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Lifted Flow Cover Inequalities for Mixed 01 Integer Programs
 Mathematical Programming
, 1996
"... We investigate strong inequalities for mixed 01 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the c ..."
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Cited by 34 (8 self)
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We investigate strong inequalities for mixed 01 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients, lifting of flow cover inequalities is computationally very demanding. We present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and computational results that justify the effectiveness of our lifting procedures. 1 Introduction A mixed integer program (MIP) with binary integer variables (BMIP) is the appropriate mathematical model for many practical optimization problems. This model is used, for example, for facility location problems, distribution problems, network design problems and more generally when fixed or concave costs are re...
LPbased approximation algorithms for capacitated facility location
 in Proc. of IPCO’04, 2004
"... In the capacitated facility location problem with hard capacities, we are given a set of facilities, F, and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. ..."
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Cited by 15 (1 self)
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In the capacitated facility location problem with hard capacities, we are given a set of facilities, F, and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set F and assign each client to an open facility so that at most ui clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance cij, 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 NPhard 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 5approximation 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 singledemand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.
A Bionomic Approach to the Capacitated pmedian Problem
, 1998
"... This paper advocates the use of the bionomic algorithm, a recently proposed metaheuristic technique, as an effective method to solve capacitated pmedian problems (CPMP). Bionomic algorithms already proved to be an effective framework for finding good solutions to combinatorial optimization problems ..."
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Cited by 10 (1 self)
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This paper advocates the use of the bionomic algorithm, a recently proposed metaheuristic technique, as an effective method to solve capacitated pmedian problems (CPMP). Bionomic algorithms already proved to be an effective framework for finding good solutions to combinatorial optimization problems, when good local optimization algorithms are available. The paper also presents an effective local search technique for the CPMP. Computational results show the effectiveness of the proposed approach, when compared to the best performing heuristics so far presented in the literature. Correspondence to: Vittorio Maniezzo, Scienze Informazione, Univ. Bologna, via Sacchi 3, 47023 Cesena, Italy. 2 1 Introduction The capacitated pmedian problem (CPMP) considered in this paper is the problem of partitioning a set of n weighted entities (objects, customers,...) into p disjoint clusters, so that the total dissimilarity within each cluster is minimized and constraints on maximum cluster capaciti...
Polyhedral Techniques in Combinatorial Optimization II: Computations
 Statistica Neerlandica
, 1995
"... The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formu ..."
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Cited by 5 (1 self)
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The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define highdimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part I of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we ...
Telecommunication and Location
, 2001
"... We review the models for telecommunication network design where there is a location problem involved. We classify the models into three classes as uncapacitated, capacitated and dynamic models. For each class, we discuss the core problem, its generalizations and the solution methods in the litera ..."
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Cited by 4 (1 self)
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We review the models for telecommunication network design where there is a location problem involved. We classify the models into three classes as uncapacitated, capacitated and dynamic models. For each class, we discuss the core problem, its generalizations and the solution methods in the literature.
Reformulation of Capacitated Facility Location Problems: How Redundant Information Can Help
 Annals of Operations Research
, 1996
"... Most facility location problems are computationally hard to solve. The standard technique for solving these problems is branchandbound. To keep the size of the branchandbound tree as small as possible it is important to obtain a good lower bound on the optimal solution by deriving strong linear ..."
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Cited by 4 (0 self)
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Most facility location problems are computationally hard to solve. The standard technique for solving these problems is branchandbound. To keep the size of the branchandbound tree as small as possible it is important to obtain a good lower bound on the optimal solution by deriving strong linear relaxations. One way of strengthening the linear relaxation is by adding inequalities that define facets of the convex hull of feasible solutions. Here we describe some simple, but computationally very useful classes of inequalities that were originally developed for relaxations of the facility location problems. Algorithms for generating violated inequalities belonging to the described classes have been implemented as system features in various branchand bound software packages, so as long as the software can recognize the relaxations for which the inequalities are developed, the inequalities will be generated "automatically". Here we explicitly add the variables and constraints that are n...
Approximation Algorithms for the Capacitated MultiItem LotSizing Problem via FlowCover Inequalities
"... We study the classical capacitated multiitem lotsizing problem with hard capacities. There are N items, each of which has specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be sat ..."
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Cited by 1 (0 self)
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We study the classical capacitated multiitem lotsizing problem with hard capacities. There are N items, each of which has specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satisfied on time. Each order incurs a timedependent fixed ordering cost regardless of the combination of items or the number of units ordered, but the total number of units ordered cannot exceed a given capacity C. On the other hand, carrying inventory from period to period incurs holding costs. The goal is to find a feasible solution with minimum overall ordering and holding costs. We show that the problem is strongly NPhard, and then propose a novel facility location type LP relaxation that is based on an exponentially large subset of the wellknown flowcover inequalities; the proposed LP can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities. Moreover, the optimal solution of the LP can be rounded to a feasible integer solution with cost that is at most twice the optimal cost; this provides a 2approximation algorithm which is the first constant approximation algorithm for the problem. We also describe an interesting onthefly variant of the algorithm that does not require solving the LP apriori with all the flowcover inequalities. As a byproduct we obtain the first theoretical proof regarding the strength of flowcover inequalities in capacitated inventory models.
Cutting Planes in Integer Programming And Mixed Integer Programming
, 1999
"... This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for i) general integer programs, ii) problems with local structure such as knapsack constraints,and iii) problems with 01 coe#cient matrices,such as set packing,are examin ..."
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This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for i) general integer programs, ii) problems with local structure such as knapsack constraints,and iii) problems with 01 coe#cient matrices,such as set packing,are examined in turn. Finally the use of valid inequalities for classes of problems with structure,such as network design,is explored. Keywords: Mixed Integer Programming,Cutting Planes Classification: MSC(199 9991,S Research carried out with financial support of the projectTMRDON) nr. ERB FMRXCT980202 of the European Community. This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's O#ce, Science Policy Programming. The scientific responsibility is assumed by the authors. 1Departmen t of Operationy ResearchLonq3 School ofEconq6qU Houghton Street,Lonet WC2A 2AE,Un,UI KinUI6I 2Kon33UyIb0qIUn trumBerlin Takustr. 7 D14195Berlin German y. 3 Fakultat fur Mathematik, IMO OttovonovUv keUnv ersitat MagdeburgUnd ersitatsplatz 2 D39106 MagdeburgGerman y. 4 CORE,UnE ersite Catholique de Louvain 1348 Louvainbb60bUy e, Belgium. Introductiz This survey is devoted to cutting planes that are useful or potentially useful in solving mixed integer programs. This topic is important because a) strengthening formulations with cutting planes is of interest independently of the algorithm used to solve the problem, and b) linear programming based branchandbound with cuts added, known as branchandcu , is now one of the most widespread and successful tools for solving mixed integer programs. The paper is divided into four sections. First we discuss ways of generating cuts for general integer program...
Incorporating a Probabilistic Choice Model in ConstraintBased Search by
"... This paper presents a local search method for constraint satisfaction problems, which is guided by some learning properties derived from a probabilistic choice model. The local search algorithm randomly selects a variable in a violated constraint and another variable from the search space. Two trial ..."
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This paper presents a local search method for constraint satisfaction problems, which is guided by some learning properties derived from a probabilistic choice model. The local search algorithm randomly selects a variable in a violated constraint and another variable from the search space. Two trials are performed, each of which assigns a different value to one of the selected variables whilst keeping the other fixed. Partial constraint propagation is performed in the process. The evaluation function for these trials is based on the overall degree of constraint violation. The trial results are accumulated with respect to individual variables. When sufficient trial history has been collected for a variable, it is analysed to infer a likely optimal value for that variable to be fixed at for a number of iterations. The algorithms presented have been applied to a demand responsive container freight rail scheduling problem. In this problem, all the decision variables are binary and relatively few feasible solutions exist. This paper will report on tests using real life data from Thailand.