## 2 The Proposed Approach
### Kriging Spatial Model for Prediction of WEEE Return  
The Kriging method is the most important spatial prediction method of random variables in spatial statistics. According to the review, this method has not been applied to the prediction of the WEEE return in reverse logistics network. In this paper, based on the second-order randomness and the spatial properties of the return amount in the recycling network, the WEEE return Kriging spatial prediction model is derived. The prediction value of WEEE return amount of an arbitrary recycling site \(\hat{Z}(s_{0})\) can be transformed into the linear combination of the known return amounts of \(n\) recycling sites, given as  
\[\hat{Z}(s_{0})=m+\sum_{i=1}^{n}\lambda_{i}[Z(s_{i})-m] \tag{16}\]  
where \(Z(s_{i})\) refers to the known return amount of recycling site \(s_{i}\), and \(\lambda_{i}\) is the corresponding weight. The key to accurately predict the unknown network return is to calculate the reasonable weight \(\lambda_{i}\). It is important for WEEE return prediction to satisfy the unbiased prediction condition and to minimize the prediction variance. The calculation of \(\lambda_{i}\) is discussed from these two aspects. In order to construct the unbiased prediction, the average prediction error should be 0, which can be expressed by  
\[E\big{[}\hat{Z}(s_{0})-Z(s_{0})\big{]}=0 \tag{17}\]  
Substituting Equation (15) into Equation (16), yields  
\[E\big{[}\hat{Z}(s_{0})-Z(s_{0})\big{]}=m+\sum_{i=1}^{n}\lambda_{i}\{m-m\}-m=0 \tag{18}\]  
Equation (18) can be further derived as  
\[E\big{[}\hat{Z}(s_{0})-Z(s_{0})\big{]}=m+\sum_{i=1}^{n}\lambda_{i}\{m-m\}-m=0 \tag{19}\]  
According to Equation (18), no matter how the weight \(\lambda_{i}\) is set, the prediction of return amount is unbiased, which means that the return prediction based on Kriging method is effective. The prediction variance minimization, the variance between the predicted value and the true value can be expressed as  
\[Var\big{[}\hat{Z}(s_{0})-Z(s_{0})\big{]}=\sigma^{2}-2\sum_{i=1}^{n}\lambda_{i }C(s_{0}-s_{i})+\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\lambda_{j}C\big{(}s_{ i}-s_{j}\big{)}=\Sigma \tag{20}\]  
where \(\sigma^{2}\) denotes the variance of the randomness \(Z(s_{0})\) defined in Equation (5), and \(\Sigma\) denotes the variance between the predicted value and the true value. To minimize \(\Sigma\), the weight \(\lambda_{i}\) could be calculated by  
\[\left[ \begin{array}{c}\lambda_{1}\\ \vdots\\ \lambda_{n}\end{array} \right]=\left[ \begin{array}{ccc}C_{11}&\cdots&C_{1n} \\ \vdots&\ddots&\vdots\\ C_{n1}&\cdots&C_{nn}\end{array} \right]^{-1}\left[ \begin{array}{c}C_{01}\\ \vdots\\ C_{0n}\end{array} \right] \tag{21}\]  
where \(C_{ij}\) is the covariance of \(s_{i}\) and \(s_{j}\), given by \(C_{ij}=Cov(s_{i}-s_{j})\). The proof of Equation (20) is shown as follows. If the prediction variance of the Kriging method is minimized, the Kriging linear calculation equation of weight \(\lambda_{i}\) can be formulated as  
\[ \begin{split}\Sigma=&\sigma^{2}-2\lambda_{1}\text{C}( s_{0}-s_{1})+\lambda_{2}\text{C}(s_{0}-s_{2})+\ldots+\lambda_{n}\text{C}(s_{0}-s_{n}) ]\\ &+\lambda_{1}[\lambda_{1}\text{C}(s_{1}-s_{1})+\lambda_{2}\text{C }(s_{1}-s_{2})+\ldots+\lambda_{n}\text{C}(s_{1}-s_{n})]\\ &+\lambda_{2}[\lambda_{1}\text{C}(s_{2}-s_{1})+\lambda_{2}\text{C }(s_{2}-s_{2})+\ldots+\lambda_{n}\text{C}(s_{2}-s_{n})]\\ &+\cdots\\ &+\lambda_{n}[\lambda_{1}\text{C}(s_{n}-s_{1})+\lambda_{2}\text{C }(s_{n}-s_{2})+\ldots+\lambda_{n}\text{C}(s_{n}-s_{n})]\end{split}  \tag{22}\]  
where \(\sigma^{2}\) denotes the variance of the randomness \(Z(s_{0})\) defined in Equation (5). In case of \(i=1\), taking the first partial derivative of the right side of Equation (21), and setting it to 0, yields  
\[