# American Institute of Mathematical Sciences

October  2010, 6(4): 811-823. doi: 10.3934/jimo.2010.6.811

## A method for optimizing over the integer efficient set

 1 USTHB - Faculty of Mathematics - Operations Research Department, Bp 32 El Alia, BEZ, Algiers, 16121, Algeria 2 UMONS, Faculté Polytechnique, 9 Rue Houdain-Mons, Mons 7000, Belgium

Received  September 2009 Revised  May 2010 Published  September 2010

In this paper, we are interested in optimizing a linear function on the set of efficient solutions of a Multiple Objective Integer Linear Programming problem ($MOILP$). We propose an exact algorithm for maximizing a linear function denoted $\phi$ on the set of efficient solutions of a $MOILP$ problem without having to enumerate explicitly all the elements of this set. Two techniques are used: the first is to reduce progressively the admissible domain by adding more constraints eliminating all the dominated points by the current solution; the second, when the new solution obtained by maximizing the function $\phi$ in the reduced area is not efficient, an exploration procedure is applied over the edges incident to this solution in order to find new alternative efficient solutions if they exist. The algorithm produces not only an optimal value of the linear function but also a subset of non-dominated solutions in the direction of $\phi$ that can be helpful in the practice.
Citation: Chaabane Djamal, Pirlot Marc. A method for optimizing over the integer efficient set. Journal of Industrial & Management Optimization, 2010, 6 (4) : 811-823. doi: 10.3934/jimo.2010.6.811
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