## 4 Conclusions  
The influence of metal slurry powder, PEG-400, KH-560, and sodium molybdate on the development of chromium-free zinc-aluminum coatings was investigated through the design of nine distinct formulations. This was achieved through the utilization of a four-factor, three-level orthogonal experiment. The extension of the analytic hierarchy process, in conjunction with index grading and scoring techniques, was employed to analyze the service performance test outcomes of the aforementioned coatings. The principal findings can be summarized as follows:  
1. The extension theory, which introduces the concept of \"domain\", has transformed the manner in which point values in the judgment matrix are treated as interval numbers. This has led to more effective combinations in consistency testing and weight vector solving, avoiding time-consuming calculations and eliminating the influence of subjective factors. As a result, the weight of evaluation indicators can be determined with greater speed and accuracy;
2. The service performance indicators of chromium-free zinc-aluminum coatings were expressed as dimensionless values, thereby eliminating the differences in numerical values that would otherwise have been caused by different orders of magnitude. A method for grading and scoring the service performance evaluation indicators of chromium-free zinc-aluminum coatings was proposed, which effectively simplified the analysis process;
3. Based on the analytic hierarchy process (AHP), formulation 3# was identified as the optimal formulation for chromium-free zinc-aluminum coatings. This method was deemed to be the most practical, reliable, reasonable, and convenient.  
However, this study has some limitations, including reliance on subjective judgment, issues with the consistency of the judgment matrix, insufficient consideration of the inter-relationships between performance indicators, and the dynamic effects of environmental factors. Future research could consider combining other multi-criteria decision-making methods to enhance the comprehensiveness and accuracy of the analysis. Additionally, incorporating electrochemical impedance spectroscopy (EIS) test indicators as performance metrics for chromium-free zinc-aluminum coatings into the grading standards could improve the accuracy and practicality of the analytic hierarchy process. Further exploration of more scientific and efficient formulation design methods is anticipated to advance the technology of chromium-free zinc-aluminum coatings.  
Figure 2: Comprehensive evaluation results of chromium-free zinc–aluminum coating performance.  
**Author Contributions:** All authors have made practical contributions to this article. Methodology, F.Z.; writing--original draft preparation, Y.X.; resources, W.T.; data curation, Y.T.; project administration, L.H.; supervision, Y.G. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Jilin Province Science and Technology Development Plan Project, grant number 20240601032 RC.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Conflicts of Interest:** The authors declare no conflicts of interest.