ETH Zuerich - Homepage
Systems Optimization (SOP)

Wichtiger Hinweis:
Diese Website wird in älteren Versionen von Netscape ohne graphische Elemente dargestellt. Die Funktionalität der Website ist aber trotzdem gewährleistet. Wenn Sie diese Website regelmässig benutzen, empfehlen wir Ihnen, auf Ihrem Computer einen aktuellen Browser zu installieren. Weitere Informationen finden Sie auf
folgender Seite.

Important Note:
The content in this site is accessible to any browser or Internet device, however, some graphics will display correctly only in the newer versions of Netscape. To get the most out of our site we suggest you upgrade to the latest Netscape.
More information

ETH Zürich - D-ITET - TIK - SOP - Downloads & Materials - Supplementary Materials - Testproblems - Zdt1
print page
this webpage might no longer be updated more...
Untitled Document


Pareto Front:

Relevant Publications:
  • E. Zitzler, K. Deb, and L. Thiele. Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation, 8(2):173-195, 2000 (PDF) (bibtex)
Reference Point: Reference Point used: (11,11)

Optimal Distributions:

(see "Maximum Hypervolume" for more plots)
of 5 points:
of 10 points:
of 20 points:
of 50 points:
Maximum Hypervolume:
µHV ValuesPlot
120 2/3

How to approximate the optimal distributions

  1. The x-values xi of the µ points are equally distributed between 0 and 1
  2. The last point is known to be an extremal value, hence only the remaining µ -1 points are optimized.
  3. All points p with 1 <= p < µ, starting with point p = 1, are optimized according to the following formulas:

    For p = 1 (leftmost point)
    The value of x1 is set to aopt(r,b), where r is the y-value of the reference point and b is x2.

    For p > 1 and p < mu (non-extremal points)
    The value of xp is set to bopt(a,c), where a is xp-1 and b is xp+1

  4. The Hypervolume-Indicator is calculated. If the value did increase less than a predefined value eps, the distribution of x is returned. Otherwise, step 3 is repeated.
© 2008 Institut TIK, ETH Zürich | Imprint | Last updated: Mon, 03 Nov 2008 18:06 | Valid XHTML 1.0! Valid CSS! Valid XHTML 1.0