Facility location with adjacent units: a simple approximation scheme (original) (raw)

A general model for the undesirable single facility location problem

2006

In this paper, a finite set in which an optimal solution for a general Euclidean problem of locating an undesirable facility in a polygonal region, is determined and can be found in polynomial time. The general problem we propose leads us, among others, to several well-known problems such as the maxisum, maximin, anticentdian or r-anticentrum problem.

Optimal Facility Location Under Various Distance Functions

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.

A Comparison of Exact and Heuristic Methods for a Facility Location Problem

International journal of simulation: systems, science & technology

We formulate a facility location problem where the demand of any single client must be allocated to a single facility and a prize is obtained by allocating the demand of a client to a certain facility, i.e. a prize-based variant of the Single Source Capacitated Facility Location Problem. For this problem we pursue both an exact approach through Integer Linear Programming and a heuristic approach based on a local search algorithm. We compare both approaches by considering 500+ instances. The heuristic approach allows to obtain a reduction of the computational time by a factor larger than 10 in 92% of instances and 100 in 64% of instances. The time reduction is obtained with a small sacrifice in the value of the objective function that is achieved, smaller than 10% in nearly 70% of cases.

Optimal Capacities in Discrete Facility Location Design Problem

DOAJ (DOAJ: Directory of Open Access Journals), 2013

Network location models comprise one of the main categories of location models. These models have various applications in regional and urban planning as well as in transportation, distribution, and energy management. In a network location problem, nodes represent demand points and candidate locations to locate the facilities. If the links network is unchangeably determined, the problem will be an FLP (Facility Location Problem). However, if links can be added to the network at a reasonable cost, the problem will then be a combination of facility location and NDP (Network Design Problem). In previous studies, capacity of facilities was considered to be a constraint while capacity of links was not considered at all. The proposed MIP model considers capacity of facilities and links as decision variables. This approach increases the utilization of facilities and links, and prevents the construction of links and facility locations with low utilization. Furthermore, facility location cost (link construction cost) in the proposed model is supposed to be a function of the associated facility (link) capacity. Computational experiments as well as sensitivity analyses performed indicate the efficiency of the model.

Improved Approximation Algorithms for the Uncapacitated Facility Location Problem

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.

An 0.828-approximation algorithm for the uncapacitated facility location problem

Discrete Applied Mathematics, 1999

The uncapacitated facility location problem in the following formulation is considered: max S⊆I Z(S) = j∈J max i∈S b ij − i∈S c i , where I and J are finite sets, and b ij , c i ≥ 0 are rational numbers. Let Z * denote the optimal value of the problem and Z R = j∈J min i∈I b ij − i∈I c i. Cornuejols, Fisher and Nemhauser (1977) prove that for the problem with the additional cardinality constraint |S| ≤ K, a simple greedy algorithm finds a feasible solution S such that (Z(S)−Z R)/(Z * −Z R) ≥ 1−e −1 ≈ 0.632. We suggest a polynomial-time approximation algorithm for the unconstrained version of the problem, based on the idea of randomized rounding due to Goemans and Williamson (1994). It is proved that the algorithm delivers a solution S such that (Z(S) − Z R)/(Z * − Z R) ≥ 2(√ 2 − 1) ≈ 0.828. We also show that there exists ε > 0 such that it is N P-hard to find an approximate solution S with (Z(S) − Z R)/(Z * − Z R) ≥ 1 − ε.

Approximate algorithms for the competitive facility location problem

Journal of Applied and Industrial Mathematics, 2011

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 a capacitated facility location problem

Proceedings of the Tenth Annual Acm Siam Symposium on Discrete Algorithms, 1999

In a recent surprising result, Korupolu, Plaxton, and Rajaraman [10,11] 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 α-approximation algorithm for this variant into one 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 3-approximation algorithm of Chudak and Shmoys [5] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than five times the value of the standard CFLP.

Heuristics for a continuous multi-facility location problem with demand regions

Computers & Operations Research, 2015

We consider a continuous multi-facility location problem where the demanding entities are regions in the plane instead of points. Each region may consist of a finite or an infinite number of points. The service point of a station can be anywhere in the region that is assigned to it. We do not allow fractional assignments, that is, each region is assigned to exactly one facility. The problem we consider can be stated as follows: given m demand regions in the plane, find the locations of q facilities and allocate regions to the facilities so as to minimize the sum of squares of the maximum Euclidean distances of the demand regions to the facility locations they are assigned to. We assume that the regions are closed polygons as any region can be approximated within any desired accuracy with a polygon. We first propose mathematical programming formulations of single and multiple facility location problems. The single facility location problem is formulated as a second order cone program (SOCP) which can be solved in polynomial time. The multiple facility location problem is formulated as a mixed integer SOCP. This formulation is weak and does not even solve medium-size problems. We therefore propose heuristics to solve larger instances of the problem. We develop three heuristics that work when the regions are polygons. When the demand regions are rectangles with sides parallel to coordinate axes, a special heuristic is developed. We compare our heuristics in terms of both solution quality and computational time.

A new approximation algorithm for the multilevel facility location problem

Discrete Applied Mathematics, 2010

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP rounding. The linear program we are using has a polynomial number of variables and constraints, being thus more efficient than the one commonly used in the approximation algorithms for this type of problems.