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Introduction
Here is the final result, three standard metrics used in every multi-objective optimization paper. Our hypervolume aligns with theory up to three decimal places, and the hybrid performs well across all three test functions, including DTLZ23 objectives. Let's start. Hello, welcome to episode five of the hybrid multi-objective metaheuristics series. Visually comparing fronts is not enough for research, especially for three objectives or more. We need metrics that measure the closeness to the true front and solution spread in a single number. Today we write the three most popular metrics.
Step 1: Hypervolume
Create a metrics.py file Hypervolume is the area dominated by the front and bounded by the reference point. The larger, the better, and it measures both convergence and spread. For two objectives, we calculate exactly, sort by F1, then sum the area of the rectangles. For three objectives or more, we estimate using Monte Carlo. Scatter 200,000 random points and calculate the percentage dominated by the front.
Step 2: IGD & Spread
IG or Inverted Generational Distance calculates the average distance of each true front point to the nearest solution. The smaller, the better, and if there is an unreachable part of the front, its value increases. Spread from DEB measures the uniformity of distances between consecutive solutions and the distance to both ends of the front. 0 means perfectly spread out.
Step 3: Validation & Comparison
Create a test underscore metrics.py file and run it. Validate first on the true front ZDT1. Hypervolume 0.8756. Very close to the theoretical 1.21 minus one-third, which is 0.8767, and IG is exactly 0. Then the three algorithms run for 150 generations. On ZDT1, Hybrid achieves Hypervolume 0.865 and IG exactly 0.0065, while NSGA2 only reaches 0.26 and 0.46. On DTLZ23 Objective, Hybrid Hypervolume is 0.58 and MO-PSO is 0.34.
Step 4: Illustration & Metric Diagram
Let's look at the graph. The left panel illustrates Hypervolume as the blue area between the hybrid front and the star reference point. The Hypervolume and IG bar charts demonstrate that hybrid performs consistently well across all test functions. Remember, this is just one trial. In the last episode, we conducted 10 trials and performed statistical tests.
Done!
Done. We have already written Hypervolume EXAC and Monte Carlo, IGD, and SPREAD, validated them, and measured the three algorithms. In the last episode, we tested on large-scale problems with 10 trials and Wilcoxon tests. Currently working on a thesis, dissertation, or research in the field of optimization and artificial intelligence? Cerdas Inovasi Digital helps with the implementation of optimization algorithms. From NSG2 and MOPSO, hybrid algorithm design, up to Hypervolume and IG metrics, statistical test comparisons of algorithms, and opening Python programming courses for engineers. Want to master the entire series, from Pareto concepts to large-scale case studies? All 6 episodes, complete with PDF step-by-step guides and source code, are available to premium tutorial subscribers. Subscribe starting at Rp69.000 per month, save 28% with the 12-month package, or pay once Rp999.000 for lifetime access. Want just one class? Buy a single class with lifetime access. Subscribe now at the premium page cerdasinovasidigital.com Contact our WhatsApp or visit cerdasinovasidigital.com Thank you for watching, see you in the next episode. Thank you for watching, see you in the next episode.
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