## 2 Methods
### Prospect Strengths and Weaknesses Degree and Choquet Integral Solution Methods
#### 2.3.3. Choquet Integral  
The matrix of prospect strengths and weaknesses mentioned above considers the psychological factors of customers and solves the problem of customer subjectivity, but in the actual customer decision-making process, the uncertainty of objective information still needs to be considered. The uncertainty of objective information includes the correlation problem between decision-making information and the weighting problem, in which the correlation problem between decision-making information also further affects its weighting problem. As a result, a mathematical tool is needed to solve the correlation problem and the weighting problem among decision-making information.  
As a mathematical tool commonly used in fuzzy set theory with the ability to deal with the interrelationships and importance of elements, the Choquet integral has demonstrated its superiority in many fields such as classification and decision support. The Choquet integral obtains a more accurate numerical representation by introducing a nonlinear weighting function to order and weigh the elements in a fuzzy set and sum them up. This feature makes the Choquet integral better able to capture the intrinsic features of the data when dealing with complex and fuzzy datasets, and also provides a new way of thinking for the recommendation of electricity sale packages.  
First, according to the application of [PERSON]'s integral theory in existing studies [15], let \(\mu_{\lambda}(D)\) be the fuzzy measure \(\lambda\) of the decision information set D, as follows:  
\[\mu_{\lambda}(D)=\left\{\begin{array}{l}\frac{1}{\lambda}[\prod\limits_{C_{ j}\in\mathrm{C}}\big{(}1+\lambda\mu_{\lambda}\big{(}D_{j}\big{)}\big{)}-1], \lambda\
eq 0\\ \prod\limits_{C_{j}\in\mathrm{C}}\big{(}\lambda\mu_{\lambda}\big{(}D_{j} \big{)}\big{)},\lambda=0\end{array}\right. \tag{10}\]  
where \(\mu_{\lambda}\big{(}D_{j}\big{)}\) is the decision information set \(D\) subset of the fuzzy measure and \(\lambda>-1\). If \(\lambda=0\), the decision information is independent of each other; if \(\lambda>0\), the decision information has a complementary correlation; if \(-1<\lambda<0\), the decision information has a redundant correlation. For any subset of decision information set \(D_{j}\), the value of \(\lambda\) can be calculated as follows:  
\[\mu_{\lambda}\big{(}D_{j}\big{)}=\frac{1}{\lambda}[\prod\limits_{D_{j}\subset \mathrm{C}}\big{(}1+\lambda\mu_{\lambda}\big{(}D_{j}\big{)}\big{)}-1] \tag{11}\]  
Referring to the idea of ascending order of [PERSON]'s integral, the degree of prospect strengths and weaknesses described in the previous section are then ranked as follows:  
\[\begin{array}{l}0\leq S^{(1)}(c_{i})\leq S^{(2)}(c_{i})\leq\cdots\leq S^{(n )}(c_{i}),\\ 0\leq I^{(1)}(c_{i})\!<\!I^{(2)}(c_{i})\leq\cdots<I^{(n)}(c_{i}).\end{array} \tag{12}\]  
Therefore, based on the fuzzy measurements with reference to the requested decision information set, the prospect strengths and weaknesses flow is defined as follows:  
\[\varphi^{>}(c_{i})=\sum\limits_{j=1}^{n}S^{(j)}(c_{i})\Big{[}\mu\Big{(}D_{(j )}\Big{)}-\mu\Big{(}D_{(j+1)}\Big{)}\Big{]} \tag{13}\]  
\[\varphi^{<}(c_{i})=\sum\limits_{j=1}^{n}I^{(j)}(c_{i})\Big{[}\mu\Big{(}D_{(j)} \Big{)}-\mu\Big{(}D_{(j+1)}\Big{)}\Big{]}, \tag{14}\]  
where \(D_{(j)}=\Big{(}D_{(l)},D_{(i+1)},\cdots,D_{(n)}\Big{)}\) and \(C_{n+1}=0\).  
From the classical SIR method, it can be observed that the ranking of electricity sale packages can be determined through the required prospect strengths and weaknesses flows, and, according to the ranking rule of SIR [48], the electricity sale packages are ranked.  
A Recommendation Method for Electricity Sale Package Based on Prospect Strengths and Weaknesses Degree and the Choquet Integral