Introduction

According to a study [1], the largest percentage of organizational knowledge (about 42%) is in the minds of specialists. One of the features of organizational management systems (OMS) is the presence of human factor influences within the object and subject of management [2,3]. Along with objective information (e.g., instrument readings), 5 categories of specialists are also widely used as sources of information in OMS [2]: "source" analysts, analysts for aggregation (generalization) of information to support decision-making at different levels of management, experts, organizers of expertise and knowledge engineers. Thus, an urgent task is to develop a mathematical apparatus for the OMS, which will allow to use qualitatively the knowledge of all these categories of specialists about the formation of an information resource by taking into account their characteristics.

Formation of an information resource of the OMS considering the peculiarities of expert knowledge

The activities of the OMS specialists involved at different levels in the processes of generating, processing (aggregating) and analyzing information (data) let's call as analytical activities. To achieve the purpose of this paper, it is necessary to identify and analyze the types of analytical activities characteristic of different hierarchical levels of the OMS.

In [1], a generalized scheme for the formation of an information resource of a certain OMS is presented, which, after shifting the emphasis to determine the place of analytical activities of personnel in it, looks like as one shown in Fig. 1. Analysis of Fig. 1 shows that at least five types of analytical activities and, accordingly, groups of specialists can be distinguished within the framework of the OMS. A "source" analyst is an analyst who carries out his/her analytical activities within one element of the OMS based on direct observations of indicators characterizing the state of the relevant element of the system and/or data on monitoring of environmental factors. As a result, such an analyst forms judgments about the state of the system element or the state of individual processes and phenomena observed in the external environment of the OMS functioning, based on his or her own ideas and experience. As a rule, such judgments are formed by determining one of the possible states of an element (or a component of the external environment) from a set of predefined states. The judgments of "source" analysts are mainly based on objective observation (monitoring) data, but the source data contain subjective distortions of information as a result of cognitive activity.

The second group includes aggregation analysts, who carry out analytical activities by generalizing (integrating) data (information) from "source" analysts. They form judgments about the state of different subsystems or groups of elements of the OMS and, accordingly, are mostly based on subjective data. Since different subsystems can be distinguished within different OMS and hierarchical links between them are possible, it is obvious that the activities of such analysts form a hierarchical procedure of aggregating information with obtaining judgments on the state of subsystems (state of the environment) of varying degrees of generalization. As a result, the subjective distortions contained in the initial judgments (results of analytical activities) may increase due to the imposition of their own cognitive distortions on the cognitive distortions of the "source" analysts. As a result, the information uncertainty about the actual state of the phenomena being analyzed may increase.

Reducing the cognitive distortions of information that are characteristic of the two groups of experts can be ensured by formalizing the process of obtaining and aggregating it to the maximum extent possible, as well as by jointly processing duplicate information from several analysts. For such information processing, tools developed within the framework of fuzzy set theory and fuzzy logic, which were specifically designed to work with expert judgments, can be effectively used. The second way is to reduce the workload on one analyst, taking into account the psychophysiological limitations of a person through the rational distribution of functional tasks among analysts.

Knowledge engineers and organizers of the examination must first familiarize themselves with the relevant subject area before building the knowledge base. For this purpose, they mostly use available open sources (the Internet, articles, lectures, ets), as well as specialized tools, such as content monitoring systems (CMSs). In this case, the cognitive bias "the effect of the illusion of truth" may be triggered [2]. Thus, it is inappropriate to form expert opinions in the OMS based directly on data from popular sources. It is more reasonable to use formalized methods of processing incomplete expert information when conducting an expert assessment. In addition, the expert evaluation organizer selects a group of experts with a sufficient level of competence in the subject area, and further work on building the LR takes into account the level of competence of the expert in each of the expert evaluation issues. In this case, a cognitive bias may be triggered -the Dunning-Kruger effect [2].

Experts decompose the subject area, divide the goals into sub-goals, determine the criteria and factors that directly affect the outcome of the examination. This process is characterized by the following cognitive distortions: the Ringelman effect, the "focusing" effect, the "survivor's error", and the "bicycle shed" effect [2]. To avoid them, it is advisable to use systems for distributed collection and processing of expert information in the work of an expert group. Expert evaluation aims to reliably determine the degree of preference between alternatives/criteria. Direct scoring and pairwise comparisons are possible. The following cognitive biases should be avoided: the distinction error, the anchoring effect, and the contrast effect [2]. You should also take into account George Miller's research [3] on the limitations of human short-term memory.

Aggregation of analysts' knowledge

Here are the heuristic requirements for aggregating data from analysts, which, unlike data from technical sources, are additionally characterized by varying degrees of reliability.

1. If different data is received from the same analyst during the same management cycle, the data is not combined and only the last value of the indicator is considered 2. If several analysts send same data messages with different degrees of reliability (confidence in their truth), the result of their combination should be characterized by greater reliability than each individual message 3. If several analysts provide data that does not match in content, the result of combining them should be characterized by no more reliability than the highest "declared" reliability in the messages 4. If the content of the data provided by the analysts differs significantly, there should be a growing possibility that the indicator actually has a different, "intermediate" value that averages the data in some way, giving more credibility to the value that tends to be more reliable than the "input" message. In this case, each message about the value of the indicator is formalized as a fuzzy set:

𝑆 = {𝜇 (𝑥 )/𝑥 }, 𝑥 ∈ 𝑋 (1)

where 𝜇 (𝑥 ) = [0,1]is the degree of belonging of the value 𝑥 to the fuzzy set 𝑆 𝑋 = {𝑥 , 𝑥 , … , 𝑥 } is a crisp set that is a carrier of the fuzzy set and contains N possible (valid) values of the indicator.

For the final formalization of the message about the value of the indicator in the form of a fuzzy set, in addition to the indicator carrier set, it is desirable to specify the degrees of membership (𝑥 ) for each of the possible values in the form of a functional dependence. There are many different ways to define a membership function for fuzzy sets, the simplest of which is a triangular function, which is described as follows:

𝜇 (𝑥 ) = ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 0, 𝑖𝑓 𝑥 ≤ 𝑎; (𝑥 − 𝑎) (𝐶 − 𝑎) , 𝑖𝑓 𝑥 < 𝑥 < 𝐶; (𝑏 − 𝑥 ) (𝑏 − 𝐶) , 𝑖𝑓 𝐶 ≤ 𝑥 < 𝑏; 0, 𝑖𝑓 𝑥 ≥ 𝑏. (2)

where To take into account the reliability of the message, it is necessary to set a numerical correspondence (scale) to the defined linguistic values of reliability (confidence in the value of the indicator). An example of such correspondence is shown in Table 1.

C

Table 1 Linguistic meanings of the confidence attribute and their corresponding numerical values

The linguistic meaning of the attribute of confidence

The numerical value of the confidence attribute (Ds) "Reliable" 1 "Probably" 0,7 "Maybe" 0,5 "Doubtful" 0,25

In the next all the values of the membership function are normalized by the corresponding numerical value of the confidence in the expression: 𝜇 (𝑥 ) = 𝐷 * 𝜇 (𝑥 ).

(3) Since this transformation of the independence function is linear, it will retain its triangular shape. Having carried out the specified simple formalization of all incoming messages from analysts, you can proceed to solving the problem of their unification. Within the framework of the theory of fuzzy sets, a significant toolkit for working with fuzzy sets has been accumulated. The operation of the algebraic sum of fuzzy sets best corresponds to the unification rules formulated above. To combine two messages A and B, the corresponding expression would look like this: 𝜇 (𝑥 ) = 𝜇 (𝑥 ) + 𝜇 (𝑥 ) − 𝜇 (𝑥 ) * 𝜇 (𝑥 ).

(4) The last operation is to obtain the combined value of the indicator and determine its reliability, for which the expression for obtaining the weighted average of the fuzzy set 𝜇 (𝑥 ) can be used:

𝑥 = ∑ 𝑥 • 𝜇 В (𝑥 ) ∑ 𝜇 В (𝑥 ) . (5)

This operation is called defuzzification of a fuzzy set. The reliability of the resulting value will be determined by the degree of membership of the weighted average obtained:

𝐷 = 𝜇 𝑥 .(6)

The proposed approach will reduce the influence of the subjective component on the result of data processing from analysts.

Aggregation of experts' knowledge

When forming the information resource of the OMS, certain features of expert knowledge should also be taken into account [6]. During the expert evaluation, cognitive distortions of data and knowledge may occur, which significantly affect its result. Human psychophysiological limitations limit the ability to process more than 9 objects at the same time [3]. Expert evaluation is time-consuming and costly, so it should be used only when absolutely necessary. If possible, it is recommended to use previously built knowledge bases (KBs), their fragments and templates (precedents) for solving similar problems in retrospect. Experts may miss assessment sessions, not answer some questions due to limited time, busy schedule, fatigue or unwillingness. Therefore, it is important to be able to process incomplete expert information [7]. At the same time, the need for generalized and systematized knowledge is gradually revealed through decomposition at smaller levels of the OMS hierarchy, and detailed information is aggregated from the bottom up to meet the information needs of users at higher management levels.

To aggregate expert knowledge, the method of goal dynamic evaluation of alternatives (MGDEA) [8,9] is used for processing of a hierarchy of goals [10]. The hierarchy is the result of decomposition of the main goal into components (subgoals), which, in turn, are also decomposed into subgoals, etc. The decomposition process stops when we get specific activities (projects) as components.

The MGDEA offers a generalized procedure for determining the degree of achievement of any hierarchy goal at a given time t. To determine the degree of achievement of a particular goal, it is necessary to analyze the degree of achievement of the goals that directly affect this goal for each subset of compatible goals [8]. Thus, the degree of achievement of the h-th goal at time t is described by the formula for di (t):

𝑑 (𝑡) = ⎩ ⎪ ⎨ ⎪ ⎧ 0, 𝑖𝑓 𝐷 (𝑡) < 𝑇 , 𝑇 , 𝑖𝑓 𝐷 (𝑡) = 𝑇 , 𝑓 𝐷 (𝑡) , 𝑖𝑓 𝑇 < 𝐷 (𝑡) < 1 − ∑ 𝑤 ( ) 1, 𝑖𝑓 1 − ∑ 𝑤 ( ) ≤ 𝐷 (𝑡) ≤ 1, ,(7)

where 𝐷 (𝑡) = sup ∑ 𝑤 ( ) 𝑑 (𝑡) ;Ti -is the threshold for achieving the i-th goal; 𝑓(𝐷 (𝑡)) -is a function of the degree of achievement of the i-th goal at time t; 𝑤 ( ) 𝑤 -is partial coefficients of influence of the j-th goal in the k-th group of compatible goals, which has a negative impact on the ith goal.

A practical example

As a practical example, the proposed mathematical tools were used to build the knowledge base of the Energy Security Strategy of Ukraine [11]. The results confirmed the applicability of the proposed approach to solving such problems. In Fig. 2 shows an example of goals' hierarchy built by means of the Solon-3 Decision Support System (DSS) [12]. No. 92 -Bringing coal production volumes in line with the needs of Ukraine's energy sector on the basis of market principles of management and competition with the determination of the term of coal use for energy needs.

Table 1 shows the threat numerical rating computed using the Solon-3 DSS.