In this paper, we propose a new exact method to optimize a stochastic linear function over an efficient set of a multi-objective stochastic integer linear programming (MOSILP) problem. The proposed method combines two approaches: the first is called the L method, while the second is an adaptation of the branch-and-bound procedure. This adaptation is reinforced by efficient cuts and tests, enabling the method to eliminate many inefficient solutions in the search tree. A didactic example is provided to detail all steps of the method. This is followed by testing the method on a set of randomly generated instances and comparing the obtained results with the most recent method. The experimental study shows that the proposed method remains competitive and delivers satisfactory results.