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COMPUTER PROBLEM-SOLVING IN
ENGINEERING AND COMPUTER SCIENCE
EGR 141
FALL 2006
PROBLEM-SOLVING EXERCISE #4
SIMULATION - SOLUTION
As we’ve previously seen, equations describing situations often contain uncertain parameters,
that is, parameters that aren’t necessarily a single value but instead are associated with a
probability distribution function. When more than one of the variables is unknown, the outcome
is difficult to visualize. A common way to overcome this difficulty is to simulate the scenario
many times and count the number of times different ranges of outcomes occur. One such
popular simulation is called a Monte Carlo Simulation. In this problem-solving exercise you will
develop a program that will perform a Monte Carlo simulation on a simple profit function.
Consider the following total profit function:
PT = nPv
Where PT is the total profit, n is the number of vehicles sold and Pv is the profit per vehicle.
PART A
Compute 5 iterations of a Monte Carlo simulation given the following information:
n follows a uniform distribution with minimum of 1 and maximum 10
Pv follows a normal distribution with a mean of $4000 and a standard deviation of $1000
Number of bins: 10
Recall that for all practical purposes we will use 3 std. deviations from the mean as the maximum
value for parameters following a normal distribution. Obviously, 5 iterations are not very many.
In fact, typically you would simulate 10,000 iterations or so to view meaningful results but I
figured that I’d give you a break . If you’d like to compute 10,000 iterations by hand for extra
credit, go ahead…
i.)
What are the ranges for the 10 bins?
n : nmin = 1 , nmax = 10
Pv: Pvmin = $1000, Pvmax = $7,000
Ptmin = nmin * Pvmin = $1000
Ptmax = nmax * Pvmax = $70,000
Range/Bin = (70000 – 1000)/10 = 6,900
Bin 1:
1000 – 7900
Bin 2:
7901 – 14800
Bin 3: 14801 – 21700
Bin 4: 21701 – 28600
Bin 5: 28601 – 35500
Bin 6: 35501 – 42400
Bin 7: 42401 – 49300
Bin 8: 49301 – 56200
Bin 9: 56201 – 63100
Bin 10: 63101 – 70000
ii.)
Fill in the table below:
Parameter
n
Pv
PT
Bin #
$ Range
Iteration 1
2
$3100
$6200
1
1000 – 7.9k
iii.)
Iteration 2
7
$4500
$31500
5
28601 – 35.5k
Iteration 3
9
$3400
$30600
5
28601 - 35.5k
Iteration 4
4
$6500
$26000
4
21701 – 28.6k
Iteration 5
5
$2200
$11000
2
7901 – 14.8k
Fill in the frequency of occurrences of each bin:
1: 1
6: 0
2: 1
7: 0
3: 0
8: 0
4: 1
9: 0
5: 2
10: 0
PART B
Write the following three functions in a module called Simulate:
Public Function GetRandomUniform(ByVal min As Integer, ByVal max as Integer) as Integer
‘ This function returns a random number from a uniform distribution between min and max.
Public Function GetRandomNormal(ByVal mean As Single, ByVal stddev as Single) as Single
‘ This function returns a random number from a normal distribution with a mean of mean and standard
‘ deviation of stddev
Public Function GetBinIndex(ByVal mini As Single, ByVal maxi As Single, ByVal numbins as _
Integer, ByVal valuetobin as Single) As Integer
‘ This function returns the Bin Index given an output minimum of mini, output maximum of maxi,
‘ numbins number of bins, and a value to bin of valuetobin
See Part D for code.
PART C
Include the module created in part B to develop a Visual Basic .NET program that will simulate
the basic profit calculation, PT = nPv, where n follows a uniform distribution, Pv follows a normal
distribution, and the user can input the number of bins and number of iterations. The user must
also input the min and max for n and the mean and standard deviation for Pv. Finally, the user
can click a button and the results will be graphed on a bar chart using the Microsoft Chart
Control. Turn in a listing of the code and a screen shot of the resulting chart using:
1. Iterations: 10000 Bins: 5 n-min: 1 n-max: 10
2. Iterations: 10000 Bins: 10 n-min: 1 n-max: 10
3. Iterations: 10000 Bins: 10 n-min: 1 n-max: 10
That’s three screen shots and a listing.
See Part D for code.
Pv-mean: 9000
Pv-mean: 9000
Pv-mean: 6000
Pv-stddev: 2000
Pv-stddev: 2000
Pv-stddev: 1000
PART D
Extend the Visual Basic .NET program developed in part C to simulate the basic profit
calculation, PT = nPv, where the user can select either a uniform or normal distribution for n
using radio buttons and then must input the appropriate parameters (min and max if they select
uniform, mean and standard deviation if they select normal) and they can similarly select either a
uniform or normal distribution for Pv with appropriate parameters depending on the selection.
Of course, the user will input the number of bins and number of iterations. Finally, the user can
click a button and the results will be graphed on a bar chart using the Microsoft Chart Control.
Turn in a listing of the code and a screen shot of the resulting chart using:
1. Iterations: 10000 Bins: 5 n-min: 1
n-max: 10 Pv-mean: 8000
2. Iterations: 10000 Bins: 10 n-mean: 9
n-stddev: 2 Pv-min: 2000
3. Iterations: 10000 Bins: 10 n- mean: 14 n- stddev: 3 Pv-min: 2000
Pv-stddev: 2000
Pv-max: 10000
Pv-max: 10000
That’s three screen shots and a listing.
Code Listing for Parts C, D and E
'Form File:
'Date:
'Description:
'
'
'
'
'
'
'
frmSIM.vb
March 31, 2005
This application runs a Monte Carlo Simulation on a
simple profit function.
This code is used for Parts C, D and E.
For Part C, the radio buttons for Pv uniform and N normal
have their visible property set to False so as not to be
accessible to the user.
For Part D and E, these buttons are reset to visible =
=True.
Private Sub cmdSIM_Click(ByVal sender As System.Object,
ByVal e As System.EventArgs)
Handles cmdSIM.Click
Dim NumBins, n, x, i As Integer
Dim Pt, Pv As Single
Dim Ptmin, Ptmax, Pvmin, Pvmax As Single
Dim nmax, nmin, nbins, cycles As Integer
Dim Bins() As Integer
'Array holds the frequency of profit total
nbins = Val(txtBins.Text)
cycles = Val(TextBox1.Text)
txtTotalProfit.Text = 0
'reset the total profit
If rdoNuni.Checked = True Then 'uniform distribution
'txtN1 is maximum
'txtN2 is minimum
nmin = Val(txtN2.Text)
nmax = Val(txtN1.Text)
Else 'rdoNnor is checked and normal distribution
'txtN1 is mean
'txtN2 is stddev
nmin = Val(txtN1.Text) - 3 * Val(txtN2.Text)
nmax = Val(txtN1.Text) + 3 * Val(txtN2.Text)
End If
If rdoPvuni.Checked = True Then ' uniform distribution
'Pv1 is maximum
'Pv2 is minimum
Pvmin = Val(txtPv2.Text)
Pvmax = Val(txtPv1.Text)
Else 'rdoPvnor is checked and normal distribution
'Pv1 is mean
'Pv2 is stddev
Pvmin = Val(txtPv1.Text) - 3 * Val(txtPv2.Text)
Pvmax = Val(txtPv1.Text) + 3 * Val(txtPv2.Text)
End If
Ptmin = nmin * Pvmin
Ptmax = nmax * Pvmax
ReDim Bins(nbins + 1)
For x = 1 To cycles
If rdoNuni.Checked = True Then 'uniform
n = GetRandomUniform(nmin, nmax)
Else 'rdoNnor is checked then normal
n = GetRandomNormal(Val(txtN1.Text), Val(txtN2.Text))
End If
If n > nmax Then 'Value must be placed in last bin
n = nmax
End If
If rdoPvuni.Checked = True Then 'uniform
'Pvmax is maximum
'Pvmin is minimum
Pv = GetRandomUniform(Pvmin, Pvmax)
Else 'rdoPvnor is checked then normal
'txtPv1 is mean
'txtPv2 is stddev
Pv = GetRandomNormal(Val(txtPv1.Text), Val(txtPv2.Text))
End If
If Pv > Pvmax Then
Pv = Pvmax
End If
'Total profit calculation
Pt = n * Pv
'Keep track of Pt for aveage calculation
txtTotalProfit.Text += Pt
i = GetBinIndex(Ptmin, Ptmax, nbins, Pt)
If i > nbins Then
MessageBox.Show("2Index is out of Bounds")
Exit For
End If
Bins(i) += 1
Next
'Now chart the results of the simulation
MSChart1.RowCount = nbins
MSChart1.ColumnCount = 1
For i = 1 To nbins
MSChart1.Row = i
MSChart1.Data = Bins(i)
'Row label will show the maximum of each bin
MSChart1.RowLabel = Ptmin + (((Ptmax - Ptmin) / nbins) * (i))
Next
'Average Total Profit calculation
txtTotalProfit.Text /= cycles
txtTotalProfit.Text = FormatNumber(txtTotalProfit.Text, 2)
End Sub
Private Sub rdoPvuni_CheckedChanged(ByVal sender As System.Object,
ByVal e As System.EventArgs) Handles rdoPvuni.CheckedChanged
If rdoPvuni.Checked = True Then
lblMax1.Visible = True
lblMin1.Visible = True
lblMean1.Visible = False
lblStddev1.Visible = False
End If
End Sub
Private Sub rdoPvnor_CheckedChanged(ByVal sender As System.Object,
ByVal e As System.EventArgs) Handles rdoPvnor.CheckedChanged
If rdoPvnor.Checked = True Then
lblMax1.Visible = False
lblMin1.Visible = False
lblMean1.Visible = True
lblStddev1.Visible = True
End If
End Sub
Private Sub cmdExit_Click(ByVal sender As System.Object,
ByVal e As System.EventArgs) Handles cmdExit.Click
End
End Sub
Private Sub rdoNuni_CheckedChanged(ByVal sender As System.Object,
ByVal e As System.EventArgs) Handles rdoNuni.CheckedChanged
If rdoNuni.Checked = True Then
lblMax2.Visible = True
lblMin2.Visible = True
lblMean2.Visible = False
lblStdDev2.Visible = False
End If
End Sub
Private Sub rdoNnor_CheckedChanged(ByVal sender As System.Object,
ByVal e As System.EventArgs) Handles rdoNnor.CheckedChanged
If rdoNnor.Checked = True Then
lblMax2.Visible = False
lblMin2.Visible = False
lblMean2.Visible = True
lblStdDev2.Visible = True
End If
End Sub
Private Sub frmSIM_Load(ByVal sender As System.Object,
ByVal e As System.EventArgs) Handles MyBase.Load
'This clears the form of the chart for startup
MSChart1.RowCount = 0
End Sub
Module Simulate
Public Function GetRandomUniform(ByVal min As Integer,
ByVal max As Integer) As Integer
'Function generates a uniformly distributed random number between min
‘and max.
Dim u As Integer
Randomize()
u = Int(min + Rnd() * (max - min + 1))
'u is a uniformly distributed random number
Return u
End Function
Public Function GetRandomNormal(ByVal Mean As Single,
ByVal StdDev As Single) As Single
'Function generates a random number on a normal distribution with
‘mean as Mean and standard deviation as StdDev.
Dim r, phi, x, z As Single
Randomize()
r = Rnd()
phi = Rnd()
z = Math.Cos(2 * 3.14159 * r) * Math.Sqrt(-2 * Math.Log(phi))
'z is a random number on the standard normal distribution
x = (z * StdDev) + Mean
'x is a random number on the normal disrtibution of interest
Return x
End Function
Public Function GetBinIndex(ByVal mini As Single, ByVal maxi As Single,
ByVal numbins As Integer, ByVal ValueToBin As Single) As Integer
'Function generates a bin index for the bin array to track frequency
'of occurance of Pt in the simulation.
Dim q As Double
Dim qc As Double
'ValueToBin is Pt
'mini is Ptmin
'maxi is Ptmax
'numbins is nbins
q = CDbl(ValueToBin - mini) * (numbins / (maxi - mini))
qc = Math.Ceiling(q)
If qc > numbins Then
MessageBox.Show("1Index is out of Bounds " & qc)
End If
'qc is the index for the Bins array
Return qc
End Function
End Module
PART E
You’re going to go to a job interview for OU Car Co. Knowing that the field is highly
competitive, you have run sales scenarios ahead of time experimenting with different numbers of
customers and vehicle profits given that in one month OU Car Co. sells between 4 and 10 cars
uniformly distributed with profits of $4,000 on the average with standard deviation of $900.
Which has a higher payoff, focusing on selling to a couple more customers or by increasing the
average sale (with the same std. dev. of $900) by retraining your sales force or do they have
basically the same effect on total sales? Support your answer.
Solution:
Note: Results for Ptave will vary due to being generated by random numbers.
Therefore, results can be in a reasonable range from what is shown here.
Baseline
First, a baseline is established, based upon the above given information.
Pv = Profit per vehicle
Fits Normal distribution with mean of $4000 and a standard deviation of $900
N = Number of vehicles sold
Fits a Uniform distribution with a maximum of 10 and a minimum of 4
Yields Ptave = Average Profit Total = $28,901.00
Case 1: 10.0 % Increase in either Pv or N
Pvmean = $4400 All other inputs remain at baseline input level
Yields Ptave = Average Profit Total = $30.881.86
Nmax = 11 All other inputs remain at baseline input level
Yields Ptave = Average Profit Total = $30,112.84
The increase in Pv yields a resulting Ptave $769.02 higher
than the increase in N.
Case 2: 20.0 % Increase in either Pvmean or Nmax
Pvmean = $4800 All other inputs remain at baseline input level
Yields Ptave = Average Profit Total = $33,686.49
Nmax = 12 All other inputs remain at baseline input level
Yields Ptave = Average Profit Total = $32,095.11
The increase in Pv yields a resulting Ptave $1591.38 higher
than the increase in N..
Continued on next page
Case 3: 30.0 % Increase in either Pv or N
Pvmean = $5200 All other inputs remain at baseline input level
Yields Ptave = Average Profit Total = $36,473.37
Nmax = 15 All other inputs remain at baseline input level
Yields Ptave = Average Profit Total = $34,115.03
The increase in Pv yields a resulting Ptave $2358.34 higher
than the increase in N.
In conclusion:
From these scenarios, an increase in either Pv or N will increase Ptave.
However, increasing Pv has a greater impact on the average total profit (Ptave) from the
baseline than a similar increase in the number of vehicles sold (N). So from these three cases
efforts to increase the average sale (Pv) would be recommended.