Comments on Reply by Romero et al
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The second counterexample provided by Professor Ogryczak (9), (10) is also invalid in the context of the paper. This is a very unusual model indeed, as there is no interaction between objective function f3 ðxÞ and the other objective functions f1 ðxÞ and f2 ðxÞ. That is, feasible increases in f3 ðxÞ do not generate any opportunity cost in f1 ðxÞ and f2 ðxÞ, and vice versa. This means the model would be broken down into two separate MOP models by any rational decision maker, neither of which form a counterexample to any of the theories or postulates in our paper.1 To put it more formally, there is no proper MCDM problem in three criteria space so no differentiable transformation surface exists and the example (9), (10) is not a valid counterexample. Professor Ogryczak tries to refute model (7) of our paper1 (model (11) in his comment) by resorting again to his counterexamples. In this sense, it was clearly stated in our paper that model (7) is proposed for situations where reference levels are not fixed at their anchor values. In fact, for examples like the one used in the P viewpoint where bi ¼ bi , it is obvious thatPthe term ÿ qi¼1 ðWi =Ki Þ pi (our notation) or the term ÿ qi¼1 vi pi (Professor Ogryczak’s notation) vanishes and model (7) turns into model (6). Therefore, none of Professors Ogryczak’s claims that reply to either of these examples are proved. With regard to Professor Ogryczak’s criticism of our model (10), the issue here of principal concern is that of utility interpretation. The object is to form a model that can both represent the classic WGP utility (metric 1) and the Chebyshev GP utility (metric 1 ). We accept the criticism that the model as stated will produce unbounded solutions. The GPSYS package used for the experimentation has facilities for basis restriction,5 which means the positive and negative deviation variables can never be in the basis simultaneously. If the solver used does not have this facility, then the condition ni pi ¼ 0 must be added to the model. In this case, when e ! 1 the utility structure becomes equivalent to that used in weighted goal programming (ie metric 1) and when e ! 0 the utility structure becomes equivalent to that of Chebyshev GP (ie metric 1). Intermediate values of e produce intermediate utility structures. It is not stated in our paper1 that a ‘perfectly balanced solution’ always exists. In fact, it is easy to find many examples where no perfectly balanced solution exists. We only claim that when conditions postulated by Ballestero and Romero2 (usual conditions in economics) hold and targets are fixed at the anchor values the GP minmax solution and the CP solution for metric 1 coincide and are efficient and perfectly balanced. A sensible line of argument is to offer counterexamples holding the commented conditions which refute our statement or to criticise the rationality underlying the conditions. Neither of these tasks are undertaken in the viewpoint. It is not argued in the paper that the small regularisation term in the reference point method should
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