Candidate of technical sciences Р. V. Tereliansky

Volgograd state technical university, Russia

PROGRAM APPLICATION OF THE NUMERICAL METHOD OF PREDICTING OF THE DYNAMICS

OF THE PRIORITIES

Developing the concept of business, economists increasingly more frequently attempt to consider not only classical indices and indicators, but also such difficultly formalized concepts as ecology, ergonomic, quality, visual attractiveness, political and social situation and other. The estimation of such criteria is connected with the analysis of the incomplete, nonparametric and weakly-structure expert’s knowledge usually. Consequently, it is necessary to develop information technologies and instrumental means for increasing the optimality of the administrative solutions at all levels of the economics[1]. The universal program’s decision support systems are the powerful instrument which is based on analysis of similar economic processes and systems [2]. The estimation of strategy (or, generally speaking, scenario, object, alternative, solution) according to many criteria means that the expert pursues more than one task, and these tasks can be the different degree of importance. It is not impossible to reduce  all sets of characteristics into the main criteria naturally, in this case. Strategies of the actors (alternatives) xi, i=1,n, can be represented by scalar, vector, matrix or even by more complex formation, in the general case. Let us examine the case, when strategy of the operating side is represented by the n-measured vector X=(x1,x2,…,xn), and the intensity of the actions of the operating side is evaluated by many local criteria of the quality K=(k1,k2,…km), the intensity of effect of which on the general system is W=(w1,w2,..wm). Any local criterion k is connected with strategy (alternative) with the mapping f=F{X,A}, furthermore, any criterion is connected with many other criteria with the mapping g=G{K,A}, where A – many fixed factors. The multicriterional task of decision making is described by the following collection of the information S=(X,K,W,F,G,P), where P – the formulation of the problem or the purpose of the research. It is appropriate to isolate the subsystem of preferences (criteria of quality) from the system S Sp=(K,W,G), which is frequently investigated separately from sets of alternatives, for example, when it is determined the collection of the limiting factors (overall sizes, weight, power, etc) in the technical task, and engineers have to solve the problem of the maximization of the correspondence of several versions of devices to the limitations, which were given. Both the properties of strategies (alternatives) being investigated and the properties of the very system of the preferences Sp can change in the course of time. Only mapping F is a function of time – in the first case, only mapping G – in the second. Furthermore, the version, when mappings G and F are functions of time is possible, too. How to find optimal solution in this case? The principle of the optimality of the solution is the mathematical model of the principle of a compromise, which was accepted in the task. The experts think that all local criteria are normalized (i.e. they have identical dimensionality or are dimensionless quantities) before the analysis of the diagram of a compromise. The alternatives, which have uniform quantitative characteristics, are led to the identical dimensionality by linear rate setting. The linear rate setting is the operation when the quantitative values fill the vector with themselves W'={w1',w2',..,wn'}, where n – number of alternatives, а w'i – the quantitative value. Then vector W' is normalized, and the vector of the priorities as a result is obtained: W={w1,w2,..,wn}, where wi=w'i/S, аnd . Moreover, if we search for the best alternative with the maximum value of this characteristic, then vector is normalized directly with these quantitative assessments, and if vice versa (the less the given quantity, the better), then each element of vector W' is substituted by the reverse to it value and rate setting occurs only after this. One of the methods of normalizing the non-parametrical local criteria is the method of paired comparisons. Let A1, A2, ...,An – set of n of elements (alternatives) and v1,v2,...,vn – accordingly their weight or intensity. Expert will carry n(n-1)/2 judgments formed the square matrix, which contains the paired comparisons, where n – the order of matrix is equal to the number of comparison elements. The matrix of paired comparisons (MPC) possesses, as a rule, the property of reverse symmetry, i.e., aij = 1/aji, since aij = vi /vj. Reverse symmetry is expressed either in the form of proper fraction or in the form of the negation of the direct estimation aij=-aji. Numerical value of the relation vi/vj for the non-parametrical parameters is expressed with the help of a certain verbal scale, which elements are correspond to the specific numerical number. It is possible to use such expressions as “equal superiority” or “significant superiority”. The number from 1 to 9 corresponds to each of such verbal estimations. not only The subjective indices of the paired estimations of elements can change in the course of time, but also their objective weights can change. For example, the estimates of object can grow, in the course of time, its mass will change, and length will decrease and so on. It is possible to select the functional idea of this regularity and to obtain a change of the vector of priorities in the time, if the change of the given metric quantity corresponds to any regularity, for the quantitative assessment. Thus, into the functional determination of the weights of alternatives is substituted the value of moment of time and the vector of priorities led to one is calculated, at the given specific moment of time. It is possible to obtain the functional idea of the dynamics of preferences through the approximating points in the obtained massif of vectors [2,3]. This approach makes it possible to use statistical information for decision making.

Fig. 1. Numerical method of predicting the dynamics of the priorities

In the figure (Fig. 1) the upper and subscripts Aij are designate, that j alternative was evaluated according to i criteria, where r – number of alternatives, p – number of criteria. Indices of the element of the vector of the priorities shows that w is the weight of j alternative by i criteria in k moment of time, where k=1,...,T, and T – a quantity of moments of time. The cardinal number of the set w(t) it makes it possible to carry out regression analysis, with a study of the behavior of system in the wide intervals of time. It is possible to obtain the functional idea of the dynamics of the priorities of alternatives according to any criterion as a result. A change in the judgments can be estimated expert with the use of the following functions. Constant increasing of one form of activity in comparison with others – a1×t+a2. A rapid increase (decrease) in the importance, which follows a slow increase (decrease) – a1×ln(t+1)+a2. A slow increase (decrease) in the importance, which follows a rapid increase (decrease)  – a 1×exp(a2×t)+ a3. Increase (decrease) in the importance to the maximum (minimum), then decrease (increase) – a1×t2+a2×t+a3. Oscillations of importance with the being increased (being decreased) amplitude – a1×ta2×sin(t+a3)+a4. Parameters ai of these function can be established so that the region of the allowed values would not fall outside the boundaries of the established intervals in temporary the section in question T. These functions reflect changes in the trend: the constant, linear, logarithmic and exponential, parabolic, and oscillating, also. The analysis of the resulting global vectors of priorities (after hierarchical convolution) makes it possible to build the forecast of the behavior of the system of preferences which was being investigated. Calculations (Fig. 1) of the local and global vectors of priorities are required many iteration for the achievement of required accuracy. The program system, which makes it possible to introduce the description of the system of the preferences in the form of hierarchy, to introduce and to edit the set of the matrices of the paired comparisons, elements of which would be the functional idea of the dynamics of priorities, was created. The eigenvectors of the positive matrices of paired comparisons are calculated with the use of an iterative (through the limit of the relation of the works of MPC to the unit vectors and the column vectors) or approximate (through the geometric mean) algorithm. The approximate algorithm of calculation gives the result with accuracy to of the order of the ranking of elements and is used for the large hierarchies and the prolonged forecast’s intervals. The analysis of the set of the obtained vectors of priorities is produced by the method of least squares. The results of analysis are represented in the form the set of graphs and table, which contains the parametric representation of the selected dependences of priorities on the time. Program product is written with the use of a system of programming Borland C++ 4.5 with the use of a library OWL 2.0 for Windows.

 

Literature:

1.                 Терелянский, П.В. Информационные технологии прогнозирования технических решений на основе нечетких и иерархических моделей : монография / П.В. Терелянский, А.В. Андрейчиков. – Волгоград : ВолгГТУ, 2007. – 204 с.

2.                 Терелянский, П.В. Информационные технологии прогнозирования технических решений на основе иерархических моделей : монография / А.В. Андрейчиков, П.В. Терелянский, О.Н. Андрейчикова.– Волгоград : ВолгГТУ, 2004. –156 с.

3.                  Терелянский, П.В. Нечеткие модели и средства для принятия решений на начальных этапах проектирования : монография / А.В. Андрейчиков, П.В. Терелянский, А.М. Шахов. – Волгоград : ВолгГТУ, 2004. –140 с.