## 3 Results and Discussion
### Multi-Criteria Decision Making with TOPSIS
#### 3.6.2 Weight Determination of Each Criteria  
The Analytical Hierarchy Process (AHP) method was employed to ascertain the relative importance of the criteria. Following the determination of the criteria, comparison  
Figure 9: Multi-criteria decision-making steps.  
matrices were generated. The scale presented in Table 5 was utilized to determine the relative importance of the criteria.  
The evaluation was conducted in accordance with the established scale comparison criteria, with the most significant criterion being the ability of the toy to withstand compressive forces exerted by both children and adults without breaking. The compressive strength of 1, representing its inherent durability, was identified as the paramount criterion. The cost of the toy was also deemed a crucial factor. While cost is not as consequential as the breakage of the part, it is nevertheless relevant when considering surface roughness and dimensional precision. Given that it is a toy part, surface roughness is not a primary concern; however, it is more significant than dimensional precision. In light of these evaluations, a table of criterion comparison values was created and is presented in Table 6.  
First, the comparison matrix, also known as the AHP Matrix, is created in Equation (2).  
\[\begin{bmatrix}1&3&6&7\\ 1/3&1&3&6\\ 1/6&1/3&1&3\\ 1/7&1/6&1/3&1\end{bmatrix} \tag{2}\]  
The values in each column are then summed and given in Equations (3)-(6).  
Sum (Column 1): \(1+1/3+1/6+1/7=1+0.333+0.167+0.143=1.643\)  
Sum (Column 2): \(3+1+1/3+1/6=3+1+0.333+0.167=4.5\)  
Sum (Column 3): \(6+3+1+1/3=6+3+1+0.333=10.333\)  
Sum (Column 4): \(7+6+3+1=17\)  
Then the sum of columns is given Equation (7).  
Then the sum of columns = [1.643, 4.5, 10.333, 17].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Criteria** & **Compressive Strength** & **Cost** & **Surface Roughness** & **Dimensional Accuracy** \\ \hline Compressive strength & 1 & 3 & 6 & 7 \\ \hline Cost & 1/3 & 1 & 3 & 6 \\ \hline Surface roughness & 1/6 & 1/3 & 1 & 3 \\ \hline Dimensional accuracy & 1/7 & 1/6 & 1/3 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Scale comparison criteria values.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Value** & **Description** \\ \hline
1 & Two criteria are equally important \\ \hline
3 & One is moderately important to the other \\ \hline
5 & One is strongly important to the other \\ \hline
7 & One is very strongly important to the other \\ \hline
9 & One is extremely important to the other \\ \hline
2, 4, 6, 8 & Intermediate values \\ \hline \hline \end{tabular}
\end{table}
Table 5: Scale comparison table.  
By dividing each value by the column total, a normalization matrix is created and given in Equation (8).  
\[\begin{bmatrix}\frac{1}{1.643}&\frac{3}{4.5}&\frac{6}{10.333}&\frac{7}{17}\\ \frac{1/3}{1.643}&\frac{1}{4.5}&\frac{3}{10.333}&\frac{6}{17}\\ \frac{1/6}{1.643}&\frac{1}{4.5}&\frac{1}{10.333}&\frac{3}{17}\\ \frac{1/7}{1.643}&\frac{1}{4.5}&\frac{1}{10.333}&\frac{1}{17}\\ \end{bmatrix}=\begin{bmatrix}0.6087&0.6667&0.5806&0.4118\\ 0.2029&0.2222&0.2903&0.3529\\ 0.1014&0.0741&0.0968&0.1765\\ 0.0870&0.0370&0.0323&0.0588\\ \end{bmatrix} \tag{8}\]  
Criterion weights are calculated by averaging each row and given in Equation (9).  
\[\text{Criterion Weight: [0.567, 0.267, 0.112, 0.054]} \tag{9}\]  
Accordingly, the criteria weights are  
* Compressive strength: 56.7%;
* Cost: 26.7%;
* Surface roughness: 11.2%;
* Dimensional accuracy: 5.4%.