# Methods for Assessing Human-Machine Performance under Fuzzy Conditions
## 7 Application of Fuzzy Relation Equations to Mathematical Modeling
### A Study of MM Skills Using FRE  
Let us consider the crisp sets X = [M], Y = {A, B, C, D, F} and Z = {S\({}_{1}\), S\({}_{2}\), S\({}_{3}\), S\({}_{4}\)}, where M denotes the imaginary notion of the _average student_ of a class; A, B, C, D, and F are the linguistic grades used for the assessment of the student performance; and S\({}_{1}\), S\({}_{2}\), S\({}_{3}\), and S\({}_{4}\) are the steps of the MM process. Without loss of generality, the validation (S\({}_{4}\)) and the implementation (S\({}_{5}\)) of the model, which are usually considered as two different steps, have been joined here in one step (S\({}_{4}\)) to make simpler the development of our assessment method. Let n be the total number of students of a class and let n\({}_{\rm i}\) be the numbers of students of the class who obtained the grade i assessing their performance, i \(\in\) Y. Then, one can represent the average student of the class as a FS on Y in the form M = {(i,\frac{n_{A}}{n}): i \(\in\) Y}. The FS M induces a FBR P(X, Y) with membership matrix P = [\(\frac{n_{A}}{n}\), \(\frac{n_{B}}{n}\), \(\frac{n_{B}}{n}\), \(\frac{n_{C}}{n}\), \(\frac{n_{F}}{n}\)]. In an analogous way, the average student of a class can be represented as a FS on Z in the form M = {(j, m(j): j \(\in\) Z}, where m: Z \(\rightarrow\) [0, 1] is the corresponding membership function. In this case, the FS M\({}^{\prime}\) induces a FBR R(X, Z) with membership matrix R = [m(S\({}_{1}\)) m(S\({}_{2}\)) m(S\({}_{3}\)) m(S\({}_{4}\))]. We consider also the FBR Q(Y, Z) with membership matrix the 5 \(\times\) 4 matrix Q = [q\({}_{\rm ij}\)], where qij = m\({}_{\rm Q}\)(i, j) with i \(\in\) Y and j \(\in\) Z and the FRE encompassed by the matrix equation P \(\subset\) Q = R. When the matrix Q is fixed and the row-matrix P is known, then the above equation always has a unique solution with respect to R, which enables the representation of the average student of a class as a fuzzy set on the set of the steps of the MM process. This is useful for the instructor for designing his/herfuture teaching plans. On the contrary, when the matrices Q and R are known, then the equation P \(\odot\) Q = R may have no solution or it may have more than one solutions with respect to P, which makes the corresponding situation more complicated. **Example 6. ** The following experiment took place at the Graduate Technological Educational Institute of Western Greece, in the city of Patras, when I was teaching a group of 60 students of the School of Technological Applications (future engineers) the use of the derivative for the maximization and minimization of a function. A written test was performed after the end of the teaching process involving two mathematical modeling problems about the construction of a channel to run water and of a cylindrical tower (see the second section of [46]). The results of the test are depicted in Table 4.  
Therefore, the average student M of the class can be represented as a fuzzy set on Y = {A, B, C, D, F} by M = {(A,\(\frac{  
conclusion is explained by the fact that students, when solving MM problems, frequently omit to check if their solutions are compatible to the restrictions imposed by the real system. Therefore, the instructor should emphasize during his/her lectures that the last two steps of the MM process (validation and implementation) are not a formality, but have their own importance for preventing several mistakes. Let us now consider the case where the membership matrices Q and R are known and we want to determine the matrix P representing the average student of the class as a fuzzy set on Y. This is a complicated case because we may have more than one solution or no solution at all. The following two examples illustrate this situation. **Example 7**. : Consider the membership matrices Q and R of the previous example and set P = [p\({}_{1}\) p\({}_{2}\) p\({}_{3}\) p\({}_{4}\) p\({}_{5}\)].