance and the average k th nearest distances are theoretically derived for k =1, 2, …, 7. As an application of the k th nearest distance, we consider a facility location problem with closing of facilities. The problem is to find the optimal regular pattern that minimizes the average distance to the nearest open facility. Assuming that facilities are closed independently and at random, we show that the triangular lattice is optimal if at least 68% of facilities are open by comparing the upper and lower bounds of the average distances.     

基于GIS的农村医疗设施空间可达性分析——以河南省兰考县为例

针对农村医疗设施空间分布的公平性问题,提出了使用GIS技术和空间可达性指标评估医疗设施的区域分布特征。以兰考县乡级以上卫生院为例,建立人口分布、医疗设施位置、行政区域等地理数据库,选择人均医疗资源分配、就医的最近距离、选择医院的机会、重力模型及改进的重力模型5个空间可达性模型,计算了各乡镇、各行政村的医疗设施可达性指标,并制作了相应的专题地图;在此基础上对医疗设施的空间布局进行了分析。空间可达性指标全面地反映医疗设施的空间分布特征,鉴别出资源分配较薄弱的区位,是农村医疗改革中设施规划和资源分配的重要依据。     

南京市社区服务设施可达性的空间格局与低收入社区空间剥夺研究

社区服务设施的空间可达性事关城市公共资源分配的社会公平与公正,是反映城市居民生活质量的重要标志。以南京市居住用地作为基本分析单元,对南京市的教育、医疗、商业和公园绿地四大类社区服务设施的空间可达性进行分析;并通过对相关低收入社区的实地调研,研究城市边缘区低收入群体的服务设施空间剥夺问题。研究表明：南京市四类服务设施的可达性水平均呈现出由城市中心区向郊区衰减的空间分异格局;较低的社区服务设施可达性不仅直接造成低收入社区居民对社区服务设施较低的满意度,导致其空间剥夺感的产生,还极大地阻碍着社区归属感的形成,导致社区居民强烈的迁居欲望,在他们较低的支付能力下进一步加剧其空间剥夺感。政府未来在规划面向低收入群体的保障性社区时,更应注重相关服务配套设施建设,做到社会与空间的公正性。     

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.     

Programming Models for Facility Dispersion: The p-Dispersion and Maxisum Dispersion Problems

The p-dispersion problem is to locate p facilities on a network so that the minimum separation distance between any pair of open facilities is maximized. This problem is applicable to facilities that pose a threat to each other and to systems of retail or service franchises. In both of these applications, facilities should be as far away from the closest other facility as possible. A mixed-integer program is formulated that relies on reversing the value of the 0–1 location variables in the distance constraints so that only the distance between pairs of open facilities constrain the maximization. A related problem, the maxisum dispersion problem, which aims to maximize the average separation distance between open facilities, is also formulated and solved. Computational results for both models for locating 5 and 10 facilities on a network of 25 nodes are presented, along with a multicriteria approach combining the dispersion and maxisum problems. The p -dispersion problem has a weak duality relationship with the (p-1)-center problem in that one-half the maximin distance in the p-dispersion problem is a lower bound for the minimax distance in the center problem for (p-1) facilities. Since the p-center problem is often solved via a series of set-covering problems, the p-dispersion problem may prove useful for finding a starting distance for the series of covering problems.     

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.     

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.     

MEDIAN,MEAN, AND OPTIMUM AS FACILITY LOCATIONS*

ABSTRACT The mean and the median are both measures of centrality. In one dimension, the median minimizes the average absolute distance from a facility to a set of customers {x_i}, and the mean minimizes the average squared distance. In two dimensions, the median minimizes the average rectangular distance, and the mean minimizes the average squared distance. This paper investigates the “location penalty” when a nonoptimal location is substituted for the optimal location. In one dimension, the average absolute distance at the mean is never more than twice the average absolute distance at the median. Surprisingly, this happens when the median and mean are close together. In two dimensions, the ratio of the average Euclidean distance at the median to the average Euclidean distance at the optimum is never more than . However, this upper bound depends on an unlikely scenario with just two customers. With three equal-sized customers, this ratio is never more than 1.12. However, if the triangle formed by the customers is rotated relative to the rectangular grid, the ratio never exceeds 1.028.     

A method of fitting the gravity model based on the Poisson distribution

"In this paper, [the authors] suggest an alternative method for fitting the gravity model. In this method, the interaction variable is treated as the outcome of a discrete probability process, whose mean is a function of the size and distance variables. This treatment seems appropriate when the dependent variable represents a count of the number of items (people, vehicles, shipments) moving from one place to another. It would seem to have special advantages where there are some pairs of places between which few items move. The argument will be illustrated with reference to data on the numbers of migrants moving in 1970-1971 between pairs of the 126 labor market areas defined for Great Britain...."     

COMPETITIVE FACILITIES: MARKET SHARE AND LOCATION WITH RANDOM UTILITY*

ABSTRACT. A new approach is proposed for calculating the expected market share. It is assumed that consumers patronize a facility according to a utility function, selecting the facility with the highest utility value. However, consumers'ratings of the utility components are stochastic by some random distribution. Therefore, the buying power of customers located at the same point is divided among several facilities. A probability that a consumer patronizes a certain facility can be calculated. Consequently, the expected market share by competing facilities can be estimated. This calculation is more than 1,000 times faster than repeating a simulation enough times to achieve a reasonable accuracy. The distance decay calculated using the new approach is approximately exponential. A procedure for finding the optimal location anywhere in the plane for a new facility that maximizes the market share is also introduced.     

公交站点可达性测度及其在停车分区中的应用

考虑公交与小汽车方式间的可达性差异,提出了依据公交服务水平调控小汽车可达性的停车分区思路,以期实现对小汽车方式的合理限制,支持城市公交优先发展。首先根据居民出行对不同类型公交站点具有不同可达性需求,将以需求点为圆心、站点时空可接近距离为半径的覆盖范围作为基本分析单元,构建基于微观个体时空约束下的公交站点可达性测度模型。其次借助ArcGIS,以昆明市公交网络数据为基础,对站点可达性空间分布特征进行可视化分析,所得结果与城市用地开发整体特征相吻合,验证了模型的合理性。最后以公交可达性水平为主要依据划分停车供给分区,从而对不同分区停车泊位供给的科学配置提供支撑。     

基于GIS的徐州市城乡交通基础设施通达性研究

以徐州市城乡交通网络数据为数据源,利用GIS空间分析技术,选取长度-半径维数、分枝维数、距离可达性、时间可达性和可达性系数5个指标对该区交通网络的空间形态、空间分布、距离关系、时间关系等进行多角度分析。结果表明,徐州市城乡交通网络长度-半径维数呈现明显的空间分异,徐州市区及周边地区交通网络密度最大,其次是东部,西北部县域交通网络密度最小,分枝维数和长度-半径维数的空间分布基本一致。区内各节点的时间和距离可达性均随距离徐州市区的增大而增大,通达性高于全区平均水平的节点有徐州市辖区各节点,接近全区平均水平的节点有邳州市、沛县和睢宁县,而丰县和新沂市的通达性水平则低于全区平均通达性水平。     

EQUILATERAL TRIANGLE VS. HEXAGON: A MOST SIMPLE DEMONSTRATION

ABSTRACT. This note presents a most simple proof of the superiority of hexagons over equilateral triangles of the same size in terms of minimization of the average distance between consumers and facilities. It also stresses that, under the same assumptions, facilities are closer to each other in a triangular grid than in an hexagonal grid and that this fact should not be neglected.     

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.     

年龄分层视角下医疗设施可达性对居民就医行为的影响——以上海市崇明岛为例

年龄是影响就医行为的重要因素,从年龄分层视角出发探讨医疗资源可达性对就医行为的影响有助于理解居民的就医行为决策机理.本文以崇明岛为案例,利用一手调查数据,结合两步移动搜寻算法和回归分析方法,探讨了医疗设施可达性对居民就医机构选择和惯常就医距离的影响,揭示了不同年龄群体之间就医行为影响机理的差异.研究发现,医疗设施可达性...