Multiple Criteria Decision Making by Multiobjective Optimization by Ignacy Kaliszewski Janusz Miroforidis & Dmitry Podkopaev

Author:Ignacy Kaliszewski, Janusz Miroforidis & Dmitry Podkopaev
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

7.5 Derivation of Efficient Variants and Variant Ranking

It is worth observing that both characterizations, Characterization A and Characterization B, assign to each variant a score, i.e., a value of the corresponding scalarizing function. These scores establish rankings of variants (see Sect. 11.​5), since variants can be ranked in decreasing or increasing order of the assigned scores.

7.6 Weight Normalization

The set of weights λ l  > 0,  l = 1, …, k, is unbounded. However, in numerical computations it is more convenient to deal with a bounded set.

Let us observe that the “intensity” of dependence of a scalarizing function on criteria is related to proportions of weights rather than to their absolute values. For instance, multiplication of all weights by a positive number does not change the properties of the scalarizing functions used in Characterization A and Characterization B, in the sense that the original and the modified scalarizing functions yield the same efficient outcomes (and thus the same efficient variants) and the same variant ranking. This observation allows us to deal with bounded sets of weights.

For any vector of weights λ l  > 0,  l = 1, …, k, we can multiply each weight by the reciprocal of their sum, i.e., by , obtaining in this way a new vector of weights λ′ l ,  l = 1, …, k. According to the above argument, each of two instances of the scalarizing function, as in Characterization A or in Characterization B, one instance with λ l  > 0,  l = 1, …, k, another with λ′ l  > 0,  l = 1, …, k, yields the same efficient outcomes (and thus an efficient variants). However, in the latter case the additional condition holds. Indeed,

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