## 3 Evidential Reasoning Algorithm
### Improved Evidential Reasoning Algorithm
#### 3.3.2 Expected Final Assessment of the Object Condition  
If the assessments of the condition of objects obtained this way are not clear enough to highlight the difference in the grades of several different objects or several different conditions of the same object, the term expected final assessment is introduced, which indicates the equivalent numerical value of the individual scores obtained by the aggregation process. Assume that \(u(H_{n})\) represents the expected final assessment rating \(H_{n}\), given the fact that \(u(H_{n+1})>u(H_{n})\) where \(H_{n+1}\) represents a more desirable assessment rating than \(H_{n}\). Expected final assessment rating \(u(H_{n})\) can be computed using the probability assignment method or a regression model with partial scores or comparisons. If the final ratings are complete (\(\beta_{H}=0\)), the expected final assessment rating of general attribute \(y\) can be computed using the following term:  
\[u(y)=\sum_{n=1}^{N}\beta_{n}u(H_{n}) \tag{25}\]  
The condition of the object represented by grade \(a\) has a more desirable rating than the condition of the object represented by grade \(b\) if the expected final assessment rating \(a\) is higher than \(b\) or \(u(y(a))>u(y(b))\). The confidence rating \(\beta_{n}\) given by (23) refers to the lower limit at which we can estimate general attribute \(y\). The upper limit of the estimate is given by the plausibility method for \(H_{n}\) or rather (\(\beta_{n}+\beta_{H}\)). The rating range to which general attribute \(y\) can be assessed is given by the interval \([\beta_{n},\beta_{n}+\beta_{H}]\). If the assessment of the observed object is complete, the interval is set to value \(\beta_{n}\); specifically, the interval of confidence ratings depends on the unassigned degree of confidence ratings \(\beta_{H}\). In any case, the value to which general attribute \(y\) can be assessed is found in the interval \(\beta_{n}\) to \((\beta_{n}+\beta_{H})\). According to the abovementioned, it is possible to define three values that unambiguously characterize the assessment of general attribute \(y\), the largest, smallest and average values of the expected final assessment rating, which are given by the following:  
\[u_{\max}(y)=\sum_{n=1}^{N-1}\beta_{n}u(H_{n})+(\beta_{N}+\beta_{H})u(H_{N}) \tag{26}\]  
\[u_{\min}(y)=(\beta_{1}+\beta_{H})u(H_{1})+\sum_{n=2}^{N}\beta_{n}u(H_{n}) \tag{27}\]  
\[u_{\text{avg}}(y)=\frac{u_{\max}(y)-u_{\min}(y)}{2} \tag{28}\]  
If all of the assessments of the general attribute \(y\) are complete, i.e., for \(\beta_{H}=0\), the relation \(u(y)=u_{\max}(y)=u_{\min}(y)=u_{\text{avg}}(y)\) is true. Comparing the assessment of two objects \(a_{i}\) and \(a_{k}\) based on their final ratings and corresponding intervals, it can be said that the rating of object \(a_{i}\) is more desirable than the rating of object \(a_{\rm k}\) if and only if \(u_{\rm min}(y(a_{i}))>u_{\rm max}(y(a_{\rm k}))\). Two objects have the same assessment if and only if \(u_{\rm min}(y(a_{i}))=u_{\rm max}(y(a_{\rm k}))\). In every other case, comparing the assessments of two objects is incomplete and unreliable. In order to increase the reliability of the comparison of two or more objects, it is necessary to increase the quality of the initial estimates in such a way as to reduce the incompleteness in the attribute estimates for the object conditions \(a_{\rm i}\) and \(a_{\rm k}\). In short, the improved evidential reasoning algorithm consists of the aggregation and interpretation of information (5). Equation (14) is used when normalizing weight values, and, with the help of Equations (5) and (14)-(16), the baseline probability ratings are determined. For the attribute aggregation process, Equations from (18-22) are used. To obtain the combined confidence ratings, it is necessary to use Equations (23) and (24). Finally, to be able to compare the condition of two or more objects, Equations from (26-28) are used.