Lecture
In the general case, DM preference is understood as some overall assessment of the quality of a management decision by the DM, based both on an objective analysis of information and on their subjective evaluation of the concept of an optimal (or best) decision. Mathematically, the concept of preference can be formulated as follows. If, when the subject of management is presented with two feasible decisions x', x'' ∈ X, the DM always chooses the feasible decision x' ∈ X rather than x'' ∈ X, then it is said that the DM chooses or gives preference, when choosing, to the first feasible decision rather than the second. In this case, the notation formally used to denote the DM's choice preference is: x' >x x'', where the sign >x denotes the DM's relation of strict preference (abbreviated as the preference relation). Here it is customary to say that on the set of feasible decisions X, a relation of strict preference of the DM, >x — is defined.
Definition
Alternatives x', x'' ∈ X are said to satisfy the relation of strict preference >x, if, from this pair of feasible decisions, the DM always chooses (gives preference to) the first of them, considering it the best (more effective). Notation: x' >xx''.
It should be noted that specifying, on the set X, a certain preference of the DM >x, does not at all mean that for any two feasible decisions x, x'' ∈ X it must necessarily hold that either x' >xx”, or x" >xx, i.e., in the set of feasible decisions X there may occur pairs of feasible decisions from which the DM cannot a priori choose the best decision. It is precisely this situation that is of the greatest interest from the point of view of decision theory, and this is a substantial difference between DT and economic-mathematical methods for solving optimization problems, where it is required to find the extremum of some numerical function depending on certain parameters.
The DM's preference relation >x> defined on the set of feasible decisions Xy is a very important tool in solving DMPs and will be used repeatedly throughout the textbook (including when considering various algorithms for constructing the Pareto set in real multi-criteria problems); at the same time, it is a special case of the well-studied mathematical concept of a binary relation. Some useful properties of binary relations in the general case are set out below.
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