Evolutionary Computation - Computational Intelligence in Time Series Forecasting - page 198

selection. Hence, scaling involves a readjustment of fitness values to sustain a
steady selective pressure in the population and to prevent the premature
convergence of the populations.
Various techniques are available for fitness scaling or fitness windowing. Let
us assume that the objective value of the worst chromosome in the population is f w ,
and that each chromosome can be assigned a fitness value proportional to the cost
difference between the chromosome i and the worst chromosome w , i.e .

r k f f (5.2)

V k

i f

i w

where f i is the objective value or raw fitness of i th chromosome, f w is the raw
fitness of the worst chromosome, k and k f are two constants. If a maximization
problem is encountered, then a positive sign is adopted in Equation (5.2), whereas
for the minimization problem negative sign is adopted. In our experiment we set
the value of k = 10 and k f = 2.
Alternatively, the fitness scaling can be implemented using linear scaling, i.e .
the linear relationship between f and V

V

a f b (5.3)

i

i

where f is the raw fitness and V the corresponding scaled fitness. The coefficients a
and b may be chosen in a number of ways; however, in all cases, the average
scaled fitness is equal to the average raw fitness, i.e . V avg = f avg .
In the following example we use V max = C mult f max , select C mult = 2, and V min =
f min . Towards the end of a GA run, this choice of C mult stretches the raw fitness
significantly. In turn, this may cause difficulty in applying the above linear
relationship, when we cannot scale to the desired multiple C mult ; in this case,
scaling is performed still keeping V avg = f avg and then stretching the fitness until the
minimum value maps to zero, i.e. f min = 0. The entire scaling procedure is
performed in three routines, namely pre-scale, scale, and scale-pop. This includes
calculation of f max , f min , f avg , etc .
Now, we check the following relation (Goldberg,1989):

f t f f . 1 (5.4)

C C

mult mult

min avg max

If the last relationship holds, then the calculation of a and b will be

a 1 f f f (5.5)

C

mult

avg max avg

b . 1 a f (5.6)

avg

Otherwise, if relationship (5.4) does not hold then calculation of a and b will be
as follows: