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Valid inequalities and facets of the capacitated plant location problem
 Mathematical Programming
, 1989
"... 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 capacitate ..."
Abstract

Cited by 11 (1 self)
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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 constant capacity K for all plants. These facet inequalities depend on K and thus differ fundamentally from the valid inequalities for the uncapacitated 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.
Inverse Spanning Tree Problems: Formulations And Algorithms
, 1996
"... Given a solution x* and an a priori estimated cost vector c, the inverse optimization problem is to identify another cost vector d so that x * is optimal with respect to the cost vector d and the deviation of d from c is minimum. In this paper, we consider the inverse spanning tree problem on an und ..."
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Given a solution x* and an a priori estimated cost vector c, the inverse optimization problem is to identify another cost vector d so that x * is optimal with respect to the cost vector d and the deviation of d from c is minimum. In this paper, we consider the inverse spanning tree problem on an undirected graph G = (N, A) with n nodes and m arcs, and where the deviation between c and d is defined by the rectilinear distance between the two vectors (that is, L 1 norm). We show that the inverse spanning tree problem can be formulated as the dual of an assignment problem on a bipartite network G 0 = (N 0 , A 0 ) with N 0 = N 1 N 2 and A 0 N 1 x N 2 . The bipartite network satisfies the property that N 1  = (n  1), N 2  = (m  n + 1), and A 0  = O(nm). In general, N 1  < < N 2 . Using this special structure of the assignment problem, we develop a specific implementation of the successive shortest path algorithm that solves the inverse spanning tree problem in O(n³) time. We also consider the weighted version of the inverse spanning tree problem where we minimize the sum of the weighted deviations of arcs and show that it can be formulated as the dual of the transportation problem. Using a cost scaling algorithm, the transportation problem can be solved in O(n² m log(nC)), where C denotes the largest arc cost in the data. Finally, we consider a minimax version of the inverse spanning tree problem and show that it can be solved in O(n²) time.