Engineering Optimization: Theory and Practice, Fourth Edition

264 Nonlinear Programming I: One-Dimensional Minimization Methods

and with one experiment left in it. This experiment will be at a distance of

L∗ 2 =

Fn− 2

Fn

L 0 =

Fn− 2

Fn− 1

L 2 (5.8)

from one end and

L 2 −L∗ 2 =

Fn− 3

Fn

L 0 =

Fn− 3

Fn− 1

L 2 (5.9)

from the other end. Now place the third experiment in the intervalL 2 so that the current

two experiments are located at a distance of

L∗ 3 =

Fn− 3

Fn

L 0 =

Fn− 3

Fn− 1

L 2 (5.10)

from each end of the intervalL 2. Again the unimodality property will allow us to

reduce the interval of uncertainty toL 3 given by

L 3 =L 2 −L∗ 3 =L 2 −

Fn− 3

Fn− 1

L 2 =

Fn− 2

Fn− 1

L 2 =

Fn− 2

Fn

L 0 (5.11)

This process of discarding a certain interval and placing a new experiment in the

remaining interval can be continued, so that the location of thejth experiment and the

interval of uncertainty at the end ofjexperiments are, respectively, given by

L∗j=

Fn−j

Fn −(j− 2 )

Lj− 1 (5.12)

Lj=

Fn −(j− 1 )

Fn

L 0 (5.13)

The ratio of the interval of uncertainty remaining after conductingjof thenpredeter-

mined experiments to the initial interval of uncertainty becomes

Lj

L 0

=

Fn −(j− 1 )

Fn

(5.14)

and forj=n, we obtain

Ln

L 0

=

F 1

Fn

=

1

Fn

(5.15)

The ratioLn/L 0 will permit us to determinen,the required number of experiments,

to achieve any desired accuracy in locating the optimum point. Table 5.2 gives

the reduction ratio in the interval of uncertainty obtainable for different number of

experiments.