## 5 Improved Radar Chart  
The conventional radar chart method is founded upon the unit circle, where the radial line extending from the center to the circumference serves as the axis of evaluation corresponding to each criterion. The normalized evaluation criteria are plotted on their corresponding axes, and the points marked on these axes are connected in sequence to form a closed polygon. This process yields the radar chart of the evaluated object, as shown in Figure 5a. However, due to the variation in index order, which results in the connection of marker points and leads to polygons of varying shapes, the conventional radar chart method fails to yield a unique final evaluation outcome. Therefore, This paper introduces an improved radar chart method, as illustrated in Figure 5b, to address this issue. The analysis procedure of the improved radar chart method encompasses the subsequent steps.  
1. The unit circle, centered at \(O\), is partitioned into n sectors according to a selected set of criteria, where each sector represents the domain corresponding to its respective criterion. By utilizing the equation \(\alpha_{i}=2\pi\alpha_{i}\), the weights (\(\omega_{i}\)) assigned to evaluation criterion \(X_{i}\) are subsequently transformed into central angles (\(\alpha_{i}\)).
2. The axis can be set as sector radius, and the corresponding criterion axis is labeled with the normalized quantitative value of each criterion.
3. The radius (\(r_{i}\)) is considered as the calibration value, while the determination of the central angle (\(\alpha_{i}\)) depends on the weight assigned. This weight is utilized for drawing a circular arc, as depicted in Figure 5b.
4. The evaluation result is derived from the criterion of eigenvectors utilizing the dimensions of both area and perimeter in the improved radar chart [69].  
By utilizing the improved radar chart, the values for area \(S_{i}\) and perimeter \(L_{i}\) can be derived, which represent the overall strengths of the evaluated object and the balance between different evaluation criteria. The evaluation object is considered better when the area in the radar chart is larger. In cases where the area remains constant, a larger perimeter  
Figure 5: Radar chart: (**a**) Traditional radar chart; (**b**) Improved radar chart.  
indicates a more balanced set of criteria. The area and perimeter of the radar chart are defined as follows.  
\[\left\{\begin{array}{l}S_{i}=\sum_{j}^{n}\frac{\pi\omega_{i}r_{j}^{2}}{n}j=1,2, \cdots,\ n\\ L_{i}=\sum_{j}^{n}\frac{2\pi\omega_{i}r_{j}}{n}j=1,2,\cdots,\ n\end{array}\right. \tag{10}\]  
where \(S_{i}\) denotes the overall merits of subject being evaluated, \(L_{i}\) signifies the balance achieved among different criteria for evaluation, \(n\) represents the number of criteria.  
Utilizing the established eigenvectors \(S_{i}\) and \(L_{i}\), the evaluation vector \(v_{i}\) can be defined as follows.  
\[\left\{\begin{array}{l}v_{i1}=S_{i}/S_{\max}\\ v_{i2}=L_{i}/(2\pi\sqrt{S_{i}/\pi})\end{array}\right. \tag{11}\]  
where \(v_{i1}\in[0,1]\) is the area evaluation value. The magnitude of the value is directly proportional to the degree of comprehensiveness in evaluating the object. Similarly, \(v_{i2}\in[0,1]\) is the perimeter evaluation value. The magnitude of the value reflects a higher level of equilibrium in the fluctuations of the evaluation object's criterion. As a result, the evaluation vector takes into account both the extent of change in the object being evaluated and the level of balance for each criterion.  
The geometric average method is adopted to construct the evaluation function. The larger the value is, the better the comprehensive performance is. The formulation of the evaluation function is shown as follows.  
\[f(v_{i1},v_{i2})=\sqrt{v_{i1}v_{i2}} \tag{12}\]  
where \(f(v_{i1},v_{i2})\) represents the comprehensive performance value.