Incorporating the threat of terrorist attacks in the design of public service facility networks

In this paper, we consider the design problem of a public service facility network with existing facilities when there is a threat of possible terrorist attacks. The aim of the system planner, who is responsible for the operation of the network, is to open new facilities, relocate existing ones if necessary, and protect some of the facilities to ensure a maximum coverage of the demand that is assumed to be aggregated at customer zones. By doing so, the system planner anticipates that a number of unprotected facilities will be rendered out-of-service by terrorist attacks. It is assumed that the sum of the fixed cost of opening new facilities, the relocation costs, and the protection costs cannot exceed a predetermined budget level. Adopting the approach of gradual (or partial) coverage, we formulate a bilevel programming model where the system planner is the leader and the attacker is the follower. The objective of the former is the maximization of the total service coverage, whereas the latter wants to minimize it. We propose a heuristic solution procedure based on tabu search where the search space consists of the decisions of the system planner, and the corresponding objective value is computed by optimally solving the attacker??s problem using CPLEX. To assess the quality of the solutions produced by the tabu search (TS) heuristic, we also develop an exhaustive enumeration method, which explores all the possible combinations of opening new facilities, relocating existing ones, and protecting them. Since its time complexity is exponential, it can only be used for relatively small instances. Therefore, to be used as a benchmark method, we also implement a hill climbing procedure employed with the same type of moves as the TS heuristic. Besides, we carry out a sensitivity analysis on some of the problem parameters to investigate their effect on the solution characteristics.     

On the complexity of the (r|p)-centroid problem in the plane

In the (r∣p)-centroid problem, two players, called leader and follower, open facilities to service clients. We assume that clients are identified with their location on the Euclidean plane, and facilities can be opened anywhere in the plane. The leader opens p facilities. Later on, the follower opens r facilities. Each client patronizes the closest facility. In case of ties, the leader’s facility is preferred. The goal is to find p facilities for the leader to maximize his market share. We show that this Stackelberg game is \(\varSigma_{2}^{P}\) -hard. Moreover, we strengthen the previous results for the discrete case and networks. We show that the game is \(\varSigma_{2}^{P}\) -hard even for planar graphs for which the weights of the edges are Euclidean distances between vertices.     

A bi-objective modeling approach applied to an urban semi-desirable facility location problem

This paper introduces a mixed-integer, bi-objective programming approach to identify the locations and capacities of semi-desirable (or semi-obnoxious) facilities. The first objective minimizes the total investment cost; the second one minimizes the dissatisfaction by incorporating together in the same function “pull” and “push” characteristics of the decision problem (individuals do not want to live too close, but they do not want to be too far, from facilities). The model determines the number of facilities to be opened, the respective capacities, their locations, their respective shares of the total demand, and the population that is assigned to each candidate site opened. The proposed approach was tested with a case study for a particular urban planning problem: the location of sorted waste containers. The complete set of (supported or unsupported) non-inferior solutions, consisting of combinations of multi-compartment containers for the disposal of four types of sorted waste in nineteen candidate sites, and population assignments, was generated. The results obtained for part of the historical center of an old European city (Coimbra, Portugal) show that this approach can be applied to a real-world planning scenario.     

Upper bounds for objective functions of discrete competitive facility location problems

Under study is the problem of locating facilities when two competing companies successively open their facilities. Each client chooses an open facility according to his own preferences and return interests to the leader firm or to the follower firm. The problem is to locate the leader firm so as to realize the maximum profit (gain) subject to the responses of the follower company and the available preferences of clients. We give some formulations of the problems under consideration in the form of two-level integer linear programming problems and, equivalently, as pseudo-Boolean two-level programming problems. We suggest a method of constructing some upper bounds for the objective functions of the competitive facility location problems. Our algorithm consists in constructing an auxiliary pseudo-Boolean function, which we call an estimation function, and finding the minimum value of this function. For the special case of the competitive facility location problems on paths, we give polynomial-time algorithms for finding optimal solutions. Some results of computational experiments allow us to estimate the accuracy of calculating the upper bounds for the competitive location problems on paths.     

γ-Robust facility relocation problem

In this paper, we consider relocating facilities, where we have demand changes in the network. Relocations are performed by closing some of the existing facilities from low demand areas and opening new ones in newly emerging areas. However, the actual changes of demand are not known in advance. Therefore, different scenarios with known probabilities are used to capture such demand changes. We develop a mixed integer programming model for facility relocation that minimizes the expected weighted distance while making sure that relative regret for each scenario is no greater than γ. We analyzed the problem structure and developed a Lagrangian Decomposition Algorithm (LDA) to expedite the solution process. Numerical experiments are carried out to show the performance of LDA against the exact solution method.     

Extensive facility location problems on networks with equity measures

This paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case.     

The location of median paths on grid graphs

In this paper we consider the location of a path shaped facility on a grid graph. In the literature this problem was extensively studied on particular classes of graphs as trees or series-parallel graphs. We consider here the problem of finding a path which minimizes the sum of the (shortest) distances from it to the other vertices of the grid, where the path is also subject to an additional constraint that takes the form either of the length of the path or of the cardinality. We study the complexity of these problems and we find two polynomial time algorithms for two special cases, with time complexity of O(n) and O(nℓ) respectively, where n is the number of vertices of the grid and ℓ is the cardinality of the path to be located. The literature about locating dimensional facilities distinguishes between the location of extensive facilities in continuous spaces and network facility location. We will show that the problems presented here have a close connection with continuous dimensional facility problems, so that the procedures provided can also be useful for solving some open problems of dimensional facilities location in the continuous case.     

A decomposition approach to solve a bilevel capacitated facility location problem with equity constraints

In this paper, we study a capacitated facility location problem with two decision makers. One (say, the leader) decides on which subset of facilities to open and the capacity to be installed in each facility with the goal of minimizing the overall costs; the second decision maker (say, the follower), once the facilities have been designed, aims at maximizing the profit deriving from satisfying the demands of a given set of clients beyond a certain threshold imposed by the leader. The leader can foresee but cannot control the follower’s behavior. The resulting mathematical formulation is a discrete–continuous bilevel optimization problem. We propose a decomposition approach to cope with the bilevel structure of the problem and the integrality of a subset of variables under the control of the leader. Such a proposal has been tested on a set of benchmark instances available in the literature.     

Location of Service Facilities for the Elderly

Senior centers provide a variety of supportive services for independent elderly adults. In many metropolitan areas, the elderly population is growing and redistributing from central cities to suburbs, where accessibility to senior centers is limited. Policy analysts need to locate senior centers to best meet changing demands for service. We present alternative hierarchical facility location models for senior centers applied to Allegheny County, Pennsylvania. We find that a model that minimizes consumer disutility and unserved demands is preferred to one that maximizes utility alone, and that the former model is well-behaved in response to changes in structural parameters.     

The minmax regret gradual covering location problem on a network with incomplete information of demand weights

The gradual covering location problem seeks to establish facilities on a network so as to maximize the total demand covered, allowing partial coverage. We focus on the gradual covering location problem when the demand weights associated with nodes of the network are random variables whose probability distributions are unknown. Using only information on the range of these random variables, this study is aimed at finding the “minmax regret” location that minimizes the worst-case coverage loss. We show that under some conditions, the problem is equivalent to known location problems (e.g. the minmax regret median problem). Polynomial time algorithms are developed for the problem on a general network with linear coverage decay functions.     

A two-level evolutionary algorithm for solving the facility location and design (1|1)-centroid problem on the plane with variable demand

In this work, the problem of a company or chain (the leader) that considers the reaction of a competitor chain (the follower) is studied. In particular, the leader wants to set up a single new facility in a planar market where similar facilities of the follower, and possibly of its own chain, are already present. The follower will react by locating another single facility after the leader locates its own facility. Both the location and the quality (representing design, quality of products, prices, etc.) of the new leader’s facility have to be found. The aim is to maximize the profit obtained by the leader considering the future follower’s entry. The demand is supposed to be concentrated at n demand points. Each demand point splits its buying power among the facilities proportionally to the attraction it feels for them. The attraction of a demand point for a facility depends on both the location and the quality of the facility. Usually, the demand is considered in the literature to be fixed or constant regardless the conditions of the market. In this paper, the demand varies depending on the attraction for the facilities. Taking variable demand into consideration makes the model more realistic. However, it increases the complexity of the problem and, therefore, the computational effort needed to solve it. Three heuristic methods are proposed to cope with this hard-to-solve global optimization problem, namely, a grid search procedure, a multistart algorithm and a two-level evolutionary algorithm. The computational studies show that the evolutionary algorithm is both the most robust algorithm and the one that provides the best results.     

Fiscal and monetary policy interactions: a game theory approach

The interaction between fiscal and monetary policy is analyzed by means of a game theory approach. The coordination between these two policies is essential, since decisions taken by one institution may have disastrous effects on the other one, resulting in welfare loss for the society. We derived optimal monetary and fiscal policies in context of three coordination schemes: when each institution independently minimizes its welfare loss as a Nash equilibrium of a normal form game; when an institution moves first and the other follows, in a mechanism known as the Stackelberg solution; and, when institutions behave cooperatively, seeking common goals. In the Brazilian case, a numerical exercise shows that the smallest welfare loss is obtained under a Stackelberg solution which has the monetary policy as leader and the fiscal policy as follower. Under the optimal policy, there is evidence of a strong distaste for inflation by the Brazilian society.     

A decomposition approach for facility location and relocation problem with uncertain number of future facilities

In this paper, we discuss two challenges of long term facility location problem that occur simultaneously; future demand change and uncertain number of future facilities. We introduce a mathematical model that minimizes the initial and expected future weighted travel distance of customers. Our model allows relocation for the future instances by closing some of the facilities that were located initially and opening new ones, without exceeding a given budget. We present an integer programming formulation of the problem and develop a decomposition algorithm that can produce near optimal solutions in a fast manner. We compare the performance of our mathematical model against another method adapted from the literature and perform sensitivity analysis. We present numerical results that compare the performance of the proposed decomposition algorithm against the exact algorithm for the problem.     

基于敌意风险分析方法的多阶段反恐设备选址模型

恐怖袭击一直是人类安全的重要威胁之一。随着当前恐怖袭击在全球的频发,对反恐问题的研究更加急迫。针对反恐设施选址问题,考虑资源分配的多阶段性以及动态性,根据贝叶斯决策理论和序贯博弈思想,构建了多阶段反恐设施的敌意风险分析决策模型。讨论在城市多个设施点离散选址的不同情况下,通过预防性和修复性的资源分配将恐怖袭击的损失降低到最小。以上海市区县网络为例进行编程仿真测试,结果表明,模型可给出不同给定资源设定下的最优选址方案,是一种有效并切实可行的分析方法。     

A two-player competitive discrete location model with simultaneous decisions

In this paper we will describe and study a competitive discrete location problem in which two decision-makers (players) will have to decide where to locate their own facilities, and customers will be assigned to the closest open facilities. We will consider the situation in which the players must decide simultaneously, unsure about the decisions of one another, and we will present the problem in a franchising environment. Most problems described in the literature consider sequential rather than simultaneous decisions. In a competitive environment, most problems consider that there is a set of known and already located facilities, and new facilities will have to be located, competing with the existing ones. In the presence of more than one decision-maker, most problems found in the literature belong to the class of Stackelberg location problems, where one decision-maker, the leader, locates first and then the other decision-maker, the follower, locates second, knowing the decisions made by the first. These types of problems are sequential and differ significantly from the problem tackled in this paper, where we explicitly consider simultaneous, non-cooperative discrete location decisions. We describe the problem and its context, propose some mathematical formulations and present an algorithmic approach that was developed to find Nash equilibria. Some computational tests were performed that allowed us to better understand some of the features of the problem and the associated Nash equilibria. Among other results, we conclude that worsening the situation of a player tends to benefit the other player, and that the inefficiency of Nash equilibria tends to increase with the level of competition.     

A projected Weiszfeld algorithm for the box-constrained Weber location problem

The Weber problem consists of finding a point in Rⁿ that minimizes the weighted sum of distances from m points in Rⁿ that are not collinear. An application that motivated this problem is the optimal location of facilities in the 2-dimensional case. A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed-point iteration.In this work we generalize the Weber location problem considering box constraints. We propose a fixed-point iteration with projections on the constraints and demonstrate descending properties. It is also proved that the limit of the sequence generated by the method is a feasible point and satisfies the KKT optimality conditions. Numerical experiments are presented to validate the theoretical results.     

Linear bilevel multi-follower programming with independent followers

This paper considers a particular case of linear bilevel programming problems with one leader and multiple followers. In this model, the followers are independent, meaning that the objective function and the set of constraints of each follower only include the leader’s variables and his own variables. We prove that this problem can be reformulated into a linear bilevel problem with one leader and one follower by defining an adequate second level objective function and constraint region. In the second part of the paper we show that the results on the optimality of the linear bilevel problem with multiple independent followers presented in Shi et al. [The kth-best approach for linear bilevel multi-follower programming, J. Global Optim. 33, 563–578 (2005)] are based on a misconstruction of the inducible region.     

A Bilevel p-median model for the planning and protection of critical facilities

The bilevel p-median problem for the planning and protection of critical facilities involves a static Stackelberg game between a system planner (defender) and a potential attacker. The system planner determines firstly where to open p critical service facilities, and secondly which of them to protect with a limited protection budget. Following this twofold action, the attacker decides which facilities to interdict simultaneously, where the maximum number of interdictions is fixed. Partial protection or interdiction of a facility is not possible. Both the defender’s and the attacker’s actions have deterministic outcome; i.e., once protected, a facility becomes completely immune to interdiction, and an attack on an unprotected facility destroys it beyond repair. Moreover, the attacker has perfect information about the location and protection status of facilities; hence he would never attack a protected facility. We formulate a bilevel integer program (BIP) for this problem, in which the defender takes on the leader’s role and the attacker acts as the follower. We propose and compare three different methods to solve the BIP. The first method is an optimal exhaustive search algorithm with exponential time complexity. The second one is a two-phase tabu search heuristic developed to overcome the first method’s impracticality on large-sized problem instances. Finally, the third one is a sequential solution method in which the defender’s location and protection decisions are separated. The efficiency of these three methods is extensively tested on 75 randomly generated instances each with two budget levels. The results show that protection budget plays a significant role in maintaining the service accessibility of critical facilities in the worst-case interdiction scenario.     

A robust model for a leader–follower competitive facility location problem in a discrete space

This paper aims at determining the optimal locations for the leader’s new facilities under the condition that the number of the follower’s new facilities is unknown for the leader. The leader and the follower have some facilities in advance. The first competitor, the leader, opens p new facilities in order to increase her own market share. On the other hand, she knows that her competitor, the follower, will react to her action and locate his new facilities as well. The number of the follower’s new facilities is unknown for the leader but it is assumed that the leader knows the probability of opening different numbers of the follower’s new facilities. The leader aims at maximizing her own market share after the follower’s new facilities entry. The follower’s objective is also to maximize his own market share. Since the number of the follower’s new facilities is unknown for leader, “Robust Optimization” is used for maximizing the leader’s market share and making the obtained results “robust” in various scenarios in terms of different numbers of the follower’s new facilities. The optimal locations for new facilities of both the leader and the follower are chosen among pre-determined potential locations. It is assumed that the demand is inelastic. The customers probabilistically meet their demands from all different facilities and the demand level which is met by each facility is computed by Huff rule. The computational experiments have been applied to evaluate the efficiency of the proposed model.     

p-Hub approach for the optimal park-and-ride facility location problem

Park and Ride facilities (P&R) are car parks at which users can transfer to public transportation to reach their final destination. We propose a mixed linear programming formulation to determine the location of a fixed number of P&R facilities so that their usage is maximized. The facilities are modeled as hubs. Commuters can use one of the P&R facilities or choose to travel by car to their destinations, and their behavior follows a logit model. We apply a p-hub approach considering that users incur in a known generalized cost of using each P&R facility as input for the logit model. For small instances of the problem, we propose a novel linearization of the logit model, which allows transforming the binary nonlinear programming problem into a mixed linear programming formulation. A modification of the Heuristic Concentration Integer (HCI) procedure is applied to solve larger instances of the problem. Numerical experiments are performed, including a case in Queens, NY. Further research is proposed.