# Service Process Problem-Solving Based on Flow Trimming
## 4 Optimal Program Selection Based on The Stochastic Dominance Rule
### The Distribution Function of the Evaluation Values of a Program in Different Evaluation Dimensions  
For the convenience of analysis, some sets and quantities involved in this section are subsequently defined.  
\(C=\{C_{1},C_{2},\ldots,C_{n}\}\): The set of evaluation dimensions, where \(C_{j}\) represents the \(j\)-th evaluation dimension, \(j\in n\).  
\(S=\{S_{1},S_{2},\ldots,S_{n}\}\): The weight vector of the evaluation dimension, where \(S_{j}\) represents the weight or importance of the evaluation dimension \(C_{j}\), \(\sum_{j=1}^{n}S_{j}=1\).The weight vector of the evaluation dimension can be given directly by decision-makers or obtained by a questionnaire survey.  
The problematic service touchpoints in the service process are denoted as \(a\), \(b\), \(c\)...  
\(a_{k}=\{a_{1},a_{2},\ldots,a_{m}\}\): The set of programs for the problematic service touchpoint \(a\), where \(a_{k}\) represents the \(k\)-th program, \(k\in m\).  
\(W_{jk}=\left(W_{j1},W_{j2},\ldots W_{jm}\right)\): The evaluation value vector of the user's satisfaction with each program corresponding to the evaluation dimension \(C_{j}\), where \(W_{jk}\) represents the evaluation value of the user's satisfaction with the program \(a_{jk}\), corresponding to the evaluation dimension \(C_{j}\). To obtain the evaluation values of users' satisfaction with different programs in a certain evaluation dimension, a questionnaire survey is required. Via the use of the Likert five-point scale, different scores reflect users' satisfaction with different programs.  
It is considered that \(W_{jk}\) is a discrete stochastic variable within the interval [\(a\),\(b\)]. When \(W_{jk}\) is a discrete stochastic variable, the distribution law can be expressed as \(P_{jk}\left(W_{jk}\right)=\rho_{jk}^{s}\). The pulse function \(\delta_{jk}(W)\) can be used to define its probability density function, namely \(f_{jk}(w)=\sum\limits_{o=1}^{o^{\prime}}\delta_{jk}\left(w-w_{jk}^{o}\right)\), where \(w_{jk}^{o}\) represents the discrete value, i.e., the user's evaluation value of the program \(a_{k}\), \(o^{\prime}\) represents the number of discrete values, and \(\rho_{jk}^{o}\) represents the probability of discrete values, i.e., the probability that the evaluation value given by the user to the program \(a_{k}\) is \(w_{jk^{\prime}}^{o}\), which satisfies \(\rho_{jk}^{o}\geq 0\) and \(\sum\limits_{o=1}^{o^{\prime}}\rho_{jk}^{o}=1\). The cumulative distribution function can be expressed as \(F_{jk}(w)=\sum\limits_{W\geq a}f_{jk}(w)\), and its mathematical expectation can be expressed as \(u_{jk=}\sum\limits_{o=1}^{o^{\prime}}w_{jk}^{o}\rho_{jk}^{o}\)Determination of the Stochastic Dominance Relation of Pairwise Programs in a Certain Evaluation Dimension  
The stochastic dominance relation of pairwise programs in a certain dimension can be judged by the distribution function relationship of the vector of the evaluation value of the user's satisfaction with the program of the pairwise programs.  
According to the stochastic dominance rule [44] and the distribution function of pairwise programs, the stochastic dominance relation matrix is established, and is denoted as \(H=\left[h_{kk^{\prime}}^{j}\right]_{m\times m^{\prime}}\), where \(h_{kk^{\prime}}^{j}\) represents the stochastic dominance relation between the programs \(a_{k}\) and \(a_{k^{\prime}}\) in the evaluation dimension \(C_{j}\).  
\[h_{kk^{\prime}}^{j}=\left\{\begin{matrix}SD,F_{jk}(W)SDF_{jk^{\prime}}(W)\\ -,others\end{matrix},\ \left(k,k^{\prime}\in m;k\
eq k^{\prime};j\in n\right)\right. \tag{1}\]  
where \(F_{jk}(W)SDF_{jk^{\prime}}(W)\) means that \(F_{jk}(W)\) is stochastically more dominant than \(F_{jk}(W)\), i.e., the evaluation value of program \(a_{k}\) is stochastically better than that of program \(a_{k^{\prime}}\) in the evaluation dimension, which is denoted as \(h_{kk^{\prime}}^{j}=SD\). If there is no stochastic dominance relation between the evaluation values of the programs \(a_{k}\) and \(a_{k^{\prime}}\), it is denoted as \(h_{kk^{\prime}}^{j}=-\).