## 6 Results and Discussion
### Algorithm Validation  
From references [25; 32; 33], it can be seen that the bat algorithm and its improved algorithms are effective and feasible in solving the TSP problem. Based on Section 5, in order to verify the effectiveness of the ordered pair method and the bat algorithms in steel mills, this section will compare the ordered pair improved bat algorithm with the Hamming distance improved bat algorithm and the Hamming distance improved particle swarm algorithm. In the mixed batch type mode, the above three algorithms are used to achieve batch scheduling of the continuous casting-hot rolling process with the goals of minimizing completion time, minimizing energy waste, and a mixed model of the two. The comparison of the three algorithms is shown in Table 7.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & **Rated Gas** & **Gas Waste** & \multirow{2}{*}{**Proportion (\%)**} & **Operation Rate** & **Completion Time** \\  & **Consumption (GJ)** & & **(GJ)** & & **(\%)** & **(min)** \\ \hline Minimum completion time & 32,522.6 & 1855.6 & 5.71 & 97.78 & 3832 \\ Minimum energy waste & 32,522.6 & 549.3 & 1.69 & 92.00 & 4073 \\ Mixed model & 32,522.6 & 1066.9 & 3.28 & 96.88 & 3868 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of energy wastage between three models.  
From the comparison between the ordered pair bat algorithm and the Hamming distance bat algorithm in Table 7, it can be seen that the Hamming distance bat algorithm solves for a long total completion time for scheduling optimization. This will increase the occupancy time of the hot rolling mill, reduce equipment turnover, and is not conducive to efficient scheduling for enterprises. Moreover, under the minimum completion time objective, the gas waste obtained by the Hamming distance bat algorithm increases by 210.7 GJ. Although the scheduling that minimizes gas waste can achieve the optimal value of 549.3 GJ, it requires an additional completion time of 201 min. In the mixed model, the minimum completion time is 215 min longer, and the energy waste also increases. Therefore, it can be considered that the ordered pair bat algorithm is superior to the Hamming distance bat algorithm, which can prove the effectiveness of the \"ordered pair\" concept.  
From the comparison of the three algorithms, it can be seen that, in terms of minimizing completion time, the Hamming distance particle swarm algorithm has a long completion time and wastes a lot of gas. From the minimum gas waste model, its completion time has increased by 428 min. In the mixed model, not only is the minimum completion time extended, but the gas consumption also increases, which is not the optimal solution to the scheduling problem.  
Overall, the ordered pair bat algorithm proposed can achieve excellent results in terms of minimum completion time and energy waste. It has practical significance for solving the batch scheduling problem of continuous casting and hot rolling in steel enterprises and can be considered as an efficient and applicable algorithm. Meanwhile, the running time of the ordered pair bat algorithm is approximately 13 s, which means that the algorithm can obtain feasible solutions for the continuous casting and hot rolling problem in a short period of time.