## 3 Methodology
### Behavioral Data Envelopment Analysis (B-DEA)  
In 1978, [PERSON] et al. introduced the primitive DEA model, referred to as the CCR model, to evaluate the technical efficiency of a decision-making unit (DMU) with the constant returns to scale assumption [33]. Because this assumption is unconvincing in many cases, the BCC model with the assumption of the variable returns to scale was proposed by [PERSON] et al. [34]. Considering \(M\) inputs (\(i=1\ldots M\)) and \(T\) outputs (\(r=1\ldots T\)) of the \(L\) DMUs (\(j=1\ldots L\)), the technical efficiency \(\psi_{k}\) of the \(kth\) DMU can be calculated by solving the following model:  
\[\textit{maximize}\ \psi_{k}=\phi+\sum_{r=1}^{T}u_{r}f_{rk}\] \[\textit{subject to}\] \[\sum_{i=1}^{M}v_{i}e_{ik}=1 \tag{1}\] \[\phi+\sum_{r=1}^{T}u_{r}f_{rj}-\sum_{i=1}^{M}v_{i}e_{ij}\leq 0, \qquad j=1,\ \ldots,L\] \[u_{r},\ v_{i}\geq 0\qquad\qquad\qquad i=1,\ \ldots,M;r=1,\ldots,T\] \[\phi\ \text{is free}\]  
where \(u_{r}\) and \(v_{i}\) represent the weight of the \(rth\) output and the \(ith\) input, respectively. The \(e_{ij}\)_and_\(f_{rj}\) represent the \(ith\) input value and \(rth\) output value of \(jth\) DMU. The \(kth\) DMU is effective if the technical efficiency (\(\psi_{k}\)) is 1. Prospect theory is a descriptive theory for individual psychological behavior, which was introduced initially by [PERSON] and [PERSON] [10]. Prospect theory is one of the widely applied theories in behavioral decision-making problems in many fields [35,36]. According to prospect theory, there are three important principles in behavioral decision-making as follows:  
* Reference dependence: Individuals' perceptions of gains and losses depend on a reference point. Figure 1: The proposed methodology. * Loss aversion: Individuals' sensitivity to losses is greater than to equal gains. * Diminishing sensitivity: Individuals tend to be risk-seeking for losses and risk-averse for gains. According to these three principles, the prospective value function is an asymmetrical S-shape curve as illustrated in Figure 2, in which the principle of reference dependence divides the prospect value function into a loss domain and a gain domain. On the other hand, the value function's slope of the gain domain is smaller than that of the loss domain according to the principle of loss aversion. In accordance with the principle of diminishing sensitivity, the prospect value function is convex in the gain domain and concave in the loss domain. The prospect value function is described in Equation (2). \[f(\Delta t)= \begin{cases}\left(\Delta t\right)^{\alpha}&,\forall\Delta t\geq 0 ;0<\alpha<1\\ -\mu(-\Delta t)^{\beta}&,\forall\Delta t<0;0<\beta<1\end{cases}  \tag{2}\]  
where \(\Delta t\) represents the value of loss or gain relative to a reference point. The parameters \(\mu,\ \alpha\), and \(\beta\) represent the loss aversion, the gain risk attitude, and the loss risk attitude of decision makers, respectively. Based on prospect theory, [PERSON] et al. proposed a new behavioral DEA model that includes the following steps [37]:  
Step 1: The normalized value of input (\(x_{ij}\)) and output (\(y_{ij}\)) are determined as Equations (3) and (4). \[x_{ij}=\frac{e_{ij}^{max}-e_{ij}}{e_{ij}^{max}-e_{ij}^{min}}\qquad\qquad j=1, \ldots,L \tag{3}\]  
\[y_{ij}=\frac{f_{ij}-f_{ij}^{min}}{f_{ij}^{max}-f_{ij}^{min}}\qquad\qquad j=1, \ldots,L \tag{4}\]  
Step 2: The positive and negative reference points are determined as Equations (5) and (6). The positive reference points (\(e_{i}^{+}\)_and_\(f_{r}^{+}\)):  
\[e_{i}^{+}=\min_{j}\bigl{(}x_{ij}\bigr{)};f_{r}^{+}=\max_{r}\bigl{(}y_{ij} \bigr{)} \tag{5}\]  
The negative reference points (\(e_{i}^{-}\)_and_\(f_{r}^{-}\)):  
\[e_{i}^{-}=\max_{j}\bigl{(}x_{ij}\bigr{)};f_{r}^{-}=\min_{r}\bigl{(}y_{ij} \bigr{)} \tag{6}\]  
Step 3: The Behavioral DEA model is constructed (7). Figure 2: The prospect value function.