# Methods for Assessing Human-Machine Performance under Fuzzy Conditions
## 5 Triangular Fuzzy Numbers  
### Preliminaries  
Fuzzy Numbers (FNs), introduced by [PERSON] [33], play an important role in fuzzy mathematics, analogous to the role of ordinary numbers in classical mathematics. A FN, e.g., A, is defined to be a FS on the set \(\mathbf{R}\) of real numbers, such that:  
* A is _normal_, i.e., there exists x in \(\mathbf{R}\) such that \(\mathrm{m_{A}(x)}=1\).
* A is _convex_, i.e., all its \(a\)-cuts \(\mathrm{A}^{a}=\{\mathrm{x}\in\mathbf{R}\colon\mathrm{m_{A}}\left(\mathrm{x}\right) \geq a\}\), with \(a\) in \([0,1]\), are closed real intervals.
* Its membership function \(\mathrm{y}=m(\mathrm{x})\) is a piecewise continuous function.  
One can define the basic arithmetic operations on FNs in two, equivalent to each other, ways:  
(i) With the help of their \(a\)-cuts and the representation-decomposition theorem of [PERSON] [34] (Theorem 2.1, p.16) stating that FS A can be completely and uniquely expressed by the family of its a-cuts in the form \(\mathrm{A}=\sum\limits_{a\in[0,1]}{a\mathrm{A}^{a}}\).  
(ii) By applying the [PERSON]'s extension principle [12] (Section 1.4, p.20), which provides the means for any function f mapping the crisp set X to the crisp set Y to be generalized to map fuzzy subsets of X to fuzzy subsets of Y.  
However, the above two general methods of the fuzzy arithmetic, requiring laborious calculations, are rarely used in practical applications, where the utilization of simpler forms of FNs is preferred.  
A _Triangular Fuzzy Number (TFN)_ (a, b, c), with a, b, and c real numbers such that a \(<\) b \(<\) c is the simplest form of a FN representing mathematically the fuzzy statement that \"the value of b lies in the interval [a, c]\". The membership function \(\mathrm{y}=\mathrm{m(x)}\) of (a, b, c) is zero outside the interval [a, c], whereas its graph in [a, c] forms a triangle with the OX axis (Figure 5). Therefore, we have:  
\[y=m(x)=\left\{\begin{array}{l}\frac{x-a}{b-a},\ x\in[a,b]\\ \frac{c-x}{c-b},\ x\in[b,c]\\ 0,\ x<a\ or\ x>c\end{array}\right.\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Grades** & **Class I** & **Class II** \\ \hline C & 10 & 0 \\ B & 0 & 20 \\ A & 50 & 40 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Student Grades.  
The coordinates (X, Y) of the Center of Gravity (COG) of the graph of the TFN A = (a, b, c), being the intersection point G of the medians AN and BM of the triangle ABC (Figure 5), are calculated by the formulas  
\[X\left(A\right)=\frac{a+b+c}{3},\ Y(A)=\frac{1}{3}. \tag{13}\]  
In fact, since M(, 0) and N(\(\frac{b+c}{2},\frac{1}{2}\)), the proof is easily obtained by calculating the equations of the medians AN and BM and by solving their linear system.  
According to the COG technique the first part of Equation (6) can be used to defuzzify the TFN A.  
The two general methods for defining arithmetic operations on FNs lead to the following simple rules for the addition and subtraction of TFNs:  
Let A = (a, b, c) and B = (a1, b1, c1) be two TFNs. Then:  
* The _sum_ A + B = (a+a1, b+b1, c+c1).
* B = A + (-B) = (a
- c1, b
- b1, c
- a1), where -B = (-c1, -b1, -a1) is defined to be the _opposite_ of B.  
On the contrary, the _product_ and the _quotient_ of A and B are FNs, which are not TFNs in general, apart from some special cases.  
The following two _scalar operations_ can also be defined:  
* k + A = (k + a, k + b, k + c), k \(\in\) R.
* kA = (ka, kb, kc), if k \(>\) 0 and kA = (kc, kb, ka), if k \(<\) 0, k \(\in\) R.  
**Remark 2**.: Another simple form of FNs that we have used as assessment tools [17] (Chapter 7) are the Trapezoidal Fuzzy Numbers (TpFNs). The membership function of a TpFN (a, b, c, d), with a, b, c, and d in **R** such that a < b \(\leq\) c < d, is zero outside the interval [a, d], whereas its graph in [a, d] forms a trapezoid with the OX axis. The TFN (a, b, d) is a special case of the TpFN (a, b, c, d) with b = c, i.e., the TpFNs are actually generalizations of the TFNs.  
For general information on FNs, we refer to the book [35].