# The Porcupine Measure for Comparing the Performance of Multi-Objective Optimization Algorithms
## 4 Regarding 2-Dimensional Attainment Surfaces
```
1:Input: The optimal POF, \(Q=\{\vec{q}_{i}:i\in\{1,\ldots,I\}\}\) with \(I\) solutions and \(\vec{q}_{i}=(q_{i1},q_{i2})\)
2:Output: An attainment surface
3:Sort \(Q\) in ascending order by \(q_{i1}\)
4:\(i\gets 1\)
5: Let \(N\) be the desired number of intersection lines
6:for\(i_{n}\gets 1,\ldots,N\)do
7:\(d\leftarrow(i_{n}-1)\times\frac{(q_{11}-q_{11})+(q_{12}-q_{12})}{N-1}\)
8:while\(d<(q_{i1}-q_{11})+(q_{12}-q_{i2})\)do
9:\(i\gets i+1\)
10:endwhile
11:if\(d=(q_{i1}-q_{11})+(q_{12}-q_{i2})\)then
12:\(\vec{\hat{q}}\leftarrow\vec{q}_{i}\)
13:elseif\(d\leq(q_{(i+1)1}-q_{11})+(q_{12}-q_{i2})\)then
14:\(\vec{\hat{q}}\leftarrow(q_{11}+(d-(q_{12}-q_{i2})),q_{i2})\)
15:else
16:\(\vec{\hat{q}}\leftarrow(q_{(i+1)1},q_{12}-(d-(q_{(i+1)1}-q_{11})))\)
17:endif
18:\(\theta\leftarrow\frac{i_{n}-1}{N-1}\times\frac{\pi}{2}\)
19:\(From\leftarrow(\hat{q}_{1}-\sin\theta,\hat{q}_{2}-\cos\theta)\)
20:\(To\leftarrow(\hat{q}_{1}+\sin\theta,\hat{q}_{2}+\cos\theta)\) // Finally, draw the generated intersection line
21: drawIntersectionLine(\(From\), \(To\))
22:endfor
```  
**Algorithm 1** Attainment-surface-shaped intersection line (ASSIL) generation. Figure 4: Attainment surfaces with outward/inward intersection lines. (**a**) Inward. (**b**) Outward. Figure 5: Attainment surface with Manhattan distance calculations. Intersection lines are spaced equally along the attainment surface. The intersection lines are rotated incrementally such that the intersection lines at the ends of the attainment surface are parallel to the objective axis. Figure 6 depicts the attainment-surface-shaped intersection line approach. The generation of the intersection lines along the shape of the attainment surface allows for an equal spacing of the intersection lines independent of the shape of the front. For all shapes that the attainment surface can assume, whether convex, concave, or mixed, the intersection lines are equally spread out. The KC measure is calculated as shown in Algorithm 2. ```
1:Input: Intersection lines for algorithms A and B to be compared
2:Output: The KC measure
3: Let \(total=0\)
4: Let \(wins_{A}=0\)
5: Let \(wins_{B}=0\)
6:for each intersection line \(l\)do
7: Let \(O\) be the strict ordering of the intersection points for algorithms \(A\) and \(B\) on intersection line \(l\)
8: Let \(O_{A}\subset O\) be the ordering of the intersection points for algorithm \(A\) on intersection line \(l\)
9: Let \(O_{B}\subset O\) be the ordering of the intersection points for algorithm \(B\) on intersection line \(l\)
10:if\(O_{A}\) is statistically significantly less than \(O_{B}\)then
11:\(wins_{A}=wins_{A}+1\)
12:elseif\(O_{B}\) is statistically significantly less than \(O_{A}\)then
13:\(wins_{B}=wins_{B}+1\)
14:endif
15:\(total=total+1\)
16:endfor
17:Return \([\frac{100 wins_{A}}{total},\frac{100 wins_{B}}{total}]\)
```  
**Algorithm 2** Algorithm for the calculation of the KC measure. An evaluation of the rotation-based and random intersection line approaches is presented using six artificially generated POF test cases based on those used by [PERSON] and [PERSON] [8]. Figure 7 depicts the six artificially generated POF test cases. Each of these artificially generated POF test cases was tested using six pof shape geometries, namely concave, convex, linear, mixed, and disconnected geometries. Figure 8 depicts the five POF shape geometries. Table 1 summarises the true KC, the KC with rotation-based and random intersection lines, and the KC with ASSIL results. Values in red indicate attainment surfaces obtained that outperformed the control method (i.e., the true KC measure) by more than 5%, while  
Figure 6: Attainment surfaces with unbiased ASSIL. (**a**) Convex POF. (**b**) Concave POF. values in blue indicate attainment surfaces that were found that were 5% worse than the control method. For each of the approaches, 1000 intersection lines were used for the calculation.