## 3 The Mathematical Model for RHF Formulation Scheme Optimization  
### Material Formulation  
This paper focuses on a RHF production line with an annual output of 250,000 tons in Shanghai Baosteel. The RHF investigated in this study features an internal diameter of 30 m, an effective hearth width of 5 m, and a material bed thickness of approximately 25 mm. The furnace rotates at a speed corresponding to 15 to 40 min per revolution, depending on the operating conditions. The operating temperature is maintained within the range of 1050 \({}^{\circ}\)C to 1300 \({}^{\circ}\)C. The raw materials from the 12 raw material bins shown in Figure 1 are mixed in a blending system and then conveyed to the pelletizing section, where they are formed into green pellets using a briquetting machine. The green pellets are subsequently dried in a pellet dryer to reduce their moisture content below 2%, and finally transported into the rotary hearth furnace via a belt conveyor. Bin 1 contains environmental dust, bin 2 contains CDQ carbon powder, bin 3 contains electric-arc furnace dust, bin 4 contains on-site iron dust, bins 5 and 6 contain LT-OG sludge, bins 7 and 8 contain electric-arc furnace OG sludge, bin 9 contains secondary blast furnace dust, bin 10 contains cold rolling sludge, and bin 11 is the recycling bin. Bins 2 to 10 are internal blending bins, containing solid wastes from steel production, which are the target materials for solid waste treatment in the RHF process  
At the production site, the material composition in each of the internal blending bins is periodically sampled and chemically analyzed to obtain detailed analytical reports. These reports cover the content of key elements such as carbon, chlorine, zinc, total iron, and ferrous oxide. The elemental composition of the external blending materials, such as the environmental dust bin and the recycling bin, can be approximated by calculating the weighted average of the current internal material compositions and the current formulation scheme. Materials from bins 1 to 11 fall through quantitative feeders and are transported by the conveyor belt to a high-intensity mixer, where they are blended into pellets. Since the RHF blending process involves only the physical blending of raw materials without chemical reactions, the effectiveness of the formulation scheme is determined by the percentage content of materials from each bin, the production plan, and the current status of the production line. In constructing a computationally tractable optimization model, the following simplifying assumptions are adopted. Firstly, the model deliberately omits stochastic factors such as raw material composition variability, pellet breakage probability, and furnace condition disturbances. This intentional simplification facilitates focused the  
Figure 1: A schematic diagram of the rotary hearth furnace blending system. development of the optimization algorithm to generate formulation schemes that satisfy actual production requirements. Secondly, during algorithm execution, it is assumed that these raw material chemical compositions exactly match the measured values and that other physical properties do not affect pellet formation. Finally, materials from Bins 1 to 11 are delivered via quantitative feeders and conveyed by the belt to a high-intensity mixer for homogenization and pelletizing. We assume that all blended materials can be successfully pelletized and that the resulting pellets maintain a sufficient mechanical strength during transportation without significant breakage. Because the RHF blending and pelletizing processes involve only physical mixing without chemical reactions, the effectiveness of the formulation scheme depends solely on the percentage content of materials from each bin, raw material chemical compositions, the production plan, and the real-time production line status.