Candidate of technical sciences Р. V. Tereliansky

Volgograd state technical university, Russia

EXTRAPOLATION OF DYNAMIC SYSTEMS OF EXPERT’S PREFERENSES FOR DECISION MAKING

The axiom about a change in the needs of humanity in the course of time is the basis of analysis of dynamic systems for decision making (systems of the prognostication of preferences). Consequently, it is possible to select such dependence, which would satisfy the law of variation in the needs. The solution of the problem of the prognostic simulation of preferences in the dynamic systems is a process of the step by step establishment of priorities. The principle of decomposition in the systems analysis provides for the structuring of the problems in the form of network or hierarchy and it is the first stage of the application of a method of the analysis hierarchy process which is widely known. In this method the elements of network are compared in pairs with respect to their action on theirs common characteristic. However, experts frequently attempt to compare the physically not comparable alternatives, similar as, for instance, “ergonomics”, “beauty” and so on. Experts are used to apply verbal estimations (the so-called scale of the static estimations, proposed by T.L. Saati [1]) during conducting of subjective paired comparisons frequently. In the most elementary form the hierarchy is built from the apex (there are purposes from the point of view of a research), through the intermediate levels (they will be criteria) to the lowest level (which is the enumeration of alternatives usually). There is a question after the hierarchical or net reproduction of problem: how to establish the priorities of criteria and to estimate each of the alternatives on the criteria and to find the most relevant alternative? How to consider to the changes in the system in the course of time? One of the methods consists in obtaining of the shear of the state of the system at the specific moments of time, interpolating the data. Extrapolating of these dependences (which are investigating and creating) is the basis for evaluating the behavior of system in the future, i.e., it is the prognosis of its state. The set of reliably known states can be obtained either by the way of the accumulation of information about the state of the separate elements of system at each moment of time with their subsequent integration or by the attraction of expert estimations. The complex system of criteria can be presented in the form to the strong or weakly-connected dominant hierarchy. Mappings of preferences are assigned from the set of the analytical functions. Those functions describe the most frequently meeting semantic (verbal) expressions which are describing paired comparison of the importance of criteria (alternatives). Functions are represented parametrically so that it would be easy to convert them (to displace, to extend and so forth). These functions describe most the frequently meeting trends (table 1). Priority analysis is reduced to the iterative calculations of the eigenvectors of matrix of the paired comparisons. The results of preferences calculated for given moment of the time of indicated functions put into the matrix. As a result subsequent convolution of these vectors we obtain the collection of the vectors of global priorities. The number of vectors will be equal to the number of moments of time, for which is comprised the prognosis. Selecting the equivalent components of the global vector of priorities, we form the massif of them. This massive is processed with the method of least squares. We obtain the analytical function of the dynamics of preferences according to any criteria after this processing. The parametric representation of preference will be the desired prognoses. For the automation of the work of the person, who makes decisions (expert on this problems), was developed the specialized program system (decision support system). The following basic building blocks are included in program system:

1) control data access system: it is intended for the dialogue with the experts and for storing the knowledge about the problem (which is investigates), about the collections of the matrices of paired comparisons and the parameters of the functions of the prognosis of priorities, and also for storing the protocols of solutions of problems;

2) the mathematical core: contains the collection of the algorithms of mathematical methods, in particular the methods of interpolation and extrapolation, the methods of calculation of the right eigenvectors of positive matrices;

3) the interface of solution of the problem: the exchange of information between the expert and the computer program. This interface gives to expert access to the knowledge, which are stored in the data base, it provides the selection of the algorithms of solution of problems;

4) the block of the agreement of the opinions of the experts: it is intended for checking the coordination of judgments and production of general opinion during the collective estimation;

5) the block of recording solution of the problem: prepares the description of the process of the solution of the problem in the form of text file.

Table 1 – Trends of a change in the priorities

Dependence of importance on the time	Verbal description of the functions	Explanations
a	Constant for all t entire value,	Relative weight does not change.
a1×t+a2	Linear increase, being increased or being decreased to a certain point value. Reciprocal value – the hyperbola.	Constant increase in the significance of one form of activity in comparison with another.
a1×ln(t+1)+ a2	Logarithmic increase to the fixed point, and then constancy.	The rapid increase (decrease). Then follows a slow increase (decrease).
a 1×exp(a2×t)+ a3	The exponential increase (or decrease, if a2 is negatively) to the fixed point and then constancy (if a2 is negative, then reciprocal value S-descriptive curve).	The slow increase (decrease), then follows a rapid increase (decrease).
a1×t^2+ a2×t+ a3	Parabola with the maximum or the minimum depending on sign a1, and then the constancy (it can be modified for the asymmetry to the right or to the left).	Increase (decrease) to the maximum (minimum), then follows decrease (increase).
a1×t^n×sin(t+a2)+a3	Oscillations	Oscillations with the being increased (being decreased) amplitude depending on n>0 (n0).

Bibliography:

1. Saaty, T. L. The Analytic Hierarchy Process/ T. L. Saaty. – Mc.Graw-Hill, 1980. – 267p.