Final reply to comments of Professor Ogryczak
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In my opinion, the readers have already received enough material to justify the validity of Conclusion 1 and the related models,3 in the case of typical OR=MS Multiple Objective Programming (MOP) models. Let me recall that the first example submitted in the Viewpoint contradicts Conclusion 1 even for two attributes in the case of discrete or, more generally, nonconvex problems (integrality requirements in this model can be replaced with continuous variables and nonlinear equations: x2i ÿ xi ¼ 0). The second counterexample submitted in the Viewpoint, although simplified, is a proper Multiple Objective Linear Programming (MOLP) model in the three criteria space and it is even characterised by the differentiable efficient frontier (which is not common in the case of MOLP models). This example contradicts Conclusion 1 in the case of MOLP models with more than two criteria (attributes) as well as it contradicts the validity of model (7)3 proposed for general case, where reference levels can take any values not necessarily being fixed at their ideal values (as shown in the Viewpoint it reduces to model (6) if the reference levels are fixed at the ideal values or larger). One may also notice that even the unique minmax solution, and therefore the efficient one, may be not perfectly equilibrated, like in the following MOLP: maxfðx1 ; x2 ; x3 Þ : x1 þ x2 þ x3 4 7; 3 4 x3 4 4; x1 5 0; x2 5 0g In my opinion, the reply already revealed enough hidden assumptions about the problems considered by Romero et al3 and the readers may now justify if dropping the regularisation term is worth such a restrictive shrinking of the class of MCDM problems to be analysed. I agree with the authors that the paper3 itself addresses important issues of preference models used by various MCDM approaches. I am both sympathetic to, and working towards, the same end. I admit very much the lexicographic model (9)3 which, although not formally proven in the paper,3 satisfies the most important properties of the RPM approaches.4 I consider also as very important the attempt to model various preference models by treating the perturbation constant e as a parameter. I do not agree, however, that model (10)3 with additional complementarity constraints (as suggested in the reply) meets this end. The model clearly applies the minmax aggregation only to negative deviations and the weighted aggregation is applied only to positive deviations. Thus, independently from the value of e, it does not model a compromise between the two different aggregations in the way suggested by Romero et al.3 Moreover, model (10)3 with additional complementarity constraints does not belong to the class of convex programming (one can easily verify this by drawing the corresponding iso-utility contours) and therefore the solution procedure suggested in the reply does not guarantee the optimality.
References 1 Ballestero E and Romero C (1991). A theorem connecting utility function optimization and compromise programming. Opns Res Lett 10: 421–427. 2 White DJ (1990). A bibliography on
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