Computation of Multi-facility Location Nash Equilibria on a Network Under Quantity Competition

We deal with the location-quantity problem for competing firms when they locate multiple facilities and offer the same type of product. Competition is performed under delivered quantities that are sent from the facilities to the customers. This problem is reduced to a location game when the competing firms deliver the Cournot equilibrium quantities. While existence conditions for a Nash equilibrium of the location game have been discussed in many contributions in the literature, computing an equilibrium on a network when multiple facilities are to be located by each firm is a problem not previously addressed. We propose an integer linear programming formulation to fill this gap. The formulation solves the profit maximization problem for a firm, assuming that the other firms have fixed their facility locations. This allows us to compute location Nash equilibria by the best response procedure. A study with data of Spanish municipalities under different scenarios is presented and conclusions are drawn from a sensitivity analysis.

     

Modeling and solving a capacitated stochastic location-allocation problem using sub-sources

We study a capacitated multi-facility location-allocation problem in which the customers have stochastic demands based on Bernoulli distribution function. We consider the capacitated sub-sources of facilities to satisfy demands of customers. In the discrete stochastic problem, the goal is to find optimal locations of facilities among candidate locations and optimal allocations of existing customers to operating facilities so that the total sum of fixed costs of operating facilities, allocation cost of the customers, expected values of servicing and outsourcing costs is minimized. The model is formulated as a mixed-integer nonlinear programming problem. Since finding an optimal solution may require an excessive amount of time depending on nonlinear constraints, we transform the nonlinear constraints of the problem to linear ones to arrive at a simple formulation of the model. Numerical results show that the LINGO 9.0 software is capable of solving small size problems. For medium and large-size problems, we propose two meta-heuristic algorithms, namely a genetic algorithm and a discrete version of colonial competitive algorithm. Computational results show that the proposed algorithms efficiently obtain effective solutions.     

A GIS-Based Optimization Framework for Competitive Multi-Facility Location-Routing Problem

In a dynamic market setting, firms need to quickly respond to shifting demographics and economic conditions. In this paper, we investigate the problem of determining the optimum set of locations for a firm, which operates a chain of facilities under competition. We consider the objective of maximizing profit, defined as gross profit margin minus logistics costs. We propose a location-routing model where revenue is realized according to probabilistic patronization of customers and routing costs are incurred due to vehicles serving the open facilities from a central depot. We propose a hybrid heuristic optimization methodology for solving this model. The optimal locations are searched for by a Genetic Algorithm while an integrated Tabu Search algorithm is employed for solving the underlying vehicle routing problem. The solution approach is tested on a real dataset of a supermarket chain. The results show that the location decisions made by the proposed methodology lead to increased market share and profit margin, while keeping logistics costs virtually unchanged. Finally, we present a GIS-based framework that can be used to store, analyze and visualize all data as well as model solutions in geographic format.     

On solving unreliable planar location problems

In this paper, we generalize conventional P-median location problems by considering the unreliability of facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. We proposed efficient solution methods to determine locations of these facilities in the unreliable location model. Space-filling curve-based algorithms are developed to determine initial locations of these facilities. The unreliable P-median location problem is then decomposed to P 1-median location problems; each problem is solved to the optimum. A bounding procedure is used to monitor the iterative search, and to provide a consistent basis for termination. Extensive computational tests have indicated that the heuristics are efficient and effective for solving unreliable location problems.Scope and purposeThis paper addresses an important class of location problems, where p unreliable facilities are to be located on the plane, so as to minimize the expected travel distance or related transportation cost between the customers and their nearest available facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. Potential application of the unreliable location problem is found in numerous areas. The facilities to be located can be fire station or emergency shelter, where it fails to provide service during some time window, due to the capacity or resource constraints. Alternatively, the facilities can be telecommunication posts or logistic/distribution centers, where the service is unavailable due to breakdown, repair, shutdown of unknown causes. In this paper, we prescribed heuristic procedures to determine the location of new facilities in the unreliable location problems. The numerical study of 2800 randomly generated instances has shown that these solution procedures are both efficient and effective, in terms of computational time and solution quality.     

The multi-period incremental service facility location problem

In this paper we introduce the multi-period incremental service facility location problem where the goal is to set a number of new facilities over a finite time horizon so as to cover dynamically the demand of a given set of customers. We prove that the coefficient matrix of the allocation subproblem that results when fixing the set of facilities to open is totally unimodular. This allows to solve efficiently the Lagrangean problem that relaxes constraints requiring customers to be assigned to open facilities. We propose a solution approach that provides both lower and upper bounds by combining subgradient optimization to solve a Lagrangean dual with an ad hoc heuristic that uses information from the Lagrangean subproblem to generate feasible solutions. Numerical results obtained in the computational experiments show that the obtained solutions are very good. In general, we get very small percent gaps between upper and lower bounds with little computation effort.     

A continuous analysis framework for the solution of location–allocation problems with dense demand

Location–allocation problems arise in several contexts, including supply chain and data mining. In its most common interpretation, the basic problem consists of optimally locating facilities and allocating customers to facilities so as to minimize the total cost. The standard approach to solving location–allocation problems is to model alternative location sites and customers as discrete entities. Many problem instances in practice involve dense demand data and uncertainties about the cost and locations of the potential sites. The use of discrete models is often inappropriate in such cases. This paper presents an alternative methodology where the market demand is modeled as a continuous density function and the resulting formulation is solved by means of calculus techniques. The methodology prioritizes the allocation decisions rather than location decisions, which is the common practice in the location literature. The solution algorithm proposed in this framework is a local search heuristic (steepest-descent algorithm) and is applicable to problems where the allocation decisions are in the form of polygons, e.g., with Euclidean distances. Extensive computational experiments confirm the efficiency of the proposed methodology.     

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.     

网络中设备位置问题分析

通过引入虚拟的质量保证协议（SLA）服务条款,提出网络中形式化的设备位置问题,并根据网络特点改进了现有的静态局部优化算法,证明了在常见的分区低服务质量敏感类型的k median问题和低服务质量敏感类型的UFL问题中的近似度上界。测试结果表明改进后的算法在较小增加现有静态算法近似度的情况下运算速度有较大的提高。     

Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem

The capacitated multi-facility Weber problem is concerned with locating I capacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a nonconvex optimization problem and difficult to solve. In this work, we focus on a multi-commodity extension and consider the situation where K distinct commodities are shipped subject to capacity constraints between each customer and facility pair. Customer locations, demands and capacities for each commodity, and bundle restrictions are known a priori. The transportation costs, which are proportional to the distance between customers and facilities, depend on the commodity type. We address several location-allocation and discrete approximation heuristics using different strategies. Based on the obtained computational results we can say that the alternate solution of location and allocation problems is a very efficient strategy; but the discrete approximation has excellent accuracy.     

Sequential versus simultaneous approach in the location and design of two new facilities using planar Huff-like models

Companies frequently decide on the location and design for new facilities in a sequential way. However, for a fixed number of new facilities, the company might be able to improve its profit by taking its decisions for all the facilities simultaneously. In this paper we compare three different strategies: simultaneous location and independent design of two facilities in the plane, the same with equal designs, and the sequential approach of determining each facility in turn. The basic model is profit maximization for the chain, taking market share, location costs and design costs into account. The market share captured by each facility depends on the distance to the customers (location) and its quality (design), through a probabilistic Huff-like model. Recent research on this type of models was aimed at finding global optima for a single new facility, holding quality fixed or variable, but no exact algorithm has been proposed to find optimal solutions for more than one facility. We develop such an exact interval branch-and-bound algorithm to solve both simultaneous location and design two-facility problems. Then, we present computational results and exhibit the differences in locations and qualities of the optimal solutions one may obtain by the sequential and simultaneous approaches.     

The min-dist location selection and facility replacement queries

We propose and study a new type of location optimization problem, the min-dist location selection problem: given a set of clients and a set of existing facilities, we select a location from a given set of potential locations for establishing a new facility, so that the average distance between a client and her nearest facility is minimized. The problem has a wide range of applications in urban development simulation, massively multiplayer online games, and decision support systems. We also investigate a variant of the problem, where we consider replacing (instead of adding) a facility while achieving the same optimization goal. We call this variant the min-dist facility replacement problem. We explore two common approaches to location optimization problems and present methods based on those approaches for solving the min-dist location selection problem. However, those methods either need to maintain an extra index or fall short in efficiency. To address their drawbacks, we propose a novel method (named MND), which has very close performance to the fastest method but does not need an extra index. We then utilize the key idea behind MND to approach the min-dist facility replacement problem, which results in two algorithms names MSND and RID. We provide a detailed comparative cost analysis and conduct extensive experiments on the various algorithms. The results show that MND and RID outperform their competitors by orders of magnitude.     

A bilevel fixed charge location model for facilities under imminent attack

We investigate a bilevel fixed charge facility location problem for a system planner (the defender) who has to provide public service to customers. The defender cannot dictate customer-facility assignments since the customers pick their facility of choice according to its proximity. Thus, each facility must have sufficient capacity installed to accommodate all customers for whom it is the closest one. Facilities can be opened either in the protected or unprotected mode. Protection immunizes against an attacker who is capable of destroying at most r unprotected facilities in the worst-case scenario. Partial protection or interdiction is not possible. The defender selects facility sites from m candidate locations which have different costs. The attacker is assumed to know the unprotected facilities with certainty. He makes his interdiction plan so as to maximize the total post-attack cost incurred by the defender. If a facility has been interdicted, its customers are reallocated to the closest available facilities making capacity expansion necessary. The problem is formulated as a static Stackelberg game between the defender (leader) and the attacker (follower). Two solution methods are proposed. The first is a tabu search heuristic where a hash function calculates and records the hash values of all visited solutions for the purpose of avoiding cycling. The second is a sequential method in which the location and protection decisions are separated. Both methods are tested on 60 randomly generated instances in which m ranges from 10 to 30, and r varies between 1 and 3. The solutions are further validated by means of an exhaustive search algorithm. Test results show that the defender's facility opening plan is sensitive to the protection and distance costs.     

Dynamic location problem under uncertainty with a regret‐based measure of robustness

In this paper, we present a dynamic uncapacitated facility location problem that considers uncertainty in fixed and assignment costs as well as in the sets of potential facility locations and possible customers. Uncertainty is represented via a set of scenarios. Our aim is to minimize the expected total cost, explicitly considering regret. Regret is understood as a measure, for each scenario, of the loss incurred for not choosing that scenario's optimal solution if that scenario indeed occurred. We guarantee that the regret for each scenario is always upper bounded. We present a mixed integer programming model for the problem and we propose a solution approach based on Lagrangean relaxation integrating a local neighborhood search and a subgradient algorithm to update Lagrangean multipliers. The problem and the solutions obtained are first analyzed through the use of illustrative examples. Computational results over sets of randomly generated test problems are also provided.     

<Emphasis Type="Italic">k</Emphasis>-most suitable locations selection

Choosing the best location for starting a business or expanding an existing enterprize is an important issue. A number of location selection problems have been discussed in the literature. They often apply the Reverse Nearest Neighbor as the criterion for finding suitable locations. In this paper, we apply the Average Distance as the criterion and propose the so-called k-most suitable locations (k-MSL) selection problem. Given a positive integer k and three datasets: a set of customers, a set of existing facilities, and a set of potential locations. The k-MSL selection problem outputs k locations from the potential location set, such that the average distance between a customer and his nearest facility is minimized. In this paper, we formally define the k-MSL selection problem and show that it is NP-hard. We first propose a greedy algorithm which can quickly find an approximate result for users. Two exact algorithms are then proposed to find the optimal result. Several pruning rules are applied to increase computational efficiency. We evaluate the algorithms’ performance using both synthetic and real datasets. The results show that our algorithms are able to deal with the k-MSL selection problem efficiently.     

Models and algorithms for competitive facility location problems with different customer behavior

Competitive facility location problems arise in the context of two non-cooperating companies, a leader and a follower, competing for market share from a given set of customers. We assume that the firms place a given number of facilities on locations taken from a discrete set of possible points. For this bi-level optimization problem we consider six different customer behavior scenarios from the literature: binary, proportional and partially binary, each combined with essential and unessential demand. The decision making for the leader and the follower depends on these scenarios. In this work we present mixed integer linear programming models for the follower problem of each scenario and use them in combination with an evolutionary algorithm to optimize the location selection for the leader. A complete solution archive is used to detect already visited candidate solutions and convert them efficiently into similar, not yet considered ones. We present numerical results of our algorithm and compare them to so far state-of-the-art approaches from the literature. Our method shows good performance in all customer behavior scenarios and is able to outperform previous solution procedures on many occasions.     

A Stackelberg game-theoretic approach to optimal real-time pricing for the smart grid

This paper proposes a Stackelberg game approach to maximize the profit of the electricity retailer (utility company) and minimize the payment bills of its customers. The electricity retailer determines the retail price through the proposed smart energy pricing scheme to optimally adjust the real-time pricing with the aim to maximize its profit. The price information is sent to the customers through a smart meter. According to the announced price, the customers can automatically manage the energy use of appliances in the households by the proposed optimal electricity consumption scheduling system with the aim to minimize their electricity bills. We model the interactions between the retailer and its electricity customers as a 1-leader, N-follower Stackelberg game. At the leader’s side, i.e., for the retailer, we adopt genetic algorithms to maximize its profit while at the followers’ side, i.e., for customers, we develop an analytical solution to the linear programming problem to minimize their bills. Simulation results show that the proposed approach is beneficial for both the customers and the retailer.     

Putting Objects in Perspective

Image understanding requires not only individually estimating elements of the visual world but also capturing the interplay among them. In this paper, we provide a framework for placing local object detection in the context of the overall 3D scene by modeling the interdependence of objects, surface orientations, and camera viewpoint. Most object detection methods consider all scales and locations in the image as equally likely. We show that with probabilistic estimates of 3D geometry, both in terms of surfaces and world coordinates, we can put objects into perspective and model the scale and location variance in the image. Our approach reflects the cyclical nature of the problem by allowing probabilistic object hypotheses to refine geometry and vice-versa. Our framework allows painless substitution of almost any object detector and is easily extended to include other aspects of image understanding. Our results confirm the benefits of our integrated approach.     

The generalized maximal covering location problem

We consider a generalization of the maximal cover location problem which allows for partial coverage of customers, with the degree of coverage being a non-increasing step function of the distance to the nearest facility. Potential application areas for this generalized model to locating retail facilities are discussed.We show that, in general, our problem is equivalent to the uncapacitated facility location problem. We develop several integer programming formulations that capitalize on the special structure of our problem. Extensive computational analysis of the solvability of our model under a variety of conditions is presented.     

Randomized local search for the discrete competitive facility location problem

Consider a finite set of consumers that two competing companies are willing to service. The companies open facilities one by one. The set of locations available to open facilities is finite. The problem is to find a facility location for the first company that maximizes its profit given that the second company also makes its decision by maximizing the profit. We propose a randomized local search scheme that employs an internal local search procedure to estimate the solutions being enumerated. Numerical experiments with random input data show that the scheme is able to find high quality approximate solutions for examples with dimension that has not been amenable to previously known algorithms.     

Discrete approximation heuristics for the capacitated continuous location–allocation problem with probabilistic customer locations

The capacitated continuous location–allocation problem, also called capacitated multisource Weber problem (CMWP), is concerned with locating m facilities in the Euclidean plane, and allocating their capacity to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a nonconvex optimization problem. In this work, we focus on a probabilistic extension referred to as the probabilistic CMWP (PCMWP), and consider the situation in which customer locations are randomly distributed according to a bivariate probability distribution. We first formulate the discrete approximation of the problem as a mixed-integer linear programming model in which facilities can be located on a set of candidate points. Then we present three heuristics to solve the problem. Since optimal solutions cannot be found, we assess the performance of the heuristics using the results obtained by an alternate location–allocation heuristic that is originally developed for the deterministic version of the problem and tailored by us for the PCMWP. The new heuristics depend on the evaluation of the expected distances between facilities and customers, which is possible only for a few number of distance function and probability distribution combinations. We therefore propose approximation methods which make the heuristics applicable for any distance function and probability distribution of customer coordinates.