## 3 Value Assessment Framework for Waste Household Appliances
### Scm\(-\)ANFIS Algorithm Design  
In this section, we present the details of the proposed SCM\(-\)ANFIS. Similar to T\(-\)S fuzzy neural networks, ANFIS also uses _if-then_ rules. Each rule consists of an antecedent (_if_) and consequent (_then_) parts. The antecedent specifies the fuzzy conditions based on the input variables, and the consequent part specifies the fuzzy conclusion based on the output variable. Let \(R^{p}\) denote a fuzzy rule, where \(p\) indicated the number of rules. The expression of the rule is as follows:  
\(R^{p}:\)_if \(x_{1}\) is \(A_{p1}\) and \(x_{2}\) is \(A_{p2}\) and...and \(x_{k}\) is \(A_{pk}\), then \(y_{p}=c_{p0}+c_{p1}x_{1}+c_{p2}x_{2}+\cdots+c_{pk}x_{k}\)_  
Here, the \"_if_\" part is fuzzy and is called the antecedent or premise. \(x_{1},x_{2},\ldots,x_{k}\) represent the input system variables, and \(A_{p1},A_{p2},\ldots,A_{pk}\) represent the corresponding fuzzy sets. The \"_then_\" part is determinate and is called the consequent or inference part. \(y\) represents the inference result (e.g., estimated value), and \(c_{p0},c_{p1},\ldots,c_{pk}\) represent different weights of input variables. Figure 3 illustrates a three-input-one-output ANFIS structure. The input represents the input variables required for the valuation of waste home appliances, such as appliance specifications, brand, weight, etc. Each input variable is typically represented by a set of fuzzy sets, which are used to capture the degree of fuzziness or membership of the input variables. The fuzzification layer converts the input variables from their actual values into membership degrees within the fuzzy sets. The product layer consists of a set of fuzzy rules. The condition part of each rule matches the membership degrees of the input variables, and an activation level is assigned to each rule, indicating the degree or weight of the rule. The normalization layer combines the activation levels of the fuzzy rules with the membership functions of the rules to calculate the matching degree of each rule. The defuzzification layer combines the activation levels of all rules to generate the aggregated fuzzy output. The summation layer converts the aggregated fuzzy output into concrete values or fuzzy sets, representing the final evaluation result.  
Before we input the data samples into the ANFIS network, they should be clustered to reduce the number of potential rules that will be generated, as shown in the following. Assuming \(n\) data samples (\(X_{i}=[x_{1},x_{2},\ldots,x_{i},\ldots,x_{n}]^{T}\)) in the training set, each data point (\(x_{i}\)) is a potential cluster center, and the density (\(D_{i}\)) of each data point is calculated first.  
\[D_{i}=\sum_{j=1}^{n}exp\left(\frac{-||x_{i}-x_{j}||^{2}}{(\frac{\tau_{a}}{2})^{ 2}}\right) \tag{9}\]  
\[D_{i}=4\sum_{j=1}^{n}exp\left(\frac{-||x_{i}-x_{j}||^{2}}{(\tau_{a})^{2}}\right) \tag{10}\]  
where \(r_{a}\) represents the radius of clustering. A large number of data points within the radius indicates that the density of the data point is high. The clustering algorithm selects the data point with the highest density as the center. When calculating the density of other data points, the influence of the selected cluster centers needs to be removed. Let \(x_{i}\) be the selected data point and \(D_{c}\) be its density. Then, Formula (10) is modified as follows:  
\[D_{i}=D_{i}-4D_{c}\sum_{j=1}^{n}exp\left(\frac{-||x_{i}-x_{c}||^{2}}{(r_{b})^{ 2}}\right) \tag{11}\]  
where \(r_{b}\) represents the radius with obvious decreased density, indicating that the density of the data points near the first cluster center (\(x_{c}\)) decreases after modification and that they are unlikely to become the next cluster center. The above steps are repeated to select the next cluster center until the desired number of cluster centers is obtained [34].