ABSTRACT We study a class of integer bilevel problems, the so-called Integer Linear Multiplicative Bilevel Problem, ILMBP, where the constraints are linear and both the upper level problem and the lower level problem are integer linear multiplicative problems. We assume that the k -linear factors of the upper level problem and the l -linear factors of the lower level problem to be non-negative. In this paper, we propose an algorithm based on branch-and-cut method for solving the Integer Linear Multiplicative Bilevel Problem. First, the continuous upper level linear multiplicative problem is solved, then the optimal integer solution obtained is tested for the optimality of the main problem by solving the lower level linear multiplicative problem. If this solution is not optimal for the problem ILMBP, a cut is added to the upper level problem and a new best integer solution is determined. The integer solution is generated by the branching process well known in the branch and bound procedure. Following, the presentation and validation of the algorithm, an example is provided to better visualize the algorithm.

ABSTRACT In various Multi-objective Programming (MOP) problems, decision-makers are often faced with a large set of efficient solutions, presenting a challenge in discerning and selecting the best solutions within this broad set. To overcome this, decision-makers can tune and optimize their preference function over this efficient set to identify the optimal solutions that align with their preferences. Most existing methods address such problems in the deterministic case, while we aim to tackle a new challenge by dealing with these problems in a stochastic environment, which increases their complexity. Indeed, we focus in this study on proposing an exact method to optimize two preference functions over the efficient set of a Multi-objective Stochastic Integer Linear Programming (MOSILP) problem in order to find the best compromise solutions without enumerating all elements of this efficient set. The proposed method is based on the combination of two techniques: the first one is called the L-shaped method, while the second one is an adaptation of the branch-and-bound strategy by reinforcing it with efficient cuts and tests, which allows the method to remove a large number of inefficient solutions within the search tree. Moreover, the method exhibits adaptability in addressing the general case involving multiple preference functions. A didactic example is presented for illustration, followed by a computational study evaluating the method’s efficacy and performance by solving randomly generated instances.