Civil Engineering Reference
In-Depth Information
this kind of algorithm are that: (1) it can be easily trapped in a local optimum; and (2) it depends heavily
on the initial solution. By allowing perturbations to move to a worse solution in a controlled fashion,
the simulated annealing algorithm is able to jump out of a local optimum and potentially find a more
promising downhill path. Moreover, the uphill moves (the acceptance of worse solutions) are carefully
controlled. Although the global solution is not guaranteed, it consistently provides solutions close to
the optimal one. Besides, with the capability of accepting uphill moves, this algorithm appears to be
independent of the initial solution. The basic elements of the simulated annealing algorithm are listed
below:
1. Configuration: a solution of the problem.
2. Move: a transition from one configuration to another.
3. Neighboring configuration: a result of a move.
4. Objective function: a measure of how good a solution is.
5. Cooling scheduling: how high the starting temperature should be, and the rules to determine (a)
when the current temperature should be lowered, (b) by how much the temperature should be
lowered, and (c) when the annealing process should be terminated.
The algorithm of the simulated annealing is described as the following pseudo-code:
Begin with an initial solution
s
and temperature
t
Determine the initial direction and step size of the move
Repeat
Perform the following loop
M
times
Pick a neighbouring configuration
s
of
s
by a random move
Let
((
f
(
s
)
f
(
s
))/
f
(
s
)
If
0 (downhill move)
Set
s
Else (uphill move)
Set
s
s
e
t
with the probability
Update the direction and step size of moves based on the ratio of accepted to rejected moves
Lower
t
until one of the terminating conditions is true
Return
s
s
Taguchi Method [9]
The Taguchi method of design of experiments begins with identifying a response and classifying the
product or process parameters as (a) control (or design) parameters, or (b) noise (or uncertain) param-
eters. Control parameters are the product characteristics whose nominal settings can be specified by the
product designer. Noise parameters are the variables that cause performance variation during the lifespan
of the product. The second step of the Taguchi method is to arrange these control and noise parameters
in the orthogonal array developed by Taguchi, and then conduct experiment and collect data. Finally,
the signal-to-noise ratios, as opposed to the ANOVA in the standard design of experiment approach, are
used to analyze the experiments. The general procedure for applying the Taguchi method to tolerance
design is as follows:
1. Consider the nominal values of dimensions with tolerances as control parameters, their corre-
sponding tolerances as noise parameters, and the direct manufacturing costs as response.
2. Select a proper orthogonal array.
3. Incorporate the “constraint” column in the standard orthogonal array to represent the tolerance
stackup constraint.
4. Calculate the tolerance stackup and response for each set (row) of parameters.
5. Select the row with the smallest response value and for which all the tolerance stackup constraints
are satisfied as the best tolerance.
