## 3 The Mathematical Model for RHF Formulation Scheme Optimization
### Objective Functions  
During the process of designing the scheme, it is necessary to comprehensively consider multiple optimization objectives to meet production requirements while achieving an optimal balance between production efficiency and product quality. The objective functions of the multi-objective optimization model are described as follows:  
1. Element content: The content of each element (C, Cl, and Zn) in the blended pellets, as well as the ratios of the aforementioned elements (C/O and Cl/Zn), needs to be maintained within specific ranges determined by the production plan. This ensures that the final product from the blending system meets specific chemical composition requirements, which is crucial for guaranteeing the performance and quality of the product. The calculation formulas are as follows: \[f_{d}(B)= \begin{cases}0&w_{d-min}\leq w_{el}\leq w_{el-max}\\ \min(abs(w_{el-min}-w_{d},w_{d-max}-w_{d}))&\text{else}\end{cases} \] (2)\[f_{1}(B)=f_{\rm C}(B)+f_{\rm CI}(B)+f_{Zn}(B)+f_{C/O}(B)+f_{\rm CI/Zn}(B) \tag{3}\]  
where \(el\) refers to the contents of three elements: C, Cl, and Zn, as well as the ratio of C to O and the ratio of Cl to Zn, \(w_{el}\) is the element content corresponding to the blended material, and \(w_{el-min}\) and \(w_{el-max}\) are the upper limit and lower limit of the allowed range, respectively. The fitness function \(f_{1}(B)\) guarantees that the formulation scheme satisfies the process requirements as much as possible within the feasible region. In extreme cases, even if no formulation scheme fully meets the allowable element content range given by Equation (2), the optimization algorithm will prioritize selecting the scheme that is closest to the target range. 2. Similarity of the formulation scheme: According to the on-site production experience of RHF, if the difference between the new formulation scheme and the current one is too large, it may lead to instability in material flow on the conveyor belt, thereby causing unpredictable fluctuations and increasing the generation of waste during the recycling process. Therefore, the new formulation scheme similar to the current one are more preferable. The formula to evaluate the similarity between formulation schemes is as follows: \[f_{2}(B)=\sum_{i=2}^{10}\ \ \ |b_{i}^{*}-b_{i}|\] (4) where \(b_{i}^{*}\) represents the proportion in the original formulation scheme. 3. Continuous operation time: After implementing a new formulation scheme, it is necessary to maintain long-duration continuous operation so as to reduce the frequency of switching between schemes and minimize manual interventions. This optimization objective aims to minimize the impact of raw material shortages on the RHF production under the current operating conditions by prioritizing the use of bins with inventory levels, while also providing buffer capacity to address unexpected material supply issues. The formula for calculating the object of continuous operation time is as follows: \[f_{3}(B)=-\min(t_{1},t_{2},\ldots,t_{11})\] (5) \[t_{i}=\frac{H_{i}}{M\times b_{i}},i=1,2,\ldots,11\] (6) where \(t_{i}\) is the time required for the remaining material in bin \(i\) to be depleted, \(H_{i}\) is the remaining material weight in bin \(i\), and \(M\) is the total output flow of all bins in the blending system. The continuous operation time of the formulation scheme is determined by the first bin to be depleted. To facilitate subsequent processing, the negative value is taken to convert this optimization objective into a minimization problem. 4. Start-stop switching: Each time a start-stop switching occurs in a bin, the conveyor belt carrying the material becomes unstable. Therefore, when optimizing the formulation scheme, it is essential to minimize start-stop switching to maintain stable operating conditions in the blending system. The calculation formula is as follows: \[f_{4}(B)=\sum_{i=1}^{11}\ \ \ c_{i},c_{i}= \begin{cases}1&b_{i}>b_{i}^{*}=0\ or\ b_{i}^{*}>b_{i}=0\\ 0&else\end{cases}