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Designing fuzzy expert methods of numeric evaluation of an object for the problems of forecasting

The research is devoted to the models and methods of determining the numeric evaluation of the object. A fuzzy model of the problem was built that allows presenting the results of experts surveys as the intervals of change in the numeric evaluation including determining the degrees of the confidence of the experts in their opinions. Heuristics were proposed to determine those experts from an expert group, the degrees of confidence of who have certain features in their assessments. The application of such mechanism allows excluding the experts who are not confident in their opinions or who display equal confidence in all the values of the intervals, determined by them. Rules of determining collective numeric evaluation of objects in the fuzzy problem of the numeric evaluation have been developed that are based on the simultaneous consideration of competence degrees of the experts as well as the degrees of their confidence in their assessments.  A forecasting model of time series was constructed, based on the fuzzy methods of determining collective numeric object assessment that provides a possibility to consider non-systematic and poor formalized external factors. The method of simultaneous usage of classic methods of forecasting and an “expert block” was designed to solve the problem of time series forecasting that provides a possibility to include the external factors to the forecasting value which have a high impact on the forecasting value but which have not systematic character or cannot be formalized. The efficiency and expedience of application of the proposed approaches to solving the practical problems of time series forecasting were proven, on the example of the task of forecasting quantitative characteristics of HIV–infected people, registered officially in the region.


Introduction
Poor structure of considerable part of modern applied problems and tasks, the lack of reliable and representative input data and the possibility to consider non-standard and unpredictable factors during their solving underlie the need to apply such an approach as the expert surveys. The problem of determining numerical evaluation of an object plays an important role among the tasks where the expert opinions are largely used to solve them. As a rule, certain complex immeasurable characteristics of the object are assessed, which requires professional knowledge and experience of the experts. The theory of decision making described a number of rules that determine collective evaluation which are based on the processing of both the expert assessments and the values that characterize degrees of expert group competence.
Subjective nature of expert assessments, in its turn, predetermines a possibility to use the apparatus of fuzzy sets prior to collective evaluation. The successful application of models and methods that are based on fuzzy logic allows improving the results of solutions of such problems by a specific criterion. System analysis of time series forecasting methods detected a class of economical, medical and sociological problems that are needed to consider not only the internal regularity of the time series but the external significant factors too, in the design of efficient methods of forecasting. Taking into account the fact that while solving the practical tasks, it is not always possible to formalize the action of external factors due to their sporadic and unpredictable nature, it would be expedient to involve the experts to evaluate them, which makes it possible to consider additional factors which can have a significant impact on the investigated phenomenon.
Thus, a development of heuristic models and forecasting methods of time series under real conditions, which allow taking into account the experts' (group of experts) assessments, is an actual and practical important task. 15

Analysis of scientific literature and the problem statement
The task of numeric evaluation of an object belongs to the problems of the theory of decision making. A lot of methods of determining the numeric evaluation of the object are known, the mathematical calculations of which are included in [1,2]. As is shown in [2,3], some of them take into account such characteristics of a person who makes a decision and experts as propensity to risk, the level of optimism, etc. The paper [4] is devoted to axiomatic and mathematical properties of expert methods of decision making, including methods for the determination of the numerical evaluation of the object that is a continuation of the classical theory of decision making.
Analysis of modern scientific publications on the theory of decision making shows that the research and application of problems of the determination of the numeric object evaluation under fuzzy conditions is rather actual. Thus, methods of processing of fuzzy expert information are offered in [5], which allow presenting subjective opinions of experts in the form of fuzzy sets. Such methods, as a rule, allow including the questions in the expert form, where words or opinions can be considered as the answers to those questions. The paper [6] contains fuzzy analogues of the methods of collective expert assessments that allow considering different types of inaccuracies and ambiguity in the experts conclusions. Possibilities and the ways of applying of fuzzy expert methods are shown in the work [7].
Also, coefficients of experts' competence, as a rule, influence the final evaluation significantly, which are aggregated evaluation of such indicators as the expert professionalism, confidence degree for the expert, the results of expertises made by him/her in the past, etc. There are different methods of defining the competence of experts. Thus, the paper [8] contains the classification and analysis of a priori methods of determining experts' competence. According to this classification, a priori methods are divided into heuristic (methods of self-assessment, mutual evaluation), experimental and documentary. The work [9] is devoted to the posteriori methods of defining the experts' competence. A separate group includes methods based on the axiom of unbiasedness which is described in the paper [10]. The method of calculation of confidence for an expert based on the method of paired comparisons is shown in the work [11].
The work [12] deals with research and design of forecasting methods of time series. As a rule, statistic and intuitive approaches are used in the design of methods of forecasting [13]. However, in the case when the forecasting is made under conditions of the action of random poor formalized factors, it would be expedient to use the results of expert surveys as the additional source of information.
It would be expedient to design such methods of time series forecasting that provide a possibility of simultaneous consideration of both the internal regularity of the time series and the impact of systematic and unsystematic poor formalized random factors on it.

The purpose and objectives of the study
The aim of the research is to develop heuristic rules of defining the numeric evaluation of the object based on fuzzy expert assessments to improve decision making in the cases when the results of expert surveys are used for forecasting and applying certain numerical characteristics of the objects.
To achieve the set goal, the following tasks were defined: -to build a mathematical model of the problem of defining the numeric object evaluation; -to develop heuristic rules of defining the numeric object evaluation that make it possible to consider certain subjective characteristics of experts and their assessments as well; -to explore a possibility of using fuzzy experts assessments to solve the problems of time series forecasting; -to carry out experimental validation of the developed rules of the numeric evaluation of the object.

Setting and analysis of the problem of the numeric evaluation of the object
Consider the problem of numeric object evaluation [2]: set the object O.
It is necessary to determine its numeric evaluation by some criterion K.
Engage a group of experts k to determine the evaluation: , their competence coefficients are, accordingly Experts' competence coefficients can be set by a person who makes decisions or defined with the help of known methods [8][9][10][11].
Consider the case when each of the experts i E (i 1, k = ) sets the following values during the numeric evaluation of the object: -the interval of changing of the object evaluation 1i 2i a , a     ; -1i 2i , 0; 1 m m Î    -degrees of confidence in their evaluation 1i a and 2i a accordingly; is the highest. It would be possible to use heuristics at the degree of preliminary processing of expert assessments that allow excluding from consideration those experts with the certain features in their degree of confidence [14].
The following heuristics are proposed: Heuristics 1. Excluding those experts from the group E who have the coefficient of confidence in their evaluations lower than the predetermined level. That is: is the predetermined level. Heuristics 2. Excluding those experts from the group E who have the coefficient of confidence in their evaluations higher than the predetermined level. That is, where ( ) is the predetermined level. Heuristics 3. Excluding those experts from the group E who are equally confident in their evaluations. That is, where ε is the predetermined value of the range of confidence degrees.
Assume that the group of experts has the following view after doing Heuristics 1-3: and define the coefficients of confidence of appropriate experts (i 1,n = ) through i a . Calculate the total competence of the group of experts E by the formula: If the compiled group of experts is incompetent (i. e. a is not high enough according to the point of view of a person who makes decisions), then further task to solve the problem of defining the numeric object evaluation would be inexpedient. In the opposite case, it is proposed to fulfill normalization of the values of experts' competence coefficients by the following formula: Experts' coefficients of competence normalized by (4) are used to determine the numeric evaluations of objects.

Fuzzy rules of defining collective evaluation
Next step during the determination of the numeric object evaluation is formation of fuzzy set . Membership functions of fuzzy sets are determined by the following rule: a , a , b , b are determined by solving the appropriate systems of linear algebraic equations.
As is seen from (5), in general case membership func- will have piecewise linear form. The following rules are proposed to determine a collective numeric evaluation of the object: Rule 1. Fuzzy set of collective numeric object evaluation is calculated in the following way: The algorithm of the calculation of membership function of the resulting fuzzy set should boil down to defining the total of the corresponding membership functions in the intervals of their monotony.
i. e., the membership function as the degree of confidence in the value of the numeric evaluation will signify the degree of confidence of the competent majority of the experts. Rule 3. Determine dephasified numeric evaluation for each expert. Let i a is the evaluation of the i-th expert. Next, perform calculations similar to the accurate case.
Rule 4. According to the accurate methods of determining collective numeric object evaluation, calculate the total of fuzzy numbers: Models and methods used frequently to solve the problem of time series forecasting include: Winters method that takes into account season components; -regression model of forecasting; -autoregression method; -method of least squares with weights; -Brown forecasting models, etc. The method of synthesis of a forecasting scheme based on the basic forecasting models with underlying mechanism of competition in a given period of time was proposed in the papers [13,15] and the resulting forecasting scheme is a convex linear combination of the winner' models with respective weight coefficients.
As shown in the papers [13,15], this approach allows increasing the quality of time series forecasting.
Classic methods and schemes that are formed on their basis, in many cases do not consider the impact of poor formalized external random factors in full, for example, during forecasting social -economic and medical indicators.
Such external factors do not have permanent or seasonal character in certain problems and should be forecasted themselves, however, they can produce significant effect on the predicted value and thus decrease the accuracy of the forecast which can be obtained by the methods of forecasting. It would be expedient to use the results of expert surveys for such problems as one of the sources of information for solving a forecasting task.
A model of time series forecasting is proposed with an "expert block" as a component of it.
where 0;1 a Î    is the value set by a person who makes decisions. This value expresses the level of impact of experts' assessments on the result of forecasting.

Experimental validation of the research results
Let us consider a problem of forecasting quantitative characteristics of HIV-infected people, officially registered in the region [13].
Values of these indicators in Zakarpatska Region in Ukraine over the period from 1987 to 2014 are given in Table 1.
To compare the results of forecasting by different methods, let us take the values from 1995 to 2009 as a learning sample. And we will compare the values from 2010 to 2014 to the results of forecasting. Table 1 Values of the main quantitative characteristics of HIV-infected people, officially registered in the region [16] Year Let us solve this problem by the method of synthesis of forecasting scheme based on basic models [15]. Let us take auto regression method, method of least squares with weights, Brown method (linear and quadratic model), Winters method as the basic models.
To evaluate the quality of forecasting models (basic and synthesized by a forecasting scheme), the mean square error (Δ) and the average relative error (μ) are used for the different steps of forecasting (τ) that are calculated by the following formulas, accordingly: ( 1 5 ) Values of forecasting errors are given in Tables 2, 3. Table 2 Values of forecasting errors for the indicator "New cases of HIV infection"  Table 3 Value of forecasting errors for the indicator "Number of HIV-infected people by the end of the year" The results for indicators "New cases of AIDS" and "Number of AIDS patients by the end of the year" are similar. Since the best forecast is obtained with the help of synthesized forecasting scheme [15], then the results of forecasting of quantitative characteristics of HIV-positive people registered officially in the region by the synthesized forecasting scheme is taken as the basis. The results of the forecasting are in Fig. 1, 2 Table 4. In spite of the forecasting errors D and m being insignificant, the real data differ significantly from the forecasted ones. Table 4 Forecasted values

Category
Step Let us solve the problem by using a proposed method at t 1 = . Compile a group of 5 experts Coefficients of their competence and the results of the expertise are given in Table 5. . As the real value of the indicator "New cases of HIV infection" in 2010 is 62, so it is evident that using the formula (13) at any ( ) 0;1 a Î will improve the forecasting results. Similarly one can determine the forecasted values of the indicator "Number of HIV-infected people by the end of the year" on the basis of expert assessments. The results of the forecasting by the proposed scheme depend on the competence of the expert group and the degree of the impact of the "expert block" on the final forecasting value, which is determined by a person who makes decisions.

Discussion of the results of the research of methods of determining the numerical evaluation of the object
Fuzzy model of the problem of determining collective numerical object evaluation is built, in which experts' conclusions are presented as the interval of change in evaluation taking into account the degrees of experts' confidence in their results.
Applying proposed heuristics for the analysis of the results of experts' surveys allows disregarding the opinions of those experts, the degrees of confidence of which have certain features. Such mechanism provides decrease of the impact on the research from those experts who are not confident in their assessments or who are not able to detect the most probable one. The choice of the rule for setting the collective numerical object evaluation depends on the features of a person who makes decisions and on the problem character.
The model of forecasting was designed during the research that is a continuation of the forecasting scheme [15]. The numeric object assessments are used in it with the aim to provide a possibility to consider the external factors in the forecasting, which have a high impact on the forecasting value, but do not have systematic character or can't be formalized. Generally, such cases occur in the problems of forecasting social and economic phenomena.
The numeric experiment was carried out to fulfill the analysis of the efficiency of using the expert methods of determining collective object evaluation that showed that the application of methods which considered only internal features of the series leads to inaccurate forecasting values in the problem of forecasting the quantity of HIV-infected people in the region by the end of the year. However, using an "expert block" for this problem allowed applying the knowledge and experience of the experts and improving the forecasting results.
The advantage of the proposed approach is that at solving the problem of time series forecasting, it could be possible to simultaneously use mathematical, fuzzy, neural network methods of their forecasting with the use of time series forecasting and the numeric object evaluations as well as the results of experts' surveys.
The proposed scheme of forecasting is sensitive to the results of experts' surveys and depends on the competence of an expert group and a person who make decisions.

Conclusions
The research of the problem of determining the numeric object evaluation was performed. The aim of the research is a development of heuristic rules of determining the numeric assessment of the object based on the fuzzy experts' evaluations to improve the efficiency of decision making for the problems where the results of experts' surveys are used and which are related to the forecasting or evaluation of certain numeric characteristics of objects. During the research, the following was achieved: 1. A fuzzy mathematical model of the problem of determining the numeric object evaluation was built that allows presenting experts' conclusions as the interval of change in the evaluation considering the degrees of experts' confidence in the results.
2. Heuristic rules of determining the numeric object evaluation were compiled, which include the cases when the experts from an expert group are not confident in their assessments that allow reducing the impact of such factors on the expertise results. 3. A model of time series forecasting was built, based on the fuzzy methods of determining collective numeric evaluation of the object, which provides the possibility to consider both internal series trends and unsystematic and poor formalized external factors.
4. Experimental validation of the developed rules was performed on the example of the problem of forecasting the quantity of HIV-infected people registered officially in Zakarpatska Region. The efficiency and expedience of inclusion of an "expert block" during determining the forecasting values were demonstrated. Therefore, the developed models and methods can be used effectively in both solving the problems of determining collective numeric evaluation of the object and forecasting the time series.

Introduction
With the development of information technologies and the globalization of the information society, the concept of a start-up develops, which is closely connected with the modern economy. Assessment of a start-up project is in the evaluation of an "idea", which may bring profits in future. The cost of such "idea" cannot be evaluated (commercially) without introducing it to the market. Yesterday's "bad idea" may turn successful today, due to the rapid changes in the industry or areas of business where it is implemented. In this regard, the problem (poorly structured now) of assessment of efficiency of start-up projects arises, the solution to which is interested for either venture funds or the startuppers themselves.
Any startup is a project at the level of "ideas", directed into the future by definition, and the project that has strategic character is directed into distant future. The future cannot be predicted at one hundred percent while the decision about the launch of implementation of the project (start-up) must be taken today. Therefore, there is an actual problem of the development of the models for the assessment of startups under conditions of information uncertainty.