Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Part I: Design Stage
I.
Define the system objectives and list any requirements.
a. The system objectives state the purpose of the design.
b. Requirements include any task the system or design must perform as well
as any constraints such as a budget or schedule.
II.
Define an option space or possible design configurations.
a. Select a few options to analyze. Since an option can include an infinite
number of choices, it is feasible to analyze the most likely choices. (Note:
However, this process may lead to a sub-optimal choice.)
b. Define the parameters of each option i.e. the numerical quantities that
specify design elements.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example

Define the system objectives and list any requirements: Suppose you need
to select the I-beam that best fulfills the following requirements:
o Member is constructed from aluminum
o I-beam is simply supported at both ends
o I-beam is uniformly loaded with a distributed force equal to 3775
N/m
o I-beam cannot deflect any more 0.60mm
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
The setup:

Parameters:
Cross Section of I-Beam
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Part II: Evaluation Measures (EMs)
I.
Based on the design objectives and requirements, create a list of evaluation
measures for your system.
a. An evaluation measure is a numerical quantity that allows one to grade
some aspect of the design.
b. It is important that the evaluation measures are independent, meaning that
two measures do not evaluate the same quality or a combination of the
same qualities.
II.
Define the details of each evaluation measure.
a. Is the EM deterministic or probabilistic?
i. A deterministic EM is one whose values are fully expressed as one
number. (For Example: The safety rating of a car is 4 stars out of 5.)
ii. A probabilistic EM is one whose values are expressed as a range.
(For Example: The length of a beam can be 2 meters +/- 10 cm.)
b. Result:
i. If the EM is deterministic, determine if higher option values are
more favorable.
ii. If the EM is probabilistic, also determine if higher option values
are more favorable. Next, determine the risk inclination.
1. There are 3 risk states, risk averse, risk neutral and risk
seeking. After selecting a risk inclination determine the
extent of the risk inclination, rho or “ρ.” If one is risk
averse then ρ is positive and if one is risk seeking then ρ is
negative, and if one is risk neutral then ρ= infinity. As a
general scale for ρ, a value of 5 is moderate and a value of
1 is extreme. This value must then be de-normalized
(multiplied by the range of the EM: High-Low). For further
details reference the graphs below which display the affect
of one’s risk inclination (rho) on the utility value:
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
List Evaluation Measures: The evaluation measures for this decision will include

Cost

Deflection
Evaluation Measure Description:

Cost is a deterministic EM since the price per beam will be used in the
analysis. For cost a lower option value is more preferable.

Deflection is a probabilistic evaluation measure. Since there are tolerances
in the beam’s dimensions, the performance of each beam needs to be
simulated. For deflection a lower option value is more preferable.
o For deflection this example will be risk averse with a rho=5.
________________________________________________________________________
CHECKLIST: Before proceeding be sure to have the following information:
 Options with parameters
 Evaluation Measures and details including:
o Deterministic / Probabilistic
o Value preference (are higher or lower option values
more preferable?)
o Risk Inclination
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Part III: Evaluation Measure Details
I.
If an EM is probabilistic then that particular factor of the design needs to
be modeled as a random variable. To complete this step a simulation must
be performed to determine the distribution. To find the distribution of the
EM:
a. Open MATLAB and change the working directory to where
EQUATION.M and LOOPY.M are saved.
b. Open “EQUATION.M” and using this file define the variables that
make up the equation which determines your EM. Then set the result
vector, RV, equal to that equation. Be sure to save the file under a new,
descriptive title. (YOURTITLE.M) Note: MATLAB is case-sensitive. It
is important to note the order of the variables because the same order
will be used in the simulation. See comments in EQUATION.M.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
5  Load  Length 4
384  Youngs  MomentI
(Youngs = Young’s Modulus and MomentI = moment of inertia)
The equation to determine the deflection of the beam is
Due to the complexity of this equation it will be broken down into two equation files.
One will determine a distribution for MomentI since the moment of inertia is a value
determined by multiple dimensions from the beam’s cross section. The second equation
will compute the rest of the deflection equation as a distribution. Then both distributions
will be combined which yields the deflection distribution.
Equation File One: Moment of Inertia
Equation File Two: Loading
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
________________________________________________________________________
c. Run “LOOPY.” Loopy is the MATLAB program that models the
variables defined in the Equation file as distributions using the Monte
Carlo Method. You will be asked:
1. “How many Variables?” The answer is equal to the number of
variables previously defined in “YOURTITLE.M.” (In Loading.m,
number of variables = 2)
2. “How many Simulations?” 10,000 to 100,000 Monte Carlo
simulations will return satisfactory results.
3. “How many bins?” After each variable is simulated a histogram of
that variable appears so that you can check that the variable was
correctly modeled after the desired distribution. The “bins” refers
to the number of bars plotted in the histogram. 50 bins will usually
produce satisfactory results.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
Below is a list of the prompts from “LOOPY”
________________________________________________________________________
Next, a menu bar of the available distribution types will appear. Select
the desired distribution. Then “LOOPY” will prompt you to input
information required to define that distribution. Enter this information
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
and then press ENTER. A histogram of the random variable will
appear. View the graph then press ENTER to continue modeling the
remaining variables. After all simulations have been run, “LOOPY”
will prompt you to type the name of “YOURTITLE.M.” In single
quotes type the name of the equation file saved (without the ‘.M’) and
then press ENTER. A final histogram of the result will be displayed
for the option. Set the Option name to equal RV (Option1=RV) press
ENTER.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
For deflection you will need to run the loopy program twice for each option. For this
example, the normal distribution will be used to model the variables. A normal
distribution will ask you to input the mean and the standard deviation of your variable.
The nominal value will be input as the mean and the +/- tolerance divided by two will
be the value for the standard deviation. Thus, if you are simulating the length of the
beam the mean value equals 3m and the standard deviation equals 0.02m or (0.04 / 2).
(See values from page 2 for further details)
o The first loopy run will be used to compute the moment of inertia for the
beam. Follow the instructions and input the appropriate information. Next
type “Moment1=RV;” and this will set the distribution of the moment of
inertia equal to the title Moment1 or the moment of inertia for option 1.
o Next run loopy again except use it to compute the load portion. After
loopy is complete set “Load1=RV;” and this will set the load portion of
the deflection equal to the title Load1 or Load of option 1.
o Finally, type “Deflection1=Load1./Moment1;” and this step will
complete the deflection equation thus returning a distribution for the
deflection of the option. (See README_MATLAB for an explanation of
the “.” operator.)
o Type “hist(Deflection1,50);” and a histogram of the distribution will be
plotted with 50 bins and it should look similar to:
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________________________________________________________________________
d. Perform the “NORMFIT” function on the simulated option. Using this
syntax: [mu,sigma,muci,sigmaci] = normfit(RV,alpha), choose a
confidence level (95% conf= 0.05=alpha); this function will return
four items:
i. mu: the mean of your sample distribution
ii. sigma: the standard deviation of your sample distribution
iii. muci: the max and min the mu is restricted to, dependant on
your confidence level.
iv. sigmaci: the max and min the sigma is restricted to, dependant
on your confidence level.
For further details simply type “help normfit” at the MATLAB
prompt.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
Normfit Results for Option 1
Repeat this process again to obtain the deflection distribution and normfit information for
the other options.
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
e. Using the information from “NORMFIT” and your risk inclination
chose a Certainty Equivalent for your option. The certainty equivalent
is a pseudo expected value for a distribution that has been adjusted
based on one’s risk inclination. For example, if the given distribution
is for deflection and the mean is 1.0mm and less deflection is
preferable, then the certainty equivalent for someone who is risk
averse could be 1.3mm. Likewise, for the same distribution, if the
person is risk seeking then his certainty equivalent may be 0.8mm.
One way to determine the certainty equivalent is to use the muci
values. If higher values are more preferable for the EM and you are
risk averse then you might want to choose the lower of the two muci
values to be on the safe side.
f. Select a HIGH and a LOW limit for the evaluation measure. HIGH is
the largest acceptable value for the EM and likewise LOW is the
smallest acceptable value. This range must encompass all of your
option scores. Any option that does not fall between the HIGH and
LOW values may be discarded.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
Continuing with Option 1 …
 The Certainty Equivalent for deflection will be taken from the
muci information. Since the example is risk averse the higher
of the two muci values will be used as the certainty equivalent.
Certainty Equivalent = 0.0005259m (Continue on and
determine a CE for each option).
 Select a HIGH and a LOW value for the evaluation measure
o HIGH= 0.0006m
o LOW= 0.00045m
In addition it should be noted that since Deflection is probabilistic, the software will
allow you to use the standard deviation of the deflection as an evaluation measure as. In
this example the standard deviation of deflection will be used as an evaluation measure
since it is important for the distribution to be tightly packed around the mean.
________________________________________________________________________
II.
If an EM is deterministic:
a. Select a HIGH and a LOW limit for the evaluation measure. HIGH is
the largest acceptable value for the EM and likewise LOW is the
smallest acceptable value.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
For the cost evaluation measure:
 Select a HIGH and a LOW value for the evaluation measure
o HIGH= $130.00
o LOW= $120.00
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
________________________________________________________________________
CHECKLIST: Before proceeding be sure to have the following information:
 For deterministic know the Evaluation Measure’s:
o HIGH
o LOW
 For probabilistic know the Evaluation Measure’s:
o HIGH
o LOW
o Certainty Equivalent
________________________________________________________________________
Part IV: Determining Option Grades
I.
To determine the option grades, a MACRO for MS Excel will be used.
Before proceeding, read the README_XL file for disclaimers.
a. Begin by double-clicking on the Excel file “Decision Making Macro.”
b. Next, go to the top tool bar and select: “Tools”  “Macro”  “Run
Macro.”
c. Select “CreateEMs” and click RUN.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
d. Follow the instruction boxes and input the necessary information.
(Have the information from the previous “Checklists” on hand to
expedite this process)
e. Once complete, SAVE AS the file with a new name descriptive of
your project.
f. Go through the newly created EM sheets and fill in the score boxes
shaded in blue. All boxes that are shaded in blue must be filled in for
the later macros to work properly. (Any boxes shaded in green are
optional and contain only reference information to benefit the
user)You should note that the Macro allows you to use sigma
(standard deviation) as an evaluation measure if the EM is
probabilistic. Refer to Appendix I for further details. As you input
values, the grade of the options should appear in the appropriate boxes.
i. The Grade of an option is a number that takes into account the
option score, your risk inclination, and the range of the option.
ii. For information on the exact formula used to compute the
option grade, please reference appendix II.
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Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
g. SAVE again.
h. Once all the evaluation measure information has been entered, select
the “Final” sheet.
i. Inputting the weights of the evaluation measures into the blue boxes:
You can choose to enter the weights yourself (arbitrarily) or you can
use the weight macro to guide you along.
i. Before proceeding, for any EM that you do not wish to include,
put 0’s (zeros) in their weight columns.
ii. To run the weighting macro: “Tools”  “Macro”  “Run
Macro.”
iii. Select “WeightCalc.”
iv. The Weight of an EM is a measure of how important that EM
is with regard to the other EMs.
v. The value of a weight is always between 0 and 1.
vi. The sum of the weights of all the EMs is equal to 1.
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vii. Below are the types of weight elicitation techniques available
in the macro:
1. Direct Methods
a. Each method follows these steps:
i. Numerically assign a rank order to the
evaluation measures. List the EMs from
most important with a value of 1 to the
least important. (This first step should
always be performed regardless of the
weight elicitation technique that you
choose to employ)
ii. Use one of the following formulas for
assigning weights
wti 
K  ri  1
K
K r
j 1
j
wti 
1
( K  ri  1) z
K
 (K  r
j 1
Rank Sum: ri is the rank of the ith EM, K
is the total number of EMs
j
 1) z
Rank Exponent: ri is the rank of the ith EM; K is the
total number of EMs; z is an undefined measure of
the dispersion in the weights
1 1
wti      
 K  j 1  rj 
1
 1 1
wt1  1    ...  / K
K
 2 3
K
wti 
1

wt K   0  0  0  ...  / K
K

1/ ri
K
 (1/ r )
j 1
Rank Order Centroid: ri is rank of the ith EM,
K is the total number of EMs
j
Rank Reciprocal: ri is the rank of the ith
EM, K is the total number of EMs
2. Indirect Methods
a. Trade Offs: (not available in the macro)
i. Rank EMs in order of their overall value
ii. You are asked if overall weight of the
second EM is as great as the weight from
the lowest to some intermediate point of
the value scale of the first EM.
iii. The third EM can now be compared to
intermediate points on either the first or
second ranked EMs and so on…
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iv. Method works well when the value
curves are firmly established and when
the value curves are continuous
b. Balance Beam Approach:
i. Establish a rank order of the overall
weights of the evaluation measures.
ii. A series of questions is posed beginning
with “Is the overall weight of the first
EM (a) greater, (b) less than, or (c) equal
to the overall weight of the second and
third EMs
iii. If answer is “less than” then the third
EM is dropped and replaced by the
fourth EM. (If “greater than” then the
fourth EM is added to the second and
third )
iv. The goal is to establish a series of
equations that define the weights for all
of the evaluation measures.
v. The least valued EM is given a weight of
1 and then you are asked to assign a
weight to the second least weighted EM
then this information is used to solve the
system of equations
vi. The results are then normalized into
weights with values between 0 and 1
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
The rank order of the evaluation measures is as follows:
1. Deflection
2. Cost
3. Deflection Deviation
Remember the numbers associated with the EMs because the macro will prompt you for
these values.
Below is a table which includes the weights determined by all of the direct weight
methods. While you only need to select one weight method the table is provided for your
edification.
Method
Cost
Deflection
Sigma of
Deflection
Rank Sum
0.3333333
0.5
Rank Exponent*
0.341081377
0.417737668
Rank Order
0.2777778
0.6111111
Rank Reciprocal
0.272727273
0.545454545
0.1666667
0.241180955
0.1111111
0.181818182
*For Rank Exponent a dispersion value of 0.5 was used.
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
And by the Indirect Method of Balance Beam:
After completing the instructions of the macro for the balance beam method the
following box will appear:
This message box contains the system of equations that you will use to determine the
weights. Since the lowest EM is set to one go to the next equation and ask yourself how
much greater Cost is than the Sigma of Deflection. For this example, Cost will equal 2.
Use these values to complete the final equation which gives Deflection a value of 3.
Finally, normalize these values into weights between 0 and 1. In the end:
 Deflection weight = 0.5
 Cost weight = 0.33
 Sigma of Deflection weight = 0.17
________________________________________________________________________
j. Go to the top tool bar and select: “Tools”  “Macro”  “Run Macro”
k. Select “MakeFinal” and click RUN. This will execute the Macro that
computes the final grades for the options.
i. The final grade is computed using the following equation:
FinalGrade  Weight i * Gradei
This equation is summed over each evaluation measure.
l. Save
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
Using the weights from Rank Exponent, the final calculated grades are:
Evaluation
Measure
Cost
Deflection
Sigma of
Deflection
Final Grade
Weight
0.3411
0.4177
IBeam1
0.6
0.519
Weighted
Grade
0.204648826
0.216799826
IBeam2
0.66
0.3535
Weighted
Grade
0.225113709
0.147676309
IBeam3
0.7
0.2845
Weighted
Grade
0.238756964
0.118842461
0.2412
0.7207
0.173820469
0.6238
0.150453745
0.5746
0.13859396
0.595269121
0.523243763
0.496193385
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Part V: Sensitivity
I.
Now that the final grades have been determined it is important to perform
a sensitivity analysis.
a. A sensitivity analysis will allow you to see if you have selected a good
weight value for an evaluation measure.
b. The analysis works by varying the weight of one EM from 0 to 1 while
adjusting the weights of the other EMs to keep their proportions the
same. The final grades of each option are re-computed then graphed.
c. To run the sensitivity analysis go to “Tools”  “Macro”  “Run
Macro”
d. Select “Sensitivity” and click RUN.
e. For each EM a new sheet containing the sensitivity plot will be created.
II.
Plot Review
a. The “Sensitivity Analysis” is primarily concerned with analyzing the
area around the chosen weight for the evaluation measure. (If the
weight of a particular evaluation measure is 0.5 then the x axis of the
plot should go from 0.4 to 0.6.) Therefore, readjust both the “x” and
the “y” axis for each new plot. To do so place the cursor over the axis
you desire to alter and right click. A menu bar should appear and select
“FORMAT AXIS”. An option box will appear and you need to select
the tab titled “SCALE”. On the scale page input the minimum and
maximum values for your graph that should be displayed and click
“OK.”
b. For each new plot look for any intersections near the chosen weight of
the varied evaluation measure.
c. If the plotted lines of the options intersect or are close together at the
point of interest you may wish to re-evaluate your weights for a clearer
decision.
d. SAVE
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
Below is the sensitivity analysis plot for the Cost evaluation measure:
Sensitivity of Cost
0.59
0.57
Final Option Grades
0.55
IBeam1
IBeam2
IBeam3
0.53
0.51
0.49
0.47
0.45
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
Weight of Cost
Note that the x and y axes on the chart were altered to display the appropriate range of
values.
________________________________________________________________________
III.
Review the final grades again and take into consideration the sensitivity
analysis and decide on the best option for the design.
Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example
After reviewing the final grades and the sensitivity analysis it is clear that I-Beam 1 is the
best option for the design.
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Appendix I
Standard Deviation Notations:
I.
For all continuous EMs, you have the option of including
sigma (standard deviation) of the distribution as an
evaluation measure.
1. While the Macro automatically makes a space for sigma,
you can eliminate its effect by giving it a weight of zero on
the “Final” page.
2. If you are unsure whether or not to include the standard
deviation the following information may be helpful:
a. First, it is important to realize that all designs are
unique. Therefore, using sigma as an evaluation
measure is situational to the specific design. (Sigma
should be considered as an EM if the tolerances
around the mean need to adhere to a certain
restriction).
b. The standard deviation may not be useful as an EM
if all points of the distribution meet your set
criteria… In general there are three criteria types.
i. All points > X
ii. All points < X
iii. X<All points <Y
1. If not all points on the distribution
meet your criteria, it may be more
useful to consider the number of
points that fail to meet the criteria as
an evaluation measure instead of the
standard deviation. To determine the
number of points that fail to meet
your set criteria use the “HISTC”
function in MATLAB. (see
README_MATLAB for details)
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Decision Making Based on the Monte Carlo Method (and Software Instruction Guide)
Appendix II
I.
Grade Determining Equations:
a. Deterministic or Risk Neutral evaluation measure
i. Higher option scores preferable:
v( x) 
x  Low
High  Low
ii. Lower option scores preferable:
v( x) 
High  x
High  Low
b. Probabilistic evaluation measure:
i. Higher option scores preferable:
v( x) 
1  exp[ ( x  Low) /  ]
1  exp[ ( High  Low) /  ]
ii. Lower option scores preferable:
v( x) 
1  exp[ ( High  x) /  ]
1  exp[ ( High  Low) /  ]
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