# A New Combination of Lagrangean Relaxation, Dantzig-Wolfe Decomposition and Benders Decomposition Methods for Exact Solution of the Mixed Integer Programming Problems

## DOI:

https://doi.org/10.26713/cma.v10i3.1116## Keywords:

Cross decomposition, Benders decomposition, Lagrangean relaxation, Dantzig Wolfe decomposition, Cutting planes, Sub-gradient, Trust region, Column generation## Abstract

The combinational technique of cross decomposition is a suitable one for exact solution of the mixed integer programming problems which uses simultaneously the advantages of Lagrangean relaxation, Dantzig-Wolfe decomposition and Benders decomposition methods for Minimization problem that each reinforces one another. The basic idea for this technique is the generation of suitable upper and lower bounds for the optimal value of the original problem at each iteration. In this paper, new cross decomposition algorithm, with the combination of Lagrangean relaxation method (the combination of three concepts of cutting-plane, sub-gradient and trust region), Dantzig-Wolfe decomposition and Benders decomposition methods are used in order to reinforce bounds and to speed up convergence. By increasing the problem scale and regarding the use of the Lagrangean relaxation method in this technique, the lower bound with more strength and efficacy, and by the aid of Dantzig-Wolfe decomposition method more suitable upper bound (if exists) and furthermore less number of iterations for achieving optimal solution is obtained. The convergence of this technique regarding the convergence of Benders decomposition method in finite iteration numbers is guaranteed.

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*Communications in Mathematics and Applications*,

*10*(3), 495–508. https://doi.org/10.26713/cma.v10i3.1116

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