Constructing preference relations for a multi-criteria decision problem

Lecture

Having singled out a set of evaluation criteria for the DMP being solved, it is not enough to be able to compare alternatives by each individual criterion. It is necessary to be able to compare them comprehensively, over the entire available set of criteria, i.e. we need to compare vectors (f1(x),f;(x), ...,fm(x)) ∈ Y for different values of x ∈ X. For this we introduce binary preference relations on the set of possible vector estimates Y. Let us consider the two relations most often encountered in practice.

Definition

It is said that vector y = (y1, y2,..., ym) ∈ UG" Pareto-dominates vector y = = (y1, y2,..., ym) ∈ 9G, if for all i = 1,2,..., m the inequality y{ > y; holds, and for at least one index value this inequality is strict. Denoted y > y1.

The Pareto dominance relation is a strict preference relation.

Another preference relation is constructed by means of a sequential coordinate-wise comparison of two vectors with each other.

Definition

It is said that vector y = (y1, y2, —,Ym) ∈ '.H"‘ lexicographically dominates vector y = (yp y2,..., ym) ∈ ChYam, if there exists a number k= 1,2,..., m, such that the following relations hold

Denoted y y. Constructing preference relations for a multi-criteria decision problem

To construct the lexicographic preference relation, the following procedure is used. First, two alternatives are compared with each other by the first criterion, and if it turns out that their estimates for this criterion coincide, then they are compared by the second criterion. If the estimates of the alternatives also coincide for the second criterion, then they are compared by the third, etc., until a difference is found for some criterion. Otherwise, the estimates will coincide for all criteria, which means these alternatives will be equivalent for the DM.

A drawback of using such a preference relation to solve a DMP is that, in fact, only the first or a few of the first criteria are taken into account, since comparison by the subsequent criteria in the list occurs only if no optimal solution was found using the preceding ones. Different variants of ranking the criteria can substantially change the decisions made.

If the lexicographic preference relation is applied to identify the most preferable manufacturer in example (**), then in the problem statement described, the «winner» will be manufacturer 3. However, if the criteria are arranged in a different order, for example putting the price-attractiveness estimate first, then the «best» will be manufacturer 1.

Definition

The preference relation > is called continuous on Y c: 9?™, if the set {(yy z) | y > z, y, z ∈ Y) is an open subset of Y .

This definition means that if y is strictly preferred to y, then a small change in each of these elements will preserve the preference relation y > y.

Note that the lexicographic preference relation is not continuous. Let us illustrate this with the following example.

Example of a violation of the continuity property by the lexicographic preference relation. Suppose there are two criteria: Y <= 9? and for the vector estimates y = (y1, y2), z = (z1, z2) it holds that y1 = zx, y2 > z2. In this connection, note that y z. will hold. However, having decreased yx by any small number, we obtain yx < z{, which gives y < 1 2.

Definitions of the concepts of binary relations and preference relations were introduced in section 7.2.
In this case it is assumed that a certain metric has been introduced in the space 9?'".
A set is called open if each of its elements belongs to the set together with some e-neighborhood of its own, where ? > 0. Here, the ?-neighborhood of an element is called the set of elements of the set such that the distance to it does not exceed ?.
A set is called open if each of its elements belongs to the set together with some e-neighborhood of its own, where ? > 0. Here, the ?-neighborhood of an element is called the set of elements of the set such that the distance to it does not exceed ?.