Page 59 - Numerical Analysis and Modelling in Geomechanics

40 A.A.JAVADI
the next step the binary strings are decoded and converted into optimisation
variable values (components of vector x) using a linear scaling. The objective
function is evaluated from the established optimisation variable values and a
measure of worth or “fitness” is evaluated. For a maximisation problem the
objective function is considered as a fitness function but for minimisation, the
inverse of the objective function or the negative of the objective function (as
used in this chapter) or the difference between a large number and the objective
function can be considered as the fitness function. A high fitness value would
indicate a better solution than a low fitness value.
The GA creates new populations from old populations. The fitter members in
the population are selected to produce new members for the next generation. There
are different methods of selection such as ranking, biased roulette wheel,
tournament, stochastic remainder sampling and stochastic uniform sampling
selection. In this study the tournament selection method was used. In this method
several individuals are chosen randomly in groups as parents and then the
individual with highest fitness is selected. In the next step, the partial strings of
the chosen individuals of the population are exchanged to improve the average
fitness of the next generation. The probability of crossover (P ) shows whether
c
selected members are used in reproduction or not. The most suitable value of P c
from literature is in the range 0.5–1. 15 Different forms of crossover can be
implemented (for example, uniform and single-point crossover), crossing two
parents at a randomly chosen point. The lowest value of P c can be used for
uniform crossover, and for single-point crossover values of 0.7–1 are
recommended. In this study the uniform crossover with P =0.5 produced good
c
convergence of the optimisation process.
Mutation is another important operator in a GA. If, for example, all the
variables in one generation have the same value, then all the new members in the
next generation will be the same and premature convergence will be achieved.
To avoid this, a mutation operator is used. The operator is based on probability
of mutation (p ) which varies in the range 0.001–0.025. The value of p m was
m
15
considered by Goldberg with the suggestion that for small population sizes (≥
35–200), a low mutation probability (p ≥ 0.0005–0.025) is appropriate. At the
m
end of each generation, convergence is checked and the procedure is repeated
until no further improvements can be made to the results.

Case study
The construction of the 635-metre long Feldmoching tunnel, U8 N-8, was started
in July 1992 and finished in February 1994. The tunnel was constructed using a
top heading and bench method. Compressed air was used to control the
groundwater and shotcrete was used as temporary support. The final concrete
lining was installed in free air after completion of the tunnel.
The excavation of the first few metres of the tunnel was started with open
trench and sheet piling which provided access to the underground work. The first