# The Porcupine Measure for Comparing the Performance of Multi-Objective Optimization Algorithms
## 5 Weighted 2-Dimensional Attainment-Surface-Shaped Intersection Lines  
As an alternative to the equally spread intersection lines used by ASSIL, intersection lines can be generated along the shape of the POF, with at least one intersection line per attainment surface line segment. Because the attainment surface segments are not all of equal lengths, a weight is associated with each intersection line to balance the KC measure result. The weighted attainment-surface-shaped intersection lines (WASSIL) generation algorithm is given in Algorithm 3. Figure 9 depicts a convex POF with WASSIL-generated intersection lines. The figure clearly shows that the intersection lines are positioned along the attainment surface, and due to the positioning, the lines are angled slightly differently from the intersection lines in Figure 2(b). The WASSIL algorithm should, for the test cases, result in a weighted KC measure result that matches the true KC measure result. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Separability** & **Modality** & **Geometry** \\ \hline ZDT1 & Separable & Unimodal & Convex \\ ZDT2 & Separable & Unimodal & Concave \\ ZDT3 & Separable & Unimodal/multimodal & Disconnected \\ ZDT4 & Separable & Unimodal/multimodal & Convex \\ ZDT6 & Separable & Multimodal & Concave \\ WFG1 & Separable & Unimodal & Convex, mixed \\ WFG2 & Non-separable & Unimodal/multimodal & Convex, disconnected \\ WFG3 & Non-separable & Unimodal & Linear, degenerate \\ WFG4 & Separable & Multimodal & Concave \\ WFG5 & Separable & Multimodal & Concave \\ WFG6 & Non-separable & Unimodal & Concave \\ WFG7 & Separable & Unimodal & Concave \\ WFG8 & Non-separable & Unimodal & Concave \\ WFG9 & Non-separable & Multimodal, deceptive & Concave \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Properties of the ZDT and WFG problems. \begin{table}
\begin{tabular}{c l c c c} \hline \hline
**Problem** & **Intersections** & \(\boldsymbol{\mathcal{V}}\)**vs.**\(\boldsymbol{\mathcal{O}}\) & \(\boldsymbol{\mathcal{V}}\)**vs.**\(\boldsymbol{\mathcal{S}}\) & \(\boldsymbol{\mathcal{S}}\)**vs.**\(\boldsymbol{\mathcal{O}}\) \\ \hline \multirow{4}{*}{WFG6} & Rotational & \((25.6,13.5)\) & \((0.0,99.9)\) & \((99.9,0.0)\) \\  & Inward & \((9.9,13.5)\) & \((0.0,100.0)\) & \((100.0,0.0)\) \\  & Outward & \((29.6,13.7)\) & \((0.0,99.9)\) & \((99.9,0.0)\) \\  & ASSIL & \((23.3,13.7)\) & \((0.0,99.9)\) & \((99.9,0.0)\) \\ \hline \multirow{4}{*}{WFG7} & Rotational & \((0.0,99.9)\) & \((0.0,99.9)\) & \((0.0,92.7)\) \\  & Inward & \((0.0,100.0)\) & \((0.0,100.0)\) & \((0.0,95.4)\) \\  & Outward & \((0.0,99.9)\) & \((0.0,99.9)\) & \((0.0,92.6)\) \\  & ASSIL & \((0.0,99.9)\) & \((0.0,99.9)\) & \((0.0,93.2)\) \\ \hline \multirow{4}{*}{WFG8} & Rotational & \((0.0,99.9)\) & \((0.0,99.9)\) & \((0.0,94.6)\) \\  & Inward & \((0.0,100.0)\) & \((0.0,100.0)\) & \((0.0,95.8)\) \\  & Outward & \((0.0,99.9)\) & \((0.0,99.9)\) & \((0.0,94.4)\) \\  & ASSIL & \((0.0,99.9)\) & \((0.0,99.9)\) & \((0.0,95.1)\) \\ \hline \multirow{4}{*}{WFG9} & Rotational & \((8.0,30.3)\) & \((0.0,99.9)\) & \((99.9,0.0)\) \\  & Inward & \((2.6,\mathbf{41.1})\) & \((0.0,100.0)\) & \((100.0,0.0)\) \\ \cline{1-1}  & Outward & \((8.8,27.9)\) & \((0.0,99.9)\) & \((99.9,0.0)\) \\ \cline{1-1}  & ASSIL & \((7.1,31.5)\) & \((0.0,99.9)\) & \((99.9,0.0)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Note that the weighted KC measure is calculated as shown in Algorithm 4. Table 5 summarises the true KC measure, the KC measure with rotation-based and random intersection lines, the KC measure with ASSIL, and the KC measure with WASSIL results. For each of the approaches, 1000 intersection lines were used for the calculation.