A hybrid multistart heuristic for the uncapacitated facility location problem

We present a multistart heuristic for the uncapacitated facility location problem, based on a very successful method we originally developed for the p-median problem. We show extensive empirical evidence to the effectiveness of our algorithm in practice. For most benchmarks instances in the literature, we obtain solutions that are either optimal or a fraction of a percentage point away from it. Even for pathological instances (created with the sole purpose of being hard to tackle), our algorithm can get very close to optimality if given enough time. It consistently outperforms other heuristics in the literature.     

A bi-objective uncapacitated facility location problem

We consider a bi-objective model for uncapacitated facility location where one objective is to maximize the net profit and the other to maximize the profitability of the investment. We first characterize the structure of the model having both a linear and a fractional objective function. In order to generate efficient solutions for the model, we develop a heuristic procedure which has computational advantages over existing methods. A numerical example is presented to illustrate the solution process and computational tests on large scale problems are also provided.     

Solving the deterministic and stochastic uncapacitated facility location problem: from a heuristic to a simheuristic

The uncapacitated facility location problem (UFLP) is a popular combinatorial optimization problem with practical applications in different areas, from logistics to telecommunication networks. While most of the existing work in the literature focuses on minimizing total cost for the deterministic version of the problem, some degree of uncertainty (e.g., in the customers’ demands or in the service costs) should be expected in real-life applications. Accordingly, this paper proposes a simheuristic algorithm for solving the stochastic UFLP (SUFLP), where optimization goals other than the minimum expected cost can be considered. The development of this simheuristic is structured in three stages: (i) first, an extremely fast savings-based heuristic is introduced; (ii) next, the heuristic is integrated into a metaheuristic framework, and the resulting algorithm is tested against the optimal values for the UFLP; and (iii) finally, the algorithm is extended by integrating it with simulation techniques, and the resulting simheuristic is employed to solve the SUFLP. Some numerical experiments contribute to illustrate the potential uses of each of these solving methods, depending on the version of the problem (deterministic or stochastic) as well as on whether or not a real-time solution is required.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

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.     

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.     

A k-product uncapacitated facility location problem

A k-product uncapacitated facility location problem can be described as follows. There is a set of demand points where clients are located and a set of potential sites where facilities of unlimited capacities can be set up. There are k different kinds of products. Each client needs to be supplied with k kinds of products by a set of k different facilities and each facility can be set up to supply only a distinct product with a non-negative fixed cost determined by the product it intends to supply. There is a non-negative cost of shipping goods between each pair of locations. These costs are assumed to be symmetric and satisfy the triangle inequality. The problem is to select a set of facilities to be set up and their designated products and to find an assignment for each client to a set of k   facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, an approximation algorithm within a factor of 2k+1$2k+1$ of the optimum cost is presented. Assuming that fixed setup costs are zero, we give a 2k-1$2k-1$ approximation algorithm for the problem. In addition we show that for the case k=2$k=2$, the problem is NP-complete when the cost structure is general and there is a 2-approximation algorithm when the costs are symmetric and satisfy the triangle inequality. The algorithm is shown to produce an optimal solution if the 2-product uncapacitated facility location problem with no fixed costs happens to fall on a tree graph.     

Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem

In this paper new MILP formulations for the multiple allocation p-hub median problem are presented. These require fewer variables and constraints than those traditionally used in the literature. An efficient heuristic algorithm, based on shortest paths, is described. LP based solution methods as well as an explicit enumeration algorithm are developed to obtain exact solutions. Computational results are presented for well known problems from the literature which show that exact solutions can be found in a reasonable amount of computational time. Our algorithms are also benchmarked on a different data set. This data set, which includes problems that are larger than those used in the literature, is based on a postal delivery network and has been treated by the authors in an earlier paper.     

An acceleration of Erlenkotter-Körkel’s algorithms for the uncapacitated facility location problem

This contribution is focused on an acceleration of branch and bound algorithms for the uncapacitated facility location problem. Our approach is based on the well-known Erlenkotters’ procedures and Körkels’ multi-ascent and multi-adjustment algorithms, which have proved to be the efficient tools for solving the large-sized instances of the uncapacitated facility location problem. These two original approaches were examined and a thorough analysis of their performance revealed how each particular procedure contributes to the computational time of the whole algorithms. These analyses helped us to focus our effort on the most frequent procedures. The unique contribution of this paper is a new dual ascent procedure. This procedure leads to considerable acceleration of the lower bound computation process and reduces the resulting computational time. To demonstrate more efficient performance of amended algorithms we present the results of extensive numerical experiments.     

An exact cooperative method for the uncapacitated facility location problem

In this paper, we present a cooperative primal-dual method to solve the uncapacitated facility location problem exactly. It consists of a primal process, which performs a variation of a known and effective tabu search procedure, and a dual process, which performs a lagrangian branch-and-bound search. Both processes cooperate by exchanging information which helps them find the optimal solution. Further contributions include new techniques for improving the evaluation of the branch-and-bound nodes: decision-variable bound tightening rules applied at each node, and a subgradient caching strategy to improve the bundle method applied at each node.     

LP-based approximation algorithms for capacitated facility location

In the capacitated facility location problem with hard capacities, we are given a set of facilities, ${\mathcal{F}}$ , and a set of clients ${\mathcal{D}}$ in a common metric space. Each facility i has a facility opening cost f _i and capacity u _i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set ${\mathcal{F}}$ and assign each client to an open facility so that at most u _i clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance c _ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.     

Lagrangian-relaxation-based solution procedures for a multiproduct capacitated facility location problem with choice of facility type

This paper presents exact and heuristic solution procedures for a multiproduct capacitated facility location (MPCFL) problem in which the demand for a number of different product families must be supplied from a set of facility sites, and each site offers a choice of facility types exhibiting different capacities. MPCFL generalizes both the uncapacitated (or simple) facility location (UFL) problem and the pure-integer capacitated facility location problem. We define a branch-and-bound algorithm for MPCFL that utilizes bounds formed by a Lagrangian relaxation of MPCFL which decomposes the problem into UFL subproblems and easily solvable 0-1 knapsack subproblems. The UFL subproblems are solved by the dual-based procedure of Erlenkotter. We also present a subgradient optimization-Lagrangian relaxation-based heuristic for MPCFL. Computational experience with the algorithm and heuristic are reported. The MPCFL heuristic is seen to be extremely effective, generating solutions to the test problems that are on average within 2% of optimality, and the branch-and-bound algorithm is successful in solving all of the test problems to optimality.     

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of “promising” variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.     

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.     

Applying GRASP to solve the multi-item three-echelon uncapacitated facility location problem

This paper considers a production–distribution problem that consists of defining the flow of produced products from manufacturing plants to clients (markets) via a set of warehouses. The problem also consists of defining the location of such warehouses that have unlimited storage capacity. This problem is known in the literature as the three-echelon uncapacitated facility location problem (TUFLP), and is known to be NP-hard when the objective function is to minimize the total cost of warehouse location and production and distribution of products. This paper proposes a Greedy Randomized Adaptive Search Procedure (GRASP) to solve the multi-item version of the TUFLP. Computational experiments are conducted using known instances from the literature. Solutions obtained using GRASP are compared against both optimal solutions and lower bounds obtained using mathematical programming. Results show that proposed algorithm performs well, obtaining good solutions (and even the optimal values) in less computational time than the mixed-integer linear programming model.     

A large-scale application of the partial coverage uncapacitated facility location problem

The traditional, uncapacitated facility location problem (UFLP) seeks to determine a set of warehouses to open such that all retail stores are serviced by a warehouse and the sum of the fixed costs of opening and operating the warehouses and the variable costs of supplying the retail stores from the opened warehouses is minimized. In this paper, we discuss the partial coverage uncapacitated facility location problem (PCUFLP) as a generalization of the uncapacitated facility location problem in which not all the retail stores must be satisfied by a warehouse. Erlenkotter's dual-ascent algorithm, DUALOC, will be used to solve optimally large (1600 stores and 13?000 candidate warehouses) real-world implemented PCUFLP applications in less than two minutes on a 500?MHz PC. Furthermore, a simple analysis of the problem input data will indicate why and when efficient solutions to large PCUFLPs can be expected.     

Approximate algorithms for the competitive facility location problem

We consider the competitive facility location problem in which two competing sides (the Leader and the Follower) open in succession their facilities, and each consumer chooses one of the open facilities basing on its own preferences. The problem amounts to choosing the Leader’s facility locations so that to obtain maximal profit taking into account the subsequent facility location by the Follower who also aims to obtain maximal profit. We state the problem as a two-level integer programming problem. A method is proposed for calculating an upper bound for the maximal profit of the Leader. The corresponding algorithm amounts to constructing the classical maximum facility location problem and finding an optimal solution to it. Simultaneously with calculating an upper bound we construct an initial approximate solution to the competitive facility location problem. We propose some local search algorithms for improving the initial approximate solutions. We include the results of some simulations with the proposed algorithms, which enable us to estimate the precision of the resulting approximate solutions and give a comparative estimate for the quality of the algorithms under consideration for constructing the approximate solutions to the problem.     

Improved approximation algorithms for capacitated facility location problems

In a surprising result, Korupolu, Plaxton, and Rajaraman [13] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8+ of the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces a solution which is within a factor of 6(1+) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron.Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn any -approximation algorithm for this variant into a polynomial-time algorithm which, at an additional cost of twice the optimum of the standard CFLP, opens at most one additional copy of any facility. This allows us to transform a recent 2-approximation algorithm of Mahdian, Ye, and Zhang [17] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than four times the value of the standard CFLP.This research was performed while the author was a postdoctoral fellow at the IBM T.J. Watson Research Center.This research was performed while the author was a Research Staff Member at the IBM T.J. Watson Research Center.A preliminary version of this paper appeared in the Proceedings of the 7th Conference on Integer Programming and Combinatorial Optimization [9].     

A facility neighborhood search heuristic for capacitated facility location with single-source constraints and flexible demand

We consider a generalization of the well-known capacitated facility location problem with single source constraints in which customer demand contains a flexible dimension. This work focuses on providing fast and practically implementable optimization-based heuristic solution methods for very large scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems.