A combined interactive approach for solving E‐convex multiobjective nonlinear programming problem

In general, there is no single optimal solution in multiobjective problems, but rather a set of non‐inferior (or Pareto optimal) solutions from which the decision maker must select the most preferred solution as the one to implement. The generation of the entire Pareto optimal solution set is not practical for most real world problems. This paper deals with a combined interactive approach for solving E‐convex multiobjective nonlinear programming problems which introduced by Youness in , where the decision space is effected by an operator $(E:R^n\to R^n)$ . This kind of convexity is called E‐convexity which has the some important applications in various branches of mathematical science. The proposed approach in this paper combines the reference direction method introduced by Narula, et al. and all of there reference point method introduced by Wierzbicki , Tchebycheff method introduced by Steuer , the satisfying trade‐off method introduced by Nakayama and combined in attainable reference point (ARP) method introduced by Wang et al. . The main development of the proposed approach is starting with weak efficient solution say $x^{**}$ corresponding to $s^{**}$ as the first step and use $s^{**}$ to improve the weighting coefficients of the augmented lexicographic Tchebycheff problem where we improve the value $\overline{s}‐\widehat{s}$ inserted in ARP by the value $\overline{s}‐s^{**}$ and hence modify the reference point in the case of unsatisfactory solution for the DM as he wishes. An illustrative numerical example is given to demonstrate the theory developed and the quality and effectiveness of the presented approach in this paper.