Multi-facility Weber Problem Research Articles

Customized Alternating Direction Methods of Multipliers for Generalized Multi-facility Weber Problem

Customized Alternating Direction Methods of Multipliers for Generalized Multi-facility Weber Problem

Fuzzy Clustering With Derivative–Free Search Algorithm for Location of Biogas Energy Systems

In the past decades, new models and algorithms have been developed for solving various types of facility location problems using different versions of the fuzzy c-means algorithm and its hybrid combinations. On the other hand, the need for renewable energy sources has become even more important than ever due to the increasing population, environmental pollution, and climate change. In this study, we proposed two revised weighted fuzzy c-means clustering-based hybrid algorithms for solving real-life biogas facility location problem. The first algorithm is Nelder-Mead simplex algorithm, and the second is center-of-gravity approach. Problem is solved as multi-facility Weber problem, and results have been analyzed.

Hybrid cell selection-based heuristic for capacitated multi-facility weber problem with continuous fixed costs

Location-allocation problem (LAP) has attracted much attention in facility location field. The LAP in continuous plane is well-known as Weber problem. This paper assessed this problem by considering capacity constraints and fixed costs as each facility has different setup cost and capacity limit to serve customers. Previous studies considered profitable areas by dividing continuous space into a discrete number of equal cells to identify optimal locations from a smaller set of promising locations. Unfortunately, it may lead to avoid choosing good locations because unprofitable areas are still considered while locating the facilities. Hence, this allows a significant increment in the transportation costs. Thus, this paper intelligently selected profitable area through a hybridization of enhanced Cell Selection-based Heuristic (CSBH) and Furthest Distance Rule (FDR) to minimize total transportation and fixed costs. The CSBH divides customer distribution into smaller set of promising locations and intelligently selected profitable area to increase possibility of finding better locations, while FDR aims to forbid the new locations of the facilities to be close to the previously selected locations. Numerical experiments tested on well-known benchmark datasets showed that the results of hybrid heuristic outperformed single CSBH and FDR, while producing competitive results when compared with previously published results, apart from significantly improving total transportation cost. The new hybrid heuristic is simple yet effective in solving LAP.

ADMM-type methods for generalized multi-facility Weber problem

<p style='text-indent:20px;'>The well-known multi-facility Weber problem (MFWP) is one of fundamental models in facility location. With the aim of enhancing the practical applicability of MFWP, this paper considers a generalized multi-facility Weber problem (GMFWP), where the gauge is used to measure distances and the locational constraints are imposed to new facilities. This paper focuses on developing efficient numerical methods based on alternating direction method of multipliers (ADMM) to solve GMFWP. Specifically, GMFWP is equivalently reformulated into a minmax problem with special structure and then some ADMM-type methods are proposed for its primal problem. Global convergence of proposed methods for GMFWP is established under mild assumptions. Preliminary numerical results are reported to verify the effectiveness of proposed methods.</p>

The continuous single-source capacitated multi-facility Weber problem with setup costs: formulation and solution methods

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 and Simulated Annealing 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.

Multi-facility green Weber problem

Locating facilities to satisfy the demands of customers is a strategic decision for a distribution system. In this article, we study the multi-facility green Weber problem (MF-GWP), an extension of the classical multi-facility Weber problem, that considers environmental concerns in a distribution system in the context of a planar facility location problem. In the MF-GWP, the vehicles are sent directly from the facilities to the assigned customers to satisfy their demands. Each customer has a deadline and the vehicles serving the customer must arrive at the location of the customer no later than the deadline. The MF-GWP determines the locations of p facilities on the plane, p > 1, allocations of customers to the facilities, and the speeds of the distribution vehicles so as to minimize the total amount of CO2 emission in the distribution system. We formulate this problem as a mixed integer second order cone programming (MISOCP) problem. This formulation turns out to be weak and therefore only small size instances can be solved to optimality within four hours. For larger size instances, a local search heuristic is proposed and some well-known heuristics developed for the multi-facility Weber problem, namely “location-allocation”, “transfer follow-up”, and “decomposition” are adapted for the MF-GWP. We use second order cone programming (SOCP) and the proposed MISOCP formulation as subproblems within the heuristics. We provide our computational experiments to compare the proposed solution methods in terms of solution quality and time. The results show that within a fixed computational time, even though the location-allocation heuristic is able to make more replications, the improvement heuristics considered, i.e., transfer or transfer followed by decomposition, usually find better solutions while using less number of replications. We also investigate how the total amount of CO2 emitted by distribution vehicles changes with respect to the number of facilities located. We argue that in several real life applications from different sectors including aviation and robotics, MF-GWP and its extensions or modifications can be used to reduce the CO2 emission or energy consumption. As an illustrative example, we show the applicability of the MF-GWP within an assembly line system, where the stations are fed by dedicated rail-guided vehicles.

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.

A variational inequality approach for constrained multifacility Weber problem under gauge

The classical multifacility Weber problem (MFWP) is one of the most important models in facility location. This paper considers more general and practical case of MFWP called constrained multifacility Weber problem (CMFWP), in which the gauge is used to measure distances and locational constraints are imposed to facilities. In particular, we develop a variational inequality approach for solving it. The CMFWP is reformulated into a linear variational inequality, whose special structures lead to new projection-type methods. Global convergence of the projection-type methods is proved under mild assumptions. Some preliminary numerical results are reported which verify the effectiveness of proposed methods.

The continuous single source location problem with capacity and zone-dependent fixed cost: Models and solution approaches

The continuous capacitated single-source multi-facility Weber problem with the presence of facility fixed cost is investigated. A new mathematical model which incorporates multi-level type capacity (or design) and facility fixed cost that is capacity-based and zone-dependent is introduced. As no data set exists for this new location problem, a new data set based on convex polygons using triangular shape is constructed. A generalised two stage heuristic scheme that combines the concept of aggregation, an exact method, and an enhanced Cooper's alternate location–allocation method is put forward. A framework that embeds Variable Neighbourhood Search is also proposed. Computational experiments show that these matheuristics produce encouraging results for this class of location problems. The proposed approaches are also easily adapted to cater for a recently studied variant namely the single-source capacitated multi-facility Weber problem where they outperform those recently published solution methods.

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.

Approximate solution methods for the capacitated multi-facility Weber problem

This work considers the capacitated multi-facility Weber problem, which is concerned with locating m facilities and allocating their limited capacities to n customers in order to satisfy their demand at minimum total transportation cost. This is a non-convex optimization problem and difficult to solve. Therefore, approximate solution methods are proposed in this article. Some of them are based on the relaxation of the capacity constraints and apply the subgradient algorithm. The resulting Lagrangian subproblem is a variant of the well-known multi-facility Weber problem and can be solved using column generation and branch-and-price approach on a variant of the set covering formulation. Others are based on the approximating mixed-integer linear programming formulations obtained by exploiting norm properties and the alternate solution of the discrete location and transportation problems. The results of a detailed computational analysis are also reported.

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.

Random Search Algorithm for the Generalized Weber Problem

In this paper, we consider the planar multi-facility Weber problem with restricted zones and non-Euclidean distances, propose an algorithm based on the probability changing method (special kind of genetic algorithms) and prove its efficiency for approximate solving this problem by replacing the continuous coordinate values by discrete ones. Version of the algorithm for multiprocessor systems is proposed. Experimental results for a high-performance cluster are given.