4.2.1. Establishing Comparison Judgment Matrix

Based on the evaluation scales in the AHP, the elements in the product hierarchy model are compared and assigned. To use mathematical methods for data processing, the data needs to be transformed into a matrix to quantify the results and determine the importance of the design elements. Supposing n influencing elements, *b*<sup>1</sup> ..., *b<sup>i</sup>* ..., *b<sup>j</sup>* ..., *bn*, the project elements are compared with each other in pairs and transformed into a judgment matrix as follows:

$$\mathbf{B} = \begin{bmatrix} 1 \cdots \mathbf{b}\_{1i} \cdots \mathbf{b}\_{1j} \cdots \mathbf{b}\_{1n} \\ b\_{i1} \cdots \mathbf{1} \cdots b\_{ij} \cdots b\_{in} \\ b\_{j1} \cdots \mathbf{b}\_{ji} \cdots \mathbf{1} \cdots b\_{jn} \\ b\_{n1} \cdots \mathbf{b}\_{ni} \cdots \mathbf{b}\_{nj} \cdots \mathbf{1} \end{bmatrix} = \begin{pmatrix} b\_{ij} \end{pmatrix}\_{n \times n} \tag{1}$$

The Perron–Fresenius theorem shows that matrix *B* has a unique nonzero eigenroot, i.e., the largest eigenroot (*λmax*) corresponds to the eigenvector (*w*).

$$B\_w = \lambda\_{\text{max}} w \tag{2}$$

The specific steps for calculating the feature vectors using the sum-product method are as follows:

Normalize the data in *b* by column.

$$
\overline{b\_{\rm ij}} = b\_{\rm ij} / \sum\_{j=1}^{n} \overline{b\_{\rm ij}}(\mathbf{i}, j, \dots, n) \tag{3}
$$

Sum the normalized matrix peers.

$$
\tilde{w}\_i = \sum\_{j=1}^n \overline{b\_{ij}} (i = 1, 2, \dots, n) \tag{4}
$$

Divide the summed vector by n to obtain the weight vector.

$$
\tilde{w}\_{\mathbf{i}} = \tilde{w}\_{\mathbf{i}} / \mathfrak{n} \tag{5}
$$

Find the maximum characteristic root.

$$
\lambda\_{\text{max}} = \frac{1}{n} \sum\_{i=1}^{n} \frac{n}{i=1} \frac{(B\_w)\_i}{w\_i} \tag{6}
$$

where (*Bw*)*<sup>i</sup>* denotes its component of the vector *Bw*.

Based on the above Equations (1)–(6), we calculated the weight values of the designed element objectives at the criterion and program levels and then ranked them in terms of importance to complete the decision on the influencing factors.
