Solving the Capacitated Multifacility Weber Problem approximately (original) (raw)
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The multi-commodity capacitated multi-facility Weber problem: heuristics and confidence intervals
IIE Transactions, 2010
The Capacitated Multi-facility Weber Problem is concerned with locating I capacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a non-convex optimization problem and difficult to solve. In this work, we focus on a multi-commodity extension and consider the situation where K distinct commodities are shipped to the customers subject to capacity and demand constraints. Customer locations, demands and capacities for each commodity are known a priori. The transportation costs, which are proportional to the distance between customers and facilities, depend on the commodity type. We first present a mathematical programming formulation of the problem. Then we propose an alternate location-allocation heuristic and a discrete approximation method which are used to statistically estimate confidence intervals on the optimal objective values. Computational experiments on randomly generated test instances are also included.
European Journal of Operational Research, 2009
The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location-allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.
Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem
Annals of Operations Research, 2018
The Multi-commodity Capacitated Multi-facility Weber Problem is concerned with locating I capacitated facilities in the plane in order to satisfy the demands of J customers for K commodities such that the total transportation cost is minimized. This is a multi-commodity extension of the well-known Capacitated Multi-facility Weber Problem and difficult to solve. In this work, we propose two branch-and-bound algorithms for exactly solving this nonconvex optimization problem. One of them considers partitioning of the allocation space while the other one considers partitioning of the location space. We have implemented two lower bounding schemes for both algorithms and tested several branching strategies. The results of an extensive computational study are also included.
Satisfying partial demand in facilities location
IIE Transactions, 2002
In this paper we consider the location of new facilities which serve only a certain proportion of the demand. The total weighted distances of the served demand is minimized. We consider the problem in the plane for the location of one facility and on a network for the location of m-facilities. Some computational experience with these models are reported. 0740-817X Ó 2002 ''IIE''
SOLVING CONSTRAINED TWO-FACILITY LOCATION PROBLEMS(ISOLDE XII)
Journal of the Operations Research Society of Japan
A general approach to optimally solve multiple facility location problems based on the t`Big [friangle Small [briangle" approach te solving single facility problems is proposed, The proposed procedure is especially effective when the solution is constrained to a given polygon such as the convex hull of demand points. The procedure is tested on the two facilities Weber problem with attraetien and repulsion (WAR) with excellent eomputational results.
Approximation algorithms for Capacitated Facility Location Problem with Penalties
In this paper, we address the problem of capacitated facility location problem with penalties (CapFLPP) paid per unit of unserved demand. In case of uncapacitated FLP with penalties demands of a client are either entirely met or are entirely rejected and penalty is paid. In the uncapacitated case, there is no reason to serve a client partially. Whereas, in case of CapFLPP, it may be beneficial to serve a client partially instead of not serving at all and, pay the penalty for the unmet demand. Charikar et. al.~\cite{charikar2001algorithms}, Jain et. al.~\cite{jain2003greedy} and Xu- Xu~\cite{xu2009improved} gave 333, 222 and 1.85261.85261.8526 approximation, respectively, for the uncapacitated case . We present (5.83+epsilon)(5.83 + \epsilon)(5.83+epsilon) factor for the case of uniform capacities and (8.532+epsilon)(8.532 + \epsilon)(8.532+epsilon) factor for non-uniform capacities.
Annals of Operations Research, 2012
Given the locations of J customers, their demands and I capacitated facilities, the Capacitated Multi-facility Weber Problem (CMWP) is concerned with locating I facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost which is proportional to the distance between them. We propose two types of branch and bound algorithms for the r distance CMWP with 1 ≤ r < ∞. One of them is an allocation space based branch and bound algorithm for which a new branching variable selection strategy and new lower bounding procedures have been proposed. The other one is new and partitions the location space. Based on extensive computational experiments we can say that the proposed algorithms are promising and perform better than the existing ones.
A Lagrangean Approach to the Facility Location Problem with Concave Costs
Journal of the Operations Research Society of Japan
We consider'the concave cost capacitated facility location pTo])lem, and develop a, composite algorithm Qf lower and upper bounding procedures. Computational results for several instarices with up to 100 customers and 25 candidate facility iocations are also presented, Our numerical experiments show that the proposed algorithm generates good selutions, The gaps between upper and lower bounds are within 1 percent fbr al1 test problems.
On the Two-Level Uncapacitated Facility Location Problem
INFORMS Journal on Computing, 1996
We study the two-level uncapacitated facility location TUFL problem. Given two types of facilities, which w e call y-facilities and z-facilities, the problem is to decide which facilities of both types to open, and to which pair of y-and z-facilities each client should be assigned, in order to satisfy the demand at maximum pro t.
Journal of Global Optimization
The continuous single-source capacitated multi-facility Weber problem (SSCMFWP) where setup cost of opening facilities is taken into account is investigated. The aim is to locate a set of facilities on the plane, to define their respective capacities which can be linked to the configuration of the processing machines used, and to allocate customers to exactly one facility with the objective being the minimisation of the total transportation and setup costs. A new nonlinear mathematical model based on the SSCMFWP is introduced where Rectilinear and Euclidean distances are used. Efficient metaheuristic approaches based on Variable Neighbourhood Search (VNS) and Simulated Annealing (SA) are also developed. The proposed metaheuristics incorporate an exact method and the commonly used Cooper's alternate location-allocation method. We also constructed a new data set to reflect the characteristic of this particular location problem as no data set is available in the literature. Computational experiments show that the proposed metaheuristics generate interesting results for this class of continuous location problems.