Capacitated Facility Research Articles

LP-based approximation algorithms for capacitated facility location

In the capacitated facility location problem with hard capacities, we are given a set of facilities, \({\mathcal{F}}\), and a set of clients \({\mathcal{D}}\) in a common metric space. Each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set \({\mathcal{F}}\) and assign each client to an open facility so that at most u i clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance c ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.

Modelling and case study of reverse logistics for construction aggregates

The paper proposes a stochastic reverse logistics model for product recovery of construction aggregates. The problem undertaken is to find an optimal network for aggregate recycling under uncertain demand and supply conditions. The proposed model is designed as multi-commodity and capacitated facility location problems. It is formulated as Mixed Integer Linear Programming (MILP) with the objective of minimisation of total cost. The model has been tested for real-life data for National Capital Region (NCR) Delhi, India. It was found that transportation cost was the single largest cost component in the reverse logistics network for aggregate recycling.

A TWO‐PHASE METHOD FOR THE CAPACITATED FACILITY PROBLEM OF COMPACT CUSTOMER SUB‐SETS

The cost optimal design of the majority of distribution and servicing systems consists of decisions on a number and on the locations of facilities from which customers’ demands are satisfied; however, there are severe difficulties in solving exact procedures because the underlying mathematical model is NP‐hard. These decisions should respect some additional conditions as a limited capacity of located facilities. The objective is to minimize the overall costs of the system and to satisfy all customers’ demands. In this paper, we enrich the set of constraints by a new requirement called sub‐pool compactness. This property of customer subset influences the quality of vehicle routes subsequently formed in a sub‐set of customers served by the same facility. This paper formulates the problem of the enriched capacitated facility location considering compactness condition, formalizes and studies the property of compactness and suggests the compound method solving this problem.

Multi‐item capacitated lot‐sizing problems with setup times and pricing decisions

AbstractWe study a multi‐item capacitated lot‐sizing problem with setup times and pricing (CLSTP) over a finite and discrete planning horizon. In this class of problems, the demand for each independent item in each time period is affected by pricing decisions. The corresponding demands are then satisfied through production in a single capacitated facility or from inventory, and the goal is to set prices and determine a production plan that maximizes total profit. In contrast with many traditional lot‐sizing problems with fixed demands, we cannot, without loss of generality, restrict ourselves to instances without initial inventories, which greatly complicates the analysis of the CLSTP. We develop two alternative Dantzig–Wolfe decomposition formulations of the problem, and propose to solve their relaxations using column generation and the overall problem using branch‐and‐price. The associated pricing problem is studied under both dynamic and static pricing strategies. Through a computational study, we analyze both the efficacy of our algorithms and the benefits of allowing item prices to vary over time. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010

A variable neighborhood search heuristic for the design of multicommodity production–distribution networks with alternative facility configurations

This paper proposes efficient heuristics to solve large-scale production---distribution network design models. The problem studied is an extension of the two-echelon multicommodity CFLPSS (capacitated facility location problem with single sourcing) considering direct shipments from manufacturing facilities, alternative facility configurations, and concave inventory holding costs. After reviewing the relevant literature, a detailed description of the production---distribution network design problem studied is provided, and the problem is formulated as a mixed-integer program. The heuristic solution approach proposed is then presented. It is a variable neighborhood search (VNS) method integrating a tabu procedure. Experiments are designed to calibrate the heuristics developed, and to compare their performance with the CPLEX solver, for problems with different realistic characteristics. Computational results are presented and discussed.

Location and design decisions of facilities in a distribution system with elastic customer demand

This paper describes a bi-level programming model that seeks to simultaneously optimize location and design decisions of facilities in a distribution system in order to realize company’s maximal total profit subject to the constraints on the facility capacity and the investment budget. In the upper-level problem, two-echelon integrated competitive/uncompetitive capacitated facility location model, which involves facility location and design, is presented. In the lower-level problem, customer is assumed to patronize store based on facility utility which is expressed by service time cost in the store and its travel cost to customer. Customer’s facility choice behavior is presented by a stochastic user equilibrium assignment model with elastic demand. Since such a distribution system design problem belongs to a class of NP-hard problem, a genetic algorithm (GA)-based heuristic procedure is presented. Finally, a numerical example is used to illustrate the application of the proposed model and some parameter sensitivity analyses are presented.

A cross entropy-based metaheuristic algorithm for large-scale capacitated facility location problems

In this paper, we present a metaheuristic-based algorithm for the capacitated facility location problem. The proposed scheme is made up by three phases: (i) solution construction phase, in which a cross entropy-based scheme is used to ‘intelligently’ guess which facilities should be opened; (ii) local search phase, aimed at exploring the neighbourhood of ‘elite’ solutions of the previous phase; and (iii) learning phase, aimed at fine-tuning the stochastic parameters of the algorithm. The algorithm has been thoroughly tested on large-scale random generated instances as well as on benchmark problems and computational results show the effectiveness and robustness of the algorithm.

The ordered capacitated facility location problem

In this paper, we analyze flexible models for capacitated discrete location problems with setup costs. We introduce a major extension with regards to standard models which consists of distinguishing three different points of view of a location problem in a logistics system. We develop mathematical programming formulations for these models using discrete ordered objective functions with some new features. We report on the computational behavior of these formulations tested on a randomly generated battery of instances.

An approximation algorithm for the k-level capacitated facility location problem

We consider the k-level capacitated facility location problem (k-CFLP), which is a natural variant of the classical facility location problem and has applications in supply chain management. We obtain the first (combinatorial) approximation algorithm with a performance factor of $k+2+\sqrt{k^{2}+2k+5}+\varepsilon$ (?>0) for this problem.

An evolutionary programming approach for solving the capacitated facility location problem with risk pooling

In this paper, we propose a genetic algorithm as an alternative technique for solving the capacitated facility location problem with risk pooling (CLMRP). The CLMRP is a joint location-inventory problem involving a single supplier and multiple retailers that face stochastic demand. Due to the stochasticity of demand associated with each retailer, risk pooling may be achieved by allowing some retailers to serve as distribution centres (DCs). This is a combinatorial optimisation problem that has been shown to be NP-hard. A genetic algorithm that is computationally very efficient is developed to solve the problem. A computational experiment is conducted to test the performance of the developed technique and computational results are reported. The algorithm can easily find optimal or near optimal solutions for benchmark test problems from the literature, where the Lagrangian relaxation approach was used.

Performance of an add-drop-interchange heuristic for the capacitated facility location problem

In this article, we present a heuristic method for solving instances of the capacitated facility location problem. In the method, first, a linear programming relaxation, a transportation problem, is solved to obtain an initial solution. This also gives a lower bound of an optimal solution of the capacitated facility location problem instance. The initial solution is improved in subsequent iterations. In each iteration all possible single-exchange add, drop or interchange of the facilities are considered. Without solving the resultant transportation problems exactly, an approximate method is used for the purpose. Some asymptotic conditions when the method would give an optimal solution are given. Average performance of the method, with respect to the quality of the solutions, has been investigated with benchmark and random instances of the problem. An optimal solution is obtained for most of the benchmark instances. For random instances also, performance of the method is satisfactory.

Stochastic single-source capacitated facility location model with service level requirements

A stochastic version of single-source capacitated facility location problem is considered. A set of capacitated facilities is to be selected to provide service to demand points with stochastic demand at the minimal total cost. The facilities have service level requirements modeled by chance constraints. For Poisson demand, the problem is proved equivalent to a known solvable deterministic problem. For Normally distributed demand, it is equivalent to a deterministic mixed integer non-linear programming problem. A hybrid heuristic of Lagrangean relaxation with a single-customer-multi-exchange heuristic is embedded within a branch-and-bound framework to find upper and lower bounds for this problem. From test instances created from benchmark problems (10–20 facilities and 50 demand nodes) and real-life data on the deterministic problem, the gap between the bounds is within 6.5% with an average of 2.5%.

A Clustering‐Based Approach to the Capacitated Facility Location Problem1

AbstractThis research develops a clustering‐based location‐allocation method to the Capacitated Facility Location Problem (CFLP), which provides an approximate optimal solution to determine the location and coverage of a set of facilities to serve the demands of a large number of locations. The allocation is constrained by facility capacities – different facilities may have different capacities and the overall capacity may be inadequate to satisfy the total demands. This research transforms this special location‐allocation problem into a clustering model. The proposed approach has two parts: (1) the allocation of demands to facilities considering capacity constraints while minimizing the cost; and (2) the iterative optimization of facility locations using an adapted K‐means clustering method. The quality of a location‐allocation solution is measured using an objective function, which is the demand‐weighted distance from demand locations to their assigned facilities. The clustering‐based method is evaluated against an adapted Genetic Algorithm (GA) alternative, which integrates the allocation component as described above but uses GA operations to search for ‘optimal’ facility locations. Experiments and evaluations are carried out with various data sets (including both synthetic and real data).

A branch and bound algorithm for the partial coverage capacitated plant location problem

In this paper, a branch and bound algorithm for the partial coverage capacitated facility locations problem is developed. We have to fix open a set of warehouses which are economically feasible. It is an extension of the algorithm given by B. M. Khumawala [10] and Sudha Arora and S.R. Arora [2]. The case when the demand of the customer may not be satisfied completely, giving rise to “opportunity demand” is also discussed. This is illustrated with the help of examples and their computational results.

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of "promising" variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.

Investigation of practical, robust and flexible decisions for facility location problems using tabu search and simulation

We investigate how robust and flexible solutions of stochastic capacitated facility location problems (CFLPs) can be obtained by combining metaheuristic optimization with Monte Carlo sampling techniques. To this end, we develop a tabu search procedure for the CFLP, and use this to solve an extensive set of stochastic versions of this problem.

Toward Finding an Optimal P2P Network Structure: A Stochastic Facility Location Allocation Approach

A stochastic model based on the Capacitated Facility Location Allocation problem is developed to find the optimal Peer-to-Peer (P2P) network structure. Given the locations of the peers and their demand distributions in a grid space, the model finds the optimal points in the space to place the main servers (brokers) that facilitate the communication between the peers by offering them indexing and searching services for all the files in the space. Although the place of each peer is fixed, they may not be online all the time. Therefore, the demand distribution for each agent is specified probabilistically and the corresponding model is solved stochastically.

Integrated design of supply chain networks with three echelons, multiple commodities and technology selection

We consider a strategic supply chain design problem with three echelons, multiple commodities and technology selection. We model the problem as a tri-echelon, capacitated facility location problem that decides on the location of plants and warehouses, their capacity and technology planning, the assignment of commodities to plants and the flow of commodities to warehouses and customer zones. We use a mixed-integer programming formulation strengthened by valid but redundant constraints and apply Lagrangean relaxation to decompose the problem by echelon. Lagrangean relaxation provides a lower bound that is calculated using an interior-point cutting plane method. Feasible solutions are generated using a primal heuristic that uses the solution of the subproblems. Unlike common practice in the literature, the decomposition does not aim at getting easy subproblems, but rather at getting subproblems that preserve most of the characteristics of the original problem. Not only does this provide a sharp lower bound but also leads to a simple and efficient primal heuristic. We can afford to have relatively difficult subproblems because the interior-point cutting plane method used to solve the Lagrangean dual makes clever and selective choices of the Lagrangean multipliers leading to fewer calls to the subproblems. Computational results indicate the efficiency of the approach in providing a sharp bound and in generating feasible solutions that are of high quality.

A cutting plane algorithm for the capacitated facility location problem

The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility. In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope. We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable. To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.

Combining Lagrangian heuristic and Ant Colony System to solve the Single Source Capacitated Facility Location Problem

The facility location problems have been applied extensively in practice. We describe a Multiple Ant Colony System (MACS) to solve the Single Source Capacitated Facility Location Problem (SSCFLP). Lagrangian heuristics have been shown to produce good solutions for the SSCFLP. A hybrid algorithm, which combines Lagrangian heuristic and Ant Colony System (ACS), LH–ACS, is developed for the SSCFLP. The performance of the proposed methods are tested on two sets of benchmark instances and compared with other heuristic algorithms in the literature. The computational results indicate that both MACS and LH–ACS are effective and efficient for the SSCFLP and competitive with other well-known algorithms.