Сandidate of economic sciences, docent

Matraeva L.V.

Tula branch of RSUTE,RUSSIA

Economic-mathematical mechanism of choice of optimal decision under uncertainly

Mathematic formulation of problem choice of optimal variants of process developing is difficulty. It's connected with not only high uncertainly which consist in forecasting future, but with wittingly opposite criterions. It’s human nature to want exclusive things, for instance, maximum profit and minimum costs, but as known it never succeeds, because minimum costs equal zero.

As the choice of any mathematical method in economic area was usually realized in the light of its implementing, then it’s essential to analyze informational situation, in which decision is made.

1. Decision making under certainly. This situation is characterized by deterministic connection between accepted decision and received result.

2. Decision making under risk. In that case each way of subject of management interaction can lead to one of the possible outcomes. And each outcome has some occurrence probability.

3. Decision making under uncertainly. In that case efficiency index depends on choosing locating strategy, models, random factors and indefinite factors, for which it's known only set of possible values. That’s why each variant is connected with set of possible outcomes, whose frequency either unknown or known with nonsufficient accuracy or meaningless.

The last variant is considered because it more commonly occurring. The first two situations are easy in choice. And there are a lot of teaching aids.

The choice of optimal decision in the third variant can be related to multicriterion problem under uncertainly and it can be described, with using game-theoretical modeling which takes into consideration lack of environment information and necessity of making decision under risk or uncertainly.

The last research’s results in application areas of above mathematical methods revealed that methods basing on different approaches give diverse results at the same information [1]. But decision-making methods basing on fuzzy model have the largest initial stability and adequacy in multicriterion problems solving under uncertainly.

This approach is considered to be the most acceptable under realization of considered method because of using classical statistical methods in our case is difficulty due to lacking of statistic data or small range of some parameters, which result from uniqueness of each research subject. In addition scientists' works (Vishnevsky R.В., Gracheva M.V.[2], Orlov A.I. [6, 7]) notice that now statistical models can’t secure high precision, which considerably exceeds precision of Delphi approach, but their using is more expensive then the last. While fuzzy logic's using will give opportunity to process and use experts’ answers in further calculations, which carry non-numeric type. Non-numeric type of expert’s answer is considered to possess greater extent of precision, as human uses images, words, but not figures.

Thus, the problem of optimal decision’s choice is multicriterion problem of alternatives' choice on basis of expert estimation of alternatives on specified criterions. To solve this problem it’s crucial to use additive convolution method.

In this case economic-mathematical model's construction can be divided into following stages:

1. Definition of alternatives' set;

2. Definition of criterions' set for alternatives' estimation;

3. Definition of criterions' relative importance;

4. Estimation of alternatives on specified criterions;

5. Economic-mathematical formulation of the choice's rule of the best alternative.

1. Definition of alternatives' set.

This stage describes “m” number of considered variants. So we have the finite set of alternatives: ₍₁₎

2. Definition of criterions' set for alternatives' estimation. This stage shows up “n” criterion, which allows estimating each variant.

, (2)

3. Definition of criterions' relative importance. Detection of considered criterions’ relative importance occurs on basis of expert survey data.

Experts have to estimate the importance of every criterion. In the process of researching there was the question about scale's type. The importance of picking out criterions can be estimated with this scale, because the scale's type sets the group of possible transformations.

The analysis of modern researching in the sphere of experts values revealed that it is possible to voice the experts in the linguistic scale. So «… Numerous experiments showed that the person answers more rightly for the qualitative questions than the quantitative ones. So, it is easier to say which of the two weights are heavier than to show their approximate weight in grams» [8].

Creating of model importance assessment process's on basis of linguistic variables includes:

- Creating dictionary, which establishes relation between linguistic description and fuzzy numbers;

- Creating transformation's box of linguistic variables to fuzzy numbers on basis of dictionary.

For estimation of criterions' relative importance the experts offered further basic ordinal scale:

- Practically unimportant (PU);

- Not very important (NVI);

- Rather important (RI);

- Important (I);

- Very important (VI).

So, for estimation of criterions' relative importance linguistic variable “W” was put in: W= {Practically unimportant; Not very important; Rather important; Important; Very important}. Let us assume that terms' values of set are defined by fuzzy numbers, which have trigonal form of membership function (Pic. 1)

Picture 1 – Terms' membership function of criterion importance.

In this case transforming of linguistic variables to fuzzy numbers looks like:

Practically unimportant = {1,0/0,0; 0,0/0,1}

Not very important = {0,0/0,0; 1,0/0,2; 0,0/0,4}

Rather important = {0,0/0,3; 1,0/0,5; 0,0/0,7}

Important = {0,0/0,5; 1,0/0,7; 0,0/0,9}

Very important = {0,0/0,8; 1,0/1,0}

After processing of expert survey's data we have the total evaluation of above criterions' relative importance in trigonal fuzzy number's form. Expert survey's data are aggregated in table 1 and the numbers of linguistic variables, which belong to each terms are calculated.

Table 1 – Aggregate matrix of linguistic variables' processing.

Terms description	Repetitions' quantity for each criterion
	1	2	3	4	5
Practically unimportant
Not very important
Rather important
Important
Very important

The calculation of complex importance estimation of j – criterion is performed by a formula:

, (3)

where _{- the membership function of unique importance’s estimation j – criterion;}

N – experts’ number;

i – sequence numbers of term i = 1,n;

m_A(i)(X) – the membership function of i – term;

_{- the algebraically sum of fuzzy subset;}

_{-the number of reiteration i – term in the result of importance j – criterion by the experts;}

₍₄₎

_{4. Estimation of alternatives on specified criterions}

_{In this case the criterions are some terms, and alternative estimations are to the right degree these terms.}

_{That is why the multitude’s A estimation of alternative variants in accordance with seted criterions C can be expressed by a formula:}

₍₅₎

_{Where i – the number of alternatives}

_{j – the number of criterions}

_{- the estimation of i – alternative on the j – criterion. It is represented the fuzzy number}

_{- considered estimation of i – alternative.}

_{The scheme of calculating the S ij index is the same that the W j index is.}

_{The experts have to estimate the extent of the j – criterion satisfying on the i – alternative.}

_{Estimating the alternatives with the criterions is the same.}

_{The linguistic variable quantity S equal «satisfactoriness» and it it seted as the multitude's values.}

_{There are {Extremely Low; Low; Average; High; Very High}. The values conversion from linguistic variable quantity to fuzzy numbers for variable quantity is the same (it is at the picture 2):}

_{Extremely Low = {1,0/0,0; 0,0/0,1}}

_{Low = {0,0/0,0; 1,0/0,2; 0,0/0,4}}

_{Average = {0,0/0,3; 1,0/0,2; 0,0/0,4}}

_{High = {0,0/0,6; 1,0/0,8; 0,0/1,0}}

_{Very High = {0,0/0,8; 1,0/1,0}}

_{Picture 2 – The terms’ membership function of satisfaction's degree alternatives.}

_{As a result of this stage the considered estimation of i – alternative (S i) is calculated.}

_{The latter one is the liner combination's result of fuzzy numbers and it has also the membership function as triangular type.}

5. Economic-mathematical formulation of the choice's rule of the best alternative.

Fuzzy set's membership functions (S_i) were obtained as result of realization on previous stages. This set (S_i) sets limits on decision variables. It's necessary as result to take one value, which should be realized. So the problem is to transform fuzzy subset to scalar. According to Zada’s offer [3] logical criterion of selection is to choice as result such basic variable value in which membership function will reach its maximum. This criterion was used successfully in applied research.

To accept alternative the degree of satisfaction to requirements shouldn’t be below level “HIGH” or below 0, 8 in transforming fuzzy number to scalar.

If no one alternatives satisfied condition, then no one considered variants are optimal. Offered economic-mathematical mechanism of choice of optimal variant has some advantages:

- Allows formalizing process of choice of optimal variant;

- Is universal: adding new alternatives or criterions doesn’t change method of calculations;

- Gives opportunity to process and use expert’s answers in further calculations, which have fuzzy form.

The list of bibliography cited

1. Andrejchikov A.V., Andrejchikova N.O. Analysis, synthesis, planning of decisions in economy. - М: Finance and statistics, 2000. - 368 p.

2. Vishnevsky R.В., Grachev M.V. Use of the device of indistinct mathematics in a problem of an estimation of efficiency of investments. - www.fuzzylogic.ru <http://www.fuzzylogic.ru>.

3. Zade L.A. Concept of a linguistic variable and its application to acceptance of the approached decisions. - М: Mir, 1976.

4. Mikrjukov V.U., The theory of interaction of economic subjects. - М: the High school book, 1999. - 96 p.

5. Processing of the indistinct information in decision-making systems / A.N.Borisov, A.V.Alexey, G.V.Merkureva, etc. - М: Radio and communication, 1989. - 304 p.

6. Orlov A.I. Organisational and economic modeling: Part 1 Non-numerical statistics. –M: 2009. – 542 p.

7. Orlov A.I. Representational theory of measurements and its application. - antorlov.euro.ru.

8. Orlov A.I. Modern stage of development of expert estimations' theory. - antorlov.euro.ru.

9. Orlov A.I. Stability in social and economic models. Saarbrücken (Germany), LAP (LAMBERT Academic Publishing),, 2011. - 436 p.