# Fuzzy System for the Quality Assessment of Educational Multimedia Edition Design
## 3 Materials and Methods  
For a predictive quality assessment of educational multimedia editions, fuzzy logic methods are employed, originally formalized by [PERSON] as an extension of classical set theory. The fundamental principle of fuzzy logic is the concept of fuzzy sets, which represent objects and parameters through membership functions [33]. These functions define the degree of membership of a particular element within a set. Another key feature of fuzzy logic is its capability for linguistic modeling, enabling the representation of natural language expressions through fuzzy sets. This approach facilitates the development of expert systems and control algorithms that effectively handle uncertainty in input data [34]. The use of expert evaluation in the quality assessment of educational multimedia editions allows one to obtain a quantitative assessment of the degree of importance of each of the criteria that form a set of values of factors influencing the process quality. In this case, the method of scale rating is used to obtain quantitative assessments of the degree of importance of each of the factors belonging to a certain set, relative to the scale of their basic (reference) values. In this case, the rating of the relative importance of each factor is expressed in points on a certain scale. The most commonly used is a 100-point scale, where the maximum possible importance corresponds to the score of 100 points, and the minimum possible is 0 (zero) points [35]. When processing the expert data, the survey results are summarized in the table, where \(H_{ji}\) is the relative importance of the parameter \(S_{i}\) from the point of view of the \(j\)-th expert, which is expressed by the corresponding score or rank value (Table 1). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Expert**} & \multicolumn{6}{c}{**Factor (Parameter)**} \\ \cline{2-7}  & \(\mathbf{S_{1}}\) & \(\mathbf{S_{2}}\) & \(\ldots\) & \(\mathbf{S_{i}}\) & \(\ldots\) & \(\mathbf{S_{n}}\) \\ \hline
1 & \(H_{11}\) & \(H_{12}\) & \(\ldots\) & \(H_{1i}\) & \(\ldots\) & \(H_{1n}\) \\
2 & \(H_{21}\) & \(H_{22}\) & \(\ldots\) & \(H_{2i}\) & \(\ldots\) & \(H_{2n}\) \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(j\) & \(H_{j1}\) & \(H_{j2}\) & \(\ldots\) & \(H_{ji}\) & \(\ldots\) & \(H_{ji}\) \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(m\) & \(H_{m1}\) & \(H_{m2}\) & \(\ldots\) & \(H_{mi}\) & \(\ldots\) & \(H_{mn}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the experts’ survey using the method of rating scale. The arithmetic mean of factor scores is determined from the expression:  
\[H_{i}=\frac{1}{m}\sum\limits_{j=1}^{m_{i}}H_{ji} \tag{1}\]  
where \(m_{i}\) is a number of experts who assessed the importance of factors \(S_{i}\). The values \(H_{ji}\) and \(H_{i}\) can be expressed quantitatively in points or ranks. In the first case, the value is the average score (average value) of the criterion \(S_{i}\), in the second case it is the average rank. Adding the numbers \(i,j\) in rows with subsequent division of the obtained result by \(m\) gives the average ranking of factors \(S_{1},S_{2},\ldots,S_{n}\), which, in turn, serves as an indicator of the generalized opinion about the importance of factors (the smaller the sum in the row \(j\) is, the more important role the factor \(i\) plays). Let \(V\) be the set of significant values related to the given problem. To establish the correspondence between the set \(V\) and the fuzzy subset \(N\) with the membership function \(\mu_{N}(v)\), the following approach is used [33; 36]:  
\[N=\{(\mu_{N}(v),v),\ v\in V\} \tag{2}\]  
where \((0\leq\mu_{N}(v)\leq 1)\). The membership function establishes the membership degree of each element of the fuzzy set to the universal set: \(N\in V\).