## 5 Conclusions  
With the process of automation in the manufacturing industry, the study of flexible workshops containing AGVs is more in line with the actual situation of modern workshops, and the scheduling results are more practical. In this paper, the green flexible job shop scheduling problem of machine and AGV integration was studied, a multi-objective mathematical model was established with the optimization goals of minimizing the makespan, minimizing the energy consumption of the workshop, and maximizing the production \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**IWOA VS.** & **Instance** & **Statistic** & \(p\)**-Value** & \(p\) **\(<a\) (\(a\) = 0.05)** \\ \hline \multirow{2}{*}{WOA(1)} & \(5\times 5\) & 0.0 & \(3.815\times 10^{-6}\) & yes \\  & \(15\times 5\) & 32.0 & \(9.452\times 10^{-3}\) & yes \\ \hline \multirow{2}{*}{WOA(2)} & \(5\times 5\) & 0.0 & \(3.815\times 10^{-6}\) & yes \\  & \(15\times 5\) & 19.0 & \(1.171\times 10^{-3}\) & yes \\ \hline \multirow{2}{*}{WOA-LEDE} & \(5\times 5\) & 37.0 & \(1.809\times 10^{-2}\) & no \\  & \(15\times 5\) & 8.0 & \(9.537\times 10^{-5}\) & yes \\ \hline \multirow{2}{*}{NSGA-II} & \(5\times 5\) & 0.0 & \(3.815\times 10^{-6}\) & yes \\  & \(15\times 5\) & 0.0 & \(3.815\times 10^{-6}\) & yes \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistical analysis of the [PERSON]’s rank-sum test. Figure 11: Gantt chart of Instance \(15\times 5\) scheduling. Figure 10: Gantt chart of Instance \(5\times 5\) scheduling. quality, and an improved whale optimization algorithm was proposed to solve it. The improvements made in the proposed whale optimization algorithm include designing encoding and decoding schemes, introducing nonlinear convergence factors and adaptive inertia weights, and improving the screw position update model to improve the optimization ability of the algorithm. In addition, the loss function of the IWOA was constructed based on fuzzy membership degree theory to obtain the optimal compromise solution of the multi-objective model. Finally, the performance of the proposed multi-objective model and algorithm was evaluated through Brandimarte benchmark instances and numerical instances, and the main conclusions are as follows:  
First, based on the results of experiments using benchmark instances, the IWOA achieved the optimal completion times for the MK01, MK02, MK04, MK07, and MK08 benchmark instances, which were 39, 26, 60, 139, and 523, respectively. The standard deviations in the MK01, MK05, MK06, MK09, and MK10 benchmark instances were the smallest at 0.50, 3.44, 1.62, 3.27, and 3.63, respectively. These results demonstrate the effectiveness of the IWOA. Furthermore, in the given two workshop instances, \(5\times 5\) and \(15\times 5\), the IWOA was able to find the optimal satisfaction values of approximately 0.747 and 0.754 in around 100 iterations. Compared to the IWOA(1), IWOA(2), WOA-LEDE, and NSGA-II algorithms, IWOA exhibited better optimization accuracy and robustness and was more competitive in large-scale cases. Finally, by comparing the results of the single-objective optimal solutions and the multi-objective optimization solutions with maximum satisfaction, it is evident that the multi-objective model constructed in this paper is necessary and effective in improving the workshop processing quality and reducing energy consumption. This is conducive to the green development of the manufacturing industry and offers significant advantages over single-objective optimization models. Moving forward, our research aims to explore further advancements in multi-objective optimization algorithms, potentially incorporating machine learning techniques for even more robust scheduling solutions. Conceptualization, T.Z., M.W. and X.G. ; Methodology, X.G. ; Software, M.W. ; Formal analysis, X.G. ; Resources, T.Z. ; Data curation, T.Z. ; Writing--original draft preparation, M.W. ; Writing--review and editing, X.G. ; Visualization, X.G. ; Supervision, T.Z. ; Project administration, T.Z. ; Funding acquisition, T.Z.