# The Porcupine Measure for Comparing the Performance of Multi-Objective Optimization Algorithms
## 4 Regarding 2-Dimensional Attainment Surfaces  
The attainment surface calculation approach developed by [PERSON] and [PERSON] [1] did not describe in detail how the intersection lines should be generated. Instead, it was only stated that a random number of intersection lines, each pointing in the direction of improvement for all the objectives, should be used. This approach worked well to construct a visualization of the attainment surface. When [PERSON] and [PERSON] [8] extended the intersection line approach to develop a quantitative comparison measure, they needed the lines to be equally distributed. If the lines were not equally distributed, as depicted in Figure 2b, certain regions of the attainment surface would contribute more than others, leading to misleading results. Figure 3 depicts two example attainment surfaces with rotation-based intersection lines. Figure 3a depicts a concave attainment surface. Visually, the rotation-based intersection lines look to be equally distributed. Figure 3b, however, depicts a convex attainment surface. Visually, the length of the attainment surface between the intersection lines is larger in the regions closer to the objective axis than in the middle regions. Clearly, the rotation-based intersection lines are not equally spaced for convex-shaped fronts when comparing the length of the attainment surface represented by each intersection line. In order to address the unequal spacing of the rotation-based intersection lines, a new approach to placing the intersection lines is proposed in this paper. To compensate for the shape of the front, the intersection lines can be generated either inwardly or outwardly positioned on a line, running from the extreme values of the attainment surface, based on the shape of the attainment surfaces being compared. Figure 4 depicts the inward and outward intersection line approaches for a convex shaped front. The regions are clearly more equally spread for the inward intersection line approach. However, the direction of the intersection lines is less desirable for comparison purposes. At the edges, the intersection lines are parallel with the opposite objective's axis. Intuitively, it is more desirable that the intersection lines should be parallel to the closest objective's axis. Another disadvantage of the inward and outward approaches is that the approach to be selected depends on the shape of the front, which is typically unknown. For attainment surfaces that are not fully convex or concave, neither approach is suitable. An alternative approach, referred to as attainment-surface-shaped intersection lines (ASSIL) in this paper, is to generate the intersection lines along the shape of the attainment surface. In order to equally spread the intersection lines, the Manhattan distance is used to  
Figure 3: Attainment surfaces with rotation-based intersection lines. (**a**) Concave POF. (**b**) Convex POF. calculate equal spacings for the intersection lines along the attainment surface. Figure 5 depicts the Manhattan distance calculation between two points on the approximated POF, which is \(\vec{q}_{3}\) and \(\vec{q}_{1}\) in this case. ASSIL can be generated using Algorithm 1.