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Joshua Knowles

Plotting attainment surfaces

3d attainment surface plots:

Knowles, J. (2005) A summary-attainment-surface plotting method for visualizing the performance of stochastic multiobjective optimizers. IEEE Intelligent Systems Design and Applications (ISDA V). In press. PDF (777KB)

Download the Plotting Source Code: plot-attainment.tar.gz (156KB)

The software download contains C source code that takes R approximation sets as input, and outputs a number of uniformly distributed points on the Jth attainment surface (where J is in the range 1,...,R). Its purpose is to allow straightforward plotting of (especially 3d) attainment surfaces in gnuplot. Although it is most useful for 2- and 3-objective problems, the code works for approximation sets of any dimension. The code can handle mixed minimization/maximization problems.

Example output:

Here, the input was a collection of 21 approximation sets (i.e. nondominated sets) from 21 runs of an optimizer on a 3-objectives mixed minimization/maximization problem. The output can be any of the 21 attainment surfaces. Here, three of these surfaces, plotted using gnuplot, are shown.

1st (best) attainment surface:

11th (median) attainment surface:

21st (worst) attainment surface:

The concept of an attainment surface in multiobjective optimization is introduced in the following two papers.

1. Carlos M. Fonseca and Peter J. Fleming. On the Performance Assessment and Comparison of Stochastic Multiobjective Optimizers. In Hans-Michael Voigt, Werner Ebeling, Ingo Rechenberg, and Hans-Paul Schwefel, editors, Parallel Problem Solving from Nature--PPSN IV, Lecture Notes in Computer Science, pages 584-593, Berlin, Germany, September 1996. Springer-Verlag.

2. Viviane Grunert da Fonseca, Carlos M. Fonseca, and Andreia O. Hall. Inferential Performance Assessment of Stochastic Optimisers and the Attainment Function. In Eckart Zitzler, Kalyanmoy Deb, Lothar Thiele, Carlos A. Coello Coello, and David Corne, editors, First International Conference on Evolutionary Multi-Criterion Optimization, pages 213-225. Springer-Verlag. Lecture Notes in Computer Science No. 1993, 2001.

Some keywords to help GOOGLE find this page are: evolutionary computation, Pareto optimization, multiobjective optimization, MOEA, EMOO, attainment surface, attainment function, approximation set, plot