# An Incentive Factor-Based Dynamic Comprehensive Evaluation on a High-Speed Railway Track
## 3 Evaluation Methods
### An Incentive Factor-Based DCE Method
The evaluation value for the evaluated object is then enlarged by an incentive model, where the optimal incentive \(v^{+}_{i}(t_{k})\) and the negative incentive \(v^{-}_{i}(t_{k})\) can be expressed as  
\[v^{+}_{i}=\left\{ \begin{array}{ll}y^{+}_{i}(t_{k})-y_{i}(t_{k})&y^{+}_{t}(t _{k})>y_{i}(t_{k})\\ 0&\mbox{others}\end{array} \right., \tag{14}\]  
\[v^{-}_{i}=\left\{ \begin{array}{ll}y_{i}(t_{k})-y^{-}_{i}(t_{k})&y_{i}(t_{k} )>y^{-}_{t}(t_{k})\\ 0&\mbox{others}\end{array} \right., \tag{15}\]  
where \(y^{+}_{i}(t_{k})\) and \(y^{-}_{i}(t_{k})\) are the optimal incentive point and the negative incentive point, respectively, for the \(i\)th evaluated object at time \(t_{k}\). Here, the data are weighted again based on incentive so that the risk, obvious or potential, can be highlighted. Suppose there is no motivation at the initial moment \(t_{1}\). Then \(v^{+}_{i}(t_{k})\) and \(v^{-}_{i}(t_{k})\) are both zero. From the values of \(v^{+}_{i}(t_{k})\) and \(v^{-}_{i}(t_{k})\), average maximum gain \(\eta^{max}\), average minimum gain \(\eta^{min}\), average gain \(\overline{\eta}\), optimal gain level \(\eta^{+}\), and inferior gain level \(\eta^{-}\) can be obtained by the following formulas:  
\[ \begin{array}{l}\eta^{max}=\max_{i}(\frac{1}{T-1}\sum_{k=1}^{T-1}(y_{i}(t_{k +1})-y_{i}(t_{k})))\\ \eta^{min}=\min_{i}(\frac{1}{T-1}\sum_{k=1}^{T-1}(y_{i}(t_{k+1})-y_{i}(t_{k})) )\\ \overline{\eta}=\frac{1}{n(T-1)}\sum_{i=1}^{n}\sum_{k=1}^{T-1}(y_{i}(t_{k+1})- y_{i}(t_{k}))\\ \eta^{+}=\overline{\eta}-k^{+}(\overline{\eta}-\eta^{min})\end{array} , \tag{16}\]where \(k^{+}\) and \(k^{-}\) represent the floating coefficients of the evaluated object. Their values can be set based on specialist experience at between 0 and 1. Generally, specialists in this field have certain expectations about the overall development status of a given evaluated object. For example, when the evaluated object is in its best, normal, or worst status, this psychological expectation can be realized by setting the floating coefficients \(k^{+}\) and \(k^{-}\). Then, \(y_{i}^{+}(t_{k})\) and \(y_{i}^{-}(t_{k})\) can be obtained as follow  
\[ \begin{array}{l}\eta^{+}=y_{i}^{+}(t_{k})-y_{i}(t_{k-1})\\ \eta^{-}=y_{i}^{-}(t_{k})-y_{i}(t_{k-1})\end{array} , \tag{18}\]  
where \(k=(1,2,3,\cdots T)\). By substituting \(y_{i}^{+}(t_{k}),y_{i}^{-}(t_{k})\) and SCE matrix \(Y\) into Equations (14) and (15), \(v_{i}^{+}(t_{k})\) and \(v_{i}^{-}(t_{k})\) are obtained. Then the DCE value can be expressed as  
\[Z_{i}(t_{k})=h^{+}v_{i}^{+}(t_{k})+y_{i}(t_{k})-h^{-}v_{i}^{-}(t_{k}), \tag{19}\]  
where \(h^{+}\) and \(h^{-}(h^{+},h^{-}>0)\) are the optimal incentive factor and the negative incentive factor, respectively; and \(h^{+}v_{i}^{+}(t_{k}))\) and \(h^{-}v_{i}^{-}(t_{k}))\) are the optimal incentive value and the negative incentive value, respectively. The incentive factors \(h^{+}\) and \(h^{-}\) are determined by the following two rules:  
Rule 1: Incentive total proportionality rule: The total proportionality of the optimal incentive and negative incentive is the direct proportion for all the evaluation objects, which can be expressed as  
\[r=\frac{h^{+}\sum_{i=1}^{n}\sum_{k=1}^{T}v_{i}^{+}(t_{k})}{h^{-}\sum_{i=1}^{n} \sum_{k=1}^{T}v_{i}^{-}(t_{k})}, \tag{20}\]  
where \(r(r\in R^{+})\) is the proportionality value, which reflects the intention of the evaluator to highlight a decision. When \(r>1\), it means the total proportion of optimal incentive is higher than that of negative incentive. When \(r<1\), it means the total proportion of optimal incentive is less than that of negative incentive. When \(r=1\), it means the total proportion of optimal incentive is equal to that of negative incentive. Rule 2: Balanced incentive rule: The sum of optimal incentive and negative incentive factors is 1, i.e.,  
\[h^{+}+h^{-}=1. \tag{21}\]  
Based on Equations (20) and (21), \(h^{+},h^{-}\) are obtained.