## 3 Evidential Reasoning Algorithm
### Improved Evidential Reasoning Algorithm
#### 3.3.1 Baseline evidential reasoning algorithm upgrade  
For the aggregation process to be objective and meaningful, it is necessary to define certain axioms for synthesis. The following axioms for the synthesis of the aggregation process were presented by [PERSON] and [PERSON] in [37]. Axiom 1: general attribute \(y\) cannot be evaluated by \(H_{n}\) if none of the baseline attributes of set \(E\) is rated as \(H_{n}\). This axiom is also referred to as the independence axiom. It refers to the instance, if \(\beta_{n,i}=0\) for every \(i=1,\ldots,L\), then \(\beta_{n}=0\). Axiom 2: general attribute \(y\) should be accurately graded by \(H_{n}\) if all baseline attributes of set \(E\) are also accurately graded by \(H_{n}\). This axiom is also referred to as the consensus axiom. It refers to the instance, if \(\beta_{k,i}=1\) and \(\beta_{n,j}=0\) for every \(I=1,\ldots,L\) and \(n=1,\ldots,N\), \(n\
eq k\), then \(\beta_{k}=1\) if \(\beta_{n}=0\) (\(n=1,\ldots,N\), \(n\
eq k\)). Axiom 3: if each baseline attribute of set \(e\) has been fully graded by a specific set of grades, then general attribute \(y\) should also be graded by the same subset of grades. This axiom is also referred to as the completeness axiom. Axiom 4: if the evaluation of some baseline attribute of set \(E\) is incomplete to a certain degree, general attribute \(y\) will also be graded as incomplete. This axiom is also referred to as the incompleteness axiom. It is possible to show that the original evidential reasoning algorithm does not fully satisfy the aforementioned axioms [22; 34; 35; 36]. To ensure their fulfillment, a new or improved evidential reasoning algorithm has been presented [37]. The new approach of the method should satisfy the aforementioned synthesis axioms and provide reliable aggregation of complete and incomplete information using the new weight normalization represented by the following equation:  
\[\sum\limits_{i=1}^{L}\omega_{i}=1 \tag{14}\]  
which satisfies the consensus axiom. With the new evidential reasoning algorithm, the remainder of the weighted probability is treated separately in terms of the relative weights of the attributes and the incompleteness of the assessment. The concept of measuring the confidence rating and reliability in the Dempster-Shafer [22] inference theory can be used to select upper and lower values of the confidence rating. The improved evidential reasoning algorithm \(m_{H,i}\), shown in (14), is divided into two parts:  
\[\overline{m}_{\text{H,i}}=1-\omega \tag{15}\]  
\[\widetilde{m}_{\text{h,i}}=\omega_{i}\left(1-\sum\limits_{i=1}^{N}\beta_{ \text{n,i}}\right) \tag{16}\]  
with the following being true:  
\[\overline{m}_{\text{H,i}}+\widetilde{m}_{\text{H,i}}=m_{\text{H,i}} \tag{17}\]  
The first part of algorithm \(\overline{m}_{H,j}\) represents a linear function of \(\omega_{i}\) and depends on the weight of the \(i\)-th attribute. If the weight of basic attribute \(e_{i}\) is zero or \(\omega_{i}=0\), \(\overline{m}_{H,i}\) has a value of 1. Otherwise, if baseline attribute \(e_{i}\) dominates the assessment or \(\omega_{i}=1\), then \(\overline{m}_{H,i}\) has a value of 0. In simpler terms, \(\overline{m}_{H,i}\) represents the degree of which other attributes take part in the general assessment. The other part of the weight probability that is not assigned to any grade is \(\widetilde{m}_{H,i}\) and it is a consequence of the incompleteness of the basic attribute assessment \(S(e_{i})\). If the assessment of the baseline attribute \(S(e_{i})\) is complete, then \(\widetilde{m}_{\text{H,i}}\) has a value of 0, otherwise, if \(S(e_{i})\) is incomplete, \(\widetilde{m}_{\text{H,i}}\) has a value proportional to \(\omega_{i}\) and will be between 0 and 1.