An LP-based heuristic for two-stage capacitated facility location problems

In this paper, a linear programming based heuristic is considered for a two-stage capacitated facility location problem with single source constraints. The problem is to find the optimal locations of depots from a set of possible depot sites in order to serve customers with a given demand, the optimal assignments of customers to depots and the optimal product flow from plants to depots. Good lower and upper bounds can be obtained for this problem in short computation times by adopting a linear programming approach. To this end, the LP formulation is iteratively refined using valid inequalities and facets which have been described in the literature for various relaxations of the problem. After each reoptimisation step, that is the recalculation of the LP solution after the addition of valid inequalities, feasible solutions are obtained from the current LP solution by applying simple heuristics. The results of extensive computational experiments are given.     

District design for arc-routing applications

In this paper we address the problem of district design for the organisation of arc-routing activities. In particular, the focus is on operations like winter gritting and road maintenance. The problem involves how to allocate the road network edges to a set of depots with given locations. The collection of edges assigned to a facility forms a district in which routes have to be designed that start and end at the facility. Apart from the ability to support good arc routing, well-designed districts for road-maintenance operations should have the road network to be serviced connected and should define clear geographical boundaries. We present three districting heuristics and evaluate the quality of the partitions by solving capacitated arc routing problems in the districts, and by comparing the solution values with a multi-depot CARP cutting plane lower bound. Our experiments reveal that based on global information about the distribution system (ie the number of facilities or districts, the average edge demand and the vehicle capacity) and by using simple guidelines, an adequate districting policy may be selected.     

Minisum collection depots location problem with multiple facilities on a network

We investigate the problem of locating a set of service facilities that need to service customers on a network. To provide service, a server has to visit both the demand node and one of several collection depots. We employ the criterion of minimizing the weighted sum of round trip distances. We prove that there exists a dominating location set for the problem on a general network. The properties of the solution on a tree and on a cycle are discussed. The problem of locating service facilities and collection depots simultaneously is also studied. To solve the problem on a general network, we suggest a Lagrangian relaxation imbedded branch-and-bound algorithm. Computational results are reported.     

Lower bounds for the two-stage uncapacitated facility location problem

In the two-stage uncapacitated facility location problem, a set of customers is served from a set of depots which receives the product from a set of plants. If a plant or depot serves a product, a fixed cost must be paid, and there are different transportation costs between plants and depots, and depots and customers. The objective is to locate plants and depots, given both sets of potential locations, such that each customer is served and the total cost is as minimal as possible. In this paper, we present a mixed integer formulation based on twice-indexed transportation variables, and perform an analysis of several Lagrangian relaxations which are obtained from it, trying to determine good lower bounds on its optimal value. Computational results are also presented which support the theoretical potential of one of the relaxations.     

A note on the m-center problem with rectilinear distances

Given n demand points on a plane, the problem we consider is to locate a given number, m, of facilities on the plane so that the maximum of the set of rectilinear distances of each demand point to its nearest facility is minimized. This problem is known as the m-center problem on the plane. A related problem seeks to determine, for a given r, the minimum number of facilities and their locations so as to ensure that every point is within r units of rectilinear distance from its nearest facility. We formulate the latter problem as a problem of covering nodes by cliques of an intersection graph. Certain bounds are established on the size of the problem. An efficient algorithm is provided to generate this set-covering problem. Computational results with this approach are summarized.     

New facets for the two-stage uncapacitated facility location polytope

The two-stage uncapacitated facility location problem is considered. This problem involves a system providing a choice of depots and plants, each with an associated location cost, and a set of demand points which must be supplied, in such a way that the total cost is minimized. The formulations used until now to approach the problem were symmetric in plants and depots. In this paper the asymmetry inherent to the problem is taken into account to enforce the formulation which can be seen like a set packing problem and new facet defining inequalities for the convex hull of the feasible solutions are obtained. A computational study is carried out which illustrates the interest of the new facets. A new family of facets recently developed, termed lifted fans, is tested with success.     

The transfer point location problem

In this paper, we introduce the transfer point location problem. Demand for emergency service is generated at a set of demand points who need the services of a central facility (such as a hospital). Patients are transferred to a helicopter pad (transfer point) at normal speed, and from there they are transferred to the facility at increased speed. The general model involves the location of p helicopter pads and one facility. In this paper, we solve the special case where the location of the facility is known and the best location of one transfer point that serves a set of demand points is sought. Both minisum and minimax versions of the models are investigated. In follow up papers we investigate the general model using the results obtained in this paper.     

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.     

The multiple location of transfer points

In this paper, we investigate the location of several transfer points to serve as collector points for customers who need the services of a facility. For example, demand for emergency services by patients is generated at a set of demand points that need the services of a central facility (such as a hospital). Patients are transferred to a helicopter pad (transfer point) at normal speed, and from there they are transferred to the facility at increased speed. The general model involves the location of multiple transfer points and one facility. Locating one transfer point when the set of demand points and the location of the facility are known was investigated in a previous paper by the authors. In this paper, we apply the results of that paper to solve the problem when the location of the facility is known. Both minisum and minimax versions of the models are investigated both in the plane and on the network.     

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.     

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.     

Lower and upper bounds for a two-stage capacitated facility location problem with handling costs

We study in this paper multi-product facility location problem in a two-stage supply chain in which plants have production limitation, potential depots have limited storage capacity and customer demands must be satisfied by plants via depots. In the paper, handling cost for batch process in depots is considered in a realistic way by a set of capacitated handling modules. Each module can be regards as alliance of equipment and manpower. The problem is to locate depots, choose appropriate handling modules and to determine the product flows from the plants, opened depots to customers with the objective to minimize total location, handling and transportation costs. For the problem, we developed a hybrid method. The initial lower and upper bounds are provided by applying a Lagrangean based on local search heuristic. Then a weighted Dantzig–Wolfe decomposition and path-relinking combined method are proposed to improve obtained bounds. Numerical experiments on 350 randomly generated instances demonstrate our method can provide high quality solution with gaps below 2%.     

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function \(f_i(z)\). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a (\(7.88+\epsilon \))-approximation local search algorithm for this problem.     

On a robustness property in single-facility location in continuous space

We consider a single-facility location problem in continuous space—here the problem of minimizing a sum or the maximum of the possibly weighted distances from a facility to a set of points of demand. The main result of this paper shows that every solution (optimal facility location) of this problem has an interesting robustness property. Any optimal facility location is the most robust in the following sense: given a suitable highest admissible cost, it allows the greatest perturbation of the locations of the demand without exceeding this highest admissible chosen cost.     

A heuristic solution method for node routing based solid waste collection problems

This paper considers a real world waste collection problem in which glass, metal, plastics, or paper is brought to certain waste collection points by the citizens of a certain region. The collection of this waste from the collection points is therefore a node routing problem. The waste is delivered to special sites, so called intermediate facilities (IF), that are typically not identical with the vehicle depot. Since most waste collection points need not be visited every day, a planning period of several days has to be considered. In this context three related planning problems are considered. First, the periodic vehicle routing problem with intermediate facilities (PVRP-IF) is considered and an exact problem formulation is proposed. A set of benchmark instances is developed and an efficient hybrid solution method based on variable neighborhood search and dynamic programming is presented. Second, in a real world application the PVRP-IF is modified by permitting the return of partly loaded vehicles to the depots and by considering capacity limits at the IF. An average improvement of 25% in the routing cost is obtained compared to the current solution. Finally, a different but related problem, the so called multi-depot vehicle routing problem with inter-depot routes (MDVRPI) is considered. In this problem class just a single day is considered and the depots can act as an intermediate facility only at the end of a tour. For this problem several instances and benchmark solutions are available. It is shown that the algorithm outperforms all previously published metaheuristics for this problem class and finds the best solutions for all available benchmark instances.     

Weber problems with high-speed lines

The Weber problem consists of finding a facility which minimizes the sum of weighted distances from itself to a finite set of given demand points.     

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.     

Discrete cooperative covering problems

A family of discrete cooperative covering problems is analysed in this paper. Each facility emits a signal that decays by the distance and each demand point observes the total signal emitted by all facilities. A demand point is covered if its cumulative signal exceeds a given threshold. We wish to maximize coverage by selecting locations for p facilities from a given set of potential sites. Two other problems that can be solved by the max-cover approach are the equivalents to set covering and p-centre problems. The problems are formulated, analysed and solved on networks. Optimal and heuristic algorithms are proposed and extensive computational experiments reported.     

A Procedure for Locating Emergency-Service Facilities for All Possible Response Distances

The problem of locating emergency-service facilities involves the assignment of a set of demand points to a set of facilities. One way to formulate the problem is to minimize the number of required facilities, given that the maximum distance between the demand points and their nearest facility does not exceed some specified value. We present a procedure for determining the numbers of such facilities for all possible values of the maximum distance. Computational results are presented for a microcomputer implementation.     

Spherical minimax location problem using the Euclidean norm: Formulation and optimization

The minimax spherical location problem is formulated in the Cartesian coordinate system using the Euclidean norm, instead of the spherical coordinate system using spherical arc distance measures. It is shown that minimizing the maximum of the spherical arc distances between the facility point and the demand points on a sphere is equivalent to minimizing the maximum of the corresponding Euclidean distances. The problem formulation in this manner helps to reduce Karush-Kuhn-Tucker necessary optimality conditions into the form of a set of coupled nonlinear equations, which is solved numerically by using a method of factored secant update with a finite difference approximation to the Jacobian. For a special case when the set of demand points is on a hemisphere and one or more point-antipodal point pair(s) are included in the demand points, a simplified approach gives a minimax point in a closed form.