Bi-Objective Median Subtree Location Problems

A number of network design problems can be built on the following premise: given an undirected tree network, T, with node set, V, identify a single subtree, t, containing nodes, v, so that the subtree is located optimally with respect to the remaining, subset of unconnected nodes {V–v}. Distances between unconnected nodes and nodes in the subtree t can be defined on paths that are restricted to lie in the larger tree T (the restricted case), or can be defined on paths in an auxiliary complete graph G (the unrestricted case). The unrestricted case represents a class of problems that is not explicitly recognized in the literature, which is of intermediate complexity relative to the widely studied restricted case, and the general problem in which the underlying graph is general. This paper presents the Median Subtree Location Problem (MSLP), formulated as a bicriterion problem that trades off the cost of a subtree, t, against the population-weighted travel distance from the unconnected nodes to nodes on the subtree where both objectives are to be minimized. Integer programs were formulated for the travel restricted and travel unrestricted cases and were tested using linear programming and branch and bound to resolve fractions. Tradeoff curves between cost and travel burden were developed for sample networks.     

Computing Approximate Solutions of the Maximum Covering Problem with GRASP

We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal.     

Efficient solution methods for covering tree problems

In recent years, interest has been shown in the optimal location of ‘extensive’ facilities in a network. Two such problems—the Maximal Direct and Indirect Covering Tree problems—were introduced by Hutson and ReVelle. Previous solution techniques are extended to provide an efficient algorithm for the Indirect Covering Tree problem and the generalization in which demand covered is attenuated by distance. It is also shown that the corresponding problem is NP-hard when the underlying network is not a tree.     

Demand Point Aggregation for Planar Covering Location Models

The covering location problem seeks the minimum number of facilities such that each demand point is within some given radius of its nearest facility. Such a model finds application mostly in locating emergency types of facilities. Since the problem is NP-hard in the plane, a common practice is to aggregate the demand points in order to reduce the computational burden. Aggregation makes the size of the problem more manageable but also introduces error. Identifying and controlling the magnitude of the error is the subject of this study. We suggest several aggregation methods with a priori error bounds, and conduct experiments to compare their performance. We find that the manner by which infeasibility is measured greatly affects the best choice of an aggregation method.     

最小树的一般算法

本文根据设计并行算法的基本原则,给出了最小树的两个对偶定理.在此基础上,建立了两种对偶的同步并行算法的雏型.这两种算法恰恰在对偶的意义下,概括了以往的最小树算法.     

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.     

带覆盖需求约束的设施选址问题

带覆盖需求约束的设施选址问题(FLPWCDL)研究:客户必须在规定的响应半径内被服务,并要求服务站能够覆盖规定的需求数量,如何选择合适的服务站,使总成本(建站成本+路线成本)最小.FLPWCDL广泛应用于应急服务、物流、便利店等服务站的选址.建立了问题的混合整数规划模型,并构造了求解FLPWCDL的Benders分解算法,计算实验显示Benders分解算法具有非常高的求解效率与求解质量.     

Covering arrays with mixed alphabet sizes

Covering arrays with mixed alphabet sizes, or simply mixed covering arrays, are natural generalizations of covering arrays that are motivated by applications in software and network testing. A (mixed) covering array A of type is a k × N array with the cells of row i filled with elements from ? and having the property that for every two rows i and j and every ordered pair of elements (e,f) ∈ ? × ?, there exists at least one column c, 1 ≤ c ≤ N, such that A_i,c = e and A_j,c = f. The (mixed) covering array number, denoted by , is the minimum N for which a covering array of type with N columns exists. In this paper, several constructions for mixed covering arrays are presented, and the mixed covering array numbers are determined for nearly all cases with k = 4 and for a number of cases with k = 5. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 413–432, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10059     

最小最大二点集覆盖问题分析及改进算法设计

本文考虑二中心问题的扩展问题-最小最大二点集覆盖问题。给定两个平面点集P₁和P₂,分别包含m和n个点,求两个圆分别覆盖P₁和P₂,并且要求两圆半径与两圆圆心距三者中的最大值最小。本文主要贡献在于分析半径变化过程中两个点集中心包之间最近距离的变化关系,其中中心包是点集所具有的一个特殊几何结构,所得到的结果改进了Huang等人之前给出的结果,并且通过该结果设计相应算法,所得到的算法复杂性是目前最好的。     

Optimal location and design of a competitive facility

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits.It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favour of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in problems in which optimal solutions might not exist.Mathematics Subject Classification (2000):90B85, 90C30, 90C29, 91B42Partially supported by Grant PB96-1416-C02-02 of the D.G.E.S. and Grant BFM2002-04525-C02-02 of Ministerio de Ciencia y Tecnología, Spain     

Cyclic Difference Packing and Covering Arrays

Let n and k be positive integers. Let C_q be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (a_ij) with entries a_ij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from C_q such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of C_q occurs in the difference list at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.     

New Linear Codes with Covering Radius 2 and Odd Basis

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r=2k+1 (the case q=3, r=4k+1 was considered earlier). New code families with r=4k are also obtained. An updated table of upper bounds on the length function for linear codes with 24, R=2, and q=3,5 is given.     

Residential location and school planning in a tightening urban economy

Facing worse fiscal plight, many municipalities in Sweden must today carefully reexamine their activities. In urban planning, this has resulted in a growing interest in how the urban development could be designed to support and facilitate the efficient use of existing public investments. This paper focuses on the school sector as being one of the most costly. A location-allocation model of the capacitated facility location type is formulated. A set of potential schools consisting of existing and new ones are considered. The school-age children are assigned to a subset of these schools so as to minimize the sum of the capital costs of this subset and the transportation costs of the children. The model is applied to the municipality of Uppsala in Sweden. Different future settlement structures proposed by the planners as well as different housing allocations generated by a separate optimization model are evaluated.     

关于同伦满态与覆叠空间

本文在点标道路连通ＣＷ空间的同伦范畴中，利用同伦推出示性了同伦满态，得出了若ｆ：Ｘ－Ｙ是同伦满态，则对π１Ｙ的任一正规子群Ｈ，升腾映射ｆ：Ｘ（ｆ－１＃（Ｈ））→■（Ｈ）也是同伦满态．     

Approximating connected facility location with buy-at-bulk edge costs via random sampling

In the connected facility location problem with buy-at-bulk edge costs we are given a set of clients with positive demands and a set of potential facilities with opening costs in an undirected graph with edge lengths obeying the triangle inequality. Moreover, we are given a set of access cable types, each with a cost per unit length and a capacity such that the cost per capacity decreases from small to large cables, and a core cable type of infinite capacity. The task is to open some facilities and to connect them by a Steiner tree using core cables, and to build a forest network using access cables such that the edge capacities suffice to simultaneously route all client demands unsplit to the open facilities. The objective is to minimize the total cost of opening facilities, building the core Steiner tree, and installing the access cables. In this paper, we devise a constant-factor approximation algorithm for this problem based on a random sampling technique.     

Optimal location of interconnected facilities on tree networks subject to distance constraints

Optimal location of interconnected facilities on tree networks is considered in the case when some of the nodes of the network contain existing facilities. The distances between the facilities must satisfy maximum constraints. Polynomial algorithms for the solution of this problem are proposed.     

Polynomial algorithms for parametric minquantile and maxcovering planar location problems with locational constraints

A location is sought within some convex region of the plane for the central site of some public service to a finite number of demand points. The parametric maxcovering problem consists in finding for eachR>0 the point from which the total weight of the demand points within distanceR is maximal. The parametric minimal quantile problem asks for each percentage α the point minimising the distance necessary for covering demand points of total weight at least α. We investigate the properties of these two closely related problems and derive polynomial algorithms to solve them both in case of either (possibly inflated) Euclidean or polyhedral distances. The research of the first author is partially supported by Grant PB96-1416-C02-02 of Ministerio de Educación y Cultura, Spain.     

重大突发事件应急设施多重覆盖选址模型及算法

为了解决应对重大突发事件过程中应急需求的多点同时需求和多次需求问题,本文研究了应对重大突发事件的应急服务设施布局中的覆盖问题:针对重大突发事件应急响应的特点,引入最大临界距离和最小临界距离的概念,在阶梯型覆盖质量水平的基础上,建立了多重数量和质量覆盖模型。模型的优化目标是满足需求点的多次覆盖需求和多需求点同时需求的要求条件下,覆盖的人口期望最大,并用改进的遗传算法进行求解;最后给出的算例证明了模型和算法的有效性,从而应急设施的多重覆盖选址模型能够为有效应对重大突发事件的应急设施选址决策提供参考依据。     

An Improvement of an Inequality Linking Packing and Covering Densities in 3-space

It is shown that if K is a compact convex set which is centrally symmetric and has a nonempty interior, then the density of the tightest lattice packing with copies of K in Euclidean 3-space divided by the density of the thinnest lattice covering of Euclidean 3-space with copies of K is greater than or equal to 1/3. This improves the previous bound in [5] of 1/4. It is possible this bound will be improved in the future, though not beyond approximately 1/2     

一类螺旋映射的增长与掩盖定理

In this paper,we deduce growth and covering theorem for f(x)by the other means,where f(x)is strongly spiral-like mapping of typeβwith orderαdefined on Unit Ball B of complex Banach space,and still give growth upper bound and distortion upper bound for subordinate mapping.