## 4 Examples of Engineering Applications
### Construction Quality Grading
#### 4.4.4 Construct Uncertain Measure Function and Matrix of Single Index  
In the process of using an unascertained measure theory to evaluate construction quality, it is necessary to construct uncertain measure function and matrix of a single index scientifically and reasonably. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Indictors** & **X\({}_{\text{1-1}}\)** & **X\({}_{\text{1-2}}\)** & **X\({}_{\text{1-3}}\)** & **X\({}_{\text{1-4}}\)** & **X\({}_{\text{2-1}}\)** & **X\({}_{\text{2-2}}\)** & **X\({}_{\text{2-3}}\)** & **X\({}_{\text{2-4}}\)** & **X\({}_{\text{3-1}}\)** & **X\({}_{\text{3-2}}\)** & **X\({}_{\text{3-3}}\)** \\ \hline IAHP method & 7 & 7 & 16 & 2 & 15 & 9 & 20 & 4 & 13 & 5 & 13 \\ CRITIC method & 6 & 19 & 20 & 6 & 9 & 13 & 17 & 4 & 1 & 3 & 5 \\ \hline Indicators & X\({}_{\text{4-1}}\) & X\({}_{\text{4-2}}\) & X\({}_{\text{4-3}}\) & X\({}_{\text{5-1}}\) & X\({}_{\text{5-2}}\) & X\({}_{\text{5-3}}\) & X\({}_{\text{5-4}}\) & X\({}_{\text{6-1}}\) & X\({}_{\text{6-2}}\) & X\({}_{\text{6-3}}\) & \\ \hline IAHP method & 19 & 17 & 10 & 1 & 11 & 6 & 3 & 21 & 18 & 12 & \\ CRITIC method & 11 & 18 & 6 & 10 & 12 & 2 & 14 & 21 & 16 & 15 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Ranking results of weight values for evaluation indicators. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Primary** **Indictors** & **Combination** **Weight** & **Secondary** **Indictors** & **Subjective** **Weight** & **Objective** **Weight** & **Combination** **Weight** \\ \hline \multirow{4}{*}{X\({}_{1}\)} & & X\({}_{\text{1-1}}\) & 0.0528 & 0.0506 & 0.0504 \\  & & X\({}_{\text{1-2}}\) & 0.0528 & 0.0378 & 0.0472 \\  & & X\({}_{\text{1-3}}\) & 0.0284 & 0.0332 & 0.0389 \\  & & X\({}_{\text{1-4}}\) & 0.0983 & 0.0506 & 0.0605 \\ \hline \multirow{4}{*}{X\({}_{2}\)} & & X\({}_{\text{2-1}}\) & 0.0301 & 0.0492 & 0.0441 \\  & & X\({}_{\text{2-2}}\) & 0.0471 & 0.0455 & 0.0477 \\  & & X\({}_{\text{2-3}}\) & 0.0174 & 0.0395 & 0.0374 \\  & & X\({}_{\text{2-4}}\) & 0.0666 & 0.0549 & 0.0546 \\ \hline \multirow{4}{*}{X\({}_{3}\)} & & \multirow{4}{*}{0.1506} & X\({}_{\text{3-1}}\) & 0.0320 & 0.0709 & 0.0503 \\  & & X\({}_{\text{3-2}}\) & 0.0639 & 0.0600 & 0.0552 \\  & & X\({}_{\text{3-3}}\) & 0.0320 & 0.0508 & 0.0451 \\ \hline \multirow{4}{*}{X\({}_{4}\)} & & X\({}_{\text{4-1}}\) & 0.0135 & 0.0478 & 0.0388 \\  & & X\({}_{\text{4-2}}\) & 0.0246 & 0.0393 & 0.0396 \\  & & X\({}_{\text{4-3}}\) & 0.0447 & 0.0506 & 0.0484 \\ \hline \multirow{4}{*}{X\({}_{5}\)} & & X\({}_{\text{5-1}}\) & 0.1358 & 0.0487 & 0.0673 \\  & & X\({}_{\text{5-2}}\) & 0.0354 & 0.0465 & 0.0449 \\ \cline{1-1}  & & X\({}_{\text{5-3}}\) & 0.0614 & 0.0648 & 0.0557 \\ \cline{1-1}  & & X\({}_{\text{5-4}}\) & 0.0960 & 0.0443 & 0.0587 \\ \hline \multirow{4}{*}{X\({}_{6}\)} & & X\({}_{\text{6-1}}\) & 0.0131 & 0.0328 & 0.0336 \\ \cline{1-1}  & & X\({}_{\text{6-2}}\) & 0.0209 & 0.0410 & 0.0390 \\ \cline{1-1}  & & X\({}_{\text{6-3}}\) & 0.0331 & 0.0412 & 0.0427 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The result of calculating the combined weights of evaluation indicators. The evaluation level space must conform to a certain order; that is, \(U_{1}>U_{2}>\cdots>U_{q}\) or \(U_{1}<U_{2}<\cdots<U_{q}\), where \(U_{t}\) represents the \(t\)th evaluation level, and its corresponding grading standard is represented as \(d_{t}\). Then \(d_{t}\) conforms to \(d_{1}>d_{2}>\cdots>d_{q}\) or \(d_{1}<d_{2}<\cdots<d_{q}\). Assuming that the measured value of the indicator at \(d_{t}\) belongs to level \(t\), when the measured value changes from \(d_{t}\) to \(d_{t+1}\), its state value at level \(t\) gradually changes from 1 to 0, and its state value at level \(t+1\) gradually changes from 0 to 1. The unconfirmed function constructed according to the above method only obtains non-zero values on two adjacent intervals, and the values are zero everywhere else.