Structural Dynamics and Vibration in Practice

1. MONTE-CARLO SIMULATION: Finding the
Volumes of the Winning Regions
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If the working point is close to a hyperplane, and thus a
hypersphere with a reasonable radius will have a portion contained
in the region of the winning team and a portion contained in the
different region.

2. MONTE-CARLO SIMULATION: Finding the
Volumes of the Winning Regions (Cont’d)
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Since evaluating volumes in higher dimensional spaces
involves complicated multiple integrations
 A much easier method is to use a MONTE-CARLO
SIMULATION
which repeatedly randomly assigns weights according to a
specified distribution for every single criterion, and recalculates
the total score for every team, then compares these scores to
identify the winner

3. MONTE-CARLO SIMULATION: Finding the
Volumes of the Winning Regions (Cont’d)
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Starting with N weights, to guarantee a uniform distribution on the
space of feasible weights (those that add up to 1), we choose
random values for N−1 weights.
Let the probability for team j to win be Pr(j). Using a uniform
distribution, this probability represents the proportion of volume of
the hyperparallelepiped inside the region of team j. Also, let N be
the total number of iterations and Xi, i=1, ... , N be the random
variables defined by

4. MONTE-CARLO SIMULATION: Finding the
Volumes of the Winning Regions (Cont’d)
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Then if N is large enough, the central limit theorem will result in
𝑖=1
𝑁
𝑋𝑖 being approximately normally distributed with:
Then 𝐴𝑣𝑖
(𝑋𝑖) would have a standard deviation:
=> The main objective is to decrease 𝜎 as much as possible. Since q + p
=1, the worst case scenario would be when q = p = 0.5 where 𝜎 would
be the largest

5. MONTE-CARLO SIMULATION: Finding the
Volumes of the Winning Regions (Cont’d)
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For this study, uniform distributions were used (for calculation
of volume proportion).
To set the endpoints of the distribution, the minimum value is
set to zero because the weights of the criteria cannot be negative.
Maximum values are hence set to double the original value. The
reason is that the owner agency’s requirements are to be taken
into consideration, i.e., the range should be centered around the
original value to represent the owner’s needs for the specific
project.
See how this affects the results. This is done for every single
criterion in turn, and the total score is calculated using these new
weights