连续设施选址:模型、方法与应用

给定度量空间和该空间中的若干顾客,设施选址为在该度量空间中确定新设施的位置使得某种目标达到最优。连续设施选址是设施选址中的一类重要问题,其中的设施可在度量空间的某连续区域上进行选址。本文对连续设施选址的模型、算法和应用方面的工作进行了综述。文章首先讨论了连续设施选址中几个重要元素,包括新设施个数、距离度量函数、目标函数;然后介绍了连续选址中的几种经典模型和拓展模型;接着概述了求解连续选址问题的常用优化方法和技术,包括共轭对偶、全局优化、不确定优化、变分不等式方法、维诺图;最后介绍了连续设施选址的重要应用并给出了研究展望。     

Multi-Service Facility Location Models

The multi-service facility (MSF) concept proposes the co-location of a range of human services under a single roof. Thus, for example, services for preschoolers, teens, and seniors might be co-located within a joint facility. MSFs are a response to economies of scale; co-location reduces the monetary costs of providing a variety of services. Yet, the spatial distributions of different societal groups are different within a city – an MSF system will not provide optimum geographical accessibility to individual groups. We introduce two p-median based location–allocation models that trade off the fixed costs of providing services and opening facilities with the travel costs of three societal groups. We observe that some mixes of single- and multi-service facilities can provide efficient service systems without unduly compromising the accessibility needs of individual groups.     

A Long-range Selection and Timing Analysis System for Facility Location: Theory

This paper describes the development of a mixed-integer mathematical programming model designed to provide strategic decisions for the selection of a set of proposed facilities to serve distinct geographic regions and, of those selected, their projected timing within a short-time implementation horizon, so that the ratio of net return to total assets is maximized over the entire economic life of the facilities considered.     

A Dual-Based Add Heuristic for Uncapacitated Facility Location

This paper presents a heuristic method for solving the uncapacitated facility-location problem (UFLP), which is similar to Erlenkotter's ‘dual ascent’ procedure. The heuristic is of the ‘add’ type, which progressively selects facilities to open according to a certain criterion derived from the analysis of the linear programming dual. Computational experience with both (static) UFLPs and dynamic UFLPs reveals that the heuristic method yields solutions in most cases superior in quality to those achieved by the dual-ascent procedure, with barely noticeable additional computation time.     

Facility Location on a Sphere

In this paper, we consider a modification of the Weber problem which consists of locating a new facility on a sphere so that the weighted sum of distances to given demand points is minimized. Two ways of measuring distance are used. One is simply the length of shortest arc on the sphere. The other norm, first suggested by Katz and Cooper, may be used to approximate squared arc distance on a hemisphere and also as a rough approximation for arc distance. Several properties of the problem are established. An iterative heuristic method for solving the problem with shortest arc distances is presented.     

离散设施选址问题研究综述

本文首先回顾了设施选址问题百年发展历史,认为其研究经历了零散研究、系统研究、不确定性研究三个阶段.离散选址问题包括中值问题、覆盖问题、中心问题、多产品问题、动态问题、多目标问题、路径选址问题、网络中心选址问题8个子问题.最后作者讨论了选址问题研究中存在的问题以及今后发展的趋势.     

两阶段选址优化问题研究

本文主要考虑如下实际问题:假设选址决策者需要建设p个设施,但是由于资金等等的影响,实际建设时会被要求先建设q个设施,其次再建设p-q个设施(设p>q),同时要求,在建设p-q个设施的时候,已经建设好的q个设施不被删除。本文建立了一个两阶段优化问题,问题的输出是两个待修建的设施的集合Fq,Fp,|Fp|=p,|Fq|=q,且Fq是Fp的子集,问题的目标是最小化这两个设施集合的费用同对应的最优费用的比值的最大值。本文给出一个近似比为9的近似算法,并对一些特殊的情况进行了讨论。所得结论对实际的选址决策具有理论意义,同时也完善已有相关研究结果。     

基于失效情景的应急设施选址问题

非常规突发事件巨大的破坏力以及发生时间、地点和规模的不确定性,使应急系统内设施有可能被破坏而失效,因此选址时必须考虑设施失效情景的发生.给出以最大限度覆盖用户需求为目标,基于失效设施数目具有不确定性情景的设施选址双层随机规划模型;通过计算模型上下界,给出减小规模的等价模型,降低了双层规划求解难度;最后实验验证了模型的合理性,并给出新增选址方案.     

A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points

Foundations of Computational Mathematics - A breakthrough paper written in 1993 by Shub and Smale unveiled the relationship between stable polynomials and points which minimize the discrete...     

Facility Location on a Polyhedral Surface

Given an orientable genus-0 polyhedral surface defined by n triangles, and a set of m point sites} on it, we would like to identify its 1-center, i.e., the location on the surface that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the surface. To compute the 1-center, we compute the furthest-site Voronoi diagram of the sites on the polyhedral surface. We show that the diagram has maximum combinatorial complexity (m n ²), and present an algorithm that computes the diagram in O(m n ²log mlog n) expected time. The 1-center can then be identified in time linear in the size of the diagram.     

Efficient Location for a Semi-Obnoxious Facility

This paper deals with a location model for the placement of a semi-obnoxious facility in a continuous plane with the twin objectives of maximizing the distance to the nearest inhabitant and minimizing the sum of distances to all the users (or the distance to the farthest user) in a unified manner. For special cases, this formulation includes (1) elliptic maximin and rectangular minisum criteria problem, and (2) rectangular maximin and minimax criteria problem. Polynomial-time algorithms for finding the efficient set and the tradeoff curve are presented.     

Branch-and-Cut-and-Price for Capacitated Connected Facility Location

We consider a generalization of the Connected Facility Location Problem (ConFL), suitable to model real world network extension scenarios such as fiber-to-the-curb. In addition to choosing a set of facilities and connecting them by a Steiner tree as in ConFL, we aim to maximize the resulting profit by potentially supplying only a subset of all customers. Furthermore, capacity constraints on potential facilities need to be considered. We present two mixed integer programming based approaches which are solved using branch-and-cut and branch-and-cut-and-price, respectively. By studying the corresponding polyhedra we analyze both approaches theoretically and show their advantages over previously presented models. Furthermore, using a computational study we are able to additionally show significant advantages of our models over previously presented ones from a practical point of view.     

A heuristic for BILP problems: The Single Source Capacitated Facility Location Problem

In the Single Source Capacitated Facility Location Problem (SSCFLP) each customer has to be assigned to one facility that supplies its whole demand. The total demand of customers assigned to each facility cannot exceed its capacity. An opening cost is associated with each facility, and is paid if at least one customer is assigned to it. The objective is to minimize the total cost of opening the facilities and supply all the customers. In this paper we extend the Kernel Search heuristic framework to general Binary Integer Linear Programming (BILP) problems, and apply it to the SSCFLP. The heuristic is based on the solution to optimality of a sequence of subproblems, where each subproblem is restricted to a subset of the decision variables. The subsets of decision variables are constructed starting from the optimal values of the linear relaxation. Variants based on variable fixing are proposed to improve the efficiency of the Kernel Search framework. The algorithms are tested on benchmark instances and new very large-scale test problems. Computational results demonstrate the effectiveness of the approach. The Kernel Search algorithm outperforms the best heuristics for the SSCFLP available in the literature. It found the optimal solution for 165 out of the 170 instances with a proven optimum. The error achieved in the remaining instances is negligible. Moreover, it achieved, on 100 new very large-scale instances, an average gap equal to 0.64% computed with respect to a lower bound or the optimum, when available. The variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality.     

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

This study presents a summary of the current state of research on the distributional Henstock-Kurzweil integral. Basic properties such as integration by parts, Hake theorem, inner product, Hölder inequality, second mean value theorem, orderings, Banach lattice, convergence theorems, fixed point theorems, are shown. This study also summarizes its applications in integral and differential equations.     

Simulated Annealing and Genetic Algorithms for the Facility Layout Problem: A Survey

The facility layout problem (FLP) has many practical applications and is known to be NP-hard. During recent decades exact and heuristic approaches have been proposed in the literature to solve FLPs. In this paper we review the most recent developments regardingsimulated annealing and genetic algorithms for solvingfacility layout problems approximately.     

Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location

In this paper we introduce an optimization problem which involves maximization of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems consisting of both mobile and stationary facilities. Let ψ be a circular path through which mobile service stations are plying, and S be a set of n stationary facilities (points) inside ψ. A demand point p is served from a mobile facility plying along ψ if the distance of p from the boundary of ψ is less than that from any member in S. On the other hand, the demand point p is served from a stationary facility p _i  ∈ S if the distance of p from p _i is less than or equal to the distance of p from all other members in S and also from the boundary of ψ. The objective is to place the stationary facilities in S, inside ψ, such that the total area served by them is maximized. We consider a restricted version of this problem where the members in S are placed equidistantly from the center o of ψ. It is shown that the maximum area is obtained when the members in S lie on the vertices of a regular n-gon, with its circumcenter at o. The distance of the members in S from o and the optimum area increases with n, and at the limit approaches the radius and the area of the circle ψ, respectively. We also consider another variation of this problem where a set of n points is placed inside ψ, and the task is to locate a new point q inside ψ such that the area of the Voronoi region of q is maximized. We give an exact solution of this problem when n = 1 and a (1 − ε)-approximation algorithm for the general case.     

A Parallel Interior Point Method and Its Application to Facility Location Problems

We present a parallel interior point algorithm to solve block structured linear programs. This algorithm can solve block diagonal linear programs with both side constraints (common rows) and side variables (common columns). The performance of the algorithm is investigated on uncapacitated, capacitated and stochastic facility location problems. The facility location problems are formulated as mixed integer linear programs. Each subproblem of the branch and bound phase of the MIP is solved using the parallel interior point method. We compare the total time taken by the parallel interior point method with the simplex method to solve the complete problems, as well as the various costs of reoptimisation of the non-root nodes of the branch and bound. Computational results on two parallel computers (Fujitsu AP1000 and IBM SP2) are also presented in this paper.