When given multiple fitness functions, it is natural to think of combining them in some way so as to produce an aggregate scalar fitness function. For example, one could use a linear combination of the form f = ć iwifi, where the parameters w1, w2, ... 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 parsimonious fitness function one can evolve solutions that satisfy both objectives (see Koza ( 1992); ?); ?); ? 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.
? 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.