Optimization of Formula for Chromium-Free Zinc–Aluminum Coatings Based on Extension Analytic Hierarchy Process

Abstract

The service performance of chromium-free zinc–aluminum coatings exhibits characteristics from multiple perspectives. Fully considering the physical properties, corrosion resistance, and economic viability of the coatings, this study incorporates the concepts of “domain” and “degree” from extenics theory into the analytic hierarchy process to optimize the formulation of chromium-free zinc–aluminum coatings. The findings reveal that the extension analytic hierarchy process takes into account the diversity of evaluation indicators, enhancing the objectivity and accuracy of the comprehensive evaluation results. Nine formulations were developed using a four-factor, three-level orthogonal experiment to evaluate the effects of metal powder, PEG-400, KH-560, and sodium molybdate on the service performance of chromium-free zinc–aluminum coatings. Utilizing an extensible hierarchical sorting weight system alongside a performance index grading and scoring method, 3# emerged with the highest score, indicating the best overall performance. The research outcomes offer innovative insights and technical support for optimizing the formulations of chromium-free zinc–aluminum coatings and other coatings.

1. Introduction

As a substitute for traditional Dacromet coatings, chromium-free zinc–aluminum coatings eliminate the toxic and carcinogenic hexavalent chromium ions produced during the coating sintering and solidification processes. Moreover, their coating hardness and wear resistance far exceed those of Dacromet products. Consequently, their development has garnered widespread attention from scholars [1,2]. Research data indicate that the reagents and dosages used in the formulation of chromium-free zinc–aluminum coatings directly affect the appearance, thickness, hardness, adhesion, corrosion resistance, and other service performance aspects of the coating. Cai et al. [3] improved the corrosion resistance of chromium-free zinc–aluminum coatings by incorporating 0.12% graphene. Li et al. [4] reported a method for preparing chromium-free zinc–aluminum coatings based on polysilazane. The results demonstrated that polysilazane forms a cross-linked structure through Si–O–Si backbones, achieving effective corrosion protection at extremely low curing temperatures below 130 °C, significantly lower than those required for current zinc–aluminum coatings. Li et al. [5] found that when the mass fraction of sodium molybdate was between 1.3% and 2.2%, the adhesion level of the coating reached ISO-1, but higher sodium molybdate concentrations reduced the adhesion performance. Li et al. [6] found that when the mass fraction of multi-walled carbon nanotubes was between 0% and 0.7%, all coatings exhibited an intact, smooth, and continuously dense appearance. Li et al. [7] noted that carbon nanotubes play a crucial role in enhancing the mechanical and corrosion resistance of organic coatings due to their excellent chemical resistance, mechanical stability, and conductivity. Liu et al. [8] introduced zirconia–graphene hybrid nanofiller structures into zinc–aluminum coatings, and the test results indicated that these nanofiller structures significantly improved the coatings’ corrosion resistance and hardness. Wang et al. [9] analyzed Al and Zn powders using scanning electron microscopy (SEM) and examined the microstructure of coatings produced under different processing parameters using optical microscopy. The mechanical and corrosion properties of the coatings were evaluated through tests on hardness, porosity, thickness, and electrochemical performance. Zhang et al. [10] found that the passivation effect on Q235 carbon steel plates was optimal when the molar ratio of MoO₄²⁻ to WO₄²⁻ in the passivation solution was 1:1. The simultaneous addition of molybdate and tungstate resulted in the formation of Fe₂(MoO₄)₃, FeWO₄, and Fe₂(WO₄)₃ in the outer passivation film, increasing the thickness of the passivation layer while filling defects in the coating. To investigate the influence of various components on the performance of chromium-free zinc–aluminum coatings, orthogonal experiments can be utilized for coating formulation design. This approach reduces the number of experiments required and ensures the objectivity and accuracy of the test results. Given that performance testing of chromium-free zinc–aluminum coatings is a comprehensive evaluation problem involving multiple indicators and dimensions, the analytic hierarchy process (AHP) is suitable for objective evaluation. However, the traditional AHP does not account for the impact of subjective judgment tendencies on the results. The judgment matrix obtained by rough estimation often contains a certain degree of ambiguity and requires continuous adjustment to pass the consistency test. To address these issues, various studies have introduced interval numbers into the AHP. Li et al. [11] used interval numbers in the AHP for risk assessment of debris flow disasters; Wen et al. [12] achieved consistency in the wind resistance capability judgment matrix of the power grid using the extension AHP; Xu et al. [13] proposed a system for mountainous tunnel planning using extension theory, transforming subjective descriptions of seismic risk into objective quantifications. Wei et al. [14] employed AHP to determine the weight coefficients of evaluation indicators at each level and arranged them in order, using specific values in the judgment matrix to effectively address the problem of subjective judgment ambiguity. Shi et al. [15] combined the concepts of sets and matter elements in extension theory with interval numbers to establish a judgment matrix that simulates the interval of human subjective judgment. This approach compensates for the extreme situations in traditional AHP matrix construction, thus eliminating the defect of inaccurate measurement. Cheng et al. [16] constructed a matter element model for slope stability analysis by applying matter element and extension theories. The research results showed that the extension AHP method effectively determines the weight coefficients of rock mass stability evaluation indicators and accurately reflects their stability rating levels. Guo et al. [17] proposed using the AHP extension model to evaluate defects in existing tunnel lining structures, determining the importance order of evaluation indicators and performing topological operations through relevant functions to obtain the evaluation level for tunnel lining defects.

In summary, the application of the analytic hierarchy process (AHP) through extension theory has been widely applied in engineering practice. However, its application in the paint industry remains unexplored. Therefore, this study employs AHP to focus on chromium-free zinc–aluminum coatings, establishing a coating performance evaluation system. By introducing extension theory, interval numbers are used instead of point values to construct a judgment matrix, and correlation functions are used to convert interval numbers into individual values for calculating weight vectors. Considering the influence of indicator weights, the performance indicators of chromium-free zinc–aluminum coatings were graded and scored. After analyzing the results of nine formula groups obtained through orthogonal experiments, a comprehensive score was derived to evaluate the overall performance of the chromium-free zinc–aluminum coating comprehensively.

2. Methods

2.1. Establishment of Coating Performance Evaluation System

The performance evaluation of chromium-free zinc–aluminum coatings is a comprehensive assessment encompassing multiple perspectives and dimensions. To evaluate the coating’s performance based on its composition and proportion, it is essential to categorize the evaluation into distinct aspects, elements, and factors. This approach forms a clear hierarchical evaluation system, enabling a comprehensive assessment of chromium-free zinc–aluminum coatings.

The research and development objective of chromium-free zinc–aluminum coatings is to provide an environmentally friendly and high-performance alternative to traditional chromium-containing coatings [18]. Specific goals include enhancing corrosion resistance and mechanical properties, meeting diverse application requirements, while minimizing environmental and health hazards. This involves compliance with rigorous environmental regulations and standards, ensuring coatings not only excel in performance but also promote safety and sustainability throughout production and use [19].

This article proposes a hierarchical structure based on the analytic hierarchy process (see Figure 1) to evaluate chromium-free zinc–aluminum coatings across three dimensions: physical properties, corrosion resistance, and economic applicability. Physical properties were analyzed based on thickness, hardness, and adhesion. Corrosion resistance was assessed through saltwater immersion and neutral salt spray tests. Economic applicability was evaluated considering material cost per kilogram and energy consumption.

2.2. Extension Analytic Hierarchy Process

Extenics is a methodology focused on exploring the expansion and innovation of concepts, designed to resolve real-world incompatible problems. Rooted in traditional set theory, the theory of extension introduces the notions of “domain” and “degree” to articulate the fuzziness and uncertainty inherent in relationships between entities [20]. Additionally, correlation functions are employed to quantify interrelations, facilitating the identification and exploitation of potential connections between entities.

Acknowledging the inherent ambiguity in judgment matrix weights within the analytic hierarchy process (AHP), scholars have proposed the concept of an interval number judgment matrix. This concept leverages extension theory to give rise to extensible AHP, offering enhanced precision in decision-making [21].

To evaluate the performance of chromium-free zinc–aluminum coatings in practical service scenarios, a preliminary assessment was conducted according to predefined standards. This evaluation yielded a quantitative score, enabling effective assessment of the coating’s performance characteristics.

$$\mathrm{T}=(\mathrm{W},{\mathrm{k}}_{\mathrm{i}},{\mathrm{u}}_{\mathrm{i}})=\left[\begin{array}{ccc}\mathrm{W}& {\mathrm{k}}_1& {\mathrm{u}}_1\\ & {\mathrm{k}}_2& {\mathrm{u}}_2\\ & ⋮& ⋮\\ & {\mathrm{k}}_{\mathrm{n}}& {\mathrm{u}}_{\mathrm{n}}\end{array}\right]$$

(1)

In the formula, T represents the performance evaluation index of the chromium-free zinc–aluminum coating layer under consideration, while u_i denotes the value of the feature k_i (i = 1, 2, …, n).

2.2.1. Constructing an Extensible Interval Number Judgment Matrix

Note that ${\mathrm{a}}_{\mathrm{i}\mathrm{j}}=⟨{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-},{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}⟩=\left\{\mathrm{x}|0<{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}<\mathrm{x}<{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}\right\}$ represents an extensible interval number. Both interval number vectors and interval number matrices consist of extensible interval numbers as their elements [22].

Based on T.L.Saaty’s pairwise comparison scale (

Table 1. The value rule of element a_ij.

Scale Value	Meaning
1	Both are equally important when compared.
3	The former is slightly more important than the latter when compared.
5	The former is significantly more important than the latter when compared.
7	The former is markedly more important than the latter when compared.
9	The former is exceedingly more important than the latter when compared.

) and the decision-making principle, a judgment matrix was constructed. Scales 2, 4, 6, and 8 were the median values between adjacent scales [23].

Let $\mathrm{A}={\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}\right]}_{\mathrm{n}\times \mathrm{n}}$ (i, j = 1, 2, …, n), ${\mathrm{a}}_{\mathrm{i}\mathrm{j}}=\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-},{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}\right]$, and ${\displaystyle \frac{1}{9}}\le {\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}\le {\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}\le 9, {\mathrm{a}}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{1}{{\mathrm{a}}_{\mathrm{j}\mathrm{i}}}}$, A is referred to as the extensible interval number judgment matrix, defined as:

$$\mathrm{A}=\left[\begin{array}{cccc}⟨1,1⟩& ⟨{\mathrm{a}}_{12}^{-},{\mathrm{a}}_{12}^{+}⟩& \cdots & ⟨{\mathrm{a}}_{1\mathrm{n}}^{-},{\mathrm{a}}_{1\mathrm{n}}^{+}⟩\\ ⟨{\displaystyle \frac{1}{{\mathrm{a}}_{12}^{+}}},{\displaystyle \frac{1}{{\mathrm{a}}_{12}^{-}}}⟩& ⟨1,1⟩& \cdots & ⟨{\mathrm{a}}_{2\mathrm{n}}^{-},{\mathrm{a}}_{2\mathrm{n}}^{+}⟩\\ \cdots & \cdots & \cdots & \cdots \\ ⟨{\displaystyle \frac{1}{{\mathrm{a}}_{1\mathrm{n}}^{+}}},{\displaystyle \frac{1}{{\mathrm{a}}_{1\mathrm{n}}^{-}}}⟩& ⟨{\displaystyle \frac{1}{{\mathrm{a}}_{2\mathrm{n}}^{+}}},{\displaystyle \frac{1}{{\mathrm{a}}_{2\mathrm{n}}^{-}}}⟩& \cdots & ⟨1,1⟩\end{array}\right]$$

(2)

2.2.2. Calculate the Weight of Extensible Interval Numbers

• Eigenvector calculation

Let $\mathrm{A}=\left[{\mathrm{A}}^{-},{\mathrm{A}}^{+}\right]$, where ${\mathsf{\xi }}^{-}$ and ${\mathsf{\xi }}^{+}$ represent, respectively, the maximum eigenvalues of the left and right matrices ${\mathrm{A}}^{-}={\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}\right]}_{\mathrm{n}\times \mathrm{n}}$ and ${\mathrm{A}}^{+}={\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}\right]}_{\mathrm{n}\times \mathrm{n}}$. Then, $\mathsf{\xi }=\left[{\mathsf{\xi }}^{-},{\mathsf{\xi }}^{+}\right]$ denotes the interval number eigenvalue of A. The eigenvector $\mathrm{x}=\left[{\mathrm{x}}^{-},{\mathrm{x}}^{+}\right]$ of A corresponding to $\mathsf{\xi }$ is represented by $\mathrm{x}=\left[\mathrm{k}{\mathrm{x}}^{-},\mathrm{m}{\mathrm{x}}^{+}\right]$, where ${\mathrm{x}}^{-}$ and ${\mathrm{x}}^{+}$ are eigenvectors of ${\mathrm{A}}^{-}$ and ${\mathrm{A}}^{+}$, respectively. The steps to find the eigenvectors ${\mathrm{x}}^{-}$ and ${\mathrm{x}}^{+}$ using the summation method are as follows:

Normalize the column vectors of a matrix:

$${\tilde{\mathrm{A}}}_{\mathrm{i}\mathrm{j}}^{-}=({\displaystyle \frac{{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}}})$$

(3)

$${\tilde{\mathrm{A}}}_{\mathrm{i}\mathrm{j}}^{+}=({\displaystyle \frac{{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}}})$$

(4)

Sum ${\tilde{\mathrm{A}}}_{\mathrm{i}\mathrm{j}}^{-}$ and ${\tilde{\mathrm{A}}}_{\mathrm{i}\mathrm{j}}^{+}$ by row:

$${\tilde{\mathrm{x}}}^{-}=(\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{j}}^{-}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}}},\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{a}}_{2\mathrm{j}}^{-}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}}},\cdots \sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{a}}_{\mathrm{n}\mathrm{j}}^{-}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}}}{)}^{\mathrm{T}}$$

(5)

$${\tilde{\mathrm{x}}}^{+}=(\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{j}}^{+}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}}},\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{a}}_{2\mathrm{j}}^{+}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}}},\cdots \sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{a}}_{\mathrm{n}\mathrm{j}}^{+}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}}}{)}^{\mathrm{T}}$$

(6)

Normalize ${\tilde{\mathrm{x}}}^{-}$ and ${\tilde{\mathrm{x}}}^{+}$ to obtain eigenvectors:

$${\mathrm{x}}^{-}=({\mathrm{x}}_1^{-},{\mathrm{x}}_2^{-},\cdots {\mathrm{x}}_{\mathrm{n}}^{-}{)}^{\mathrm{T}}$$

(7)

$${\mathrm{x}}^{+}=({\mathrm{x}}_1^{+},{\mathrm{x}}_2^{+},\cdots {\mathrm{x}}_{\mathrm{n}}^{+}{)}^{\mathrm{T}}$$

(8)

• Consistency check

Calculate the values of k and m based on the following formula:

$$\mathrm{k}=\sqrt{\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{1}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{+}}}}$$

(9)

$$\mathrm{m}=\sqrt{\sum _{\mathrm{j}=1}^{\mathrm{n}}{\displaystyle \frac{1}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}^{-}}}}$$

(10)

If $0\le \mathrm{k}\le 1\le \mathrm{m}$, the consistency of the extensible interval number judgment matrix A can be confirmed. Otherwise, adjustments to the judgment matrix are necessary to meet the consistency requirements.

• Weight calculation

Assuming that the K-th layer contains n_K elements, each element in the K-th layer has a weight of an interval number.

Assuming the K-th layer contains n_K elements, each element in the K-th layer is assigned a weight represented as an interval number:

$${\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}=⟨\mathrm{k}{\mathrm{x}}^{-},\mathrm{m}{\mathrm{x}}^{+}⟩=⟨{{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{-},{{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{+}⟩$$

(11)

In the formula: i = 1, 2, …, n_K.

2.2.3. Interval Weight Univalued

Given two interval numbers, $\mathrm{z}=⟨{\mathrm{z}}^{-},{\mathrm{z}}^{+}⟩$ and $\mathrm{r}=⟨{\mathrm{r}}^{-},{\mathrm{r}}^{+}⟩$, the likelihood that z ≥ r is:

$$\mathrm{V}(\mathrm{z}\ge \mathrm{r})=\sup ({\mathrm{k}}_{\mathrm{z}}(\mathrm{u})\wedge {\mathrm{k}}_{\mathrm{r}}(\mathrm{u}))$$

(12)

V (z ≥ r) is the relative importance of z over r.

The correlation function of u with respect to the extended interval number $\mathrm{z}=⟨{\mathrm{z}}^{-},{\mathrm{z}}^{+}⟩$ is denoted as ${\mathrm{k}}_{\mathrm{z}}(\mathrm{u})$:

$${\mathrm{k}}_{\mathrm{z}}(\mathrm{u})=\left\{\begin{array}{c}{\displaystyle \frac{2(\mathrm{u}-{\mathrm{z}}^{-})}{{\mathrm{z}}^{+}-{\mathrm{z}}^{-}}},\mathrm{u}\le {\displaystyle \frac{{\mathrm{z}}^{+}+{\mathrm{z}}^{-}}{2}}\\ {\displaystyle \frac{2({\mathrm{z}}^{+}-\mathrm{u})}{{\mathrm{z}}^{+}-{\mathrm{z}}^{-}}},\mathrm{u}\ge {\displaystyle \frac{{\mathrm{z}}^{+}+{\mathrm{z}}^{-}}{2}}\end{array}\right.$$

(13)

The correlation function of u with respect to the extended interval number $\mathrm{r}=⟨{\mathrm{r}}^{-},{\mathrm{r}}^{+}⟩$ is denoted as ${\mathrm{k}}_{\mathrm{r}}(\mathrm{u})$:

$${\mathrm{k}}_{\mathrm{r}}(\mathrm{u})=\left\{\begin{array}{c}{\displaystyle \frac{2(\mathrm{u}-{\mathrm{r}}^{-})}{{\mathrm{z}}^{+}-{\mathrm{z}}^{-}}},\mathrm{u}\le {\displaystyle \frac{{\mathrm{r}}^{+}+{\mathrm{r}}^{-}}{2}}\\ {\displaystyle \frac{2({\mathrm{r}}^{+}-\mathrm{u})}{{\mathrm{r}}^{+}-{\mathrm{r}}^{-}}},\mathrm{u}\ge {\displaystyle \frac{{\mathrm{r}}^{+}+{\mathrm{r}}^{-}}{2}}\end{array}\right.$$

(14)

The likelihood that $\mathrm{z}\ge \mathrm{r}$, denoted as $\mathrm{V}(\mathrm{z}\ge \mathrm{r})$, can be calculated using the following formula:

$$\mathrm{V}(\mathrm{z}\ge \mathrm{r})={\displaystyle \frac{2({\mathrm{z}}^{+}-{\mathrm{r}}^{-})}{({\mathrm{z}}^{+}-{\mathrm{z}}^{-})+({\mathrm{r}}^{+}-{\mathrm{r}}^{-})}}$$

(15)

According to Equation (15), calculate the importance weight of ${\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}=⟨{{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{-},{{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{+}⟩$ compared to ${\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}}=⟨{{\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}}}^{-},{{\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}}}^{+}⟩$, and obtain:

$$\left\{\begin{array}{l}{\mathrm{P}}_{\mathrm{j}}=1\\ {\mathrm{P}}_{\mathrm{i}}=\mathrm{V}({\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}\ge {\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}})={\displaystyle \frac{2({{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{+}-{{\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}}}^{-})}{({{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{+}-{{\mathrm{S}}_{\mathrm{i}}^{\mathrm{K}}}^{-})+({{\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}}}^{+}-{{\mathrm{S}}_{\mathrm{j}}^{\mathrm{K}}}^{-})}}\end{array}\right.$$

(16)

In the formula: i, j = 1, 2, …, n_K.

Sort the n_K elements of layer K, obtain weight vectors, and normalize them:

$$\mathrm{P}=({\mathrm{P}}_1^{\mathrm{K}},{\mathrm{P}}_2^{\mathrm{K}}\cdots ,{\mathrm{P}}_{{\mathrm{n}}_{\mathrm{K}}}^{\mathrm{K}}{)}^{\mathrm{T}}$$

(17)

2.2.4. Calculate the Total Ranking Weight of the Hierarchy

The relative total objective combination ranking weight vector of K−1 layers containing n_K−1 elements is denoted as:

$${\mathrm{W}}^{(\mathrm{K}-1)}=[{\mathrm{w}}_1^{(\mathrm{K}-1)},{\mathrm{w}}_2^{(\mathrm{K}-1)},\cdots ,{\mathrm{w}}_{{\mathrm{n}}_{\mathrm{K}-1}}^{(\mathrm{K}-1)}{]}^{\mathrm{T}}$$

(18)

The K-th layer contains n_K elements, and their ranking weight vector based on a single criterion for specific factors in the K−1st layer is:

$${\mathrm{P}}_{\mathrm{i}}^{(\mathrm{K})}=[{\mathrm{w}}_{1\mathrm{i}}^{(\mathrm{K})},{\mathrm{w}}_{2\mathrm{i}}^{(\mathrm{K})},\cdots ,{\mathrm{w}}_{{\mathrm{n}}_{\mathrm{K}}\mathrm{i}}^{(\mathrm{K})}{]}^{\mathrm{T}}$$

(19)

The ranking weight vector of n_K elements in the K-th layer relative to the total target is:

$$[{\mathrm{w}}_1^{(\mathrm{k})},{\mathrm{w}}_2^{(\mathrm{k})},\cdots ,{\mathrm{w}}_{{\mathrm{n}}_{\mathrm{k}}}^{(\mathrm{k})}{]}^{\mathrm{T}}=\left[{\mathrm{P}}_1^{(\mathrm{k})},{\mathrm{P}}_2^{(\mathrm{k})},\cdots ,{\mathrm{P}}_{\mathrm{k}-1}^{(\mathrm{k})}\right]{\mathrm{W}}^{(\mathrm{k}-1)}$$

(20)

2.3. Grading and Scoring of Evaluation Indicators

2.3.1. Evaluation Index Grading Method

A list of grading standards for various performance indicators of chromium-free zinc–aluminum coatings has been developed based on a literature review and references to the preparation processes detailed in

Table 2. Performance grading standard for chromium-free zinc–aluminum coatings.

Level	I	II	III	IV	V
Thickness (μm)	>10	9.5~10	9~9.5	8.5~9	8~8.5
Hardness (H)	9	8	7	6	5
Spalling Area (%)	<5	5~15	15~35	35~65	>65
Saline Soaking Time (d)	<35	30~35	25~30	20~25	<20
Neutral Salt Fog Time (h)	>1200	1100~1200	1000~1100	900~1000	<900
Cost ($/kg)	<8.41	8.41~9.11	9.11~9.81	9.81~10.51	>10.51
Energy consumption (kw·h)	<230	230~240	240~250	250~260	>260

.

2.3.2. Scoring Method for Evaluation Indicators

Due to the different units of the above indicators, it is necessary to convert their observed values into a unified score BBB for calculation. It is worth noting that these indicators can be further categorized into benefit indicators (expected to be large) and cost indicators (expected to be small). To ensure the accuracy of the scoring, different calculation methods will be used for each category, as shown in Formulas (21) and (22).

$$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{t} \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s} {\mathrm{B}}_{\mathrm{i}}={\mathrm{L}}_1+\left|{\mathrm{C}}_{\mathrm{i}}-{\mathrm{C}}_1\right|({\mathrm{L}}_2-{\mathrm{L}}_1)/\left|{\mathrm{C}}_2-{\mathrm{C}}_1\right|$$

(21)

$$\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t} \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s} {\mathrm{B}}_{\mathrm{i}}={\mathrm{L}}_1+\left|{\mathrm{C}}_2-{\mathrm{C}}_{\mathrm{i}}\right|({\mathrm{L}}_2-{\mathrm{L}}_1)/\left|{\mathrm{C}}_2-{\mathrm{C}}_1\right|$$

(22)

In the formula, B_i represents the score of indicator i; C_i represents the value of the evaluated indicator; L₁ is the lower limit of the score for the segment where C_i is located; L₂ is the upper limit of the score for the segment where C_i is located; C₁ is the value within the score range of C_i corresponding to L₁; C₂ is the value within the score range of C_i corresponding to L₂. The corresponding scores for each evaluation level are shown in

Table 3. Index score corresponding to evaluation grade.

Level	I	II	III	IV	V
Score	90~100	75~90	60~75	40~60	0~40

.

3. Results and Discussion

3.1. Paint Formulation Design

The composition of chromium-free zinc–aluminum coatings typically includes metal slurry powder, chromium-free passivation solution, and other additives [24]. The metal slurry powder is composed of zinc powder, aluminum powder, and a wetting dispersant; the chromium-free passivation solution is principally constituted of a binder, a passivation agent, and deionized water. The chromium-free zinc–aluminum coating can be prepared by preparing the aforementioned mother liquor with auxiliary agents, such as defoamers, leveling agents, and thickeners, with the objective of improving the coating performance. The individual components of the chromium-free zinc–aluminum coating are each responsible for a specific role, and the selection of different materials will directly affect the final service performance of the coating. This article determines the reagents and dosage that should be used for each component based on the literature [25,26], and then designs and optimizes several main influencing factors in chromium-free zinc–aluminum coatings, with the objective of identifying the optimal formulation of coatings under the influence of each factor.

A four-factor, three-level orthogonal experiment was designed based on the degree of influence of each component on the performance of chromium-free zinc–aluminum coatings. The four main factors (A, B, C, and D) in the experiment were zinc–aluminum powder, PEG-400, KH-560, and sodium molybdate. The results of the L9 (3⁴) orthogonal experimental design are presented in

Table 4. Orthogonal experimental factor level table.

	Factor (%)	Zn-Al Powder (A)	PEG-400 (B)	KH-560 (C)	Molybdate (D)
Level
1		25	15	10	2
2		30	20	15	3
3		35	25	20	4

.

Coatings were prepared according to the nine formulations generated from the orthogonal experiment. The specific procedures are detailed below. Zinc and aluminum powders were gradually added to the wetting dispersant, with the stirring speed controlled from 100 r/min to 800 r/min to ensure uniform dispersion. After adding the dispersant, defoamer, and leveling agent, the mixture was stirred for 3 h to obtain solution A. Separately, sodium molybdate was dissolved in deionized water, followed by the addition of other components and pH adjustment to approximately 3.5. KH-560 was then added, and the solution was sealed and stored for 48 h to hydrolyze, resulting in solution B. Solutions A and B were then combined, and the stirring speed was increased from 600 r/min to 900 r/min, stirring for 6 h to produce the chromium-free zinc–aluminum coating. During the coating process using a centrifuge-integrated machine, both forward and reverse centrifuge speeds were set to 500 r/min, with a centrifugation time of 15 s in each direction. The coated samples were cured at 80 °C for 20 min, then baked at 300 °C for 30 min, followed by cooling to room temperature, resulting in the chromium-free zinc–aluminum coating. The results of the coating performance tests conducted on the nine distinct component coatings are presented in

Table 5. Observations of various evaluation indicators.

Serial Number	Thickness (μm)	Hardness (H)	Spalling Area (%)	Saltwater Soak (d)	Salt Spray (h)	Cost ($/kg)	Energy Consumption (kw·h)
1#	9.6	7	3	32	1052	9.73	255.8
2#	10.3	7	7	29	1040	10.35	251.4
3#	10	8	9	32	970	9.39	236.2
4#	9.9	7	4	31	1086	9.66	255.9
5#	9.5	8	10	28	982	10.54	240.8
6#	10.4	7	9	31	1050	10.03	248.6
7#	8.1	8	8	29	1105	10.13	259
8#	8.4	8	6	33	1003	9.58	253.1
9#	8.8	7	6	29	974	9.82	233.6

.

3.2. Calculation of Evaluation Index Weights

It is necessary to retrieve relevant literature and consult with experts in the paint industry in order to determine the applicable judgement matrix for extensible interval numbers. The results are presented in

Table 6. O-A extension interval number judgment matrix and weights.

O-A	A1	A2	A3	Weight
A1	<1, 1>	<2, 4>	<5, 7>	3.517
A2	<0.250, 0.500>	<1, 1>	<1, 6>	3.529
A3	<1.393, 1.700>	<3.167, 6>	<7, 14>	1.000

,

Table 7. A₁-C extension interval number judgment matrix and weights.

A1-C	C1	C2	C3	Weight
C1	<1, 1>	<1.500, 2.500>	<3, 5>	6.500
C2	<0.400, 0.667>	<1, 1>	<2, 4>	7.600
C3	<0.200, 0.333>	<0.250, 0.500>	<1, 1>	1.000

,

Table 8. A₂-C extension interval number judgment matrix and weights.

A2-C	C4	C5	Weight
C4	<1, 1>	<1, 8>	2.921
C5	<0.125, 1>	<1, 1>	1.000

and

Table 9. A₃-C extension interval number judgment matrix and weights.

A3-C	C6	C7	Weight
C6	<1, 1>	<2, 8>	5.937
C7	<0.125, 0.500>	<1, 1>	1.000

.

The extensible interval judgment matrix data presented above should be substituted into Formulas (3)–(5), respectively, in order to calculate the eigenvectors x⁻ and x⁺. The resulting calculations are presented in

Table 10. Feature vector x⁻ and x⁺.

O-A		A1-C		A2-C		A3-C
x−	x+	x−	x+	x−	x+	x−	x+
0.688	0.585	0.557	0.542	0.694	0.694	0.778	0.778
0.213	0.296	0.316	0.328	0.306	0.306	0.222	0.222
0.099	0.119	0.128	0.131

.

In accordance with the prescribed Formulas (9) and (10), the values of k and m were ascertained, respectively. Following an examination of the consistency of the extended interval number judgement matrix, the results are presented in

Table 11. k value and m value.

	O-A	A1-C	A2-C	A3-C
k	0.819	0.916	0.782	0.882
m	1.155	1.011	1.179	1.106

. The calculated result satisfies the constraints $0\le \mathrm{k}\le 1\le \mathrm{m}$, and all four extended interval number judgment matrices passed the consistency test.

In accordance with the calculated feature vectors x⁻ and x⁺, they are substituted into Formulas (11), (16), and (17) for calculation purposes. The outcomes of the single-layer weights are presented in Table 6, Table 7, Table 8 and Table 9.

The total ranking weight is calculated in accordance with the Formula (20), and the results are presented in

Table 12. Weight and score of road performance index.

	Thickness	Hardness	Spalling Area	Saltwater Soak	Salt Spray	Cost	Energy Consumption
Weight	0.188	0.220	0.029	0.327	0.112	0.106	0.018
1#	78.00	67.50	94.00	81.00	67.80	62.70	48.40
2#	100.00	67.50	87.00	72.00	66.00	46.00	52.70
3#	100.00	82.50	84.00	81.00	54.00	69.90	80.70
4#	87.00	67.50	92.00	78.00	72.90	64.20	48.20
5#	75.00	82.50	82.50	69.00	56.40	40.80	60.30
6#	100.00	67.50	84.00	78.00	67.50	55.20	62.10
7#	8.00	82.50	85.50	72.00	75.75	52.40	41.50
8#	32.00	82.50	88.50	84.00	40.60	66.00	50.53
9#	52.00	67.50	88.50	72.00	54.80	60.90	84.60

.

3.3. Performance Evaluation of Chromium-Free Zinc–Aluminum Coating

The results of the performance evaluation of the chromium-free zinc–aluminum coating prepared by each component are presented in Figure 2. The calculation results demonstrate that the 3 # formula exhibited the highest comprehensive score and the optimal coating performance.

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:

• 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;
• 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;
• 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 interrelationships 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.