By Andranick S. Tanguiane

Aggregation is the conjunction of knowledge, aimed toward its compact represen tation. Any time whilst the totality of information is defined when it comes to normal ized signs, traditional counts, regular representatives and attribute dependences, one at once or in some way bargains with aggregation. It comprises revealing the main major features and precise positive aspects, quanti tative and qualitative research. for that reason, the knowledge turns into adaptable for additional processing and handy for human belief. Aggregation is typical in economics, records, administration, making plans, approach research, and lots of different fields. for this reason aggregation is so vital in info seasoned cessing. Aggregation of personal tastes is a specific case of the overall challenge of ag gregation. It arises in multicriteria decision-making and collective selection, whilst a suite of possible choices should be ordered with admire to contradicting standards, or a number of person evaluations. despite the fact that, inspite of obvious similarity the issues of multicriteria decision-making and collective selection are a little various. certainly, an development in a few requisites on the price of irritate ing others isn't the similar because the pride of pursuits of a few members to the bias of the remaining. within the former case the reciprocal compensations are thought of inside a undeniable entirety; within the latter we infringe upon the rights of autonomous contributors. furthermore, in multicriteria decision-making one usu best friend takes into consideration target components, while in collective selection one has to check subjective critiques which can't be measured properly.

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**Example text**

On the other hand, numerical representations allow us to study preferences in habitual functional terms. With regard to these reasons the present chapter introduces the concept of preference from the standpoint of analogy of preferences and real-value functions. The results of this chapter are published in Russian in (Tanguiane 1979a, 24 2 PREFERENCES AND GOAL FUNCTIONS 1979b, 1980b, 1988). 2 Binary Relations and Orderings Let us introduce basic definitions. 1. DEFINITION. A binary relation S on a set X is defined to be a set S of ordered pairs (x,y), where x,y E X, or, which is equivalent, a subset S of the Cartesian product X x X.

We also retain the unanimity axiom A3 as quite evident. Therefore we shall reject alternately axiom A2 (universality), A4 (independence of irrelevant alternatives), and later A5 (prohibition of dictatorship). Giving up the universality principle (A2) in the present chapter, we assume that individual preferences are fixed (a partial restriction of the universality condition we also consider in chapter 7). 4 in a quite general case both individual and collective orderings can be represented by numerical goal functions.

8. EXAMPLE. Let X be the set of real numbers except the interval (0; 1). Let P be the strict order "to be greater than" on X, which is obviously continuous. Let a dense subset S to be X except the two boundary points 0 and 1 (fig. 29). It is easy to see that there is neither element s E S such that 0< s < 1, nor even element s E S such that 0 ~ s ~ 1. 9. EXAMPLE. Let X be the set of real numbers. 8: P = {(x,y) : x,y E X, x> Y + I}, 2 PREFERENCES AND GOAL FUNCTIONS 38 ° which is obviously continuous.