Creation of a methodology for assessing the impact of the intellectual capital management mechanism on key indicators of sustainable development of an enterprise

. Annotation. The sustainable development of an enterprise is increasingly dependent on the creation of effective intellectual capital management systems, on the applied intellectual capital management mechanism. In turn, this implies the creation of a reliable mathematical tool for monitoring and evaluating this efficiency, as well as assessing the impact of the mechanism for managing the intellectual capital of an enterprise on key indicators of its sustainable development. The article presents a description of the creation of a methodology and a model for such an assessment. The fuzzy-multiple approach is used as the main one. The article presents a cognitive model of the relationship between assessment indicators, a procedure for assessing the relationship, models for assessing the influence of factors, the procedure for identifying and evaluating key results, identifying critical factors, and calculating the final complex


Introduction
The intellectual capital of enterprises is becoming an increasingly important factor that directly and strongly affects the final results of an enterprise, including its sustainable development and competitiveness [1][2].This naturally raises the question of the need to develop a methodology for assessing the impact of the mechanism of managing the intellectual capital of an enterprise on key indicators of its sustainable development [3][4].
At present, to solve such problems, various mathematical solutions are used, built on the basis of a deterministic approach, a statistical approach, or a fuzzy-set approach.A comparative analysis of models built on the basis of these approaches makes it possible to choose the approach that in a particular situation provides the greatest acceptability of solutions and fully takes into account the specified criteria and limitations.Solving the problem of assessing the impact of the mechanism of managing the intellectual capital of an enterprise on key indicators of sustainable development is currently one of the most important.Previous studies have made it possible to formulate the role of the intellectual capital of an enterprise in its development, including sustainable development [5][6], to develop a mechanism for the strategic management of enterprise personnel for the purpose of its innovative development [1,7], as well as a conceptual model for managing the intellectual capital of an enterprise [8].However, the assessment of the influence of the mechanism for managing the intellectual capital of an enterprise on the key indicators of its sustainable development remains a poorly resolved issue [8][9].This determines the relevance of the article.

Materials and methods
The choice of a model begins with the specification of the priority criteria that the model must meet and the definition of constraints.Limitations are most often due to the use of the model in an enterprise with specific industry specifics.The main criteria for building the model were used:  adequacy and complexity of the model,  speed of preparation of initial data for making managerial decisions  the possibility to quickly adapt the model when the initial data changes.In the process of preparatory studies, a comparative analysis of models built on the basis of a deterministic approach, a statistical approach, and a fuzzy-set approach was carried out.According to the results of the comparative analysis, the choice was made in favor of the fuzzy-set approach [10][11].It should be noted that this approach is actively used today to model complex situations in various industry specifics [12][13][14][15].
The choice of the fuzzy-set approach was determined by the following arguments:  in the literature and in practice, there are no typical (standardized) methods for designing (modeling) fuzzy systems, they are exclusively individual and include a unique set of characteristics that accumulate (reflect) in the specifics of the implemented approaches;  impossibility of correct mathematical analysis of fuzzy systems by existing methods;  removes the problem of incompatibility with a large number of estimates  meets the requirements of the decision maker to the speed, logic of understanding all the meanings, relationships and analysis procedures [16][17][18].This allows using fuzzy set theory methods for a fairly reliable assessment of the relationship between indicators that reflect the maturity (development) of the intellectual capital management mechanism approaches (implicit indicators) and key performance results (explicit indicators).
The disadvantage of fuzzy set theory methods is that the use of a fuzzy-set approach, as compared with a deterministic and probabilistic one, does not lead to an increase in the accuracy of estimates [19][20].This shortcoming was taken into account when constructing the methodology and using it to analyze specific data.

The cognitive model
A cognitive model that reflects the interaction between indicators that reflect the maturity (development) of approaches to the intellectual capital management mechanism (implicit indicators) and key performance results (explicit indicators) is presented in the form of graph G (V, E) (see Figure 1).Figure 1 shows the influence of implicit factors on explicit indicators through the vertices of the graph (A, B and C) and the edges of the graph  ,  , ,  , In this case, the weights of the edges  ,  , ,  , set the degree of influence and show the influence on the impulse level in the direction of its propagation.This interaction can be represented as a vertex (set  = {  },  = 1,3) of a graph that correspond to interaction factors, and edges (set  = {  },  = 1,3) determine the direction of influence one factor to another.

Data
For the purposes of our study (assessment of the influence of the mechanism of intellectual capital management on the key indicators of sustainable development of the enterprise), the following implicit factors and explicit indicators were chosen.
The node A is represented by a set of qualitative assessments of the maturity of approaches (implicit factors), node B is a set of assessments of the perception of stakeholders, and node C is a set of final performance indicators.The data of group A "Implicit factors" were obtained on the basis of expert assessments according to previously selected criteria.Perception indicators (group B) are determined based on the results of a special preliminary study [8].Group C "Indicators of performance" contains the actual key strategic results achieved by year.
A set of specific factors and indicators is given in Table 1.
Table 1.Initial data for model calculations.

Indicators of perception (group B)
Customer Satisfaction Index 0.6 0.6 0.7 0.72 Qualitative indicators (implicit factors) (group A) affect the final performance of the enterprise (group C) indirectly, through indicators (group B).Thus, the initiation and dissemination of the impulse generated by goal-setting and enhanced by activities (Implicit factors, group A), aimed at maximizing the involvement of stakeholders (Indicators of perception, group B) and focusing the impulse on the final results (Indicators of performance, group C) is realized.As a result, a kind of "intellectual impact impulse" is realized, reflecting the effectiveness of managerial efforts to develop and involve the intellectual capital of an enterprise.
This approach adequately accompanies the implementation of the principle of minimizing the passive accumulation of the intellectual potential of an enterprise, ensures maximum consideration of industry specifics in the approaches being formed, and acts as a kind of rating that reflects the intensity of the influence or impulse of intellectual impact on performance.
With a high performance of actions, it is expected to obtain high final indicators, if the effectiveness of actions is insufficient, then the final results should be low.

Assessment method
The procedure for assessing the relationship between indicators reflecting the maturity (development) of the intellectual capital management mechanism approaches (implicit indicators) and key performance results (explicit indicators) can be carried out according to the following algorithm (see Figure 2).The following is a brief description of the content of each stage.

Stage 1. Data normalization
To perform the actions of this stage, it is required, using one-to-one correspondence operations, to transform the data so that they correlate with the numbers of the segment from 0 to 1, since the value of the membership function of any fuzzy sets is in this range.To perform operations of one-to-one correspondence, it is advisable to operate with the growth rate of the initial data.In addition, for each year of the analysis of the factors of blocks A, B, C, it is required to find the maximum number.The normalization operation involves the normalization of growth rates, the data of each year of analysis must be divided for each block into the values of the line Maximum.

Stage 2. Assessing the impact of implicit factors on perception indicators and perception indicators on key results (construction of JAB and JBC matrices).
A fuzzy assessment of the influence of an implicit factor on the main performance indicators of an organization is obtained based on the rules, algorithms and procedures of fuzzy logic.The rule of fuzzy implication according to J. Gauguin [21][22] was taken as a basis, since it is it that satisfies the logic of the connection of indicators of the cognitive model within the framework of the constructed causal model [3].

Stage 3. Evaluation of the influence of implicit factors on key results (construction of the JAC matrix).
The matrix JAC shows the degree of influence of implicit indicators (assessment of the maturity of detailed approaches) on the key performance indicators of the enterprise.With the help of this matrix, it will be possible to estimate the costs of improving the implicit factor based on changes in the organization's key performance indicators.Note that all indicators , ,  to find the values of the connection strength are taken by us based on the current state of affairs in the corresponding organization at different points in time.
The final impact evaluation matrix is found by the rule of minimax matrix multiplication.Minimax matrix multiplication involves performing the usual matrix transformations, in which the multiplication operations are replaced by the operations of finding the minimum, and the addition operations are replaced by the operation of calculating the maximum.
The method for calculating the JAC matrix involves calculations using the operation of finding the minimum of table elements located on the main diagonal Further, applying the maximum operation to each row of the table with the result recorded in the Max column and finding the maximum value among the row values of the Max column.

Stage 4. Identification of key results with insufficient use of the potential of implicit factors.
To solve this problem, it is initially necessary to have an idea of a comprehensive assessment of the strength of the influence of implicit factors on key results.Let us consider this issue in more detail from the point of view of the possibilities of fuzzy sets For a comprehensive assessment of the strength of the influence of implicit factors on key results, the Mamdani algorithm can be used [23][24].This algorithm involves the following actions.
1) Determining the complete set of Сi values for an arbitrary individual key result Хi.
Five subsets of Ci values are usually distinguished (see Table 2).2) Choosing the form of the membership function The membership function determines the level of membership of an element x in the set A. The membership function is described as a curve indicating how each point in the input space is mapped to a membership degree between the numbers 0 and 1.In relation to this study, a Gausson curve of the form was chosen: where b is the coordinate of the maximum, c is the concentration coefficient It should be noted that the distribution density represented by formula ( 4) is a normal distribution law with the parameters mx = b = 0.5 and σx = c = 0.25.The mathematical representation of this law is given by the formula: Therefore, after choosing the form of the membership function of the form (4) and comparing this form with the linguistic variables βi, we get a complete picture of the behavior of the membership function on the interval [0, 1] (see Figure 3).3) Calculation of initial value of the complex indicator (CI) This calculation in relation to the situation under consideration is carried out according to formula 6: where ri -criterion weight, xi -current value of the criterion, n -number of criteria 4) Calculation of the total value of the complex indicator.The final value of the complex indicator can be found using the expression (follows from a direct proportion) of the form:

Stage 5. Evaluation of the impact of the recorded key results on the formation of new values of implicit factors (solution of the inverse problem of constructing the matrices JСB, JBA, JCA).
The inverse problem, the construction of the matrices JСB, JBA, JCA will be solved in accordance with the approach outlined in the justification of stages 2 and 3. To do this, we find the elements of the matrices JСB, JBA using binary Gauguin transformations, and the elements of the matrix JСA are defined as the convolution of the matrices JСB, JBA using the rules of minimax multiplication (stage 3).
Then for the matrix JСB we have where   = min( 1; ⁄ ) ; ,  = 1, The matrix JCA shows the degree of influence of the fixed key results on the implicit performance indicators of the enterprise.With the help of this matrix, it will be possible to estimate the costs of improving the implicit factor based on changes in the organization's key performance indicators.

Stage 6. Identification of critical implicit factors based on finding a complex assessment and a fuzzy conclusion using the Mamdani algorithm.
The task of determining critical implicit indicators can be solved on the basis of the approach described in the justification of the provisions of stage 4. Since this approach involves finding a comprehensive assessment of the state of implicit factors that provide the most complete and effective impact on key results, at the first stage it is required to find the elements of this vector for the year of analysis under consideration.To this end, we first need to determine the elements of the transposed matrix    .Next, you need to determine the elements of the vector of implicit factors that characterize the most complete impact on key results.To do this, we calculate the convolution of the matrices   Т and JCA using the rules of minimax matrix multiplication.Let's calculate the complex indicator.To do this, we will establish for each implicit indicator Xi the full set of its values Аi (see Table 3) The results of calculating the complex indicator are shown in  0,1 0,3 0,5 0,7 0,9 1.1-1.5 1,000 0,167 0,0000 0,0000 0,0000 0,0000 0,3062 0,0460 2.1-2.4 0,935 0,167 0,0000 0,0000 0,0000 0,0014 0,8646 0,1301 3.1-3.5 0,864 0,167 0,0000 0,0000 0,0000 0,0410 0,8599 0,1340 4.1-4.40,929 0,34 0,0000 0,0000 0,0000 0,0021 0,9079 0,2783 5.1-5.5 0,929 0,167 0,0000 0,0000 0,0000 0,0021 0,9079 0,1367 This stage involves the construction of weakly structured situations using linear dynamic models.Formally, in a linear dynamic model based on a cognitive map, a factor is defined as a variable that takes values from a certain numerical scale.According to the results of the study, the model has the following form: factors, the calculation of the weights of all arcs of the graph G(V,E), as well as the normalization of data, we present the final normalized results of cognitive modeling (see Table 5 and figure 4).It follows from the graphs in Figure 4 that in 2023, key results tend to move from the "medium" to the "high" gradation.

Conclusions
The article presents the results of the development and testing of a methodology for assessing the impact of the intellectual capital management system of an enterprise on its sustainable development.The approach includes the analysis and evaluation of a group of implicit management indicators that have a significant impact on the resulting indicators.The goals in the priority areas of sustainable development were adopted as the resulting indicators.To assess the impact of these indicators on target performance indicators, it is proposed to use the mathematical apparatus of the theory of fuzzy sets and the Mamdani algorithm.The formed approach to the analysis and evaluation of the influence of the intellectual capital management mechanism on the key results of the enterprise is designed to increase the effectiveness of management efforts aimed at ensuring the sustainable development of the enterprise.
Management of the effectiveness of the involvement of intellectual capital in the activities of the enterprise 60 65 70 75

Stage 7 .
Building a cognitive model of the impulse of innovative changes with an assessment of the duration of transient processes and fuzzy inference rules using the Mamdani algorithm.
value of the potential strength of the influence of factor A on B, equal to CPAB wBC -the value of the potential strength of the influence of factor B on C, equal to CPBC wAC -the value of the potential strength of the influence of factor A on C, equal to CPAC Without dwelling on the results of intermediate calculations, including the determination of the elements of the vector B, which characterizes the full use of the potential of implicit E3S Web of Conferences 431, 07037 (2023) ITSE-2023 https://doi.org/10.1051/e3sconf/202343107037

Fig. 4 .
Fig. 4. Graphs of the main results of cognitive modeling

Table 2 .
Gradation of subset C levels

Table 3 .
Gradation of subset A levels

Table 4 .
The process of calculating a complex indicator according to the Mamdani algorithm

Table 5 .
Normalized results of cognitive modeling