# Methods for Assessing Human-Machine Performance under Fuzzy Conditions
## 6 Grey System Theory
### Applications of the Assessment Methods with TFNs and GNs
(ii) Use of GNs: Here, in Step R1, we have 51 GNs equal to C \(\in\)[60; 74], 24 GNs equal to B \(\in\)[75; 84] and 30 GNs equal to A \(\in\)[85; 100]. The mean value of those GNs is R1* = \(\frac{1}{105}\) (51C + 24B + 30A) \(\in\)[70.57, 83.71]. Therefore, W(R1*) = \(\frac{70.57+83.71}{2}\) = 77.14, etc. Remarks 4:
1. As we show in the previous section, the use of GNs provides in general the same assessment outcomes with the use of TFNs. However, observe that, to obtain the mean value M* \(\in\)[m1, m2], one needs to calculate two components only, in contrast to the mean value M (a, b, c) where the calculation of three components is needed. Consequently, the method with GNs reduces the required computational burden. 2. Another promising area for applying the above assessment methods is the fuzzy control systems (e.g., see [43], [44] (Paragraph 5.3), [45]). Traditional controllers, often implemented as PID (proportional-integral-derivative) controllers, are based on mathematical models in which the control system is described using one or more differential equations that define the system response to its inputs. They are the products of decades of development and theoretical analysis and are highly effective in general. However, in certain cases, the mathematical model of the control process may not exist, or may be too \"expensive\" in terms of computer processing power and memory. In such cases, a system based on empirical rules may be more effective. Fuzzy controllers consist of _input_, _processing_ and _output stages_. The input stage maps sensors, switches and other inputs to the appropriate membership functions (usually triangular, although trapezoidal and bell curves are also used) and truth values. The processing stage is based on a collection of logic rules in the form of IF-THEN statements. A result is generated from each rule and all these results are properly combined. For example, such a rule for a thermostat is \"IF the temperature is low, THEN the heater is high\". Typical fuzzy controllers have dozens of rules. Finally, the output stage converts the combined result back into a specific control output value. In many cases, fuzzy control can be used to improve existing traditional control systems by adding an extra layer of intelligence to the current control method. Example 5: Six different trainers ranked with scores from 0 to 100 the performance of five athletes as follows: A1 (Athlete 1): 43, 48, 49, 49, 50, and 52; A2: 81, 83, 85, 88, 91, and 95; A3: 76, 82, 89, 95, 95, and 98; A4: 86, 86, 87, 87, and 88; and A5: 35, 40, 44, 52, 59, and 62. The mean value of the 6 \(\times\) 5 = 30 given numerical scores is approximately equal to 72.07, demonstrating a good (C) mean performance of the athletes. For reasons of comparison of the two methods, the system's mean performance is also calculated using GNs. The given numerical scores provide 14 GNs equal to A, 4 equal to B, 1 equal to C, 4 equal to D and 7 equal to F. Their mean value is M* = \(\frac{1}{30}(14\)A + 4B + C + 4D + 7F) \(\in\) [60.33, 79.63]. Therefore, Equation (18) gives that W(M*) = 69.98, which shows that the five athletes demonstrate a good performance, but the exact score corresponding to their mean performance is the mean value 72.02 of the given numerical scores calculated in the traditional way. In conclusion, the assessment methods using GNs and TFNs, although they give a satisfactory approximation of the system's mean performance, are practically useful only when the individual performance is evaluated by qualitative grades and not by numerical scores, because, in this case, the calculation of the mean value of those grades in the traditional way is not possible.