## 6 Illustrative Example  
Nowadays, companies are paying more attention to supply chain management in order to achieve success in the competitions and keep the customers satisfied. The supply chain is a very complex field that involves a large number of participants. The goal of a complete supply chain is to find an optimal solution from the point of view of all participants, which is a rather complicated task. Under the pressure of global competition, enterprises are more inclined to improve the ability of supply chain management to enhance the competitive advantage. The goal of the decision is to minimize the total cost of the supply chain, and how to choose the right supplier is the key to minimize the total cost of the supply chain. Therefore, supplier selection is of prime importance in a supply chain which has a key role in determining the success or failure of a business [33; 34; 35]. In order to select suitable suppliers, a dealer's investment department selects five suppliers \(X_{i}(i=1,2,3,4,5)\) in accordance with their own needs in the market. The Investment Department invited a group of experts and scholars in related fields to evaluate the five suppliers in the following four indicators, namely \(C_{1}:\) Quality, \(C_{2}:\) Rate of Return, \(C_{3}:\) Delivery ratio, \(C_{4}:\) Service and Technology, and index weighted vector is \(w=(0.15,0.3,0.2,0.35)^{\text{T}}\). According to their own professional knowledge and experience, relevant experts give the evaluation information \(h_{ij}(i=1,2,\cdots,5,j=1,2,3,4)\) of each supplier under each index. Furthermore, the probabilistic interval-valued hesitant fuzzy decision matrix \(H=\left(h_{ij}\right)_{5\times 4}\) is shown in Table 1.  
Step 1: Since the four attribute indexes \(C_{j}(j=1,2,3,4)\) are all benefit indicators, it is not necessary to standardize the original probabilistic interval-valued hesitant fuzzy decision matrix. Step 2: The decision information of each supplier in the attribute index set is integrated by using the GPIVHFOWA operator and the GPIVHFOWG operator (without generality, let additive operator \(g(t)=-\ln(t)\). That is, the PIVHFOWA operator and the PIVHFOWG operator, respectively, which compute comprehensive attribute information of each supplier. The calculation results are shown in Tables 2 and 3. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(C_{1}\) & \(C_{2}\) \\ \hline \(X_{1}\) & \{([0.6,0.8],0.5),([0.4,0.6],0.2),([0.0,0.5],0.3)\} & \{([0.6,0.8],0.4),([0.5,0.7],0.3),([0.2,0.4],0.3)\} \\ \(X_{2}\) & \{([0.5,0.7],0.6),([0.4,0.6],0.1),([0.2,0.4],0.3)\} & \{([0.4,0.6],0.5),([0.3,0.5],0.1),([0.1,0.3],0.4)\} \\ \(X_{3}\) & \{([0.7,0.9],0.2),([0.4,0.6],0.4),([0.0,0.2],0.4)\} & \{([0.6,0.8],0.3),([0.3,0.5],0.5),([0.2,0.4],0.2)\} \\ \(X_{4}\) & \{([0.4,0.6],0.6),([0.3,0.5],0.2),([0.2,0.4],0.2)\} & \{([0.4,0.6],0.4),([0.2,0.4],0.4),([0.0,0.2],0.2)\} \\ \(X_{5}\) & \{([0.6,0.8],0.3),([0.2,0.4],0.4),([0.1,0.3],0.3)\} & \{([0.7,0.9],0.6),([0.6,0.8],0.2),([0.3,0.5],0.2)\} \\ \hline \multicolumn{2}{c}{\(C_{3}\)} & \multicolumn{1}{c}{\(C_{4}\)} \\ \hline \(X_{1}\) & \{([0.2,0.4],0.5),([0.1,0.3],0.4),([0.0,0.2],0.1)\} & \{([0.7,0.9],0.3),([0.4,0.6],0.3),([0.2,0.4],0.4)\} \\ \(X_{2}\) & \{([0.5,0.7],0.3),([0.4,0.6],0.5),([0.0,0.2],0.2)\} & \{([0.7,0.9],0.6),([0.5,0.7],0.3),([0.0,0.2],0.1)\} \\ \(X_{3}\) & \{([0.4,0.6],0.7),([0.1,0.3],0.2),([0.0,0.2],0.1)\} & \{([0.7,0.9],0.4),([0.2,0.4],0.5),([0.1,0.3],0.1)\} \\ \(X_{4}\) & \{([0.6,0.8],0.5),([0.5,0.7],0.2),([0.3,0.5],0.3)\} & \{([0.5,0.7],0.5),([0.3,0.5],0.3),([0.2,0.4],0.2)\} \\ \(X_{5}\) & \{([0.5,0.7],0.4),([0.4,0.6],0.4),([0.1,0.3],0.2)\} & \{([0.7,0.9],0.5),([0.4,0.6],0.1),([0.2,0.4],0.4)\} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Probabilistic interval-valued hesitant fuzzy decision matrix \(H=\left(h_{ij}\right)_{5\times 4}\). Step 4: The five suppliers are sorted according to the value of the score function. The greater the score function value is, the better the supplier's comprehensive condition is.