## 4 Empirical Tests
### Materials  
In these empirical tests, assumed that there is 1000 L milk that needs packaging, and the DMUs are as follows (see Figure 4):  
Figure 4a shows the picture of the real products of composite box. The packaging materials of composite box is consisted of cardboard made in China, low density polyethylene (LDPE made in China), and auxiliary materials, and the structure is three-layers composite structure (print layer/cardboard/LDPE). Packaging 1000 L of milk needs 4000 stand-up composite box of 250 mL each. The primary data was based on the study carried out by Chinese researcher [PERSON] et al. [28] and the CLCD-China-ECER 0.8 database. All the above these data represent average cardboard and LDPE industry information.  
Figure 4b shows the picture of the real products of PrePack. The packaging materials of PrePack are made by LDPE film co-extrusion (made in China), and the structure is a three-layers composite structure. Packaging 1000 L of milk needs 5000 pouched PrePack of 200 mL each. The weight of the PrePack was measured and found 3.55 g and required 17.75 kg of polyethylene packaging in total. The primary data of PrePack was based on the study that was carried out by Chinese researcher [PERSON] et al. [29] and [PERSON] et al. [30]. These data represent average LDPE film industry information.  
Figure 4c shows the picture of the real products of Tetra Pak. The packaging materials of Tetra Pak is consisted of cardboard (made in China), LDPE (made in China) and aluminum laminate (Al made in China). The structure is a six-layers composite structure (PE/cardboard-PE-Al-PE/PE). Packaging 1000 L of milk needs 1000 stand-up Tetra Pak, 1000 mL each. The weight of the Tetra Pak was measured and found to be 28.56. The 1000 L of milk required 28.56 kg of cardboard -PE-Al laminate and cardboard-PE-Al laminate was made from 21.93 kg cardboard, 5.29 kg polyethylene, and 1.34 kg aluminum foil. The primary data of Tetra Pak was also based on the study carried out by Chinese researcher [PERSON] et al. [29] and [PERSON] et al. [30]. All the above these data represent the average industry level. The detailed information of three types of packaging systems are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c}  
In this section, we constructed two virtual DMUs using the cubic-Hermite interpolation function. The purposes are dual. On the one hand, when the number of DMUs is insufficient, adding virtual units can improve the accuracy of the model calculation and reduce errors. On the other hand, the evaluation results of virtual DMUs can provide directions for how to improve the inefficient DMU. The reason for choosing the cubic-Hermite interpolation function method is that it not only satisfies the same function value of the interpolation polynomial as the original function value, but also satisfies the derivative value at the node which is equal to the derivative value of the original function. The function of cubic-Hermite interpolation for n nodes (\(x_{1}\), \(x_{2}\),..., \(x_{n}\)) is as follows [31]:  
\[H_{{}_{3}}\left(x\right)=y_{{}_{k}}a_{{}_{k}}+y^{{}^{\prime}}_{{}_{k}}\ \beta_{{}_{k}}\]  
\[y_{{}_{k}}=y\left(x_{{}_{k}}\right),y^{{}^{\prime}}_{{}_{k}}=y^{{}^{\prime}} \left(x_{{}_{k}}\right),\ \ a_{{}_{k}}=\left(1+2\frac{x-x_{{}_{j}}}{x_{{}_{k}}-x_{{}_{j}}} \right)\left(\frac{x-x_{{}_{k}}}{x_{{}_{j}}-x_{{}_{k}}}\right)^{2},\ \beta_{{}_{k}}=\left(x-x_{{}_{k}} \right)\left(\frac{x-x_{{}_{j}}}{x_{{}_{k}}-x_{{}_{j}}}\right)^{2} \tag{19}\]  
where \(k\) and \(j\) are the two nodes.