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CHVATAL CLOSURES FOR MIXED INTEGER PROGRAMMING PROBLEMS
, 1990
"... Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. T ..."
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Cited by 61 (0 self)
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Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cuttingplane proofs is that for a polyhedron P and integral vector w, if max(wx]x ~ P, wx integer} = t, then wx ~ t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that #x be integer for some other integral vector #. The cuttingplane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cuttingplane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvátal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.
The convex hull of two core capacitated network design problems
 MATHEMATICAL PROGRAMMING
, 1993
"... The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs ..."
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Cited by 43 (0 self)
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The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.
Capacitated facility location: separation algorithms and computational experience
 Mathematical Programming
, 1998
"... We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamili ..."
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Cited by 20 (2 self)
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We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NPhard in general. For the wellknown subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of all these heuristic is based on constructive proofs that two specific conditions are necessary for the effective capacity inequalities to be facet defining. The proofs show precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 small and medium size problems.
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...
A Comprehensive Model for the Concurrent Determination of Aisles and Load Stations for AisleBased Material Handling Systems
, 1994
"... We have developed a set of procedures that convert a block layout into a material handling layout by simultaneously determining the aisles and the locations of the load/unload stations for carrier based material handling systems. The procedures can be applied for several vehicle dispatching policies ..."
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We have developed a set of procedures that convert a block layout into a material handling layout by simultaneously determining the aisles and the locations of the load/unload stations for carrier based material handling systems. The procedures can be applied for several vehicle dispatching policies and for unidirectional and bidirectional aisles. We will report on computational results for standard mixed integer and linear programming solutions as well briefly discuss our heuristic solution procedures and software implementation. Determining the set of aisles and the locations of the stations on the aisles for an aislebased unit load material handling system is an important problem faced in facilities design. Existing research has addressed either the problem of 1) determining the aisle selection set when the station location is given, 2) the problem of determining the station location set when the aisle set is provided, 3) while accounting for the transportation costs only and ignor...