## 6 Illustrative Example
The results of the specific sorting are shown in Table 5. By using the PIVHFOWA operator and the PIVHFOWG operator, the suppliers with optimal conditions are all \(X_{5}\). In order to explore the similarities and differences of decision results, we will use the PIVHFEOWA operator and the PIVHFEOWG operator to deal with the above problems and get the score function of each vendor's attribute value, the calculation results are shown in Table 4. Then, according to the value of the score function, five suppliers are sorted, and the results are shown in Table 5. As shown in Table 5, after using the PIVHFEOWA operator and the PIVHFEOW \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Operators** & \(\mathbf{X_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{X_{4}}\) & \(\mathbf{X_{5}}\) \\ \hline PIVHFOWA & [0.1336,0.2079] & [0.1325,0.1999] & [0.1227,0.1915] & [0.1219,0.1898] & [0.1570,0.2260] \\ PIVHFOW & [0.1044,0.1905] & [0.1207,0.1904] & [0.1088,0.1829] & [0.1051,0.1831] & [0.1495,0.2180] \\ PIVHFEOWA & [0.1313,0.2062] & [0.1312,0.1989] & [0.1215,0.1905] & [0.1208,0.1888] & [0.1560,0.2251] \\ PIVHFEOWG & [0.1074,0.1938] & [0.1218,0.1917] & [0.1098,0.1842] & [0.1057,0.1841] & [0.1506,0.2192] \\ IVHFOWA & [0.3678,0.5919] & [0.3439,0.5407] & [0.3231,0.5286] & [0.3223,0.5259] & [0.4325,0.6390] \\ IVHFOWG & [0.2775,0.5443] & [0.3076,0.5184] & [0.2719,0.5047] & [0.2544,0.5036] & [0.4079,0.6144] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Score functions of each supplier are calculated with different operators. \begin{table}
\begin{tabular}{c c} \hline \hline
**Suppliers** & **Comprehensive Decision Attribute Values** \\ \hline \(X_{1}\) & \{([0.5845,0.8045],0.4250),([0.3840,0.5896],0.3000),([0.1350,0.3816],0.2750) \\ \(X_{2}\) & \{([0.5584,0.7774],0.5000),([0.4104,0.6133],0.2500),([0.0630,0.2314],0.2500) \\ \(X_{3}\) & \{([0.6243,0.8375],0.4000),([0.2463,0.4487],0.4000),([0.0986,0.2997],0.2000) \\ \(X_{4}\) & \{([0.4809,0.6851],0.5000),([0.3188,0.5232],0.2750),([0.1672,0.3693],0.2250) \\ \(X_{5}\) & \{([0.6531,0.8618],0.4500),([0.4453,0.6547],0.2750),([0.1991,0.4004],0.2750) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comprehensive attribute values of each supplier are calculated with the probabilistic interval-valued hesitant fuzzy ordered weighted averaging (PIVHFOWA) operator. \begin{table}
\begin{tabular}{c c} \hline \hline
**Operators** & **Ranking Results** \\ \hline PIVHFOWA operator & \(X_{5}\succ X_{1}\succ X_{2}\succ X_{3}\succ X_{4}\) \\ PIVHFOWG operator & \(X_{5}\succ X_{2}\succ X_{1}\succ X_{3}\succ X_{4}\) \\ PIVHFEOWA operator & \(X_{5}\succ X_{1}\succ X_{2}\succ X_{3}\succ X_{4}\) \\ PIVHFEOWG operator & \(X_{5}\succ X_{2}\succ X_{1}\succ X_{3}\succ X_{4}\) \\ IVHFOWA operator & \(X_{5}\succ X_{1}\succ X_{2}\succ X_{3}\succ X_{4}\) \\ IVHFOWG operator & \(X_{5}\succ X_{2}\succ X_{1}\succ X_{3}\succ X_{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Ranking results of the five suppliers with different operators. aggregation, the suppliers with the best comprehensive performance are \(X_{5}\). Therefore, the results obtained by using the four operators proposed in this paper are the same, which shows that the generalized probabilistic hesitant fuzzy information aggregation operators proposed in this paper has inherent consistency. In order to illustrate the rationality and reliability of multi attribute group decision making method in this paper, the interval-valued hesitant fuzzy ordered weighted averaging (IVHFOWA) operator and interval hesitant fuzzy ordered weighted geometric (IVHFOWG) operator (Equations (27) and (28)) are compared and analyzed in [28]. Using the method of [28] to deal with the above supplier selection problem, the steps are as follows:  
* Step 1\({}^{\prime}\): Same as step 1. * Step 2\({}^{\prime}\): Based on the original decision matrix \(H=\left(h_{ij}\right)_{5\times 4}\), the comprehensive attribute values \(h_{i}(i=1,2,3,4,5)\) of each supplier are calculated by using Equations (27) and (28).