Analysis of decision maker's preferences structure with standard aggregative functions

If decision maker preferences structure is under the conditions of rationality (axioms of Edgeworth-Pareto), then any object determined as optimal with any multicriteria aggregative function must be in Pareto set X_P, e.g. set of non-dominated. All objects in multicriteria tasks are characterized by n features, which are evaluated by functionsfi(x), ..., f_n(x). E.g. an object x has an assigned vector y(x) = f1(x),f₂(x), .,fn(x)), and a set of objects Х has an image - a set of vectors Y (if all functions. f(x) are numerical, then Y с R). The decision maker forms for every feature a criteriaf(x) - extr, j = 1, ., n. At that, the component-wise vector domination relation in Y induces the Pareto dominance relation, and the set of non-dominated vectors Y_P с Y determines X_P с X. Because the step of formalization of preferences precedes the step of choosing of an optimal object as one of non-dominated vectors y(x), the choice results may differ from decision maker's intuitive expectations. We consider an algorithm of testing the decision maker preferences for compliance with his expectation. The decision maker chooses any object x e X. The first test is about if this object can be optimal. The object can be optimal if it belongs to conv_P(Y) -Pareto's envelope of the set Y. This envelope is a union of 2 Pareto sets Y_P for all possible combinations of criteria directions. If y(x) i conv_P(Y), then object cannot be optimal with any combination of conditionsf(x) - extr. Therefore, if the decision maker wants it to be optimal, he must add some more features and form the addition criteria for them (because | Y_P | increases with n). If y(x) e conv_P(Y), the conditions with which the object х can be optimal, are exist. Any of three aggregative functions F_O can describe the decision maker's preferences expressed as the vector dominance relation: additive function, multiplicative function or maxmin Germeyer function. All of them are monotone in respect of this relation. They use a vector parameter w > 0, which components fix the relative criteria importance for the decision maker. If y(x) e convP(Y), then one of these functions can be used with some vector w to choose y(x) as optimal, and we can calculate that vector w. The type of function is determined by the y(x) location in the envelope of Y: if y(x) econv_P(Y) n conv_L(Y), then we can use the additive function, if ln(y(x)) e conv_P(ln(Y)) n conv_L(ln(Y)), then we can use the multiplicative function, and otherwise - maximin function. Here conv_L(Y) is the linear convex envelope of Y. So, if y(x) e conv_P(Y), our second test makes for decision maker a pair O, w> to choose the specified object x as optimal. The decision maker may perform the preference test for concrete vector w, or for concrete order of preference of criteria. There is the large set of {w}, components of which satisfy a given order. In this case, we must test if there is a standard aggregative function F_O that can describe the given preferences. If yes, we declare that the decision maker's preference structure can be represented by this standard function. Otherwise, the decision maker can change his preferences by using the calculated vector w instead his own or another decision method. If the decision maker agrees, then the automatic preferences configuration, which will provide the desired result, is possible.

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