![](bg5.png)
the probabilities of crossover and mutation are 1.0
and 0.5, respectively.
Inputs: Graph G=(V,E), edge cost C(e) and node
degree deg(v)
Output: binary strings representing the Euler route
Step1) Initialization
Randomly create an initial generation of N
binary string P
0
= { X
1
, X
2
,…, X
N
}, X = {
x
1
, x
2
,…, x
k
}
and initialize the generation number gen
to 0.
* X is a chromosome, k = |V|
Step 2) Population Evolution
WHILE ( gen < MAX_GEN )
BEGIN
4.3 Simple Genetic Algorithm (SGA)
The simple genetic algorithm adopts a one-point
crossover in Step 2.1 instead of orthogonal
crossover.
4.4 Another Orthogonal Genetic
Algorithm using Orthogonal
Array in the Creation of the
Initial Population of
Chromosomes (OGA
2
)
This algorithm uses an orthogonal array also in Step
1 in Fig.5 instead of random number, and the
utilization method is shown in 3.4.
4.5 Results
Fig.6-1 shows the relationship between the number
of obtained Eulerian graphs and the number of
generations. On the other hand, Fig.7-1 shows the
relationship in the case where orthogonal array is
used in the creation of initial population of
chromosomes. Fig.6-2 shows the relationship
between the obtained minimum weight of the
Postman route and the number of generations.
Fig.7-2 shows the relationship in the case where
orthogonal array is used in the creation of initial
population of chromosomes. These results mean that
our orthogonal genetic algorithm shows better
performance, especially in L9(3
4
). SGA almost
never finds the solutions for Problem 3 where the
number of edges is 7.
For reader’s information, we show the
relationship between the number of generations and
the computation time required in 3 algorithms ( SGA,
OGA, and OGA
2
) in Tables 5 and 6.
In this mixed Chinese postman problem we
treated, in less than 10
4
generations we can obtain a
solution in graphs with nodes of less than 11.
However, we can’t obtain a solution in 2 or 3 days
for the larger sizes.
For reference we will show the data in the case of
non directed graphs G’’=(V’’, E’’), where |V’’|= 20,
|E’’|=30, total weight=178. The Chinese Postman
problem for non directed graphs belongs in Class P.
Figs.8-1 and 8-2 show the relationship between two
numbers of obtained Eulerian graphs and
generations, and the relationship between the
obtained minimum weight of the Postman route and
the number of generations, respectively.
Figure 5: A orthogonal genetic algorithm.
5 CONCLUSION
In order to investigate the salient feature of
orthogonal design, we designed a genetic algorithm
adopting an orthogonal crossover operation in the
mixed Chinese Postman Problem and evaluated the
salient ability. The orthogonal design shows better
performance, even for graph scales where simple
genetic algorithms almost never find the solution.
The experimental results show that, for problems of
non practical sizes, the orthogonal genetic algorithm
using the orthogonal array L
9
(3
4
) can find close-to-
optimal solutions within a moderate number of
generations. This optimal scale of orthogonal array
was confirmed for the multimedia multicast routing
problem of practical size (Zhang and Leung, 1999).
However, this orthogonal design is not yet effective
for the mixed Chinese Postman Problem of practical
sizes. For more effective computation, our one
possible extension of this research can be considered
as to incorporate the orthogonal array into the
DO N/2 times
BEGIN
Step 2.1) Orthogonal Crossover
Randomly select n parents strings from
P
gem
and perform orthogonal crossover
on them to generate m offspring o
1
,
o
2
,…, o
m
.
Step 2.2) Mutation
To perform mutation of offspring, flip
every bit in this string with a small
probability p.
Step 2.3) Select
Calculate the offspring fitness f, and
sort them by f, and choose n for the
next generation.
END
Step 3) Increment the generation number gem by 1.
END
ON ABILITY OF ORTHOGONAL GENETIC ALGORITHMS FOR THE MIXED CHINESE POSTMAN PROBLEM
43