Fractional location problems

1997

We present new approximation algorithms for several facility location problems. In each facility location problem that we study, there is a set of locations at which w e m a y build a facility such a s a w arehouse, where the cost of building at location i is fi; furthermore, there is a set of client locations such a s stores that require to be serviced by a facility, and if a client at location j is assigned to a facility at location i, a cost of cij is incurred. The objective i s t o determine a set of locations at which to open facilities so as to minimize the total facility and assignment costs. In the uncapacitated case, each facility can service an unlimited number of clients, whereas in the capacitated case, each facility can serve, for example, at most u clients. These models and a number of closely related ones have been studied extensively in the Operations Research literature. We shall consider the case in which the assignment costs are symmetric and satisfy the triangle inequality. For the uncapacitated facility location, we give a polynomial-time algorithm that nds a solution within a factor of 3.16 of the optimal. This is the rst constant performance guarantee known for this problem. We also present approximation algorithms with constant performance guarantees for a number of capacitated models as well as a generalization in which there is a 2-level hierarchy of facilities. Our results are based on the ltering and rounding technique of Lin & Vitter. We also give a randomized variant of this technique that can then be derandomized to yield improved performance guarantees.

Applied Mathematical Modelling, 2014

This paper considers the multi level uncapacitated facility location problem (MLUFLP). A new mixed integer linear programming (MILP) formulation is presented and validity of this formulation is given. Experimental results are performed on instances known from literature. The results achieved by CPLEX and Gurobi solvers, based on the proposed MILP formulation, are compared to the results obtained by the same solvers on the already known formulations. The results show that CPLEX and Gurobi can optimally solve all small and medium sized instances and even some large-scale instances using the new formulation.

1997

We present new approximation algorithms for several facility location problems. In each facility location problem that we study, there is a set of locations at which w e m a y build a facility such a s a w arehouse, where the cost of building at location i is fi; furthermore, there is a set of client locations such a s stores that require to be serviced by a facility, and if a client at location j is assigned to a facility at location i, a cost of cij is incurred. The objective i s t o determine a set of locations at which to open facilities so as to minimize the total facility and assignment costs. In the uncapacitated case, each facility can service an unlimited number of clients, whereas in the capacitated case, each facility can serve, for example, at most u clients. These models and a number of closely related ones have been studied extensively in the Operations Research literature. We shall consider the case in which the assignment costs are symmetric and satisfy the triangle inequality. For the uncapacitated facility location, we give a polynomial-time algorithm that nds a solution within a factor of 3.16 of the optimal. This is the rst constant performance guarantee known for this problem. We also present approximation algorithms with constant performance guarantees for a number of capacitated models as well as a generalization in which there is a 2-level hierarchy of facilities. Our results are based on the ltering and rounding technique of Lin & Vitter. We also give a randomized variant of this technique that can then be derandomized to yield improved performance guarantees.

International Journal of Computational Geometry & Applications, 2000

We present efficient algorithms for two problems of facility location. In both problems we want to determine the location of a single facility with respect to n given sites. In the first we seek a location that maximizes a weighted distance function between the facility and the sites, and in the second we find a location that minimizes the sum (or sum of the squares) of the distances of k of the sites from the facility.

Information Processing Letters, 1999

In the k-level uncapacitated facility location problem, we have a set of demand points where clients are located. The demand of each client is known. Facilities have to be located at given sites in order to service the clients, and each client is to be serviced by a sequence of k different facilities, each of which belongs to a distinct level. There are no capacity restrictions on the facilities. There is a positive fixed cost of setting up a facility, and a per unit cost of shipping goods between each pair of locations. We assume that these distances are all nonnegative and satisfy the triangle inequality. The problem is to find an assignment of each client to a sequence of k facilities, one at each level, so that the demand of each client is satisfied, for which the sum of the setup costs and the service costs is minimized.

SIAM Journal on Computing, 2003

We consider the uncapacitated facility location problem. In this problem, there is a set of locations at which facilities can be built; a fixed cost f i is incurred if a facility is opened at location i. Furthermore, there is a set of demand locations to be serviced by the opened facilities; if the demand location j is assigned to a facility at location i, then there is an associated service cost proportional to the distance between i and j, c ij . The objective is to determine which facilities to open and an assignment of demand points to the opened facilities, so as to minimize the total cost. We assume that the distance function c is symmetric and satisfies the triangle inequality. For this problem we obtain a (1 + 2/e)-approximation algorithm, where 1 + 2/e ≈ 1.736, which is a significant improvement on the previously known approximation guarantees.

Brazilian Journal of Operations & Production Management, 2017

The objective in terms of the facility location problem with limited distances is to minimize the sum of distance functions from the facility to its clients, but with a limit on each of these distances, from which the corresponding function becomes constant. The problem is applicable in situations where the service provided by the facility is insensitive after given threshold distances. In this paper, we propose a polynomial-time algorithm for the discrete version of the problem with capacity constraints regarding the number of served clients. These constraints are relevant for introducing quality measures in facility location decision processes as well as for justifying the facility creation.

Computers & Operations Research, 2017

We consider the two-level uncapacitated facility location problem with single assignment constraints (TUFLP-S), an extension of the uncapacitated facility location problem. We present six mixed-integer programming models for the TUFLP-S based on reformulation techniques and on the relaxation of the integrality of some of the variables associated with location decisions. We compare the models by carrying out extensive computational experiments on large, hard, artificial instances, as well as on instances derived from an industrial application in freight transportation.

2016

We consider the problem of optimally locating a single facility anywhere in a network to serve both on-network and off-network demands. Off-network demands occur in a Euclidean plane, while on-network demands are restricted to a network embedded in the plane. On-network demand points are serviced using shortest-path distances through links of the network (e.g., on-road travel), whereas demand points located in the plane are serviced using more expensive Euclidean distances. Our base objective minimizes the total weighted distance to all demand points. We develop several extensions to our base model, including: (i) a threshold distance model where if network distance exceeds a given threshold, then service is always provided using Euclidean distance, and (ii) a minimax model that minimizes worst-case distance. We solve our formulations using the " Big Segment Small Segment " global optimization method, in conjunction with bounds tailored for each problem class. Computational experiments demonstrate the effectiveness of our solution procedures. Solution times are very fast (often under one second), making our approach a good candidate for embedding within existing heuristics that solve multi-facility problems by solving a sequence of single-facility problems.

arXiv (Cornell University), 2021

We study a general class of quadratic capacitated p-location problems facility location problems with single assignment where a non-separable, non-convex, quadratic term is introduced in the objective function to account for the interaction cost between facilities and customer assignments. This problem has many applications in the field of transportation and logistics where its most well-known special case is the single-allocation hub location problem and its many variants. The non-convex, binary quadratic program is linearized by applying a reformulation-linearization technique and the resulting continuous auxiliary variables are projected out using Benders decomposition. The obtained Benders reformulation is then solved using an exact branch-and-cut algorithm that exploits the underlying network flow structure of the decomposed separation subproblems to efficiently generate strong Pareto-optimal Benders cuts. Additional enhancements such as a matheuristic, a partial enumeration procedure, and variable elimination tests are also embedded in the proposed algorithmic framework. Extensive computational experiments on benchmark instances (with up to 500 nodes) and on a new set of instances (with up to 1,000 nodes) of four variants of single-allocation hub location problems confirm the algorithm's ability to scale to large-scale instances.