Capacitated Multi-facility Research Articles

Global optimization algorithm for multi-period design and planning of centralized and distributed manufacturing networks

This paper addresses the design and planning of manufacturing networks considering the option of centralized and distributed facilities, taking into account the potential trade-offs between investments and transportation. The problem is formulated as an extension of the Capacitated Multi-facility Weber Problem, which involves the selection of which facilities to build in each time period, and their location in the continuous two-dimensional space, in order to meet demand and minimize costs. The model is a multi-period GDP, reformulated as a nonconvex MINLP. We propose an accelerated version of the Bilevel Decomposition by Lara et al. (2018) that finds stronger bounds in the decomposition scheme. We benchmark the performance of our algorithm against the original Bilevel Decomposition and commercial global solvers and show that our approach outperforms the others in all instances tested. Additionally, we illustrate the applicability of the proposed model and solution framework with a biomass supply chain case study.

Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem

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.

Global optimization algorithm for capacitated multi-facility continuous location-allocation problems

In this paper we propose a nonlinear Generalized Disjunctive Programming model to optimize the 2-dimensional continuous location and allocation of the potential facilities based on their maximum capacity and the given coordinates of the suppliers and customers. The model belongs to the class of Capacitated Multi-facility Weber Problem. We propose a bilevel decomposition algorithm that iteratively solves a discretized MILP version of the model, and its nonconvex NLP for a fixed selection of discrete variables. Based on the bounding properties of the subproblems, $$\epsilon $$ -convergence is proved for this algorithm. We apply the proposed method to random instances varying from 2 suppliers and 2 customers to 40 suppliers and 40 customers, from one type of facility to 3 different types, and from 2 to 32 potential facilities. The results show that the algorithm is more effective at finding global optimal solutions than general purpose global optimization solvers tested.

Discretization Based Heuristics for the Capacitated Multi-facility Weber Problem with Convex Polyhedral Barriers

The Capacitated Multi-facility Weber Problem addresses optimally locating I capacitated facilities in the plane and satisfying demand of J customers so as to minimize the total transportation cost. It assumes that facilities can be located anywhere on the plane and customers are directly connected to them. This study considers the case where there exist convex polyhedral barriers blocking passage and locating facilities inside. Then, the distances between facilities and customers have to be measured by taking into account the polyhedral barriers. The resulting problem is non-convex and difficult to solve. We propose several discretization based heuristic procedures which are especially designed for the problem. The performance of the suggested methods are tested on an extensive set of randomly generated test instances which are derived from standard test instances. Our results imply that the suggested heuristics yield very accurate and efficient solutions for this problem.

The capacitated multi-facility weber problem with polyhedral barriers: Efficient heuristic methods

The Capacitated Multi-facility Weber Problem investigates the optimal locations of I capacitated facilities in the plane to satisfy the demand of J customers so that the total transportation cost is minimized. Facilities can be placed without any restrictions and customers are directly served without interruptions. In this work, we focus on the case with polyhedral barriers in which neither locating a facility nor travelling is permitted. Then, the transportation costs are dependent not only on the direct distances between facilities and customers but also on the location and size of the polyhedral barriers in presence. This results in a non-convex optimization problem which is difficult to solve. We offer two alternate location-allocation heuristics and two discrete approximation heuristics that are specially tailored for this problem. An extensive set of computational experiments are performed on the randomly generated test instances. Our results indicate that suggested heuristic methods are very efficient and yield promising results for this difficult problem.

Beam search heuristics for the single and multi-commodity capacitated Multi-facility Weber Problems

The Multi-facility Weber Problem (MWP) is concerned with locating I uncapacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. It is a non-convex optimization problem and difficult to solve. In this work, we focus on the capacitated extensions of the MWP which are Capacitated MWP (CMWP) and multi-commodity CMWP (MCMWP). Both the CMWP and MCMWP impose capacity restrictions on facilities. Indeed, the MCMWP is a natural extension of the CMWP and considers the situation where K distinct commodities are shipped subject to limitations on the total amount of goods sent from facilities to the customers. Customer locations, demands and capacities for each commodity are known a priori. The transportation costs, which depend on the commodity type, are proportional to the distance between customers and facilities. We first introduce branch and bound algorithms for both the CMWP and the MCMWP then we propose beam search heuristics for these problems. According to our computational experiments on standard and randomly generated test instances, we can say that the new heuristics perform very well.

New heuristic methods for the single-source capacitated multi facility Weber problem

In this paper, a generalized variant of the multi facility Weber problem is proposed, where each customer must be satisfied by a single capacitated facility. To solve the problem, the allocation phase of the alternate location–allocation (ALA) is strengthened by a priority-based approach. Computational experiments are implemented on a set of random test instances, along with a set of instances taken from the literature. Computational results compared with GAMS/BARON indicate that the priority-based ALA solves large-scale instances within promising computational time and outperforms GAMS/BARON in terms of solution quality. This success rate is achieved in nearly 80 % of studied cases. Finally, some guidelines for future research are presented.

Heuristics for the single source capacitated multi-facility Weber problem

The Single Source Capacitated Multi-facility Weber Problem (SSCMWP) 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 such that each customer satisfies all its demand from exactly one facility. The SSCMWP is a non-convex optimization problem and difficult to solve. In the SSCMWP, customer locations, customer demands and facility capacities are known a priori. The transportation costs are proportional to the distance between customers and facilities. We consider both the Euclidean and rectilinear distance cases of the SSCMWP. We first present an Alternate Location and Allocation type heuristic and its extension by embedding a Very Large Scale Neighborhood search procedure. Then we apply a Discrete Approximation approach and propose both lower and upper bounding procedures for the SSCWMP using a Lagrangean Relaxation scheme. The proposed heuristics are compared with the solution approaches from the literature. According to extensive computational experiments on standard and randomly generated test sets, we can say that they yield promising performance.

Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem

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 lr 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.

Solving the multi-commodity capacitated multi-facility Weber problem using Lagrangean relaxation and a subgradient-like algorithm

The Multi-commodity Capacitated Multi-facility Weber Problem (MCMWP) is concerned with locating I-capacitated facilities in the plane in order to satisfy the demands of J customers for K commodities so that the total transportation cost is minimized. We propose a Lagrangean relaxation scheme and a subgradient-like algorithm based on the relaxation of the capacity and commodity bundle constraints. The resulting subproblem is a variant of the well-known Multi-facility Weber Problem and it can be solved by using column generation and branch-and-price on an equivalent set covering formulation, which is accurate but extremely inefficient. Therefore, we devise different strategies to increase the efficiency. They mainly benefit from the effective usage of the lower and upper bounds on the optimal value of the Lagrangean subproblem. On the basis of extensive computational tests, we can say that they increase the efficiency considerably and result in accurate Lagrangean heuristics.

Single-Source Capacitated Multi-Facility Weber Problem—An iterative two phase heuristic algorithm

Multi-Facility Weber Problem (MFWP), also known as continuous location–allocation problem, entails determining the locations of a predefined number of facilities in a planar space and their related customer allocations. In this paper, we focus on a new variant of the problem known as Single-Source Capacitated MFWP (SSCMFWP). To tackle the problem efficiently and effectively, an iterative two-phase heuristic algorithm is put forward. At the phase I, we aim to determine proper locations for facilities, and during the phase II, assignment of customers to these facilities is pursued. As an alternative solution method, a simulated annealing (SA) algorithm is also proposed for carrying out the phase I. The proposed algorithms are validated on a comprehensive set of test instances taken from the literature. The proposed iterative two-phase algorithm produces superior results when assessed against the proposed SA algorithm as well as a general MINLP Solver known as BARON. The latter is applied to produce optimal solutions for small sized instances and generate upper bound for medium ones.

The multi-commodity capacitated multi-facility Weber problem: heuristics and confidence intervals

The Capacitated Multi-facility Weber Problem (CMWP) is concerned with locating I capacitated facilities so as 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 is difficult to solve. This work focuses on a multi-commodity extension and considers 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. A mathematical programming formulation of the problem is presented and two alternate location-allocation heuristics and a discrete approximation method are proposed and subsequently used to statistically estimate confidence intervals on the optimal objective function values. Computational experiments on standard and randomly generated test instances are also presented.

Comparison of Exact Algorithms for Rectilinear Distance Single-Source Capacitated Multi-Facility Weber Problems

Problem statement: The objective of this study is to develop efficient exact algorithms for a single source capacitated multi-facility location problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the two dimensional plane to satisfy the demand of n customers with minimum total transportation cost which is proportional to the rectilinear distance between the facilities and their customers. Approach: Two exact algorithms are proposed and compared. The first algorithm, decomposition algorithm, uses explicit branching on the allocation variables and then solve for location variable corresponding to each branch as the original Mixed Integer Programming (MIP) formulation with nonlinear objective function of the problem. For the other algorithm, the new formulation of the problem is first created by making use of a well-known condition for the optimal facility locations. The problem is considered as a p-median problem and the original formulation is transformed to a binary integer programming problem. The classical exact algorithm based on this formulation which is branch-and-bound algorithm (implicit branching) is then used. Results: Computational results show that decomposition algorithm can provide the optimum solution for larger size of the studied problem with much less processing time than the implicit branching on the discrete reformulated problem. Conclusion: The decomposition algorithm has a higher efficiency to deal with the studied NP-hard problems but is required to have efficient MIP software to support.

An Efficient Algorithm for Capacitated Multifacility Location Problems

In this paper, a squared-Euclidean distance multifacility location problem with inseparable demands under balanced transportation constraints is analyzed. Using calculus to project the problem onto the space of allocation variables, the problem becomes minimizing concave quadratic integer programming problem. The algorithm based on extreme point ranking method combining with logical techniques is developed. The numerical experiments are randomly generated to test efficiency of the proposed algorithm compared with a linearization algorithm. The results show that the proposed algorithm provides a better solution on average with less processing time for all various sizes of problems.

Heuristic Methods for Location-Allocation Problems

Heuristic Methods for Location-Allocation Problems