A dual-based optimization procedure for the two-echelon uncapacitated facility location problem

The two-echelon uncapacitated facility location problem (TUFLP) is a generalization of the uncapacitated facility location problem (UFLP) and multiactivity facility location problem (MAFLP). In TUFLP there are two echelons of facilities through which products may flow in route to final customers. The objective is to determine the least-cost number and locations of facilities at each echelon in the system, the flow of product between facilities, and the assignment of customers to supplying facilities. We propose a new dual-based solution procedure for TUFLP that can be used as a heuristic or incorporated into branch-and-bound procedures to obtain optimal solutions to TUFLP. The algorithm is an extension of the dual ascent and adjustment procedures developed by Erlenkotter for UFLP. We report computational experience gained by solving over 420 test problems. The largest problems solved have 25 possible facility locations at each echelon and 35 customer zones, implying 650 integer variables and 21,875 continuous variables.     

Facility location when demand is time dependent

In this article we investigate the problem of locating a facility among a given set of demand points when the weights associated with each demand point change in time in a known way. It is assumed that the location of the facility can be changed one or more times during the time horizon. We need to find the time “breaks” when the location of the facility is to be changed, and the location of the facility during each time segment between breaks. We investigate the minisum Weber problem and also minimax facility location. For the former we show how to calculate the objective function for given time breaks and optimally solve the rectilinear distance problem with one time break and linear change of weights over time. Location of multiple time breaks is also discussed. For minimax location problems we devise two algorithms that solve the problem optimally for any number of time breaks and any distance metric. These algorithms are also applicable to network location problems.     

Determination of efficient points in multiple-objective location problems

This article is devoted to an MCDM problem connected with locational analysis. The MCDM problem can be formulated so as to minimize the distance between a facility and a given set of points. The efficient points of this problem are candidates for optimal solutions to many location problems. We propose an algorithm to find all efficient points when distance is measured by any polyhedral norm.     

Replacing continuous demand with discrete demand in a competitive location model

Location models commonly represent demand as discrete points rather than as continuously spread over an area. This modeling technique introduces inaccuracies to the objective function and consequently to the optimal location solution. In this article this inaccuracy is investigated by the study of a particular competitive facility location problem. First, the location problem is formulated over a continuous demand area. The optimal location for a new facility that optimizes the objective function is obtained. This optimal location solution is then compared with the optimal location obtained for a discrete set of demand points. Second, a simple approximation approach to the continuous demand formulation is proposed. The location problem can be solved by using the discrete demand algorithm while significantly reducing the inaccuracies. This way the simplicity of the discrete approach is combined with the approximated accuracy of the continuous-demand location solution. Extensive analysis and computations of the test problem are reported. It is recommended that this approximation approach be considered for implementation in other location models. © 1997 John Wiley & Sons, Inc.     

A note on equity across groups in facility location

An equity model between groups of demand points is proposed. The set of demand points is divided into two or more groups. For example, rich and poor neighborhoods and urban and rural neighborhoods. We wish to provide equal service to the different groups by minimizing the deviation from equality among groups. The distance to the closest facility is a measure of the quality of service. Once the facilities are located, each demand point has a service distance. The objective function, to be minimized, is the sum of squares of differences between all pairs of service distances between demand points in different groups. The problem is analyzed and solution techniques are proposed for the location of a single facility in the plane. Computational experiments for problems with up to 10,000 demand points and rectilinear, Euclidean, or general ?_p distances illustrate the efficiency of the proposed algorithm. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Algorithms for solving the conditional covering problem on paths

Consider the conditional covering problem on an undirected graph, where each node represents a site that must be covered by a facility, and facilities may only be established at these nodes. Each facility can cover all sites that lie within some common covering radius, except the site at which it is located. Although this problem is difficult to solve on general graphs, there exist special structures on which the problem is easily solvable. In this paper, we consider the special case in which the graph is a simple path. For the case in which facility location costs do not vary based on the site, we derive characteristics of the problem that lead to a linear‐time shortest path algorithm for solving the problem. When the facility location costs vary according to the site, we provide a more complex, but still polynomial‐time, dynamic programming algorithm to find the optimal solution. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A stochastic gradual cover location problem

Covering models assume that a point is covered if it is within a certain distance from a facility and not covered beyond that distance. In gradual cover models it is assumed that a point is fully covered within a given distance from a facility, then cover gradually declines, and the point is not covered beyond a larger distance. Gradual cover models address the discontinuity in cover which may not be the correct approach in many situations. In the stochastic gradual cover model presented in this article it is assumed that the short and long distances employed in gradual cover models are random variables. This refinement of gradual cover models provides yet a more realistic depiction of actual behavior in many situations. The maximal cover model based on the new concept is analyzed and the single facility location cover problem in the plane is solved. Computational results illustrating the effectiveness of the solution procedures are presented. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Exact algorithms for integrated facility location and production planning problems

We consider a class of facility location problems with a time dimension, which requires assigning every customer to a supply facility in each of a finite number of periods. Each facility must meet all assigned customer demand in every period at a minimum cost via its production and inventory decisions. We provide exact branch‐and‐price algorithms for this class of problems and several important variants. The corresponding pricing problem takes the form of an interesting class of production planning and order selection problems. This problem class requires selecting a set of orders that maximizes profit, defined as the revenue from selected orders minus production‐planning‐related costs incurred in fulfilling the selected orders. We provide polynomial‐time dynamic programming algorithms for this class of pricing problems, as well as for generalizations thereof. Computational testing indicates the advantage of our branch‐and‐price algorithm over various approaches that use commercial software packages. These tests also highlight the significant cost savings possible from integrating location with production and inventory decisions and demonstrate that the problem is rather insensitive to forecast errors associated with the demand streams. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

An algorithm for the solution of a location problem with metric constraints

In many location problems, the solution is constrained to lie within a closed set. In this paper, optimal solutions to a special type of constrained location problem are characterized. In particular, the location problem with the solution constrained to be within a maximum distance of each demand point is considered, and an algorithm for its solution is developed and discussed.     

A heuristic for capacity expansion planning with multiple facility types

A capacity expansion model with multiple facility types is examined, where different facility types represent different quality levels. Applications for the model can be found in communications networks and production facilities. The model assumes a finite number of discrete time periods. The facilities are expanded over time. Capacity of a high-quality facility can be converted to satisfy demand for a lower-quality facility. The costs considered include capacity expansion costs and excess capacity holding costs. All cost functions are nondecreasing and concave. An algorithm that finds optimal expansion policies requires extensive computations and is practical only for small scale problems. Here, we develop a heuristic that employs so-called distributed expansion policies. It also attempts to decompose the problem into several smaller problems solved independently. The heuristic is computationally efficient. Further, it has consistently found near-optimal solutions.     

无线传感器网络DV-Hop节点定位改进算法

针对无线传感器网络中距离无关类节点定位算法定位误差较大的问题,提出了一种改进型DV-Hop节点定位算法。通过设置待定位节点到信标节点间的最小跳数门限,降低了定位累积误差;改进了平均每跳距离估计方法,并利用信标节点测量的位置误差作为修正值,对每跳距离的估计值进行修正;待定位节点仅选取与之较近的信标节点计算位置,降低了距离误差。仿真结果显示,在信标节点比例和网络节点总数相同的条件下,改进算法性能明显优于DV-Hop算法。     

Polynomial algorithms for center location on spheres

When locating facilities over the earth or in space, a planar location model is no longer valid and we must use a spherical surface. In this article, we consider the one-and two-center problems on a sphere that contains n demand points. The problem is to locate facilities to minimize the maximum distance from any demand point to the closest facility. We present an O(n) algorithm for the one-center problem when a hemisphere contains all demand points and also give an O(n) algorithm for determining whether or not the hemisphere property holds. We present an O(n³ log n) algorithm for the two-center problem for arbitrarily located demand points. Finally, we show that for general p, the p center on a sphere problem is NP-hard. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 341–352, 1997     

Dominating sets for rectilinear center location problems with polyhedral barriers

In planar location problems with barriers one considers regions which are forbidden for the siting of new facilities as well as for trespassing. These problems are important since they model various actual applications. The resulting mathematical models have a nonconvex objective function and are therefore difficult to tackle using standard methods of location theory even in the case of simple barrier shapes and distance functions. For the case of center objectives with barrier distances obtained from the rectilinear or Manhattan metric, it is shown that the problem can be solved in polynomial time by identifying a dominating set. The resulting genuinely polynomial algorithm can be combined with bound computations which are derived from solving closely connected restricted location and network location problems. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 647–665, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10038     

An algorithm for optimal shipments with given frequencies

This article deals with the problem of minimizing the transportation and inventory cost associated with the shipment of several products from a source to a destination, when a finite set of shipping frequencies is available. A mixed-integer programming model—shown to be NP-hard—is formulated for that problem. The computational complexity of some similar models applied to different problems is also investigated. In particular, whereas the capacitated plant location problem with operational cost in product form is NP-hard, the simple plant location problem with the same characteristics can be solved in polynomial time. A branch-and-bound algorithm is finally worked out, and some computational results are presented. © 1996 John Wiley & Sons, Inc.     

Minimax and maximin facility location problems on a sphere

The problem dealt with in this article is as follows. There are n “demand points” on a sphere. Each demand point has a weight which is a positive constant. A facility must be located so that the maximum of the weighted distances (distances are the shortest arcs on the surface of the sphere) is minimized; this is called the minimax problem. Alternatively, in the maximin problem, the minimum weighted distance is maximized. A setup cost associated with each demand point may be added for generality. It is shown that any maximin problem can be reparametrized into a minimax problem. A method for finding local minimax points is described and conditions under which these are global are derived. Finally, an efficient algorithm for finding the global minimax point is constructed.     

A facility reliability problem: Formulation,properties, and algorithm

Having a robustly designed supply chain network is one of the most effective ways to hedge against network disruptions because contingency plans in the event of a disruption are often significantly limited. In this article, we study the facility reliability problem: how to design a reliable supply chain network in the presence of random facility disruptions with the option of hardening selected facilities. We consider a facility location problem incorporating two types of facilities, one that is unreliable and another that is reliable (which is not subject to disruption, but is more expensive). We formulate this as a mixed integer programming model and develop a Lagrangian Relaxation‐based solution algorithm. We derive structural properties of the problem and show that for some values of the disruption probability, the problem reduces to the classical uncapacitated fixed charge location problem. In addition, we show that the proposed solution algorithm is not only capable of solving large‐scale problems, but is also computationally effective. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Duality in constrained multi‐facility location models

We consider the ??_p‐norm multi‐facility minisum location problem with linear and distance constraints, and develop the Lagrangian dual formulation for this problem. The model that we consider represents the most general location model in which the dual formulation is not found in the literature. We find that, because of its linear objective function and less number of variables, the Lagrangian dual is more useful. Additionally, the dual formulation eliminates the differentiability problem in the primal formulation. We also provide the Lagrangian dual formulation of the multi‐facility minisum location problem with the ??_pb‐norm. Finally, we provide a numerical example for solving the Lagrangian dual formulation and obtaining the optimum facility locations from the solution of the dual formulation. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 410–421, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10010     

A robust sensor covering and communication problem

We consider the problem of placing sensors across some area of interest. The sensors must be placed so that they cover a fixed set of targets in the region, and should be deployed in a manner that allows sensors to communicate with one another. In particular, there exists a measure of communication effectiveness for each sensor pair, which is determined by a concave function of distance between the sensors. Complicating the sensor location problem are uncertainties related to sensor placement, for example, as caused by drifting due to air or water currents to which the sensors may be subjected. Our problem thus seeks to maximize a metric regarding intrasensor communication effectiveness, subject to the condition that all targets must be covered by some sensor, where sensor drift occurs according to a robust (worst‐case) mechanism. We formulate an approximation approach and develop a cutting‐plane algorithm to solve this problem, comparing the effectiveness of two different classes of inequalities. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 582–594, 2015     

Facility location and capacity acquisition: An integrated approach

The facility location and capacity acquisition decisions are intertwined, especially within the international context where capacity acquisition costs are location dependent. A review of the relevant literature however, reveals that the facility location and the capacity acquisition problems have been dealt with separately. Thus, an integrated approach for simultaneous optimization of these strategic decisions is presented. Analytical properties of the arising model are investigated and an algorithm for solving the problem is devised. Encouraging computational results are reported. © 1995 John Wiley & Sons, Inc.     

One-facility location with rectilinear tour distances

This article concerns the location of a facility among n points where the points are serviced by “tours” taken from the facility. Tours include m points at a time and each group of m points may become active (may need a tour) with some known probability. Distances are assumed to be rectilinear. For m ≤ 3, it is proved that the objective function is separable in each dimension and an exact solution method is given that involves finding the median of numbers appropriately generated from the problem data. It is shown that the objective function becomes multimodal when some tours pass through four or more points. A bounded heuristic procedure is suggested for this latter case. This heuristic involves solving an auxiliary three-point tour location problem.