## 2 Methods
### Prospect Strengths and Weaknesses Degree and Choquet Integral Solution Methods
#### 2.3.2 Comprehensive Prospect Strengths and Weaknesses Degree
Therefore, \(f\), with respect to the fuzzy measure \(u\) of the Choquet integral, can be defined as follows:  
\[\int fd\mu=\sum_{i=1}^{n}f\Big{(}c_{(i)}\Big{)}\left[\mu\Big{(}U_{(i)}\Big{)}- \mu\Big{(}U_{(i+1)}\Big{)}\right] \tag{5}\]  
where \(0\leq f\Big{(}c_{(1)}\Big{)}\leq\cdots\leq f\Big{(}c_{(n)}\Big{)}\), \(U_{(i)}=\Big{(}c_{(I)},\cdots,c_{(n)}\Big{)}\), and \(U_{(n+1)}=\varnothing\). Since the individual Choquet integral only considers the correlation of each decision information and ignores the correlation between customers, a Choquet integral operator is proposed: the appendix calculates all subset fuzzy measures \(u\) of the customer set by each customer weight, and the Choquet integral operator of (2) is applied to assemble the decision information later. In this paper, we use Reference [47]'s formula to assemble the probabilistic linguistic information of each customer. \[ \begin{array}{ll}H_{ij}&=E_{k}\Big{(}h_{1j},h_{2j},\cdots,h_{kj}\Big{)}= \otimes_{I=1}^{k}\Big{(}L_{(I)}(p)\Big{)}^{\mu(U_{(I_{0})})-\mu(U_{(I_{0})})} \\ &=\Big{(}L_{(1)}(p)\Big{)}^{\mu(U_{(1)})-\mu(U_{(2)})}\otimes\Big{(}L_{(2)}(p) \Big{)}^{\mu(U_{(2)})-\mu(U_{(3)})}\otimes\cdots\otimes\Big{(}L_{(k)}(p)\Big{)} ^{\mu(U_{(4)})}\\ &=g^{-1}(\underset{q_{(I)}\in(L_{(I)})}{\bigcup}\left\{\overset{k}{\underset{I =1}{\prod}}\left(\eta_{(I)}\right)\overset{k}{\underset{I=1}{\prod}}\left(p_{( I)}\right)^{\mu(U_{(I_{0})})-\mu(U_{(I_{0})})}\right\})\end{array}  \tag{6}\]  
where \(L_{(1)}(p)\leq L_{(2)}(p)\leq\cdots\leq L_{(k)}(p)\), \(u(U)\) is the fuzzy measure of the subset of decision maker set, and, \(U_{(I)}=\{(I),(I+1),\cdots,(k)\}\), \(U_{(k+1)}=\varnothing\), \(g^{-1}(\bullet)\) are the conversion functions of linguistic terms and their affiliation. In the process of customer decision matrix standardization, to ensure that all the language term sets under each decision information have the same length. In this paper, the maximum set of linguistic terms is added repeatedly, and the probability of all added linguistic elements is 0. The standardized customer group decision matrix is defined as \(H_{k}=\left(h_{kij}\right)_{m\
eq n}\). At the same time, in order to eliminate the influence of the psychological factors of customers in the decision-making process, the SIR method is introduced considering the two-by-two comparisons between electricity sale packages, so the prospect value function can be rewritten as follows:  
\[ \begin{array}{ll}v^{+}\big{(}d\big{(}L_{ij}(p),L_{ij}(p)\big{)}\big{)}&= \left\{\begin{array}{ll}0,other\\ d\big{(}L_{ij}(p),L_{ij}(p)\big{)}^{\alpha},L_{ij}(p)>L_{tj}(p)\\ \end{array}\right.\\ v^{-}\big{(}d\big{(}L_{ij}(p),L_{tj}(p)\big{)}\big{)}&=\left\{\begin{array} []{ll}0,other\\ -\theta d\big{(}L_{ij}(p),L_{tj}(p)\big{)}^{p},L_{ij}(p)>L_{tj}(p)\end{array}\right. \end{array}  \tag{7}\]  
In this formula, \(I,t=1,2,\cdots,m\), \(i\
eq t,j=1,2,\cdots,n\), and \(0<\alpha,\beta<1\) denote the risk preference coefficients of an individual facing gains and losses, respectively. Combined with the Prospect Theory, the customer's mental weight function offers a comprehensive framework for understanding how individuals evaluate potential gains and losses. The Prospect Theory, which was developed to explain human decision-makingunder risk, suggests that people perceive gains and losses differently and are often more sensitive to changes near reference points. The customer's mental weight function, as defined in this context, can be specified as follows:  
\[ \begin{cases}\kappa^{+}(p_{i})=\frac{p_{i}^{\eta}}{\left[p_{j}^{\eta}+\left(1-p_{ j}\right)^{\eta}\right]^{\frac{1}{\eta}}}\\ \kappa^{-}(p_{i})=\frac{p_{i}^{\eta}}{\left[p_{j}^{\delta}+\left(1-p_{j}\right) ^{\delta}\right]^{\frac{1}{\delta}}}\end{cases}  \tag{8}\]  
Among them, \(\eta\) and \(\delta\) denote the individual risk appetite and aversion coefficients, respectively.