# Fuzzy System for the Quality Assessment of Educational Multimedia Edition Design
## 3 Materials and Methods
The fuzzy set \(V\) is represented, taking into account the conditions of discreteness and finiteness of the base scale, divided into certain parts or intervals:  
\[N=(\mu_{N}(v_{1})/v_{1},\mu_{N}(v_{2})/v_{2},\ldots,\mu_{N}(v_{n})/v_{n})= \sum\limits_{i=1}^{n}\mu_{N}(v_{i})/v_{i} \tag{3}\]  
In a simplified record, the fuzzy set \(V\) has the form: \(N=\sum\limits_{i=1}^{n}\mu_{i}/v_{i}\). At the same time, the symbol \"/\" in expression (3) does not indicate the division operation, but only conditionally attaches the membership function \(\mu_{N}(v_{i})\) to the element \(v_{i}\), and the sign \(\sum\) conditionally denotes the set of pairs \(\mu_{N}(v_{i})\) and \(v_{i}\). Finally, membership functions serve as identifiers of input values of linguistic variables in a fuzzy format. The set of values of the input variable \(v\) corresponds to the membership function \(\mu(v)\). It should be noted that linguistic variables are variables whose values are represented by words or word combinations of a natural language, for example, \"Edition format\", \"Edition type\", \"Edition volume\", etc. (provided that they have a linguistic, not numerical, value). The set of values of a linguistic variable constitutes its term-set, a random element of which is called a term. For example, the terms \"small\", \"medium\", \"large\" constitute the term-set of the linguistic variable \"Edition format\". The process of creating a design for a multimedia edition will be considered as the function \(Q\), the arguments of which are the factors of this process:  
\[Q=F(S_{1n},\ S_{2n},\ldots,\ S_{n_{m}}) \tag{4}\]  
where \(n_{m}\) is a number of factors of the \(m\)-th process. According to expression (4), the technological process is represented as a procedure with a set of input variables \(S_{i}\)\((i=\overline{1,n})\) and one output--the variable \(Q\):  
\[\big{|}S_{i},\overline{S_{i}}\big{|},\ i=\overline{1,n};\quad\big{|}\underline {Q},\overline{Q}\big{|} \tag{5}\]The input set contains factors that are qualitative variables. Considering this, it is necessary to establish the set and the limits of the value assignment. \[V=\left\{v^{(1)},v^{(2)},\ldots,v^{(j)}\right\}, \tag{6}\]  
where \(v^{(k)},\;\;k=\overline{1,j}\)--conditional units with the weight coefficient \(j\). To determine the variable \(Q\) according to expression (4), the following expression is obtained:  
\[Q=\left\{q^{(1)},q^{(2)},\ldots,q^{(g)}\right\} \tag{7}\]  
Universal sets, represented by expressions (4)-(6) establish the domains of input and output linguistic variables and ensure the fulfilment of the dependency (4). In parallel, linguistic variables are assessed by means of a natural language, such as \"small\", \"medium\", \"large\", etc. The process of constructing the membership function of linguistic variables of the educational multimedia edition design can be described in more detail. For the formalized representation of the linguistic term \(Q\), a universal fuzzy set \(V=\left\{v_{1},v_{2}\ldots,v_{n}\right\}\) is used with linguistic variables and ranks \(r_{Q}(v_{i})\) in the range \(v_{i}(i=1,\ldots,n)\). Thus, the formalized representation of the linguistic term \"quality of multimedia edition\" \(Q\), according to [20], has the form:  
\[Q_{F}=\left\{\frac{\mu_{Q}(v_{1})}{v_{1}},\frac{\mu_{Q}(v_{2})}{v_{2}},\ldots,\frac{\mu_{Q}(v_{n})}{v_{n}}\right\} \tag{8}\]  
where \(Q_{F}\subset V\); \(\mu_{Q}(v_{i})\) is a membership degree of the element \(v_{i}\in V\) to the set \(Q_{F}\). A set of values \(\mu_{Q}(v_{i})\) is connected by logical operations \(\wedge\) and \(\vee\). This technique allows one to obtain a quantitative expression of the desired membership function of a linguistic term \(Q\). Then the distribution of membership degrees will be as follows:  
\[\frac{\mu_{1}}{r_{1}}=\frac{\mu_{2}}{r_{2}}=\ldots\frac{\mu_{n}}{r_{n}} \tag{9}\]  
where \(\mu_{i}=\mu_{Q}(v_{i})\); \(r_{i}=r_{Q}(v_{i})\) for all \(i=1,\ldots,n\). This takes into account the condition: \(\mu_{1}+\mu_{2}+\ldots+\mu_{n}=1\).