## 3 Construction of a Model based on the ViP Principle and the AHP
### Analytic Hierarchy Process (AHP)  
The analytic hierarchy process (AHP) was proposed by the American operations research scientist [PERSON] in the early 1970s [15]. It is a decision analysis method that uses hierarchical weighting. The significance of indicators in the hierarchy is assessed using values ranging from 1 to 9 based on the scaling principle of the AHP, with their reciprocals serving as satisfaction values. The higher the satisfaction value, the more significant the indicator. The weights of design factors are determined using the analytic hierarchy process as outlined below [16]:  
(1) Formulating a judgment matrix for the indicator layer \(A\) is performed as follows:  
\[A=(a_{ij})_{n\times n} \tag{1}\]  
where \(a_{ij}\) represents the significance of \(i\) compared to indicator \(j\). \[a_{ij}>\ 0\ \text{and}\ a_{ij}=\frac{1}{a_{ji}}(i,\ j=1,\ 2,\ \ldots,\ n) \tag{2}\]  
(2) Solving the maximum eigenvalue using the consistency test is performed as follows:  
\[CI=\frac{\lambda\text{max}\ -n}{n-1} \tag{3}\]  
\[CR=\frac{CI}{RI} \tag{4}\]  
In the equation, \(\lambda\text{max}\) represents the maximum eigenvalue of the judgment matrix. \(RI\) represents the random average consistency index (as shown in Table 1). When \(CR<0.10\), the consistency of the judgment matrix is considered acceptable; otherwise, the judgment matrix should be modified appropriately. (3) The robustness of the data is determined using three different calculation methods based on Equations (5)-(7), and the three results are summed and averaged \(\omega_{4}\) as follows:  
\[\mathbf{AW}=\lambda\text{max}\mathbf{W} \tag{5}\]  
In the equation above, the eigenvector method is used, and \(w\) represents the weight vector. (4) The calculation of the weight vector using the geometric average method is performed as follows:  
\[W_{i}=\frac{(\prod\limits_{j=1}^{n}a_{ij})^{\frac{1}{n}}}{\sum\limits_{i=1}^{ n}(\prod\limits_{j=1}^{n}a_{ij})^{\frac{1}{n}}},\ i=1,\ 2,\ \ldots,\ n \tag{6}\]  
(5) The calculation of the weight vector using the arithmetic mean method is performed as follows: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**N** & **1** & **2** & **3** & **4** \\ \hline RI & 0 & 0 & 0.52 & 0.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Random mean consistency indicators. \[W_{i}=\frac{1}{n}\sum_{j=1}^{n}\frac{a_{ij}}{\sum\limits_{k=1}^{n}a_{kj}},\ i=1,\ 2,\ \ldots,\ n \tag{7}\]  
The AHP establishes clear user preferences for the future requirements of this product type, simplifies the complexity of design factors, optimizes the existing design methods, and provides new insights (as shown in Figure 2). The operational steps of the analytic hierarchy process mainly comprise the following steps:  
1. Establish a hierarchical structure model: Divide the decision-making goals, factors to be considered (decision-making criteria), and decision-making objects into the highest level, the middle level, and the lowest level according to their interrelationships, and draw a hierarchical structure diagram. The highest level is the purpose of decision making and the problem to be solved; the lowest level is the alternatives when making decisions; and the middle level is the factors to be considered and the criteria for decision making. Figure 2: ViP design flowchart with the introduction of the AHP. 2. Construct all judgment matrices in each level: Use the paired comparison method and the scale of 1 to 9 to construct a pairwise comparison matrix of each factor in each level to the previous level. The paired comparison method is a pairwise comparison method in which two indicators are compared; the 1 to 9 scale refers to using a number between 1 and 9 to scale the relative importance of different indicators. 3. Calculate the relative weight of the compared element to the criterion from the judgment matrix and conduct a consistency test: The necessity of conducting a consistency test on the matrix is that the judgment matrices we construct are all forward and reciprocal matrices.