Lecture
The mathematical formulation of a multicriteria decision-making problem is stated as a problem of optimizing the vector criterion f(x) = (f1(x), 2(x), ...,m(x)), also called the vector estimate of the decision, defined on the set of alternatives (or options), i.e., on the set of all possible decisions X. The individual components fk(x), k = 1,2,m, expressing the DM's desire to achieve one of the m stated goals, are called partial criteria. The role of the functions fk(x) is most often played by quality indicators (technical, economic, etc., reliability and safety indicators), or by financial quantities (project implementation cost, profit, expenses, etc.). Depending on the content of the problem, the functions fk(x) are called criteria of optimality, criteria of effectivenessy objective functions, indicators (or criteria) of quality. Optimization of the partial criteria fk(x) involves one of two options:
Both types of optimization are equivalent, since the minimum of any criterion can easily be turned into a maximum by a linear transformation of the scale with a negative coefficient (for example, by multiplying fk(x) by -1). In practice, the transformation M - fk(x) is usually performed, where the constant M is chosen to exceed the maximum value of the function fk(x), i.e., the transformed criterion M - fk(x) remains non-negative. Such a transition from minimum to maximum is convenient for jointly describing all types of criteria fk(x) defined on the set of decisions X, since there is no need to study each of the two types of optimization separately. Therefore, in what follows, for simplicity and clarity (without loss of generality) we shall assume that the DM's goal is — to achieve the maximum over all components of the vector f(x).
Note!
We shall assume that all partial criteria fk(x) have been transformed so that they require maximization according to the DM's goals.
Note that in some monographs on decision theory, conversely, the partial criteria fk(x) are transformed so that they require minimization according to the DM's goals.
All m criteria fk(x) are combined into a single vector, even though they are, as a rule, given in different scales and units of measurement. Moreover, some of them may be quantitative, while others — qualitative. Criteria can also be treated as fuzzy numbers. None of these considerations prevent us from viewing the decision estimates (f1(x), f2(x),... ,f„,(•*)) as elements of the linear vector space 'R"'.
Definition
All possible values of the vector estimates (y1, y2 ym) = (f1(x),f2(x), ...,fm(x))
form the set Y of possible estimates. This set lies in the m-dimensional vector space R™, which is called the criterion space. Y ⊂ R'".
The need for a joint, i.e., vector, consideration of the m criteria f1,(x),f2(x), ...,fm(x) is dictated by the requirement of a correct definition of the concept of dominance, i.e., of specifying an order relation in the criterion space. The main difference between a multicriteria decision-making problem and a single-criterion one lies precisely in the different properties of the order relations in the spaces R and Rm.
If the DM pursues a single goal that can be characterized quantitatively (for example, profit), then the corresponding criterion space R — is a linearly ordered set . This means that any estimates f(x1) and f(x2) of two alternatives xi and x2 are always related either by f(x1) <f(x2), or by f(x2) 2)). However, the multidimensional space R'n for m > 2 is not a linearly ordered set.
Note!
The practical meaning of the absence of linear ordering for m > 2 lies in the fact that the DM's local goals, corresponding to different components fk(x) of the vector criterion, most often turn out to be mutually contradictory.
In other words, the functions fk(x) in most situations attain their maximum values at different points of the set of alternatives X. For example, an increase in the quality of the product being purchased almost always means a higher price, i.e., the DM's two goals (buy more cheaply and acquire a high-quality product) contradict each other. Therefore, the option ideal from the DM's point of view — a simultaneous maximum of all criteriafk(x) — is practically never attainable on the set of alternatives that exists in reality. In the case considered — the cheapest and the highest-quality products are almost always different.
However, the absence of linear ordering in 'R?'" does not yet mean a complete absence of order. Indeed, let us consider the order relation y > y * in the space, based on the binary order relation y1, > y* in R', i.e., defined by the relation 
Definition
The vector estimate y is called dominant with respect to y*, and y* — is called, respectively, dominated, if this pair of estimates satisfies the order relation (9.1): y > y*.
The meaning of the concept of dominance — it can be interpreted as a relation of strict preference between vector criteria. Indeed, the vector criterion of optimality in a multicriteria choice problem expresses the very same interests of the DM as the previously introduced preference relation on the set of alternatives X. Therefore, the order relation on the set Y of possible estimates y =f(x) is interrelated with the preference relation on the set of alternatives X. The latter (the DM's preference on the set of options X) is formalized mathematically as follows.
Definition
Alternatives xk and xk are said to be satisfying the strict preference relation >, if, from the pair of solutions x} and xk the DM always chooses (gives preference to) the first of them, considering it the best (more efficient). Notation: x) > xk.
This definition makes it possible to extend the notion of dominance, introduced for vector evaluations, to alternatives.
Definition
Alternative x is called dominant with respect to x*, and x* is called, correspondingly, dominated, if this pair of solutions x and x* satisfies the strict preference relation: x > x*.
Thus, the notion of Pareto dominance as a whole combines two definitions of dominance in two different spaces (in the criterial space and in the space of alternatives).
Attention!
Pareto dominance — is a collection of order relations on the sets of alternatives X and their vector evaluations Y, characterizing the DM's preferences. Identifying dominant and dominated alternatives makes it possible to discard from consideration solutions that are manifestly not effective from the DM's standpoint (not satisfying the formulated criteria), and thereby — reduce the number of alternatives being compared.
A question arises about the relationship between the two definitions of dominance — on the sets of alternatives X and vector evaluations Y.
This relationship is formulated mathematically in the form of the Pareto axiom (Vilfredo Pareto (1848—1923) — an Italian economist and sociologist ).
Pareto axiom
Any pair of alternatives x and x*} whose vector evaluations (y,, y2,.... ym) = (/,(x), 1>(x),„(x)) and (y*, y2, .... y*) = (/,(x*),/2(x*), (x*)) satisfy the order relation y > y*, also satisfies the strict preference relation of the alternatives x > x*.
In accordance with formula (9.1), the vector evaluations (y,, y2,... , ym) satisfy here the set of inequalities y, > y*, y2 > y2,.... ym > y*m, with one of these inequalities (for some index k) — being strict: yk > yk.
The Pareto axiom expresses the fact of mutual consistency between the DM's preferences and optimality criteria. In other words, DM judgments that do not satisfy the Pareto axiom are considered illogical (contradictory). From the standpoint of common sense, a violation of the Pareto axiom is explained first of all by the incompleteness of the set of criteria. Indeed, if from two alternatives x and x* the DM did not prefer the alternative x, which has superiority in one of the criteria yk and is not inferior in all the other criteria, — then there must exist a criterion not included in the considered set of m criteria that prevented the DM from making the choice x > x*. In this case, it is only necessary to supplement the list of criteria. Therefore, in practice the Pareto axiom is always considered to be satisfied.
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