Background
Ordinary aggregated indices method allows comprehensive estimation of complex (multi-attribute) objects’ quality. Examples of such complex objects (decision alternatives, variants of a choice, etc.) may be found in diverse areas of business, industry, science, etc. (e.g., large-scale technical systems, long-time projects, alternatives of a crucial financial/managerial decision, consumer goods/services, and so on). There is a wide diversity of qualities under evaluation too: efficiency, performance, productivity, safety, reliability, utility, etc.
The essence of the aggregated indices method consists in an aggregation (convolution, synthesizing, etc.) of some single indices (criteria) q(1),...,q(m), each single index being an estimation of a fixed quality of multiattribute objects under investigation, into one aggregated index (criterion) Q=Q(q(1),...,q(m)).
In other words, in the aggregated indices method single estimations of an object, each of them being made from a single (specific) “point of view” (single criterion), is synthesized by aggregative function Q=Q(q(1),...,q(m)) in one aggregated (general) object's estimation Q, which is made from the general “point of view” (general criterion).
Aggregated index Q value is determined not only by single indices’ values but varies depending on non-negative weight-coefficients w(1),...,w(m). Weight-coefficient (“weight”) w(i) is treated as a measure of relative significance of the corresponding single index q(i) for general estimation Q of the quality level.
Summary
It is well known that the most subtle and delicate stage in a variant of the aggregated indices method is the stage of weights estimation because of usual shortage of information about exact values of weight-coefficients. As a rule, we have only non-numerical (ordinal) information, which can be represented by a system of equalities and inequalities for weights, and/or non-exact (interval) information, which can be represented by a system of inequalities, which determine only intervals for the weight-coefficients possible values. Usually ordinal and/or interval information is incomplete (i.e., this information is not enough for one-valued estimation of all weight-coefficients). So, one can say that there is only non-numerical (ordinal), non-exact (interval), and non-complete information (NNN-information) I about weight-coefficient.
As information I about weights is incomplete, then weight-vector w=(w(1),...,w(m)) is ambiguously determined, i.e., this vector is determined with accuracy to within a set W(I) of all admissible (from the point of view of NNN-information I) weight-vectors. To model such uncertainty we shall address ourselves to the concept of Bayesian randomization . In accordance with the concept, an uncertain choice of a weight-vector from set W(I) is modeling by a random choice of an element of the set. Such randomization produces a random weight-vector W(I)=(W(1;I),...,W(m;I)), which is uniformly distributed on the set W(I).
Mathematical expectation of random weight-coefficient W(i;I) may be used as a numerical estimation of particular index (criterion) q(i) significance, exactness of this estimation being measured by standard deviation of the corresponding random variable. Since such estimations of single indices significance are determined on the base of NNN-information I, these estimations may be treated as a result of quantification of the non-numerical, inexact and incomplete information I.
An aggregative function Q(q(1),...,q(m)) depends on weight-coefficients. Therefore, random weight-vector (W(1;I),...,W(m;I)) induces randomization of an aggregated index Q, i.e., its transformation in the corresponding randomized aggregated index Q(I). The looked for average aggregated estimation of objects’ quality level may be identified now with mathematical expectation of corresponded random aggregated index Q(I). The measure of the aggregated estimation's exactness may be identified with the standard deviation of the correspondent random index.
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