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98     I.K. Nikolos et al.
                              The DE algorithm implements real encoding for the values of the objective
                           function’s parameters. In order to obtain a starting point for the algorithm,
                           an initialization of the population takes place. Often the only information
                           available is the boundaries of the parameters. Therefore the initialization is
                           established by randomly assigning values to the parameters within the given
                           boundaries

                               (0)      (U)    (L)    (L)
                              x   = r · x  − x     + x  ,i =1,...,n pop ,j =1,...,n param ,  (22)
                               i,j      j      j      j
                           where r is a uniformly distributed random value within range [0, 1]. DE’s
                           mutation operator is based on a triplet of randomly selected different individ-
                           uals. A new parameter vector is generated by adding the weighted difference
                           vector between the two members of the triplet to the third one (the donor).
                           In this way a perturbed individual is generated. The perturbed individual
                           and the initial population member are then subjected to a crossover opera-
                           tion that generates the final candidate solution

                                   ⎧  (G)        (G)   (G)
                                   ⎨ x   + F · x    − x
                             (G+1)    C i ,j     A i ,j  B i ,j  if (r ≤ C r ∨ j = k) ∀ j =1,...,n param
                           x i,j  =
                                              x                         otherwise ,
                                   ⎩           (G)
                                               i,j
                                                                                           (23)
                                  (G)
                           where x    is called the “donor”, G is the current generation,
                                  C i ,j
                                     i =1,...,n pop ,j =1,...,n param
                                     A i ∈ [1,...,n pop ] ,B i ∈ [1,...,n pop ] ,C i ∈ [1,...,n pop ]
                                                                                           (24)
                                     A i  = B i  = C i  = i
                                     C r ∈ [0, 1] ,F ∈ [0, 1+] ,r ∈ [0, 1] ,
                           and k a random integer within [1, n param ], chosen once for all members of
                           the population. The random number r is seeded for every gene of each chro-
                           mosome. F and C r are DE control parameters, which remain constant during
                           the search process and affect the convergence behaviour and robustness of the
                           algorithm. Their values also depend on the objective function, the character-
                           istics of the problem and the population size.
                              The population for the next generation is selected between the current
                           population and the final candidates. If each candidate vector is better fitted
                           than the corresponding current one, the new vector replaces the vector with
                           which it was compared. The DE selection scheme is described as follows (for
                           a minimization problem)
                                              ⎧
                                                   (G+1)          (G+1)         (G)
                                              ⎨ X      if f obj X      ≤ f obj X
                                       (G+1)      i             i              i
                                     X      =                                              (25)
                                       i
                                                 X               otherwise .
                                              ⎩    (G)
                                                   i
                              A new scheme [42] to determine the donor for mutation operation is
                           used, for accelerating the convergence rate. In this scheme, donor is ran-
                           domly selected (with uniform distribution) from the region within the “hyper-
                           triangle”, formed by the three members of the triplet. With this scheme the
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