Lagrangean heuristics applied to a variety of large capacitated plant location problems

A series of performance improving heuristics are developed and embedded into the procedure of Lagrangean heuristics. The principles and ideas of these heuristics are used to build a general framework with which a variety of large Capacitated Plant Location Problems (CPLP) are solved, including the multi-capacitated case. A significant improvement over the results discussed in the literature is recorded. The single-source multi-Capacitated Plant Location Problem, which has not been addressed in the literature, is also tackled with encouraging results.     

Lagrangian heuristics for the two-echelon,single-source,capacitated facility location problem

Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries. In this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities. Each facility in the second echelon has limited capacity and can be supplied by only one facility (or depot) in the first echelon. Each customer is serviced by only one facility in the second echelon. The number and location of facilities in both echelons together with the allocation of customers to the second-echelon facilities are to be determined simultaneously. We propose a mathematical model for this problem and consider six heuristics based on Lagrangian relaxation for its solution. To solve the dual problem we make use of a subgradient optimization procedure. We present numerical results for a large suite of test problems. These indicate that the lower-bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation. Furthermore, the overall solution time for the heuristics are less than the time to solve the LP relaxation.     

An algorithm for solving large capacitated warehouse location problems

In this paper we present a lower bound for the capacitated warehouse location problem based upon lagrangean relaxation of a mixed-integer formulation of the problem. Feasible solution exclusion constraints are used together with problem reduction tests derived from both the original problem and the lagrangean relaxation.By incorporating the lower bound and the reduction tests into a tree search procedure we are able to solve problems involving up to 500 potential warehouse locations and 1000 customers.     

A cut-and-solve based algorithm for the single-source capacitated facility location problem

In this paper, we present a cut-and-solve (CS) based exact algorithm for the Single Source Capacitated Facility Location Problem (SSCFLP). At each level of CS’s branching tree, it has only two nodes, corresponding to the Sparse Problem (SP) and the Dense Problem (DP), respectively. The SP, whose solution space is relatively small with the values of some variables fixed to zero, is solved to optimality by using a commercial MIP solver and its solution if it exists provides an upper bound to the SSCFLP. Meanwhile, the resolution of the LP of DP provides a lower bound for the SSCFLP. A cutting plane method which combines the lifted cover inequalities and Fenchel cutting planes to separate the 0–1 knapsack polytopes is applied to strengthen the lower bound of SSCFLP and that of DP. These lower bounds are further tightened with a partial integrality strategy. Numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the CS approach in searching an optimal solution in a reasonable time. Computational results on large sized instances are also presented.     

A repeated matching heuristic for the single-source capacitated facility location problem

Facility location models form an important class of integer programming problems, with application in many areas such as the distribution and transportation industries. An important class of solution methods for these problems are so-called Lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust. The general facility location problem can be divided into a number of special problems depending on the properties assumed. In the capacitated location problem each facility has a specific capacity on the service it provides. We describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility. The approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied. The method generates feasible solutions in each iteration in contrast to Lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution. Numerical results show that the approach produces solutions which are of similar and often better than those produced using the best Lagrangean heuristics.     

Efficient solution of large scale Lyapunov and Riccati equations arising in model order reduction problems

Model order reduction of large–scale linear time–invariant systems is an omnipresent task in control and simulation of complex dynamical processes. The solution of large scale Lyapunov and Riccati equations is a major task, e.g., in balanced truncation and related model order reduction methods, in particular when applied to semi–discretized partial differential equations constraint control problems. The software package LyaPack has shown to be a valuable tool in the task of solving these equations since its introduction in 2000. Here we want to discuss recent improvements and extensions of the underlying algorithms and their implementation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heuristics for the capacitated plant location model

The paper generalizes heuristics for the uncapacitated plant location model to the capacitated case. The heuristics are ADD, DROP, SHIFT, ALA (Alternate Location Allocation) and VSM (Vertex Substitution Method). The generalizations take place within an unifying framework based on elementary ADD and DROP operations.Heuristics from the literature are discussed and compared to the procedures developed in this paper.     

Easy capacitated facility location problems,with connections to lot-sizing

In this note, we consider the capacitated facility location problem when the transportation costs of the instance satisfy the Monge property. We show that a straightforward dynamic program finds the optimal solution when the demands are polynomially bounded. When demands are not polynomially bounded, we give a fully polynomial-time approximation scheme by adapting an algorithm and analysis of Van Hoesel and Wagelmans (2001).     

Improved approximation algorithms for capacitated facility location problems

In a surprising result, Korupolu, Plaxton, and Rajaraman [13] 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 -approximation algorithm for this variant into a polynomial-time algorithm 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 2-approximation algorithm of Mahdian, Ye, and Zhang [17] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than four times the value of the standard CFLP.This research was performed while the author was a postdoctoral fellow at the IBM T.J. Watson Research Center.This research was performed while the author was a Research Staff Member at the IBM T.J. Watson Research Center.A preliminary version of this paper appeared in the Proceedings of the 7th Conference on Integer Programming and Combinatorial Optimization [9].     

A facility neighborhood search heuristic for capacitated facility location with single-source constraints and flexible demand

We consider a generalization of the well-known capacitated facility location problem with single source constraints in which customer demand contains a flexible dimension. This work focuses on providing fast and practically implementable optimization-based heuristic solution methods for very large scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems.     

An LP-based heuristic for two-stage capacitated facility location problems

In this paper, a linear programming based heuristic is considered for a two-stage capacitated facility location problem with single source constraints. The problem is to find the optimal locations of depots from a set of possible depot sites in order to serve customers with a given demand, the optimal assignments of customers to depots and the optimal product flow from plants to depots. Good lower and upper bounds can be obtained for this problem in short computation times by adopting a linear programming approach. To this end, the LP formulation is iteratively refined using valid inequalities and facets which have been described in the literature for various relaxations of the problem. After each reoptimisation step, that is the recalculation of the LP solution after the addition of valid inequalities, feasible solutions are obtained from the current LP solution by applying simple heuristics. The results of extensive computational experiments are given.     

Solving capacitated facility location problems by Fenchel cutting planes

In this paper, we apply the Fenchel cutting planes methodology to Capacitated Facility Location problems. We select a suitable knapsack structure from which depth cuts can be obtained. Moreover, we simultaneously obtain a primal heuristic solution. The lower and upper bounds achieved by our procedure are compared with those provided by Lagrangean relaxation of the demand constraints. As the computational results show the Fenchel cutting planes methodology outperforms the Lagrangean one, both in the obtaining of the bounds and in the effectiveness of the branch and bound algorithm using each relaxation as the initial formulation.     

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.     

On solving large instances of the capacitated facility location problem

We present two sets of results pertaining to the solution of capacitated facility location problems that are large, especially with regard to the number of customers. One set of results relates to customer aggregation, while another set of results concerns the judicious selection of variable-upper-bounding (VUB) constraints to include in the initial integer-programming formulation.     

Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with a constant capacity for all plants. These facet inequalities depend on the capacity and thus differ fundamentally from the valid inequalities for the uncapacited version of the problem.We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.This research was partially supported by Grant # ECS-8316224 from the National Science Foundation's Program in Systems Theory and Operations Research.     

ADD-heuristics' starting procedures for capacitated plant location models

This paper is concerned with heuristics for capacitated plant location models where locations have different capacities. In this case ADD-heuristics normally lead to bad solutions. We present some starting procedures (priority rules) in order to overcome this difficulty. Finally, we report numerical results, including comparisons between ADD-heuristics with starting procedures and DROP-heuristics.     

A heuristic lagrangean algorithm for the capacitated plant location problem

Lagrangean techniques have been widely applied to the uncapacitated plant location problem, and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account. In our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one. Several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems, the heuristics consisting of a two phase procedure. The first (location phase) defines a set of multipliers from the analysis of the dual LP relaxation, and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems. Several heuristics have been studied for this second phase, based either on a decomposition of knapsack type subproblems through a definition of a set of penalties, or of looking into the duality gap and trying to reduce it. Computational experience is reported.     

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.     

A comparison of heuristics and relaxations for the capacitated plant location problem

Approaches proposed in the literature for the Capacitated Plant Location Problem are compared. The comparison is based on new theoretical and computational results. The main emphasis is on relaxations. In particular, dominance relations among the various relaxations found in the literature are identified. In the computational study, the relaxations are compared as a function of various characteristics of the test problems. Several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics, both in computing time and in the quality of the solutions found.