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.     

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.     

A capacitated competitive facility location problem

We consider a mathematical model similar in a sense to competitive location problems. There are two competing parties that sequentially open their facilities aiming to “capture” customers and maximize profit. In our model, we assume that facilities’ capacities are bounded. The model is formulated as a bilevel integer mathematical program, and we study the problem of obtaining its optimal (cooperative) solution. It is shown that the problem can be reformulated as that of maximization of a pseudo-Boolean function with the number of arguments equal to the number of places available for facility opening. We propose an algorithm for calculating an upper bound for values that the function takes on subsets which are specified by partial (0, 1)-vectors.     

LP-based approximation for uniform capacitated facility location problem

In this paper, we study uniform hard capacitated facility location problem. The standard LP for the problem is known to have an unbounded integrality gap. We present constant factor approximation by rounding a solution to the standard LP with a slight $(1+ϵ)$ violation in the capacities.Our result shows that the standard LP is not too bad.Our algorithm is simple and more efficient as compared to the strengthened LP-based true approximation that uses the inefficient ellipsoid method with a separation oracle. True approximations are also known for the problem using local search techniques that suffer from the problem of convergence. Moreover, solutions based on standard LP are easier to integrate with other LP-based algorithms.The result is also extended to give the first approximation for uniform hard capacitated $k$-facility location problem violating the capacities by a factor of $(1+ϵ)$ and breaking the barrier of 2 in capacity violation. The result violates the cardinality by a factor of $\frac{2}{1+ϵ}$.     

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].     

Kernel search for the capacitated facility location problem

The Capacitated Facility Location Problem (CFLP) is among the most studied problems in the OR literature. Each customer demand has to be supplied by one or more facilities. Each facility cannot supply more than a given amount of product. The goal is to minimize the total cost to open the facilities and to serve all the customers. The problem is $\mathcal{NP}$ -hard. The Kernel Search is a heuristic framework based on the idea of identifying subsets of variables and in solving a sequence of MILP problems, each problem restricted to one of the identified subsets of variables. In this paper we enhance the Kernel Search and apply it to the solution of the CFLP. The heuristic is tested on a very large set of benchmark instances and the computational results confirm the effectiveness of the Kernel Search framework. The optimal solution has been found for all the instances whose optimal solution is known. Most of the best known solutions have been improved for those instances whose optimal solution is still unknown.     

Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy space of each player consists of the bases of a player-specific matroid defined on the set of resources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility location problems, where players have to jointly pay for opened facilities. The goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate two classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium and separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource. We find optimal basic and separable protocols that guarantee the price of stability/price of anarchy to grow logarithmically/linearly in the number of players. These results extend our previous results (cf. von Falkenhausen & Harks, 2013), where optimal basic and separable protocols were given for the case of symmetric matroid games without delays.     

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.     

Accelerating Benders' decomposition for the capacitated facility location problem

Discrete facility location problems are attractive candidates for decomposition procedures since two types of decisions have to be performed: on the one hand the yes/no-decision where to locate the facilities, on the other hand the decision how to allocate the demand to the selected facilities. Nevertheless, Benders' decomposition seems to have a rather slow convergence behaviour when applied for solving location problems. In the following, a procedure will be presented for strengthening the Benders' cuts for the capacitated facility location problem. Computational results show the efficiency of the modified Benders' decomposition algorithm. Furthermore, the paretooptimality of the strengthened Benders' cuts in the sense of [Magnanti and Wong 1990] is shown under a weak assumption.This paper was written when the author was at the Institute for Operations Research, University of St. Gallen, Switzerland, and partly supported by Schweizerischer Nationalfond zur Förderung der wissenschaftlichen Forschung (Grant 12-30140.90).     

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.     

A branch-and-price algorithm for the capacitated facility location problem

The capacitated facility location problem (CFLP) is a well-known combinatorial optimization problem with applications in distribution and production planning. It consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs. A number of solution approaches based on Lagrangean relaxation and subgradient optimization has been proposed for this problem. Subgradient optimization does not provide a primal (fractional) optimal solution to the corresponding master problem. However, in order to compute optimal solutions to large or difficult problem instances by means of a branch-and-bound procedure information about such a primal fractional solution can be advantageous. In this paper, a (stabilized) column generation method is, therefore, employed in order to solve a corresponding master problem exactly. The column generation procedure is then employed within a branch-and-price algorithm for computing optimal solutions to the CFLP. Computational results are reported for a set of larger and difficult problem instances.     

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.     

Scatter search for the single source capacitated facility location problem

This paper considers the Single Source Capacitated Facility Location Problem (SSCFLP). We propose a Scatter Search approach to provide upper bounds for the optimal solution of the problem. The proposed approach uses GRASP to initialize the Reference Set. Solutions of the Reference Set are combined using a procedure that consists of two phases: (1) the initialization phase and (2) the improvement phase. During the initialization phase each client is assigned to an open facility to obtain a solution that is then improved with the improvement phase. Also, a tabu search algorithm is applied. In order to evaluate the proposed approach we use different sets of test problems. According to the results obtained we observe that the method provides good quality solutions with reasonable computational effort.     

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.     

Integrality gaps for colorful matchings

We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali–Adams “lift-and-project” technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Füredi (1981). We further leverage this to show upper bounds on the performance of the Sherali–Adams hierarchy when applied to the natural LP relaxation of the BCM problem.     

A tabu search heuristic procedure for the capacitated facility location problem

A tabu search heuristic procedure is developed, implemented and computationally tested for the capacitated facility location problem. The procedure uses different memory structures. Visited solutions are stored in a primogenitary linked quad tree. For each facility, the recent move at which the facility changed its status and the frequency it has been open are also stored. These memory structures are used to guide the main search process as well as the diversification and intensification processes. Lower bounds on the decreases of total cost are used to measure the attractiveness of the moves and to select moves in the search process. A specialized network algorithm is developed to exploit the problem structure in solving transportation problems. Criterion altering, solution reconciling and path relinking are used to perform intensification functions. The performance of the procedure is tested through computational experiments using test problems from the literature and new test problems randomly generated. It found optimal solutions for almost all test problems from the literature. As compared to the heuristic method of Lagrangean relaxation with improved subgradient scheme, the tabu search heuristic procedure found much better solutions using much less CPU time.     

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.     

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 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.