Page 90 - 49A Field Guide to Genetic Programming

76 9 Multi-objective Genetic Programming

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example, one could use a linear combination of the form f = w i f i , where
i
the parameters w 1 , w 2 , . . . are suitable constants. A MOO problem can then
be solved by using any single-objective optimisation technique with f as a
fitness function. This method has been used frequently in GP to control
bloat. By combining program fitness and program size to form a parsimo-
nious fitness function one can evolve solutions that satisfy both objectives
(see Koza (1992); Zhang and M¨uhlenbein (1993, 1995); Zhang, Ohm, and
M¨uhlenbein (1997) and Section 11.3.2).
A semi-linear aggregation of fitness and speed was used in (Langdon
and Poli, 1998b) to improve the performance of GP on the Santa Fe Trail
Ant problem. There, a threshold was used to limit the impact of speed to
avoid providing an excessive bias towards ants that were fast but could not
complete the trail.
A fitness measure which linearly combines two related objectives, the
sum of squared errors and the number of hits (a hit is a fitness case in which
the error falls below a pre-defined threshold), was used in (Langdon, Barrett,
and Buxton, 2003) to predict biochemical interactions in drug discovery.
Zhang and Bhowan (2004) used a MO GP approach for object detection.
Their fitness function was a linear combination of the detection rate (the
percentage of small objects correctly reported), the false alarm rate (the
percentage of non-objects incorrectly reported as objects), and the false
alarm area (the number of false alarm pixels which were not object centres
but were incorrectly reported as object centres).
O’Reilly and Hemberg (2007) used six objectives for the evolution of
L-systems which developed into 3-D surfaces in response to a simulated
environment. The objectives included the size of the surface, its smoothness,
its symmetry, its undulation, the degree of subdivision of the surface, and
the softness of its boundaries.
(Koza, Jones, Keane, and Streeter, 2004) used 16 different objectives
in the process of designing analogue electrical circuits. In the case of an
amplifier circuit these included: the 10dB initial gain, the supply current, the
offset voltage, the gain ratio, the output swing, the variable load resistance
signal output, etc. These objectives were combined in a complex heuristic
way into a scalar fitness measure. In particular, objectives were divided
into groups and many objectives were treated as penalties that were applied
to the main fitness components only if they are outside certain acceptable
tolerances.

9.2 Keeping the Objectives Separate

Since selection does not depend upon how the members of the population
are represented, the MOO techniques developed for other evolutionary al-
gorithms can be easily adapted to GP.