# Methods for Assessing Human-Machine Performance under Fuzzy Conditions
## 6 Grey System Theory
### Applications of the Assessment Methods with TFNs and GNs  
The first application in this section concerns the assessment of _Case-Based Reasoning (CBR)_ systems. CBR is the method of solving new problems based on the solution of analogous problems solved in the past (_past cases_). A case collection can be a powerful resource to use when handling a new problem. A CBR system, usually designed and functioned with the help of proper software, allows the collection of cases to develop incrementally, while its maintenance is an easy task for domain experts. The CBR approach has gained much attention and found many applications over the last 30-40 years, because, as an intelligent systems method, it enables information managers to increase efficiency and reduce cost by substantially automating processes. The steps of the CBR process involve:  
* R\({}_{1}\): _Retrieve_ the most suitable past case to the new problem. * R\({}_{2}\): _Reuse_ the information of the retrieved case to solve the new problem. * R\({}_{3}\): _Revise_ the solution. * R\({}_{4}\): _Retain_ the part of the solution that could be useful for future problems. Steps R\({}_{1}\), R\({}_{2}\) and R\({}_{3}\) are not linear, characterized by a backward-forward flow. A simple flow diagram of the CBR process is presented in Figure 6:  
More information and a detailed functional diagram illustrating the four steps of the CBR process are presented in [42]. **Example 4**. : Consider a couple of CBR systems with a collection of 105 and 90 past cases, respectively. The designers have supplied both systems with a mechanism evaluating the degree of success of the past cases for solving new analogous problems. Table 3 depicts the performance of the past cases in the three first steps of the CBR process. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**First system**} \\ \hline
**Steps** & **F** & **D** & **C** & **B** & **A** \\ \hline R\({}_{1}\) & 0 & 0 & 51 & 24 & 30 \\ R\({}_{2}\) & 18 & 18 & 48 & 21 & 0 \\ R\({}_{3}\) & 36 & 30 & 39 & 0 & 0 \\ \hline \hline \multicolumn{5}{c}{**Second system**} \\ \hline
**Steps** & **F** & **D** & **C** & **B** & **A** \\ \hline R\({}_{1}\) & 0 & 18 & 45 & 27 & 0 \\ R\({}_{2}\) & 18 & 24 & 48 & 0 & 0 \\ R\({}_{3}\) & 36 & 27 & 27 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Assessment of the past cases of the CBR systems. Figure 6: A simplified flow-chart of the CBR process. Here, we compare the mean performance of the two systems by applying both methods with TFNs and GNs. _(i) Use of TFNs:_ The data in Table 3 show that in Step R1 we have for the first system 51 TFNs equal to C(60, 67, 74), 24 TFNs equal to B(75, 79.5, 85) and 30 TFNs equal to A(85, 92.5, 100). The calculation of the mean value of those TFNs gives that R1 = \(\frac{1}{105}\)(51C + 24B + 30A) = \(\frac{1}{105}\) (3060, 3417, 3774) + (1800, 1908, 2016) + (2550, 2775, 3000) = \(\frac{1}{105}\) (7410, 8100, 8790) \(\approx\) (70.57, 77.14, 83.71). Therefore, from Equation (15), one gets that X(R1) = 77.14, which shows that the first system demonstrates a very good (B) performance at Step R1. In the same way, one calculates for the first system the mean values R2 = \(\frac{1}{105}\) (18F + 18D+ 48C + 21B) \(\approx\) (51, 60.07, 69.14) with X(R2) = 60.07 and R3 = \(\frac{1}{105}\) (36F + 30D + 39C) \(\approx\) (36.57, 48.86, 61.14) with X(R2) = 48.86, thus obtaining the analogous conclusions for the system's performance at Steps R2 and R3 of the CBR process. The overall system's performance can be assessed by the mean value R = \(\frac{1}{3}\) (R1 + R2 + R3) \(\approx\) (52.71, 62.02, 71.33). Since X(R) = 62.02, the system demonstrates a good (C) mean performance. A similar argument gives for the second system the values R1 = (62.5, 68.25, 74), R2 \(\approx\) (45.33, 55.17, 65), R3 = (33, 46.25, 59.5) and R \(\approx\) (46.94, 56.56, 66.17). Therefore, analogous conclusions can be obtained for its mean performance at each step of the CBR process and for its overall mean performance, which is characterized as fair (D).