## 2 The Proposed Approach
### Spatial Structure Analysis of WEEE Recycling Sites  
In this paper, the variogram, first proposed in spatial statistics, is used to describe the spatial variation structure among the random variables of return amount. Denoting the variogram by \(\gamma(h)\), the limited variance of \(\left[Z(s+h)-Z(s)\right]\) can be formulated as  
\[\begin{split}& Var[Z(s+h)-Z(s)]=E\Big{\{}\Big{\{}\left[Z(s+h)-Z(s) \right]-E[Z(s+h)-Z(s)]\big{\}}^{2}\Big{\}}\\ &=E\Big{\{}\left[Z(s+h)-Z(s)\right]^{2}\Big{\}}=2\gamma(h)\end{split} \tag{3}\]  
Under the assumption of second-order randomness, the properties of the spatial structure of recycling sites in WEEE reverse logistics are summarized as follows.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenarios** & **Ore Quantity Prediction** & \begin{tabular}{c} **WEEE Return Amount** \\ **Prediction** \\ \end{tabular} \\ \hline Objective & \begin{tabular}{c} The storage capacity of \\ ore in different mines in a \\ certain area. \\ \end{tabular} & \begin{tabular}{c} Return amount of WEEE \\ in each recycling site in a \\ certain area. \\ \end{tabular} \\ \hline Randomness & \begin{tabular}{c} The storage capacity of a \\ certain ore in different \\ mines is uncertain and \\ random. \\ \end{tabular} & \begin{tabular}{c} The return amount of \\ WEEE \\ in each recycling site in a \\ certain area. \\ \end{tabular} \\ \hline \begin{tabular}{c} Second order \\ randomness \\ \end{tabular} & \begin{tabular}{c} Mean value is \\ constant. \\ \end{tabular} & \
\begin{tabular}{c} The total amount of ore is \\ stable. \\ There is a correlation \\ among the ore quantity \\ in different mines, \\ which is weaker as the \\ distance gets greater. \\ \end{tabular} &
\begin{tabular}{c} The return amount is \\ stable in a certain period. \\ The WEEE return \\ amounts of different sites \\ are correlated, which is \\ weaker as the \\ gets greater. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Demonstration of WEEE return in accordance with the second-order randomness hypothesis.  
_Property 1_: In the whole area \(\mathbf{D}\), the mathematical relationship between the spatial variogram and return amounts of different recycling sites is given by  
\[\gamma(h)=\sigma^{2}-C(h) \tag{4}\]  
where \(\sigma^{2}\) and \(C(h)\) denote the variance and covariance, formulated by Equations (5) and (6) respectively.  
\[\sigma^{2}=Var[Z(s)]=E\Big{\{}[Z(s)-m]^{2}\Big{\}} \tag{5}\]  
\[C(h)=Cov[Z(s+h),Z(s)]=E[Z(s+h)-m][Z(s)-m]=E[Z(s+h)Z(s)]-m^{2} \tag{6}\]  
The proof of Property 1 is given as follows.  
\[\begin{array}{ll}2\gamma(h)&=E\Big{\{}[Z(s)-m]^{2}\Big{\}}=E\Big{\{}\Big{\{} \big{[}Z(s+h)-m]-[Z(s)-m]\big{\}}^{2}\Big{\}}\\ &=E\Big{\{}[Z(s+h)-m]^{2}-2[Z(s)-m][Z(s+h)-m]+[Z(s)-m]^{2}\Big{\}}\\ &=E\Big{\{}[Z(s+h)-m]^{2}\Big{\}}-2E\{[Z(s)-m][Z(s+h)-m]\}+E\Big{\{}[Z(s)-m]^{ 2}\Big{\}}\\ &=\sigma^{2}-2C(h)+\sigma^{2}=2\sigma^{2}-2C(h)\end{array} \tag{7}\]  
_Property 2_: In the whole area \(\mathbf{D}\), the mathematical relationship between the spatial variogram and return amount of a single recycling site is given by  
\[\gamma(h)=\sigma^{2}[1-\rho(h)] \tag{8}\]  
where \(\rho(h)\) is the spatial correlation coefficient, defined by \(\rho(h)=0.5C(h)/\sigma^{2}\).  
The proof of Property 2 is given as follows.  
According to Equation (3) and the definition of \(\rho(h)\), it can be derived that  
\[\begin{array}{ll}\gamma(h)&=\sigma^{2}-C(h)=\sigma^{2}\big{[}1-0.5C(h)/ \sigma^{2}\big{]}\\ &=\sigma^{2}[1-\rho(h)]\end{array} \tag{9}\]