A hierarchical location model for public facility planning

In this article, we present a discrete hierarchical location model for public facility planning. The main features of the model are: an accessibility maximization objective; several levels of demand and of facilities; a nested hierarchy of facilities (i.e. a facility of a given level can serve demand of equal and lower levels); maximum and minimum capacity constraints; and user-to-facility assignment constraints. The latter include single-assignment and closest-assignment constraints, as well as a new type of constraints called path-assignment constraints. Their purpose is to enforce some desirable properties for the spatial pattern of assignments. If they are not included, model solutions are difficult to interpret and to explain in a public facility planning context, therefore being less likely to be accepted by the users. The usefulness of the model is illustrated through a real-world application to school network planning.     

Outcomes of voting and planning in single facility location problems

For a given set of users located at the vertices of a network N, we consider the location of a single facility. Two different decision making procedures, voting among the users and minimizing total distance travelled by the users, are compared.Provided that a voting solution exists, it is shown that the two solutions sets coincide if N is a so-called cactus. For general networks, the outcomes of the two procedures are compared in terms of the cyclic structure of N and the number of users.     

An integrated approach to dynamic plant location and capacity planning

We consider a dynamic capacitated plant location problem in which capacities of opened plants are determined by acquisition and/or disposal of multiple types of facilities. We determine the opening schedule of plants, allocations of customers' demands and plans for acquisition and/or disposal of plant capacities that minimise the sum of discounted fixed costs for opening plants, delivery costs of products, and acquisition and operation costs of facilities. The dynamic capacitated plant location problem is formulated as a mixed integer linear program and solved by a heuristic algorithm based on Lagrangian relaxation and a cut and branch algorithm which uses Gomory cuts. Several solution properties of the relaxed problem are found and used to develop efficient solution procedures for the relaxed problem. A subgradient optimisation method is employed to obtain better lower bounds. The heuristic algorithm is tested on randomly generated test problems and results show that the algorithm finds good solutions in a reasonable amount of computation time.     

Online facility location with facility movements

In the online facility location problem demand points arrive one at a time and the goal is to decide where and when to open a facility. In this paper we consider a new version of the online facility location problem, where the algorithm is allowed to move the opened facilities in the metric space. We consider the uniform case where each facility has the same constant cost. We present an algorithm which is 2-competitive for the general case and we prove that it is 3/2-competitive if the metric space is the line. We also prove that no algorithm with smaller competitive ratio than \({(\sqrt{13}+1)/4\approx 1.1514}\) exists. We also present an empirical analysis which shows that the algorithm gives very good results in the average case.     

Robust facility location

Let A be a nonempty finite subset of the plane representing the geographical coordinates of a set of demand points (towns, ), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for aA if the facility is located at xS is proportional to dist(x,a) — the distance from x to a — and that demand of point a is given by _a , minimizing the total transportation cost TC(,x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector is not known, and only an estimator circ; can be provided. Moreover the errors in such estimation process may be non-negligible. We propose a new model for this situation: select a threshold value B>0 representing the highest admissible transportation cost. Define the robustness of a location x as the minimum increase in demand needed to become inadmissible, i.e. (x)=min{|–circ;|:TC(,x)>B,0} and find the x maximizing to get the most robust location. Acknowledgment.The authors acknowledge the constructive remarks of the referees. The research of the first author has been supported in part by grant BFM2002-04525-C02-02 of MCYT, Spain. The research of the second author has been supported in part by a grant of the Deutsche Forschungsgemeinschaft.     

平方度量动态设施选址问题的近似算法

研究了单阶段度量设施选址问题的推广问题平方度量动态设施选址问题. 研究中首先利用原始对偶技巧得到 9-近似算法, 然后利用贪婪增广技巧将近似比改进到2.606, 最后讨论了该问题的相应变形问题.     

Single facility location and relocation problem with time dependent weights and discrete planning horizon

In this paper a single facility location problem with multiple relocation opportunities is investigated. The weight associated with each demand point is a known function of time. We consider either rectilinear, or squared Euclidean, or Euclidean distances. Relocations can take place at pre-determined times. The objective function is to minimize the total location and relocation costs. An algorithm which finds the optimal locations, relocation times and the total cost, for all three types of distance measurements and various weight functions, is developed. Locations are found using constant weights, and relocations times are the solution to a Dynamic Programming or Binary Integer Programming (BIP) model. The time horizon can be finite or infinite.     

An approximation algorithm for the dynamic facility location problem with submodular penalties

In this paper, we study the dynamic facility location problem with submodular penalties （DFLPSP）. We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.     

The use of state space relaxation for the dynamic facility location problem

Dynamic facility location is concerned with developing a location decision plan over a given planning horizon during which changes in the market and in costs are expected to occur. The objective is to select from a list of predetermined possible facility sites the locations of the facilities to use in each period of the planning horizon to minimise the total costs of operating the system. The costs considered here include not only transport and operation/maintenance charges but also relocation costs arising from the opening and closing of facilities as required by the plan.The problem is formulated in terms of dynamic programming but for simplicity with restrictions on the numbers of facilities that can be opened in a given period. The problem was solved using both dynamic programming and a branch and bound approach using state space relaxation. These two approaches are contrasted with different data and with different assumptions to compare the influence of alternative factors on the computational efficiency of both solution methods.     

Covering models and optimization techniques for emergency response facility location and planning: a review

With emergencies being, unfortunately, part of our lives, it is crucial to efficiently plan and allocate emergency response facilities that deliver effective and timely relief to people most in need. Emergency Medical Services (EMS) allocation problems deal with locating EMS facilities among potential sites to provide efficient and effective services over a wide area with spatially distributed demands. It is often problematic due to the intrinsic complexity of these problems. This paper reviews covering models and optimization techniques for emergency response facility location and planning in the literature from the past few decades, while emphasizing recent developments. We introduce several typical covering models and their extensions ordered from simple to complex, including Location Set Covering Problem (LSCP), Maximal Covering Location Problem (MCLP), Double Standard Model (DSM), Maximum Expected Covering Location Problem (MEXCLP), and Maximum Availability Location Problem (MALP) models. In addition, recent developments on hypercube queuing models, dynamic allocation models, gradual covering models, and cooperative covering models are also presented in this paper. The corresponding optimization techniques to solve these models, including heuristic algorithms, simulation, and exact methods, are summarized.     

Soft-capacitated facility location game

We study the soft-capacitated facility location game which is an extension of the facility location game of Pa1 and Tardos. We propose a 6-approximate cross-monotonic cost-sharing method. Numerical tests indicate that the method is effective.     

Competitive facility location models

Two classes of competitive facility location models are considered, in which several persons (players) sequentially or simultaneously open facilities for serving clients. The first class consists of discrete two-level programming models. The second class consists of game models with several independent players pursuing selfish goals. For the first class, its relationship with pseudo-Boolean functions is established and a novel method for constructing a family of upper and lower bounds on the optimum is proposed. For the second class, the tight PLS-completeness of the problem of finding Nash equilibriums is proved.     

Cooperative facility location games

The location of facilities in order to provide service for customers is a well-studied problem in the operations research literature. In the basic model, there is a predefined cost for opening a facility and also for connecting a customer to a facility, the goal being to minimize the total cost. Often, both in the case of public facilities (such as libraries, municipal swimming pools, fire stations, … ) and private facilities (such as distribution centers, switching stations, … ), we may want to find a ‘fair’ allocation of the total cost to the customers—this is known as the cost allocation problem. A central question in cooperative game theory is whether the total cost can be allocated to the customers such that no coalition of customers has any incentive to build their own facility or to ask a competitor to service them. We establish strong connections between fair cost allocations and linear programming relaxations for several variants of the facility location problem. In particular, we show that a fair cost allocation exists if and only if there is no integrality gap for a corresponding linear programming relaxation; this was only known for the simplest unconstrained variant of the facility location problem. Moreover, we introduce a subtle variant of randomized rounding and derive new proofs for the existence of fair cost allocations for several classes of instances. We also show that it is in general NP-complete to decide whether a fair cost allocation exists and whether a given allocation is fair.     

Noncooperative facility location games

A noncooperative game theoretical approach is considered for the multifacility location problem. It turns out that the facility location game is a potential game in the sense of Monderer and Shapley and some properties of the game are studied.     

Flow location (FlowLoc) problems: dynamic network flows and location models for evacuation planning

In this paper we combine two modeling tools to predict and evaluate evacuation plans: (dynamic) network flows and locational analysis. We present three exact algorithms to solve the single facility version 1-FlowLoc of this problem and compare their running times. After proving the $\mathcal{NP}$ -completeness of the multi facility q-FlowLoc problem, a mixed integer programming formulation and a heuristic for q-FlowLoc are proposed. The paper is concluded by discussing some generalizations of the FlowLoc problem, such as the multi-terminal problem, interdiction problem, the parametric problem and the generalization of the FlowLoc problem to matroids.     

Decision/forecast horizon for a single facility dynamic location/relocation problem

Procedures to solve finite horizon dynamic location/relocation problems have been reported in the literature by many authors. This paper provides several decision/forecast horizon results for a single facility dynamic location/relocation problem; these results are helpful in finding optimal initial decisions for the infinite horizon problem by using information only for a finite horizon.     

Competitive facility location and design problem

We develop a spatial interaction model that seeks to simultaneously optimize location and design decisions for a set of new facilities. The facilities compete for customer demand with pre-existing competitive facilities and with each other. The customer demand is assumed to be elastic, expanding as the utility of the service offered by the facilities increases. Increases in the utility can be achieved by increasing the number of facilities, design improvements, or locating facilities closer to the customer.     

Continuous facility location on graphs

Mathematical Programming - We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well...     

Competitive facility location under attrition

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.

     

The obnoxious p facility network location problem with facility interaction

Three heuristics are proposed to solve the maximin formulation for siting p facilities on a network considering a pollution dispersion equation and facility interaction. Initially, the single facility problem is approached by building up polygons which model pollution spread about the nodes of the network. This is extended in the first heuristic to the p facility problem. The second combines both the p-maximin and p-maxisum objectives in a lexicographic manner. The third is based on recent developments of Simulated Annealing. The proposed heuristics are evaluated for up to six facilities on a set of randomly generated networks having 20 to 40 nodes and low or medium arc density.