# Service Process Problem-Solving Based on Flow Trimming
## 4 Optimal Program Selection Based on The Stochastic Dominance Rule
### Program Ranking
#### 4.3.1 Calculation of the Overall Priority Degree Function of Each Program  
For the evaluation dimension \(C_{j}\), there are three relationships between any two programs \(a_{k}\) and \(a_{k^{\prime}}\):  
1. The program \(a_{k}\) is strictly dominant over the program \(a_{k^{\prime}}\). At this time, \(F_{jk}(W)SDF_{jk}(W)\)_and_\(u_{jk}\geq u_{jk^{\prime}}+\epsilon_{j}\)
2. The program \(a_{k}\) is weakly dominant over the program \(a_{k^{\prime}}\). At this time, \(F_{jk}(W)SDF_{jk}(W)\)_and_\(u_{jk^{\prime}}<u_{jk}<u_{jk^{\prime}}+\epsilon_{j}\)
3. The program \(a_{k}\) is not dominant over the program \(a_{k^{\prime}}\). There is no \(F_{jk}(W)SDF_{jk}(W)\) at this time.  
where \(\epsilon_{j}\) represents the preference threshold value of the evaluation dimension \(C_{j}\) and is related to the expected difference of the evaluation values of the pairwise programs. Here, the average value of the expected difference of the evaluation values of the pairwise programs is taken as the preference threshold [43], the calculation formula for:  
\[\epsilon_{j}=\frac{2}{m(m-1)}\sum_{k=1}\sum_{k^{\prime}=1;k\
eq k^{\prime}}d_{ kk^{\prime}}^{j}(j\in n) \tag{3}\]  
where \(d_{kk^{\prime}}^{j}\) represents the expected absolute difference between the pairwise programs for the evaluation dimension \(C_{j}\), as follows.  
\[\begin{split} d_{kk^{\prime}}^{j}&=\left\{\begin{matrix}u _{jk}-u_{jk^{\prime}},u_{jk}\geq u_{jk^{\prime}}\\ 0,u_{jk}<u_{jk^{\prime}}\end{matrix}\right.\\ &\left(k,k^{\prime}\in m;k\
eq k^{\prime};j\in n\right)\end{split} \tag{4}\]Based on the stochastic dominance relation between the two programs and the value of \(\varepsilon_{j}\), the priority degree function \(gj(ak,ak^{\prime})\) of the program \(a_{k}\) relative to the program \(a_{k^{\prime}}\) for the evaluation dimension \(C_{j}\) is constructed:  
\[g_{j}(a_{k},a_{k^{\prime}})=\left\{\begin{array}{c}1,F_{jk}(W)SDF_{jk}(W)\;and \;u_{jk}\geq u_{jk^{\prime}}+\varepsilon_{j},\\ \frac{u_{jk}-u_{jk^{\prime}}}{\varepsilon_{j}},F_{jk}(W)SDF_{jk}(W)\;and\;u_{jk }<u_{jk^{\prime}}+\varepsilon_{j},\\ 0,other,\\ (k,k^{\prime}\in m;k\
eq k^{\prime};j\in n)\end{array}\right. \tag{5}\]  
where \(g_{j}(a_{k},a_{k^{\prime}})\in[0,1]\). If the value of \(g_{j}(a_{k},a_{k^{\prime}})\) is larger, it means that the program \(a_{k}\) is better than the program \(a_{k^{\prime}}\).  
According to the simple weighting method, the overall priority degree matrix \(G=\big{[}g_{j}(a_{k},a_{k^{\prime}})\big{]}_{m\times m}\) for the comparison of the pairwise programs can be established, where \(g_{j}(a_{k},a_{k^{\prime}})\) is the overall priority degree of the program \(a_{k}\) relative to the program \(a_{k^{\prime}}\).  
\[g(a_{k},a_{k^{\prime}})=\sum_{j=1}^{n}s_{j}g_{j}(a_{k},a_{k^{\prime}})\big{(}k,k^{\prime}\in m;k\
eq k^{\prime};j\in n\big{)} \tag{6}\]  
\(g(a_{k},a_{k^{\prime}})\) represents the total credibility that the program \(a_{k}\) is better than the program \(a_{k^{\prime}}\). The larger the value of \(g(a_{k},a_{k^{\prime}})\), the greater the degree to which the program \(a_{k}\) is dominant over the program \(a_{k^{\prime}}\).