22nd Annual Review of Progress in Applied Computational Electromagnetics
March 12-16, 2006 - Miami, FL ©2006 ACES
A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna
Design
Randy L. Haupt
The Pennsylvania State University
Applied Research Laboratory
P. O. Box 30
State College, PA 16804-0030
haupt@ieee.org
Abstract: This paper describes a genetic algorithm (GA) that optimizes chromosomes
containing a mix of continuous and binary encoded variables. Uniform crossover from the binary
GA and a mutation rate akin to that of the continuous GA are fundamental parts of the algorithm.
Parameter selection as well as an application to antenna design are presented.
Keywords: Genetic algorithm, optimization, microstrip antenna
1. Introduction
Most optimization problems work with continuous values. If the variables have integer
values, then special algorithms must be used. If continuous values and integers are part of the
same optimization problem, then it is known as mixed integer optimization, and popular
approaches, such as branch and bound, are used. The genetic algorithm and particle swarm
optimization have also been used for mixed integer optimization.
This paper presents a GA formulation that simultaneously works with continuous,
integer, and binary values in one chromosome. Operators for his new algorithm is described in
the next section and applications to a circularly polarized patch antenna designs are shown in the
following section. The beauty of this algorithm is that it can optimize on any type of variable
value without a change in the algorithm.
2. Mixed Integer/Binary GA
The initial population matrix of the GA is given by
P
ª a11
«a
« 21
« #
«
¬ aR1
a12 " a1C º
»
a22
»
% # »
»
" aRC ¼
591
(1)
22nd Annual Review of Progress in Applied Computational Electromagnetics
March 12-16, 2006 - Miami, FL ©2006 ACES
where 0 d arc d 1 . Each row is a chromosome and the values are created by a uniform random
number generator. A cost function evaluates each chromosome and returns a cost.
cost
f
> a
m1
, am 2 ," , amC @
(2)
Inside the cost function, the variables may be converted to a new continuous range by
xmax xmin amn xmin
(3)
roundup ^ xmax xmin amn ` xmin
(4)
xn
or converted to an integer by
xn
where roundup rounds to the next highest integer and x values are integers. In some cases, the
value is converted to binary. One possibility is to round the value.
round ^amn `
(5)
quantize ^amn `
(6)
xn
Another possibility is to quantize the value.
xn
The cost function does all the scaling, quantizing, and rounding, so the GA can operate
independent of the type of variable. Uniform crossover works well for the binary GA, so it is
implemented here. Two parents are selected and a random binary mask is created. If the mask
has a one in the column, then the offspring receives the variable value in parent#1. If it has a
zero, then the offspring receives the variable value in parent#2.
parent #1 am1
parent # 2 an1
1
mask
offspring am1
am 2
an 2
0
an 2
am3
an 3
1
am3
am 4
an 4
1
am 4
am 5
an 5
0
an 5
am 6
an 6
1
am 6
am 7
an 7
0
an 7
am8
an8
0
an8
am 9
an 9
1
am 9
(7)
This type of crossover results in a diversity of values if the values are binary, but only
interchanges values between chromosomes if the values are integer or continuous. Consequently,
the mutation must be responsible for creating diversity within the population for continuous and
integer values.
One possible approach to mutation is to randomly select variables in the population and
replace them with uniform random values. The mutated chromosome ( chromc ) is created from
the selected chromosome ( chrom ) by
chromc
> ar1
ar 2
arc3
ar 4
592
ar 5
ar 6
ar 7
ar 8
arc9 @
(8)
22nd Annual Review of Progress in Applied Computational Electromagnetics
March 12-16, 2006 - Miami, FL ©2006 ACES
where the primed values are uniform random numbers. Another approach is to add a random
correction factor. The correction factor may be created by multiplying each element within a
chromosome by a random number ( 1 d E rm d 1 ) and multiplying the entire chromosome by a
mutation factor ( 0 d D r d 1 ).
chromc
D r > E r1ar1 E r 2 ar 2
E r 3 ar 3 E r 4 ar 4
E r 5 ar 5 E r 6 ar 6
E r 7 ar 7
E r 8 ar 8 E r 9 ar 9 @ (9)
Now, the mutated chromosome is given by
chromc rem ^chrom chrom c `
(10)
where rem is the remainder function (digits to the left of the decimal point are dropped).
This algorithm was tested on two cost functions to try to determine an appropriate
population size and D r . In both cases, the GA quit after 400 function evaluations and reported
the best results. The first test functions is
f1 x 6
¦x
2
n
(11)
n 1
with a minimum of zero at xn 0 . The results shown in Figure 1 were averaged over 100 runs
for population sizes between 8 and 96 and mutation rates between 0.01 and 0.3. The best results
occurred when the population size was 8 and the mutation rate was 0.1.
Figure 1. GA results for f1 averaged over 100 runs for various population sizes and
mutation rates.
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22nd Annual Review of Progress in Applied Computational Electromagnetics
March 12-16, 2006 - Miami, FL ©2006 ACES
The second test functions is
6
f 2 x 60 ¦ ª¬ xn2 10 cos 2S xn º¼
(12)
n 1
with a minimum of zero at xn 0 . The results shown in Figure 2 were averaged over 500 runs
for population sizes between 8 and 96 and mutation rates between 0.005 and 0.3. The best results
occurred when the population size was 40 and the mutation rate was 0.01.
Figure 2. GA results for f 2 averaged over 500 runs for various population sizes and
mutation rates.
3. Applications to Antenna Design
The goal is to design a rectangular patch for circular polarization at 10 GHz using FEKO.
Input variables for the cost function are
x, y position of probe feed
Lx , Ly
patch length in x and y directions
h substrate thickness (either 1.575mm or 3.15mm)
H r relative dielectric constant of the substrate (either 2.2 or 2.33)
The cost function returns the following value
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22nd Annual Review of Progress in Applied Computational Electromagnetics
cost
March 12-16, 2006 - Miami, FL ©2006 ACES
­ ET EM
½
S
°
°
max ®
, ‘ET ‘EM , s11 ¾
2
°¯ ET EM
¿°
(13)
The first two quantities in (13) equal zero for circular polarization, and s11 is zero for a
perfect match to 50: . When the patch is circularly polarized and perfectly matched, then
cost = 0.
The optimization was performed using the best population size and mutation rate for both
test functions in the previous section. Results were averaged over 5 runs and are summarized
in Table 1. Figure 3 shows the best of five independent results for a population size of 8 and
D r 0.10 and for a population size of 40 and D r 0.01 . Since the optimizations were
terminated after 400 function calls, the number of generations to convergence is different for
different population sizes.
Table 1. Cost statistics after 5 independent runs.
maximum minimum average
population size = 8
0.48995
0.055435 0.22332
mutation rate = 0.10
population size = 40
0.51665
0.30441
0.40838
mutation rate = 0.01
Nelder-Mead downhill
1.0921
0.27444
0.53891
simplex algorithm
Figure 3. Convergence results for the two best runs with a population size of 40 and
D r 0.01 , and a population size of 8 and D r 0.10 . The top two dashed curves are the
population average and the bottom two curves are the best population cost. The GA was
terminated after 400 function evaluations.
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22nd Annual Review of Progress in Applied Computational Electromagnetics
March 12-16, 2006 - Miami, FL ©2006 ACES
The best chromosome translates to the optimum design values of
x, y 1.262mm,3.451mm Lx , Ly 8.618mm,10.138mm
h 1.575mm
H r 2.2
The resulting patch is right-hand elliptically polarized with an axial ratio of 1.09 and s11
0.055 .
4. Conclusions
Many antenna designs have variables with integer values and with continuous values.
This paper presented a version of a GA that works with values between zero and one and uses
binary uniform crossover and continuous mutation. The patch antenna design is a multiobjective
optimization using continuous and binary variables.
References
[1] Y. Li and M. Gen, "Nonlinear mixed integer programming problems using genetic algorithm
and penalty function,"
[2] Z. Gaing, "Constrained optimal power flow by mixed-integer particle swarm optimization,"
[3] R.L. Haupt and Sue Ellen Haupt, Practical Genetic Algorithms, 2nd edition, New York: John
Wiley & Sons, 2004.
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