A Novel Ideal Point Method for Uncertain Random Multi-Objective Programming Problem Under PE Criterion

There are two kinds of methods for uncertain random multi-objective programming (URMOP) problem now. One is to convert the URMOP problem into deterministic multi-objective programming (DMOP) problem directly, and then solves the DMOP problem, which neglects the nature of the uncertainty and randomness. The other is to use the linear weighting method (LVM) to convert the URMOP problem into the uncertain random single-objective programming (URSOP) problem, and then convert it into the deterministic single-objective programming (DSOP) problem, which can be solved directly. However, the LVM has limited application range and low reliability. In this paper, we propose a new method named ideal point method (IPM) for solving the URMOP problem. First, we define the ideal point of URMOP. Based on different modules, we propose three different IPMs named SD-IPM, SWS-IPM, and WMM-IPM. It is then proved that under the PE criterion, the three IPMs can transform the URMOP problem into its equivalent URSOP problem, that is, the optimal solution of the transformed URSOP problem is proved to be the Pareto efficient solution of the original URMOP problem. Then, the URSOP problem can be transformed into its equivalent DSOP problem, which can be solved directly. The example discusses the differences and application range of the IPMs and other methods. The influences of weights are discussed simultaneously.