Forward Backward Transformation - Differential Evolution: A Topic for Global Permutation-Based Combinatorial Optimization - page 37

To ensure that each child solution differs from the parent solution, both the expo-
nential and binomial schemes take at least one value from the mutated donor solution
v i , G +1 .

3.2.1 Tuning Parameters

Outlining an absolute value for CR is difficult. It is largely problem dependent. However
a few guidelines have been laid down [31]. When using binomial scheme, intermediate
values of CR produce good results. If the objective function is known to be separable,
then CR = 0 in conjunction with binomial scheme is recommended. The recommended
value of CR should be close to or equal to 1, since the possibility or crossover occurring
is high. The higher the value of CR , the greater the possibility of the random number
generated being less than the value of CR , and thus initiating the crossover.
The general description of F is that it should be at least above 0.5, in order to provide
sufficient scaling of the produced value.
The tuning parameters and their guidelines are given in Table 3.1

Ta b l e 3 . 1 . Guide to choosing best initial control variables

Control Variables Lo Hi Best? Comments

F: Scaling Factor 0 1.0+ 0.3-0.9 F ≥ 0.5

CR: Crossover probability 0 1 0.8 − 1.0 CR = 0, seperable

CR = 1, epistatic

3.3 Discrete Differential Evolution

The canonical DE cannot be applied to discrete or permutative problems without modi-
fication. The internal crossover and mutation mechanism invariably change any applied
value to a real number. This in itself will lead to in-feasible solutions.
The objective then becomes one of transformation, either that of the population or
that of the internal crossover / mutation mechanism of DE. For this chapter, it was de-
cided not to modify in any way the operation of DE strategies, but to manipulate the
population in such a way as to enable DE to operate unhindered.
Since the solution for the population is permutative, a suitable conversion routine was
required in order to change the solution from integer to real and then back to integer
after crossover. The population was generated as a permutative string. Two conversions
routines were devised, one was Forward transformation and the other Backward trans-
formation for the conversion between integer and real values. This new heuristic was
termed Discrete Differential Evolution (DDE) [28].
The basic outline DDE is given below.

1. Initial Phase
a) Population Generation : An initial number of discrete trial solutions are gener-
ated for the initial population.