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value is then multiplied by F , the predefined scaling factor. This is added to the value
indexed by r 3 .
However, this solution is not arbitrarily accepted in the solution. A new random
number is generated, and if this random number is less than the value of CR ,then
the new value replaces the old value in the current solution. Once all the values in
the solution are obtained, the new solution is vetted for its fitness or value and if this
improves on the value of the previous solution, the new solution replaces the previous
solution in the population. Hence the competition is only between the new child solution
and its parent solution.
[32] have suggested ten different working strategies. It mainly depends on the prob-
lem on hand for which strategy to choose. The strategies vary on the solutions to be
perturbed, number of difference solutions considered for perturbation, and finally the
type of crossover used. The following are the different strategies being applied.
Strategy 1: DE / best / 1 / exp:
u i , G +1 = x best , G + F
( x r 1 , G
x r 2 , G )
x r 2 , G
x r 3 , G
Strategy 2: DE / rand / 1 / exp:
u i , G +1 = x r 1 , G + F
λ x best , G
x r 1 , G
Strategy 3: DE / rand
best / 1 / exp: u i , G +1 = x i , G +
+ F
( x r 1 , G
x r 2 , G )
x r 1 , G
x r 4 , G
Strategy 4: DE / best / 2 / exp:
u i , G +1 = x best , G + F
x r 2 , G
x r 3 , G
x r 1 , G
x r 4 , G
Strategy 5: DE / rand / 2 / exp:
u i , G +1 = x 5 , G + F
x r 2 , G
x r 3 , G
Strategy 6: DE / best / 1 / bin:
u i , G +1 = x best , G + F
( x r 1 , G
x r 2 , G )
x r 2 , G
x r 3 , G
Strategy 7: DE / rand / 1 / bin:
u i , G +1 = x r 1 , G + F
λ x best , G
x r 1 , G
Strategy 8: DE / rand
best / 1 / bin:
u i , G +1 = x i , G +
+ F
( x r 1 , G
x r 2 , G )
x r 1 , G
x r 4 , G
Strategy 9: DE / best / 2 / bin:
u i , G +1 = x best , G + F
x r 2 , G
x r 3 , G
x r 1 , G
x r 4 , G
Strategy 10: DE / rand / 2 / bin:
u i , G +1 = x 5 , G + F
x r 2 , G
x r 3 , G
The convention shown is DE/x/y/z. DE stands for Differential Evolution, x repre-
sents a string denoting the solution to be perturbed, y is the number of difference solu-
tions considered for perturbation of x ,and z is the type of crossover being used (exp:
exponential; bin: binomial).
DE has two main phases of crossover: binomial and exponential. Generally a child
solution u i , G +1 is either taken from the parent solution x i , G or from a mutated donor
solution v i , G +1 as shown: u j , i , G +1 = v j , i , G +1 = x j , r 3 , G + F
x j , r 2 , G .
The frequency with which the donor solution v i , G +1 is chosen over the parent solu-
tion x i , G as the source of the child solution is controlled by both phases of crossover.
This is achieved through a user defined constant, crossover CR which is held constant
throughout the execution of the heuristic.
The binomial scheme takes parameters from the donor solution every time that the
generated random number is less than the CR as given by rand j [0 , 1] < CR , else all
parameters come from the parent solution x i , G .
The exponential scheme takes the child solutions from x i , G until the first time that
the random number is greater than CR ,asgivenby rand j [0 , 1] < CR , otherwise the
parameters comes from the parent solution x i , G .
x j , r 1 , G
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