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Improved integral simplex using decomposition for the set partitioning problem

Improved integral simplex using decomposition for the set partitioning problem Integral simplex using decomposition (ISUD) is a method that efficiently solves set partitioning problems. It is an iterative method that starts from a known integer solution and moves through a sequence of integer solutions, decreasing the cost at each iteration. At each iteration, the method decomposes the original problem into a reduced problem (RP) and a complementary problem (CP). Given an integer solution to RP (that is also solution to the original problem), CP finds a descent direction having the minimum ratio between its cost and the number of its positive variables. We loop on until an optimal or near-optimal solution to the original problem is reached. In this paper, we introduce a modified model for CP. The new model finds a descent direction that minimizes the ratio between the cost of the direction and an overestimation of the number of variables taking one in the next solution. The new CP presents higher chances of finding improved integer solutions without branching. We present results for the same large instances (with up to 570,000 columns) as the ones previously used to test ISUD. For all the instances, optimality is always reached with a speedup factor of at least five. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png EURO Journal on Computational Optimization Springer Journals

Improved integral simplex using decomposition for the set partitioning problem

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European Operational Research Societies
Subject
Business and Management; Operations Research/Decision Theory; Operations Management; Operations Research, Management Science; Optimization
ISSN
2192-4406
eISSN
2192-4414
DOI
10.1007/s13675-018-0098-6
Publisher site
See Article on Publisher Site

Abstract

Integral simplex using decomposition (ISUD) is a method that efficiently solves set partitioning problems. It is an iterative method that starts from a known integer solution and moves through a sequence of integer solutions, decreasing the cost at each iteration. At each iteration, the method decomposes the original problem into a reduced problem (RP) and a complementary problem (CP). Given an integer solution to RP (that is also solution to the original problem), CP finds a descent direction having the minimum ratio between its cost and the number of its positive variables. We loop on until an optimal or near-optimal solution to the original problem is reached. In this paper, we introduce a modified model for CP. The new model finds a descent direction that minimizes the ratio between the cost of the direction and an overestimation of the number of variables taking one in the next solution. The new CP presents higher chances of finding improved integer solutions without branching. We present results for the same large instances (with up to 570,000 columns) as the ones previously used to test ISUD. For all the instances, optimality is always reached with a speedup factor of at least five.

Journal

EURO Journal on Computational Optimization – Springer Journals

Published: May 18, 2018

References

  • On the set-covering problem
    Balas, E; Padberg, MW
  • On the set-covering problem: II. An algorithm for set partitioning
    Balas, E; Padberg, MW
  • An improved primal simplex algorithm for degenerate linear programs
    El Hallaoui, I; Metrane, A; Desaulniers, G; Soumis, F
  • The integral basis method for integer programming
    Haus, U; Köppe, M; Weismantel, R
  • Column generation in the integral simplex method
    Rönnberg, E; Larsson, T
  • An integral simplex algorithm for solving combinatorial optimization problems
    Thompson, GL
  • Integral simplex using decomposition for the set partitioning problem
    Zaghrouti, A; Soumis, F; El Hallaoui, I