## 3 A Trade-Off between the Mean and Spread  
According to utility theory, it is intuitive that a higher mean (\(\mu_{A}\)) has a higher utility (\(u(A)\)) and a lower spread (\(\sigma_{A}\)), correspondingly, has a higher utility. For simplicity, we assume the equations are linear as:  
\[u(A)=a+b\mu_{A} \tag{2}\]  
\[u(A)=c-d\sigma_{A} \tag{3}\]  
and  
\[\frac{du}{d\mu_{A}}>0,\;\frac{du}{d\sigma_{A}}<0\]  
Next, we derive the relationship between the mean and spread according to our assumptions. Let two points be \(\mu_{A_{1}}\) and \(\mu_{A_{2}}\), then \(u(A_{1})\), \(u(A_{2})\), \(\sigma_{A_{1}}\), and \(\sigma_{A_{2}}\) can be derived by Equations (2) and (3) to form the relationship between \(\mu_{A}\) and \(\sigma_{A}\), as shown in Figure 2.  
Figure 2: The relationship between the mean and the spread.  
According to Figure 2, we know that having an alternative with a larger spread is the same as having a smaller mean and vice versa. Hence, the trade-off between the mean and the spread can be defined as:  
\[\Delta\mu_{A}=\theta\Delta\sigma_{A} \tag{4}\]  
where \(\theta<0\) is an indifference ratio between the mean and the spread.  
Next, we develop a criterion to incorporate the mean and the spread according to Equation (4). This is because if we can transform the spread to the mean, we can easily compare any imprecise data using the rule for the higher mean with the higher ranking order. Based on this concept, we can transform the difference between the spreads into the difference between the means and let:  
\[\Delta\mu_{A}=\mu_{\Delta}(A_{1},A_{2})=\mu_{A_{1}}-\mu_{A_{2}} \tag{5}\]  
\[\Delta\sigma_{A}=\sigma_{\Delta}(A_{1},A_{2})=\sigma_{A_{2}}-\sigma_{A_{1}} \tag{6}\]  
Then, we can obtain the equation to be the criterion to rank the imprecise data as:  
\[\Delta\Psi_{A}=\Psi(A_{1},A_{2})=\mu_{\Delta}(A_{1},A_{2})+\lambda\sigma_{ \Delta}(A_{1},A_{2}) \tag{7}\]  
where \(\Delta\Psi_{A}\) denotes the preference difference between two imprecise numbers and \(\lambda=-\theta\) indicates the trade-off ratio between the mean and the spread.  
Equation (7) indicates that the grades of the utility in \(A_{1}\) are superior to \(A_{2}\), so we can conclude that  
\[\Delta\Psi_{A}>0\text{, if }A_{1}\succ A_{2} \tag{8}\]  
\[\Delta\Psi_{A}<0\text{, if }A_{1}\prec A_{2} \tag{9}\]  
\[\Delta\Psi_{A}=0\text{, if }A_{1}\approx A_{2} \tag{10}\]  
As the mean and spread are monotone functions, it is obvious that \(\Delta\Psi_{A}\) satisfies the transitive axiom (i.e., if \(A_{1}\succ A_{2}\) and \(A_{2}\succ A_{3}\), then \(A_{1}\succ A_{3}\)). In addition, our simulation indicates \(\theta\) is approximately equal to \(-0.25\), and the method of our experiment is listed in Appendix A.  
Let us consider the defuzzifying problem of the proposed method as follows. Let \(A_{i}\) be a status quo alternative. The defuzzifying value of \(A_{i}\) is its \(\mu_{A_{1}}\), and the defuzzifying value of \(A_{j}\) can be derived as:  
\[A_{j}=\mu_{A_{1}}+\lambda\sigma_{\Delta}(A_{1},A_{2}) \tag{11}\]  
In addition, this paper uses the Mellin transform to calculate the mean and spread/variance in any distribution quickly. Next, we describe details of the Mellin transform, and link it to the \(n\)th moment to calculate the mean and spread.