This document discusses statistical analysis methods for simulation results, including confidence limits and descriptive statistics. It then provides an example of using Crystal Ball software to simulate a profit analysis model with uncertain variables. The simulation defines volume, price, and variable cost as probability distributions and runs 5,000 trials. Statistical analysis of the results shows an 81.61% probability that the company will break even.

1. STATISTICAL ANALYSIS OF
SIMULATION RESULTS
MANAGEMENT SCIENCE – CHAPTER 14

2. STATISTICAL ANALYSIS OF
SIMULATION RESULTS
In general, the outcomes of a simulation model are statistical measures such
as averages. These statistical results are typically subjected to additional
statistical analysis to determine their degree of accuracy.
One of the most frequently used tools for the analysis of the statistical
validity of simulations results is confidence limits. Confidence limits can be
developed within Excel for the averages resulting from simulation models in
several different ways. Recall that the statistical formulas for 95% confidence
limits are:
where the mean and s is the sample standard deviation from a sample of size
n from any population.

3.  Exhibit 14.9 shows Excel spreadsheet for
machine breakdown example (from Exhibit
14.8), with the upper and lower confidence
limits for average repair cost in cells L13 and
L14. Cell L11 contains average repair cost
(for each incidence of a breakdown),
computed by using the formula Cell L12
contains the sample standard deviation,
computed by using the formula The upper
confidence limit is computed in cell L13 by
using the formula shown on the formula bar
at the top of the spreadsheet, and the lower
control limit is computed similarly.
Confidence limits plus several additional
statistics can also be obtained by using the
“Data Analysis” option from the “Data” menu.
Select the “Data Analysis” option from the
“Data” menu at the top of the spreadsheet, and
then from the resulting menu select
“Descriptive Statistics.” This will result in a
Descriptive Statistics dialog box like the one
shown on the right in Exhibit 14.10.

4. CRYSTAL BALL
MANAGEMENT SCIENCE – CHAPTER 14

5. CRYSTAL BALL is a risk analysis and
forecasting program that uses
Monte Carlo simulation to forecast a
statistical range of results possible for a
given situation.

6. SIMULATION OF A PROFIT
ANALYSIS MODEL
In Chapter 1 we used a simple example for the Western Clothing Company to
demonstrate break-even and profit analysis. In that example, Western Clothing
Company produced denim jeans.
The price (p) for jeans was $23, the variable cost was $8 per pair of jeans, and
the fixed cost was $10,000.
Given these parameters, we formulated a profit (Z) function as follows:
Where:
Z = profit
p = price
v = volume
Cf = fixed cost
VC = variable cost

7. First, we will assume that volume is
actually volume demanded and
that it is a random variable defined
by a normal probability
distribution.
Furthermore, we will assume that the price is
not fixed but is also uncertain and defined by a
uniform probability distribution (from $20 to
$26) and that variable cost is not a constant
value but defined by a triangular probability
distribution.
The Distribution menu window
will again appear, and this time
we click on “Uniform
Distribution.” This results in the
Uniform Distribution window
shown in Exhibit 14.13.

8. Next, click on “Minimum” or use the Tab key to
move to the “Minimum” display at the bottom of
the window and enter 20, the lower limit of the
uniform distribution specified in the problem
statement. Next, activate the “Maximum” display
window and enter 26. Then click on the “Enter”
button to configure the distribution graph in the
window. Finally, click on “OK” to exit this window.
Click “Define Forecast” at the top of the spreadsheet and
this will result in the “Define Forecast” window. The
heading “Profit(Z) ” will already be entered from the
spreadsheet. Click on the “Units” display and enter
“dollars.”, and click “OK” to exit this window.
To run the simulation, click on “Run
Preferences” at the top of the
spreadsheet in Exhibit 14.16 to activate
window shown in Exhibit 14.17. Enter
the number of simulations for the
simulation run.

9. Next, click on “Sampling” at the top of the
window to activate the window shown in Exhibit
14.18. In this window, we must enter the seed
value for a sequence of random numbers for
the simulation, which is always 999. Click on
“OK” and then go back to the spreadsheet.
From the spreadsheet menu (Exhibit 14.19),
we click on “Start,” which will run the
simulation. Exhibit 14.19 shows the
simulation window with the simulation
completed for 5,000 trials and the
frequency distribution for this simulation.

10. A statistical summary report for this simulation
can be obtained by clicking on “View” at the
top of the “Forecast” window and then
selecting “Statistics” from the drop-down
menu. This results in the window shown in
Exhibit 14.20.
The frequency chart that shows the
location of the new lower limit and the
“Certainty” of zero profit is shown as
81.61% at the bottom of the window as
shown in Exhibit 14.21. Thus, there is a
.8161 probability that the company will
break even.

11. PREPARED BY:
MIKEE AGATHA NOMBRE, BSA1A
THANK YOU AND
HAVE A GREAT
DAY AHEAD!