Biomedical Engineering Reference
In-Depth Information
solution point that reflects preferences presumably incorporated in the selection of a
single set of weights. Using the weighted sum method to solve the optimization
problem given in ( 22 ) entails selecting scalar weights w j and minimizing the
following composite objective function:
X
F . b /
D
w j f j . b /
(23)
j
D
1;m
If all of the weights are positive, as assumed in this study, then minimizing
( 23 ) provides a sufficient condition for Pareto optimality, which means that its
minimum is always Pareto optimal [ 25 ]. The weighting for an individual objective
can be determined by either fixed weights or random weights. A strategy of
randomly assigning weights is used to search for an optimum solution through
diverse directions [ 26 , 27 ]. To provide decision makers with flexible and diversified
solutions, this study adopts random weights calculated by using
r j
w j
D
;
j
D
1; :::; m
(24)
r 1 C
r 2 C
:::
C
r m
where r j is a random positive integer.
The genetic algorithm scheme searching for optimal solutions is based on
four operators supported by an elitist strategy that always preserves a core of
best individuals of the population whose genetic material is transferred into the
next generations [ 28 , 29 ]. A new population of solutions P t C 1
is generated from
the previous P t
using the genetic operators: Selection, Crossover, Mutation and
Deletion.
The optimization scheme includes the following steps:
Coding: the design variables expressed by real numbers are converted to binary
numbers, forming a string, and each binary string is looked as an individual;
Initializing: the individuals which consist of an initial population P 0
are produced
randomly within each allowable interval;
Evaluation: the fitness of each individual is evaluated using the objective function
given in ( 23 ), and individuals are ranked according to their fitness value;
Selection: definition of the elite group that includes individuals highly fitted.
Selection of the progenitors is the mechanism that defines the process in which the
chromosomes are mated before applying crossover on them. We apply a procedure
that randomly chooses one parent from the best-fitted group (elite) and another from
the least fitted one. Transfer of the whole population P t to an intermediate step where
they will join the offspring determined by the crossover operator;
Crossover: One offspring per each pair of selected parents is considered in the
present work. The value of each gene in the offspring chromosome coincides with
the value of the same gene in one of the parents depending on a given probability.
The new individuals created by crossover will join the original population.
Mutation: the implemented mutation is characterized by changing a set of bits of
the binary string corresponding to one variable of a randomly selected chromosome
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