The probabilistic 1-maximal covering problem on a network with discrete demand weights

We discuss the probabilistic 1-maximal covering problem on a network with uncertain demand. A single facility is to be located on the network. The demand originating from a node is considered covered if the shortest distance from the node to the facility does not exceed a given service distance. 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 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.     

A single facility stochastic location problem undera-distance

In this paper we consider a stochastic facility location model in which the weights of demand points are not deterministic but independent random variables, and the distance between the facility and each demand point isA-distance. Our objective is to find a solution which minimizes the total cost criterion subject to a chance constraint on cost restriction. We show a solution method which solves the problem in polynomial order computational time. Finally the case of rectilinear distance, which is used in many facility location models, is discussed.     

Multi-objective competitive location problem with distance-based attractiveness and its best non-dominated solution

The multi-objective competitive location problem (MOCLP) with distance-based attractiveness is introduced. There are m potential competitive facilities and n demand points on the same plane. All potential facilities can provide attractiveness to the demand point which the facility attractiveness is represented as distance-based coverage of a facility, which is “full coverage” within the maximum full coverage radius, “no coverage” outside the maximum partial coverage radius, and “partial coverage” between those two radii. Each demand point covered by one of m potential facilities is determined by the greatest accumulated attractiveness provided the selected facilities and least accumulated distances between each demand point and selected facility, simultaneously. The tradeoff of maximum accumulated attractiveness and minimum accumulated distances is represented as a multi-objective optimization model. A proposed solution procedure to find the best non-dominated solution set for MOCLP is introduced. Several numerical examples and instances comparing with introduced and exhaustive method demonstrates the good performance and efficiency for the proposed solution procedure.     

An interactive approach for multiobjective decision making

We develop an interactive approach for multiobjective decision-making problems, where the solution space is defined by a set of constraints. We first reduce the solution space by eliminating some undesirable regions. We generate solutions (partition ideals) that dominate portions of the efficient frontier and the decision maker (DM) compares these with feasible solutions. Whenever the decision maker prefers a feasible solution, we eliminate the region dominated by the partition ideal. We then employ an interactive search method on the reduced solution space to help the DM further converge toward a highly preferred solution. We demonstrate our approach and discuss some variations.     

Mixed integer programming-based solution procedure for single-facility location with maximin of rectilinear distance

In this paper, we study the 1-maximin problem with rectilinear distance. We locate a single undesirable facility in a continuous planar region while considering the interaction between the facility and existing demand points. The distance between facility and demand points is measured in the rectilinear metric. The objective is to maximize the distance of the facility from the closest demand point. The 1-maximin problem has been formulated as an MIP model in the literature. We suggest new bounding schemes to increase the solution efficiency of the model as well as improved branch and bound strategies for implementation. Moreover, we simplify the model by eliminating some redundant integer variables. We propose an efficient solution algorithm called cut and prune method, which splits the feasible region into four equal subregions at each iteration and tries to eliminate subregions depending on the comparison of upper and lower bounds. When the sidelengths of the subregions are smaller than a predetermined value, the improved MIP model is solved to obtain the optimal solution. Computational experiments demonstrate that the solution time of the original MIP model is reduced substantially by the proposed solution approach.     

Interactive procedure for a multiobjective stochastic discrete dynamic problem

Multiple objectives and dynamics characterize many sequential decision problems. In the paper we consider returns in partially ordered criteria space as a way of generalization of single criterion dynamic programming models to multiobjective case. In our problem evaluations of alternatives with respect to criteria are represented by distribution functions. Thus, the overall comparison of two alternatives is equivalent to the comparison of two vectors of probability distributions. We assume that the decision maker tries to find a solution preferred to all other solutions (the most preferred solution). In the paper a new interactive procedure for stochastic, dynamic multiple criteria decision making problem is proposed. The procedure consists of two steps. First, the Bellman principle is used to identify the set of efficient solutions. Next interactive approach is employed to find the most preferred solution. A numerical example and a real-world application are presented to illustrate the applicability of the proposed technique.     

Gearhart–Koshy acceleration for affine subspaces

The method of cyclic projections finds nearest points in the intersection of finitely many affine subspaces. To accelerate convergence, Gearhart & Koshy proposed a modification which, in each iteration, performs an exact line search based on minimising the distance to the solution. When the subspaces are linear, the procedure can be made explicit using feasibility of the zero vector. This work studies an alternative approach which does not rely on this fact, thus providing an efficient implementation in the affine setting.     

Optimizing capacity,pricing and location decisions on a congested network with balking

In this paper, we consider the problem of making simultaneous decisions on the location, service rate (capacity) and the price of providing service for facilities on a network. We assume that the demand for service from each node of the network follows a Poisson process. The demand is assumed to depend on both price and distance. All facilities are assumed to charge the same price and customers wishing to obtain service choose a facility according to a Multinomial Logit function. Upon arrival to a facility, customers may join the system after observing the number of people in the queue. Service time at each facility is assumed to be exponentially distributed. We first present several structural results. Then, we propose an algorithm to obtain the optimal service rate and an approximate optimal price at each facility. We also develop a heuristic algorithm to find the locations of the facilities based on the tabu search method. We demonstrate the efficiency of the algorithms numerically.     

The capacitated maximal covering location problem with backup service

The maximal covering location problem has been shown to be a useful tool in siting emergency services. In this paper we expand the model along two dimensions — workload capacities on facilities and the allocation of multiple levels of backup or prioritized service for all demand points. In emergency service facility location decisions such as ambulance sitting, when all of a facility's resources are needed to meet each call for service and the demand cannot be queued, the need for a backup unit may be required. This need is especially significant in areas of high demand. These areas also will often result in excessive workload for some facilities. Effective siting decisions, therefore, must address both the need for a backup response facility for each demand point and a reasonable limit on each facility's workload. In this paper, we develop a model which captures these concerns as well as present an efficient solution procedure using Lagrangian relaxation. Results of extensive computational experiments are presented to demonstrate the viability of the approach.     

The expropriation location problem

In this paper, we consider the location of a new obnoxious facility that serves only a certain proportion of the demand. Each demand point can be bought by the developer at a given price. An expropriation budget is given. Demand points closest to the facility are expropriated within the given budget. The objective is to maximize the distance to the closest point not expropriated. The problem is formulated and polynomial algorithms are proposed for its solution both on the plane and on a network.     

Finding the optimal solution to the Huff based competitive location model

In this paper we solve the gravity (Huff) model for the competitive facility location problem. We show that the generalized Weiszfeld procedure converges to a local maximum or a saddle point. We also devise a global optimization procedure that finds the optimal solution within a given accuracy. This procedure is very efficient and finds the optimal solution for 10,000 demand points in less than six minutes of computer time. We also experimented with the generalized Weiszfeld algorithm on the same set of randomly generated problems. We repeated the algorithm from 1,000 different starting solutions and the optimum was obtained at least 17 times for all problems.     

A BSSS algorithm for the single facility location problem in two regions with different norms

Suppose the plane is divided by a straight line into two regions with different norms. We want to find the location of a single new facility such that the sum of the distances from the existing facilities to this point is minimized. This is in fact a non-convex optimization problem. The main difficulty is caused by finding the distances between points on different sides of the boundary line. In this paper we present a closed form solution for finding these distances. We also show that the optimal solution lies in the rectangular hull of the existing points. Based on these findings then, an efficient big square small square (BSSS) procedure is proposed.     

An interactive algorithm to find the most preferred solution of multi-objective integer programs

In this paper, we develop an interactive algorithm that finds the most preferred solution of a decision maker (DM) for multi-objective integer programming problems. We assume that the DM’s preferences are consistent with a quasiconcave value function unknown to us. Based on the properties of quasiconcave value functions and pairwise preference information obtained from the DM, we generate constraints to restrict the implied inferior regions. The algorithm continues iteratively and guarantees to find the most preferred solution for integer programs. We test the performance of the algorithm on multi-objective assignment, knapsack, and shortest path problems and show that it works well.     

A minimum length covering subgraph of a network

This paper concerns the problem of locating a central facility on a connected networkN. The network,N, could be representative of a transport system, while the central facility takes the form of a connected subgraph ofN. The problem is to locate a central facility of minimum length so that each of several demand points onN is covered by the central facility: a demand point atv _i inN is covered by the central facility if the shortest path distance betweenv _i and the closest point in the central facility does not exceed a parameterr _i . This location problem is NP-hard, but for certain special cases, efficient solution methods are available.     

Biobjective center – balance graph location model *

In the paper, a new algorithm for finding the absolute center of the graph is proposed. Later on problem of the absolute balance point of the graph is investigated the objective function of which is defined as the difference in the distance from the located facility to the farthest and the nearest demand point. An algorithm for finding it is presented By following the balance criterion it may happen, that the balance point is located too far from the demand points. Conversely, if we are interested not only in the distance to the farthest demand point, but (what is more realistic) also in the structure of the distances, a criterion of balance is appropriate. Consequently, we investigate the center balance constrained model that simultaneously used center and balance criteria. By considering the center and the balance objective in a multiobjective framework, efficient solutions are generated by minimizing former objective such that latter satisfies an upperbound     

Improving solution of discrete competitive facility location problems

We consider discrete competitive facility location problems in this paper. Such problems could be viewed as a search of nodes in a network, composed of candidate and customer demand nodes, which connections correspond to attractiveness between customers and facilities located at the candidate nodes. The number of customers is usually very large. For some models of customer behavior exact solution approaches could be used. However, for other models and/or when the size of problem is too high to solve exactly, heuristic algorithms may be used. The solution of discrete competitive facility location problems using genetic algorithms is considered in this paper. The new strategies for dynamic adjustment of some parameters of genetic algorithm, such as probabilities for the crossover and mutation operations are proposed and applied to improve the canonical genetic algorithm. The algorithm is also specially adopted to solve discrete competitive facility location problems by proposing a strategy for selection of the most promising values of the variables in the mutation procedure. The developed genetic algorithm is demonstrated by solving instances of competitive facility location problems for an entering firm.     

Computing Approximate Solutions of the Maximum Covering Problem with GRASP

We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal.     

A discrete competitive facility location model with variable attractiveness

We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff's gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the locations of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time.     

Optimal location and design of a competitive facility

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits.It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favour of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in problems in which optimal solutions might not exist.Mathematics Subject Classification (2000):90B85, 90C30, 90C29, 91B42Partially supported by Grant PB96-1416-C02-02 of the D.G.E.S. and Grant BFM2002-04525-C02-02 of Ministerio de Ciencia y Tecnología, Spain     

Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.