Optimal M/G/1 server location on a tree network with continuous link demands

This paper represents two major extensions of Hakimi's one-median problem specialized on a tree network T:(i) queueing delay is explicitly included in the objective function, and (ii) probabilistic demands for service can originate continuously along a link as well as discretely at a node. Calls for service occur as a time-homogeneous Poisson process. A single mobile server resides at a facility located on T. The server, when available, is dispatched immediately to any demand that occurs. When a customer finds the server busy with previous demand, it is entered into a first-come first-served queue. One desires to locate a facility on T so as to minimize the average response time, which is the sum of mean queueing delay and mean travel time. Convexity properties of the average response time and related functions allow us to develop an efficient two-stage algorithm for finding the optimal location. We also analytically trace the trajectory of the optimal location when the Poisson arrival rate is varied. A numerical example is constructed to demonstrate the algorithm as well as the trajectory results.     

The minisum location problem on an undirected network with continuous link demands

In the one-median problem, a facility is to be located on a network minimizing total travel distances from the facility to customer demands restricted to nodal locations. In reality, however, demands do occur on links of a network. Thus, aggregation of or restriction to nodal demands may not be a satisfactory approximation. In this paper, we generalize the one-median problem to a network with discrete nodal as well as general continuous link demands. Properties of the total travel distance, as a function of the facility's location, are examined. We introduce an exact and a heuristic procedure to find an optimal location for the facility. An efficient algorithm is developed when the network is a tree.     

The probabilistic gradual covering location problem on a network with discrete random demand weights

We study the gradual covering location problem on a network with uncertain demand. A single facility is to be located on the network. Two coverage radii are defined for each node. The demand originating from a node is considered fully covered if the shortest distance from the node to the facility does not exceed the smaller radius, and not covered at all if the shortest distance is beyond the larger radius. For a distance between these two radii, the coverage level is specified by a coverage decay function. It is assumed that demand weights are independent discrete random variables. The objective of the problem is to find a location for the facility so as to maximize the probability that the total covered demand weight is greater than or equal to a pre-selected threshold value. We show that the problem is NP-hard and that an optimal solution exists in a finite set of dominant points. We develop an exact algorithm and a normal approximation solution procedure. Computational experiment is performed to evaluate their performance.     

Approximation algorithms for hierarchical location problems

We formulate and (approximately) solve hierarchical versions of two prototypical problems in discrete location theory, namely, the metric uncapacitated k-median and facility location problems. Our work yields new insights into hierarchical clustering, a widely used technique in data analysis. For example, we show that every metric space admits a hierarchical clustering that is within a constant factor of optimal at every level of granularity with respect to the average (squared) distance objective. A key building block of our hierarchical facility location algorithm is a constant-factor approximation algorithm for an “incremental” variant of the facility location problem; the latter algorithm may be of independent interest.     

Maximum Dispersion and Geometric Maximum Weight Cliques

We consider a facility location problem, where the objective is to “disperse” a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.     

多站时差无源定位传感器网络优化

在可移动无源传感器网络中,观测器与目标的相对几何关系对定位精度有重要影响。为提高对运动目标的定位跟踪精度,提出一种基于时差无源定位几何稀释精度的移动平台实时布站方法。首先推导出二维时差无源定位方法下的带有基线长度和基线偏角的GDOP表达式,将其作为目标优化函数,使用加权离散搜索优化算法求解网内各观测器每一时刻的最佳观测位置,并在此最佳位置对目标进行量测,完成目标运动分析。该方法通估计和优化相结合实现移动平台无源传感器网络的实时优化部署,仿真证明该算法一定程度上解决了时差无源定位算法的定位模糊问题,提高了对运动目标的跟踪精度。     

Geometric complexity of some location problems

Given a set ofn demand points with weightW _i ,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An (n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.Supported in part by the National Science Foundation under Grants ECS 83-40031 and DCR 84-20814.     

Approximation Algorithms for Access Network Design

Abstract. We consider the problem of designing a minimum cost access network to carry traffic from a set of endnodes to a core network. Trunks are available in K types reflecting economies of scale . A trunk type with a high initial overhead cost has a low cost per unit bandwidth and a trunk type with a low overhead cost has a high cost per unit bandwidth. We formulate the problem as an integer program. We first use a primal—dual approach to obtain a solution whose cost is within O(K ² ) of optimal. Typically the value of K is small. This is the first combinatorial algorithm with an approximation ratio that is polynomial in K and is independent of the network size and the total traffic to be carried. We also explore linear program rounding techniques and prove a better approximation ratio of O(K) . Both bounds are obtained under weak assumptions on the trunk costs. Our primal—dual algorithm is motivated by the work of Jain and Vazirani on facility location [7]. Our rounding algorithm is motivated by the facility location algorithm of Shmoys et al. [12].     

基于权重的目标覆盖控制算法

针对无线传感器网络中随机部署无法实现对重要性不同的目标的优化覆盖控制问题,利用目标重叠域和贪婪算法设计一种基于目标权重的最优部署算法。以概率感知模型的传感器节点作为研究对象,通过标定目标权重确定目标重叠域,采用贪婪算法选取节点的最优部署范围,根据指标函数的最小值确定节点的部署位置。实验结果表明,所提出的算法能够实现对离散目标的最优覆盖监测,而且能保证监测节点网络的连通性。     

Stochastic problem of competitive location of facilities with quantile criterion

A stochastic problem of facility location formulated as a discrete bilevel problem of stochastic programming with quantile criterion was proposed. Consideration was given there to a pair of competitive players successively locating facilities with the aim of maximizing their profits. For the case of discrete distribution of the random consumer demands, it was proposed to reduce the original problem to the deterministic problem of bilevel programming. A method to calculate the value of the objective function under fixed leader strategy and procedures to construct the upper and lower bounds of the optimal value of the objective function were proposed.     

Maximum Dispersion and Geometric Maximum Weight Cliques

We consider a facility location problem, where the objective is to disperse a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.     

An Approximation Algorithm for the Fault Tolerant Metric Facility Location Problem

We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to r _j facilities. We give the first non-trivial approximation algorithm for this problem, having an approximation guarantee of 3 · H _k , where k is the maximum requirement and H _k is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primal–dual algorithm of [5] for the metric facility location problem, reduces the maximum residual requirement by one.     

An Approximation Algorithm for the Fault Tolerant Metric Facility Location Problem

We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to r _j facilities. We give the first non-trivial approximation algorithm for this problem, having an approximation guarantee of 3 · H _k , where k is the maximum requirement and H _k is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primal–dual algorithm of [5] for the metric facility location problem, reduces the maximum residual requirement by one.     

A multi-objective approach for determining optimal air compressor location in a manufacturing facility

Determining the optimal location of an air compressor in a manufacturing facility is a challenging problem that can offer significant energy savings. A novel simulation-optimization model is proposed to increase energy efficiency in a facility by determining optimal air compressor location. The optimization strategy is based on an objective function that minimizes the total energy consumption of the air compressor – hence, the energy cost for the facility – while considering the user's preference for the air compressor location. The proposed mathematical model first integrates the facility's characteristics based on user inputs, divides the facility into zones, and generates a rectilinear zone-to-zone distance matrix within the facility. The user location preference is incorporated into the proposed model via a five level user-preference index, assigned using preferential locations as suggested by twenty-two experienced facility managers. A sensitivity analysis is conducted to determine the relationship between the selected user preference level and the resulting energy consumption at each location in the facility. A simulation-driven analysis is performed using a real-life facility layout and typical compressed air equipment with corresponding nameplate data. In order to investigate and demonstrate the effectiveness of the proposed approach, the derived optimal zones are compared with five zones, including the most energy efficient zone, least energy efficient zone, and three other zones selected at random. The results of our study reveal that the proposed method achieves significant energy reductions while maintaining the user's desired air compressor location.     

Geometric complexity of some location problems

Given a set ofn demand points with weightW _i ,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An Ω(n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.     

Exact optimal solutions of the minisum facility and transfer points location problems on a network

We consider hierarchical facility location problems on a network called Multiple Location of Transfer Points (MLTP) and Facility and Transfer Points Location Problem (FTPLP), where q facilities and p transfer points are located and each customer goes to one of the facilities directly or via one of the transfer points. In FTPLP, we need to find an optimal location of both the facilities and the transfer points while the location of facilities is given in MLTP. Although good heuristics have been proposed for the minisum MLTP and FTPLP, no exact optimal solution has been tested due to the size of the problems. We show that the minisum MLTP can be formulated as a p‐median problem, which leads to obtaining an optimal solution. We also present a new formulation of FTPLP and an enumeration‐based approach to solve the problems with a single facility.     

Risk management in uncapacitated facility location models with random demands

In this paper we consider a location-optimization problem where the classical uncapacitated facility location model is recast in a stochastic environment with several risk factors that make demand at each customer site probabilistic and correlated with demands at the other customer sites. Our primary contribution is to introduce a new solution methodology that adopts the mean–variance approach, borrowed from the finance literature, to optimize the “Value-at-Risk” (VaR) measure in a location problem. Specifically, the objective of locating the facilities is to maximize the lower limit of future earnings based on a stated confidence level. We derive a nonlinear integer program whose solution gives the optimal locations for the p facilities under the new objective. We design a branch-and-bound algorithm that utilizes a second-order cone program (SOCP) solver as a subroutine. We also provide computational results that show excellent solution times on small to medium sized problems.     

A Linear Time Algorithm for Computing Minmax Regret 1-Median on a Tree Network

In a model of facility location problem, the uncertainty in the weight of a vertex is represented by an interval of weights, and the objective is to minimize the maximum “regret.” The most efficient algorithm previously known for finding the minmax regret 1-median in a tree network with nonnegative vertex weights takes O(nlogn) time. We improve it to O(n), settling the open problem posed by Brodal et al. (Oper. Res. Lett. 36:14–18, 2008).     

Incorporating uncertainty in optimal decision making: Integrating mixed integer programming and simulation to solve combinatorial problems

We introduce a novel methodology that integrates optimization and simulation techniques to obtain estimated global optimal solutions to combinatorial problems with uncertainty such as those of facility location, facility layout, and scheduling. We develop a generalized mixed integer programming (MIP) formulation that allows iterative interaction with a simulation model by taking into account the impact of uncertainty on the objective function value of previous solutions. Our approach is generalized, efficient, incorporates the impact of uncertainty of system parameters on performance and can easily be incorporated into a variety of applications. For illustration, we apply this new solution methodology to the NP-hard multi-period multi-product facility location problem (MPP-FLP). Our results show that, for this problem, our iterative procedure yields up to 9.4% improvement in facility location-related costs over deterministic optimization and that these cost savings increase as the variability in demand and supply uncertainty are increased.     

A bi-objective reverse logistics network analysis for post-sale service

Reverse logistics, induced by various forms of return, has received growing attention throughout this decade. Reverse logistics network design is a major strategic issue. This paper addresses the analysis of reverse logistic networks that deal with the returns requiring repair service. A problem involving a manufacturer outsourcing to a third-party logistics (3PLs) provider for its post-sale services is proposed. First, a bi-objective optimization model is developed. Two objectives, minimization of the overall costs and minimization of the total tardiness of cycle time, are addressed. The facility capacity option at each potential location is treated as a discrete parameter. The purpose is to find a set of non-dominated solutions to the facility capacity arrangement among the potential facility locations, as well as the associated transportation flows between customer areas and service facilities. Then, a solution approach is designed for solving this bi-objective optimization model. The solution approach consists of a combination of three algorithms: scatter search, the dual simplex method and the constraint method. Finally, computational analyses are performed on trial examples. Numerical results present the trade-off relationship between the two objectives. The numerical results also show that the optimization for the first objective function leads to a centralized network structure; the optimization for the second objective function results in a decentralized network structure.