# Methods for Assessing Human-Machine Performance under Fuzzy Conditions
## 7 Application of Fuzzy Relation Equations to Mathematical Modeling
### Fuzzy Relation Equations  
_Fuzzy Relation Equations (FREs)_ are associated with the composition of _Fuzzy Binary Relations (FBRs)_. A FBR is defined as follows:  
**Definition 1**.: Let X and Y be two crisp sets. Then, a FBR R(X, Y) is a FS on the Cartesian product X \(\times\) Y of the form: R(X, Y) = {(r, mR(r): r = (x, y) \(\in\) X \(\times\) Y}, where mR: X \(\times\) Y \(\rightarrow\) [0, 1] is the corresponding membership function. When X = {x\({}_{1}\),......, \({}_{n}\)} and Y = {y\({}_{1}\),......, y\({}_{m}\)}, a FBR R(X, Y) can be represented by a n X m matrix R = [r\({}_{ij}\)], where r\({}_{ij}\) = m\({}_{R}\) (x\({}_{i}\), y)\({}_{j}\)), with i = 1,..., n and j = 1,..., m. The matrix R is called the membership matrix of the FBR R(X, Y).  
The basic ideas of fuzzy relations, which were introduced by [PERSON] [47] and further investigated by other researchers, are extensively covered in the book [48].  
Definition 2: Consider two FBR P(X, Y) and Q(Y, Z) with a common set Y. Then, the standard composition of these relations, denoted by P(X, Y) \(\subset\) Q(Y, Z) produces a FBR R(X, Z) with membership function m\({}_{\rm R}\) defined by:  
\[\rm m_{R}(x_{i},\ z_{j})=\max_{y\in Y}\ min\ \big{[}m_{P}(x_{i},\ y),\ m_{Q}(y,\ z_{j}) \big{]},\ i=1,\ldots,n,\ j=1,\ldots,m. \tag{19}\]  
This composition is usually referred to as _max-min composition_. Compositions of FBR are conveniently performed in terms of their membership matrices. In fact, if P = [\(\rm p_{ik}\)] and Q = [\(\rm q_{kj}\)] are the membership matrices of the FBRs P(X, Y) and Q(Y, Z)m respectively, then by Equation (19)m we get that the membership matrix of R(X, Y) = P(X, Y) \(\subset\) Q(Y, Z) is the matrix R = [\(\rm r_{ij}\)], with  
\[\rm r_{ij}=\max_{k}\rm min(p_{ik},\ q_{kj}). \tag{20}\]  
Note that the same elements of P and Q are used in the calculation of m\({}_{\rm R}\) as would be used in the regular multiplication of matrices, but the product and sum operations are here replaced with the min and max operations, respectively.  
Definition 3: Consider the FBRs P(X, Y), Q(Y, Z) and R(X, Z), defined on the sets X = {x\({}_{\rm i}\): i \(\in\) Nn}, Y = {y\({}_{\rm j}\): j \(\in\) Nm }, Z = {z\({}_{\rm k}\): k \(\in\) Ns}, where Nt = {1,2, \(\ldots\).t}, for t = n, m, k, and let P = [\(\rm p_{ij}\)], Q = [\(\rm q_{jk}\)] and R = [\(\rm r_{ik}\)] be the membership matrices of P(X, Y), Q(Y, Z) and R(X, Z), respectively. Assume that the above three relations constrain each other in such a way that P \(\subset\) Q = R, where \(\subset\) denotes the max-min composition. This means that, for each i in Nn and each k in Ns,  
\[\rm r_{ik}=\max_{j\in J}\ min\ (p_{ij},\ q_{jk}). \tag{21}\]  
Therefore, the matrix equation P \(\subset\) Q = R encompasses nXs simultaneous equations of the form of Equation (21). When two of the components in Equation (21) are given and one is unknown, these equations are referred as FRE. The notion of FRE was first proposed by [PERSON] [49] and later further investigated by other researchers (e.g., [50; 51; 52]).