# The Porcupine Measure for Comparing the Performance of Multi-Objective Optimization Algorithms
## 7 Conclusions  
This article investigated shortcomings that may have led to the lack of adoption of attainment-surface-based quantitative performance measurements for multi-objective optimization algorithms. It was shown that the quantitative measure proposed by [PERSON] and [PERSON] was biased against convex Pareto-optimal fronts (POFs) when using rotational intersection lines. The attainment-surface-shaped intersection lines (ASSIL) generation approach was proposed. The ASSIL generation approach was shown not to be biased against any attainment surface shape. An algorithm for an \(M\)-dimensional attainment-surface-based quantitative measure, named the porcupine measure, was presented. Additionally, a computationally optimized implementation of the porcupine measure was introduced and analyzed. The results indicated that the optimized implementation performed as well as the naive implementation. The porcupine measure allows for a quantitative comparison between \(M\)-dimensional approximated POFs through the use of attainment surfaces. The porcupine measure provides additional information on an algorithm's performance when compared to another algorithm, which was previously not quantifiable. A thorough comparison comparing the _state-of-the-art_ multi-objective optimization algorithms using the porcupine measure is left as future work. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Naive**} & \multicolumn{4}{c}{**Optimized**} \\  & & **Mean** & \(\sigma\) & **Min** & **Max** & **Mean** & \(\sigma\) & **Min** & **Max** \\ \hline \multirow{6}{*}{WFG6} & OMOPSO & 10.21 & 2.398 & 6.651 & 15.76 & 10.55 & 2.356 & 7.035 & 15.89 \\  & SMPSO & 45.3 & 3.61 & 37.82 & 53.57 & 45.62 & 3.754 & 38.12 & 54.17 \\ \cline{2-10}  & OMOPSO & 11.93 & 4.704 & 6.098 & 22.65 & 12.29 & 4.854 & 5.991 & 23.65 \\  & VEPSO & 16.05 & 5.158 & 5.421 & 25.88 & 16.53 & 5.414 & 5.566 & 26.5 \\ \cline{2-10}  & VEPSO & 1.02 & 0.8952 & 0.01929 & 3.795 & 1.046 & 0.9204 & 0.007599 & 3.983 \\  & SMPSO & 20.61 & 6.907 & 8.904 & 37.24 & 20.94 & 7.022 & 8.945 & 37.69 \\ \hline \multirow{6}{*}{WFG7} & OMOPSO & 38.86 & 3.64 & 31.77 & 44.27 & 39.28 & 3.516 & 32.12 & 44.38 \\  & SMPSO & 3.892 & 0.9721 & 1.684 & 5.522 & 3.888 & 1.018 & 1.641 & 5.507 \\ \cline{2-10}  & OMOPSO & 74.41 & 2.681 & 67.75 & 78.68 & 74.72 & 2.743 & 67.8 & 79.15 \\  & VEPSO & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ \cline{2-10}  & VEPSO & 0.001361 & 0.007456 & 0.0 & 0.04084 & \(9.506\times 10^{-5}\) & 0.0005206 & 0.0 & 0.002852 \\  & SMPSO & 73.39 & 2.598 & 68.5 & 77.94 & 73.69 & 2.682 & 68.92 & 79.15 \\ \hline \multirow{6}{*}{WFG8} & OMOPSO & 41.21 & 4.789 & 31.08 & 49.28 & 41.55 & 4.954 & 30.95 & 49.24 \\  & SMPSO & 3.049 & 0.948 & 1.443 & 5.066 & 3.106 & 0.9568 & 1.54 & 5.088 \\ \cline{2-10}  & OMOPSO & 70.18 & 3.918 & 63.04 & 77.36 & 70.49 & 3.847 & 63.4 & 77.46 \\ \cline{2-10}  & VEPSO & \(6.884\times 10^{-5}\) & 0.0002857 & 0.0 & 0.001467 & 0.0 & 0.0 & 0.0 & 0.0 \\ \cline{2-10}  & VEPSO & 0.1489 & 0.2362 & 0.0 & 1.023 & 0.1458 & 0.223 & 0.0 & 0.869 \\ \cline{2-10}  & SMPSO & 62.88 & 3.841 & 56.14 & 72.58 & 63.15 & 3.861 & 56.65 & 73.03 \\ \hline \multirow{6}{*}{WFG9} & OMOPSO & 15.89 & 2.462 & 11.12 & 21.29 & 15.87 & 2.47 & 11.48 & 21.66 \\  & SMPSO & 37.58 & 3.03 & 32.14 & 43.94 & 37.68 & 3.048 & 31.95 & 44.27 \\ \cline{2-10}  & OMOPSO & 20.7 & 2.526 & 16.86 & 25.28 & 20.61 & 2.515 & 16.9 & 26.05 \\ \cline{2-10}  & VEPSO & 23.59 & 1.801 & 19.21 & 26.88 & 24.0 & 1.795 & 19.61 & 27.51 \\ \cline{2-10}  & VEPSO & 1.206 & 0.7576 & 0.2238 & 3.162 & 1.219 & 0.8411 & 0.2526 & 3.605 \\ \cline{2-10}  & SMPSO & 21.91 & 5.093 & 12.24 & 31.96 & 21.8 & 5.307 & 11.12 & 32.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._  
**Author Contributions:** Conceptualization, C.S. and A.E. ; methodology, C.S. and A.E. ; formal analysis, C.S. ; investigation, C.S. ; resources, C.S. ; data curation, C.S. ; writing-original draft preparation, C.S. ; writing-review and editing, A.E.