DECISION MODELING
WITH
MICROSOFT EXCEL
Chapter 9
Monte Carlo Simulation
Part 2
Copyright 2001
Prentice Hall Publishers and
Ardith E. Baker
An Inventory Control Example:
Foslins Housewares
_________can be used for models in which the question
is “How much of this should we do?”
We will now use an _________________model to provide
an illustration of simulation.
THE OMELET PAN PROMOTION:
HOW MANY PANS TO ORDER?
In Foslins, certain sections of the _____________
department have just suffered their second consecutive
bad year.
Due to competition, the gourmet cooking, glassware,
stainless flatware, and contemporary dishes sections of
Foslins are not generating enough ________to justify the
amount of floor space.
To fight back, the chief buyer ____________the sections
that are in trouble to create a store within a store.
With these changes, plus the store’s ______________for
quality and service, she feels that Foslins can effectively
compete.
An “International Dining Month” _____________will be
featured in October to introduce the new facility.
Five specially made __________(each from a different
country) will be featured on sale. For example, a copper
omelet pan from France, a set of 12 long-stem wine
glasses from Spain, etc.
All items must be ordered 6 months in___________. Any
unsold items after October will be sold to a discount
chain at a reduced price. If they run out, a more
__________item from the regular line will be substituted.
Consider the special omelet pans:
Buying price: $22.00
Selling price: $35.00
Discounted price: $15.00 (at the end of October)
Regular pans:
Buying price: $32.00
Normal selling price: $65.00
Selling price if substituted: $35.00
Now, without knowing the demand for this special
product, how many pans should be ordered in advance?
For example, suppose you order 1000 pans and the
demand turned out to be ________pans.
In this situation, you would be 100 pans short and would
have to buy 100 _______pans and sell them at the sale
price in order to make up the_________.
The net profit would be:
$35(1100) - $32(100) - $22(1000) = $12,300
In general, let y = number of pans ordered and
D =___________. Then for D > y,
Profit = 35D – 32(D – y) – 22y
Profit = 3D + 10y
In another scenario, suppose you order ______pans and
the demand turned out to be 200 pans.
In this situation, you would have an _______of 800 pans
and would have to sell the addition pans at $15 each and
take a___________.
The net profit would be:
$35(200) + $15(800) - $22(1000) = -$3000
In general, for D < y,
Profit = 35D + 15(y – D) – 22y
Profit = 20D - 7y
For D = y, the two formulas are identical.
This spreadsheet __________an order of 11 omelet pans
and a random demand of 8 (i.e., y = 11 and D = 8).
Since the order quantity is ________than demand, (y >D)
there are 3 extra pans. Thus, the profit is $83.
PROFIT VERSUS ORDER QUANTITY
Now, assume that demand has the following
_______________:
Prob {demand = 8} = 0.1
Prob {demand = 9} = 0.2
Prob {demand = 10} = 0.3
Prob {demand = 11} = 0.2
Prob {demand = 12} = 0.1
Prob {demand = 13} = 0.1
Note: These demands have been chosen artificially
___________in order to simplify the example.
To generate random ___________for this probability
distribution in Crystal Ball, enter this ____________
distribution in a two-column format for Crystal Ball.
Now, click on
cell B5. Next,
click on the
Define
Assumptions
icon,
choose
Custom
distribution
and click OK.
In the resulting dialog, click on the Data button
Now, enter the spreadsheet cell range where the discrete
distribution was placed and click OK.
After clicking OK, the distribution will be displayed:
Click OK again to accept these settings.
To determine the ______________through the use of the
simulations, click on cell B11 and then click on the
Define Forecast icon.
If not already selected, choose _____as the window size
and When Stopped (Faster). Click OK.
In order to use _________to calculate the average profit,
first generate a number of trials, setting y = 11.
The ________that results on any given trial depends on
the value of demand that was generated on that trial.
The average profit over all trials is the_______________.
To do this, click on the Run Preferences icon
and
change the Maximum Number of Trials to________.
Click OK.
Next, click on the Start Simulation
icon and after
Crystal Ball has run the 500 iterations, the following
dialog will appear:
Click OK and Crystal Ball will automatically produce a
histogram.
To look at the statistics from the____________, go to the
Crystal Ball View menu and choose Statistics.
Based on these trials, the best _________of the expected
profit of ordering 11 omelet pans is $123.24.
Note that since both the ________and the average profit
are random variables, running the simulation again will
most likely result in a different ____________profit.
Expected Value versus Order Quantity. To calculate the
_____expected profit using the spreadsheet and Crystal
Ball, simply enter each _______in cell B5 (one at a time),
run the simulation and then record the average profit.
These average _________will then be multiplied by their
respective____________. The sum of these values will
give the true expected profit.
Simulated versus Expected Profits. For any particular
order____________, the average profit generated by the
spreadsheet simulator does not ______the true expected
profit. The implication of this fact on the ___________of
making a decision is interesting.
The __________expected profits and simulated average
profits for order sizes of 9, 10,11, and 12 pans are shown
below:
Based on the____________, your decision would be to
order 10 pans for the true expected profit or 11 pans for
the simulated average profit.
The previous example illustrates that_____________, in
general, is not guaranteed to achieve________________.
A simple way to increase the ____________of achieving
optimality is to increase the number of trials.
With simulation, your decision may be wrongly identified
if care is not taken to simulate a ____________number of
trials.
In a real problem you would not ______calculate the true
expected profit and use simulation to calculate an
average profit.
Use simulation when
it is computationally _______________or not even
possible to calculate the expected profit associated
with the alternative decisions,
or when it is important to ______the variability of the
performance measure for various solutions.
RECAPITULATION
To summarize:
1. A spreadsheet simulator takes _____________and
decisions as _________and yields a performance
measure(s) as output.
2. Each _________of the spreadsheet simulator will
generally yield a different value for the
performance___________.
3. The performance measure (for an order size of 11)
was_____. The 500 trials taken together combine
to produce a measure of ___________of the order
size: average profit.
More information can be obtained from the simulation.
Suppose you want to know how often a ___________
occurred (i.e., demand exceeded quantity ordered), click
on cell B9 and add it as a “__________” cell. Then, rerun
the simulation.
The results
show that a
shortage
occurred in 95
of the 500
trials (20.6%).
Indicators of ___________are also important products of
simulation studies. Management usually seeks _______
which have low variability.
4. Increasing the number of simulation ___________
will usually improve the accuracy of the ________
of the expected value of the performance
measure.
5. In a_____________, we can never be sure that the
optimal decision has been found. Although, a
95% or 99% _____________interval can be
calculated.
6. Management must assess four main factors :
a. Does the model capture the real________________?
b. Are the _________of the starting and ending conditions
of the simulation properly accounted for?
c. Have enough ____been performed for each decision so
that the average value of the measure(s) of performance
is a good indication of the true expected value?
d. Have enough _________been evaluated so that you can
believe that the best answer found is “close enough” to
the optimum?
Simulation of Foslins’ Model with a
More Realistic Demand Assumption
Previously with the Foslins model, we had assumed a
simplified __________distribution. However, a more
realistic model of demand is the _________distribution
with a mean of 1000 and a standard deviation of 100.
The Foslins’ Spreadsheet: Simulating Demand More
Realistically. To simulate___________, modify the Run
Preferences dialog to indicate the larger number of
_______________.
In addition, use the same 1000 __________values for
demand in each of the upcoming simulations (for
different order quantities for comparison). This is
known as setting the __________value.
The Foslins’ Spreadsheet: Simulating Demand More
Realistically. In the spreadsheet, change the quantity
ordered to 1020.
Also, change the random distribution of ________from a
customized one to the normal distribution with a mean
of 1000 and a standard deviation of 100.
To simulate 1000 trials, click on the Run Preferences
icon and change the number of iterations.
Now, specify the seed value:
First, click on __________in the Run Preferences dialog.
Check the use
Same Sequence
… option and
enter an Initial
Seed Value of
_______(or any
other number).
Click OK.
Note that the _________do not change. We want to
compare the average profit for different order quantities
but the _______set of random demands. Profit will differ
only because of different order____________.
_______________is the process of decreasing variability
in simulation results.
It is an important technique in reducing the amount of
________________necessary to obtain valid simulation
results.
With simulation, using the same __________of random
variables provides us with complete control of the
random_____________.
THE EFFECT OF ORDER QUANTITY
Click on the Start Simulation
icon to begin the
simulation. After 1000 iterations, the following dialog
will open.
Click OK to bring up a
_________of the profit
values.
Here is the resulting_____________.
You can view the
statistics by
clicking on the
View – Statistics
________option.
The average
profit for an
order quantity of
1020 pans is
$12,270.44.
You can determine how the __________profit varies for a
range of order quantities and a given set of demands by
entering different order quantities in cell B3.
First, place a _______order quantity in cell B3.
Click on the Reset Simulation
icon and in the
resulting dialog, click OK. Now, you can begin the
simulation again by clicking on the Start Simulation
icon.
Finding the ____________quantity is an iterative
procedure -- change the order quantity and re-run the
simulation. Then, compare the profits to find the
______________profit.
For example, suppose you ran 1000 iterations at each of
the following order quantities:
Expected
Order
Quantity
Profit
$12,268.95
1010
$12,270.29
1013
$12,270.70
1015
$12,270.76
1016
$12,270.77
1017
$12,270.71
1018
$12,270.44
1020
$12,268.34
1025
1030
$12,264.60
So, this iterative process has shown that an order
quantity of 1017 gives a maximum profit of $12,270.77.
However, this quantity that _____________profit may not
maximize the true ___________profit. Indeed, it may be
impossible to guarantee that the optimal solution will be
found using______________-.
Remember, the ______________solution is theoretical as
opposed to a real-world concept. At best, an optimal
solution represents a “____________” for the real-world
problem.
Since the average profit for order quantities between
1015 and 1018 only varied by a couple of________, even
if we didn’t get the exact optimal order quantity, we are
confident that we are “________________.”
The Probability Distribution of Profit. To find out more
about the solution suggested by simulation (ordering
1017 pans), we can look at the ____________distribution.
This histogram
visually
describes the
______________
that Foslins will
be taking in
ordering 1017
pans.
The ________distribution means that there is a definite
probability that the profit will _________the mean profit.
The area to the left means that there is some chance of
__________a profit below the expected profit. However,
there is little chance of _______________money.
Midwest Express:
Airline Overbooking Model
Here is a _______management example from the service
industry. The airlines were the first to pioneer the use of
these tools.
American Airlines (AA) started the practice of _________
off the value of a seat on a given flight when more
customers showed up than it had seats available.
AA estimates that ______________alone adds over $200
million per year to its bottom line.
Other areas besides overbooking that are practiced in
the revenue management area include___________, seat
allocation among the various fare classes, and control of
the entire ___________of flight legs.
Midwest Express is headquartered in Milwaukee,
Wisconsin and was started by the large consumer
products company Kimberly Clark, which has large
operations in nearby Appleton, Wisconsin.
After reviewing the __________data on the percentage of
no-shows, it was found that the average no-show rate for
Flight 227 from Milwaukee to San Francisco is 15%.
The aircraft (MD88) has a _________of 112 seats in a
single cabin. As all service is considered premium,
there is no First Class/Coach cabin distinction.
Demand for this primarily __________route is strong and
the average fare is $400.
If only 112 reservations are accepted, then the flight will
almost certainly go out with ______seats because of the
no-shows. These no-shows represent an ____________
cost for Midwest Express.
On the other hand, if more _____________than seats are
accepted, then even after accounting for the no-shows,
there will still be a risk of ____________________seats.
The normal procedure in the event that a customer is
denied boarding, is to put the “______” customers on
the next available flight and provide them with some
_____________(a free flight in the future, a voucher for a
free meal and a hotel, etc.).
On the average, the compensation usually costs
Midwest Express around __________.
The more ____________that are accepted, the less likely
there will be empty seats, but the ______________that a
customer will be denied boarding is increased.
The goal is to find the
_______overbooking level
that maximizes_________.
This model assumes that
the fares are fully
____________(if they
don’t show, they don’t
pay).
You can perform multiple trials using ________since the
value returned will change every time the spreadsheet is
recalculated by pressing _________.
In order to see the __________values returned for each
iteration, set @Risk to Monte Carlo by clicking on the
Change @Risk Settings
icon and then on the
Sampling tab.
In the Standard Recalc
section, click on Monte
Carlo.
Now, use simulation to determine the ________
overbooking level. To see which value maximizes
Midwest Express’_________, consider the range of
reservations from 112 to 146 in increments of 2.
To do this, enter the following formula into cell B10:
=RiskSimtable({112,114,116,118,120,…,144,146})
Now, click on the
@Risk icon. In the
resulting Simulations
Settings dialog, click on
the _________tab. Change
the number of iterations to
1000 and the number of
simulations to 18.
Now, click on cell B19 and then click on the Add Output
Cell
icon to add this cell to the simulation.
Click on the Run Simulation
simulation.
icon to start the
The Simulating dialog will appear in which
you can observe the status of the
simulation.
@Risk automatically displays the results of these 18
simulations:
Click on the Merge Sim#’s button.
The following information will be displayed.
As shown in Sim#1, if only 112 reservations are
accepted, the avg. profit would be $38,080.
As the no. of reservations are increased, the avg. profit
increases then starts to decline.
The max. profit of $43,901 occurs at 134 reservations.
Capacity Balancing
Now, let’s explore the assertion that ___________should
be “__________” throughout a manufacturing plant.
MODELING A WORK CELL
Consider the following work cell:
Raw Material
WS1
WIP
WS2
This cell takes a___________, processes it at the first
work station (WS1), holds it in a temporary _______area
if the 2nd work station is busy, and then processes it at
WS2.
The completed part is used in assembly at the rate of 3
per hour.
The goals are to
Meet the ___________for this part on the assembly
line,
Keep work in ________(WIP) between the two work
stations down
Minimize the ________of the work stations subject
to achieving the first two goals.
SIMULATING BALANCED CAPACITY
Since the assembly area needs the part at a rate of 3 per
hour, set the _______of both work stations at 3 per hour.
The Capacities of WS1 and WS2 are_____________.
However, because of processing time_________, a work
station might be able to process anywhere from 1 to 5
units per hour.
Suppose that during any given hour, a work station will
have the capability of _____________1, 2, 3, 4, or 5 units
with equal probability (discrete uniform distribution).
Then the __________output of that work station will be 3
units per hour provided it always has something on
which to work.
Assume that sufficient raw material will always be
available to WS1 so that it will never be_____________.
However, because the processing times are variable at
WS1, there may be times when WS2 is ______for lack of
material.
The Initial Conditions. The following spreadsheet
displays the first 16 simulated hours of operation of the
work cell.
The initial conditions at the beginning of the first hour of
the simulation are no work in process (WIP), and WS1
and WS2 are idle.
Now, enter a _______uniform random number generator
in cell C10 (WS1 Output) and cell D10 (Potential WS2
Output) to produce one of 5 possible values with
equal____________.
First, type in the data in a _______format for Crystal Ball.
Now, to enter the discrete uniform random number
____________for WS1 Output and Potential WS2 Output
in Crystal Ball, click on cell C10 and enter
= CD.Custom($C$2:$D$6)
With the ________still on cell C10, click on Excel’s Copy
icon and then highlight the range C10:D25. Next, click
on Excel’s _________icon to copy the uniform discrete
distribution to all 32 of these cells.
Highlight cells G10, G13, G16, G19, G22, and G25.
Click on the Define Forecast
icon and change the
Window size to Large and the Display to When Stopped
(faster) and click OK.
Repeat this step for each of the 6 _________cells, or use
the Copy/Paste feature with the forecast cells.
Click on the Run Preferences
icon and change the
Maximum Number of Trials to 1000 and click OK.
Now click on the Start Simulation
icon and after
Crystal Ball has run the 1000 iterations, it automatically
produces an individual histogram for each of the 6
forecast cells.
To better view these, click on Run – Open Trend Chart.
Crystal Ball
plots the
distribution of
Average ____
over the first
16 hours by
using the six
selected
hours that we
indicated.
The center band indicates the ______ + 25% and the next
darker band is a 95% confidence interval.
This shows that the Average WIP seems to be ________
slightly over the first 16 hours.
Beyond the Initial Conditions. Now, expand the
simulation beyond the 16 hours by copying the _______
down for another 184 hours. Re-run Crystal Ball using
the same basic steps as in the previous example.
The resulting _____
of avg. WIP for the
first 200 hours of
operation shows a
__________growth.
The longer the cell
is in operation, the
greater the amount
of WIP that
_____________.
Here, the average ____________and the average service
rate are equal when the capacities of the two work
stations are_________. Hence, the unexpected growth.
SIMULATING UNBALANCED CAPACITY
Now, let’s add ____________to WS2 so that its average
production rate is 3.5 units per hour (using a discrete
uniform distribution between 2 and 5 units).
Using the data from the previous spreadsheet, add the
following distribution for ___________________2:
Now, change the formulas in cells D10:D209 to be:
= CD.Custom($G$2:$H$5)
Select six forecast cells, spread ________across the 200
hours and run 1000 iterations.
The resulting graph of average WIP over time shows a
much __________average WIP.
These results suggest that there is no ___________
growth in the Average WIP with the steady-state value
lying somewhere between 5 and 7 units.
In conclusion, the capacity of the two work stations
should not be ____________(equal output rates).
If WIP is to be kept to reasonable levels, then the
_______________work station (WS2) should have a
somewhat greater capacity.
By running the simulation for longer periods of time, the
effect of the initial ____________can be overcome, and
the true long-term behavior can be discerned.
In general, simulation results are useful only when care
is taken in the experimental design to eliminate
____________effects such as starting or ending
conditions.
Simulations that are too short may give ______________
results.
Optimization Under Uncertainty
We can now combine the optimization ______that we’ve
discussed with the ability to do ____________simulation.
Let’s demonstrate the use of optimization under
_________with OptQuest (available with Crystal Ball Pro)
on two very common examples – portfolio allocation and
project______________.
Suppose that we can choose to invest in one of three
stocks – Intel, Microsoft, and Proctor & Gamble, or in a
money market account. Let
W represent the fraction invested in______________
X represent the fraction invested in Intel stock
Y represent the fraction invested in_______________
Z represent the fraction invested in P&G stock
Historically, based on the last 9 years, the average
_____________return has been:
46.6%
Intel
62.1%
Microsoft
20.8%
Procter & Gamble
5.2%
Money Market
The constraints are:
No more than 50% of the ___________is to be in any
one asset.
The sum of the decision variables must be_______
In order to optimize the portfolio, we can either
maximize _________subject to a constraint to keep
the risk at some satisfactory level or
minimize risk subject to a ____________to keep the
return at some satisfactory level.
In this example, we will be minimizing risk. Set up a
spreadsheet as shown below.
Here are the Solver results.
These show
that you should
______in 0.51%
of Intel, 20.7%
of Microsoft,
50% of P&G,
and 28.8% of
money market
accounts.
The minimum
____________
is 0.0127.
Now let’s use Crystal Ball with___________. Perform the
following steps:
1. Change cells C11:F11 to numbers rather than
__________. Highlight cells C11:F11 and click
on Edit – Copy and then click Edit – Paste
Special – Values and click OK.
2. Click on the Define Assumption
icon to
specify the distribution for cells C11:F11.
Specify each as a ________distribution with the
following parameters, respectively:
Mean
46.6%
62.1%
20.8%
5.2%
Standard
Deviation
.5646
.3999
.1334
.0141
3. Click on the Define Forecast
icon and select
cell H17 (Portfolio_________).
4. Click on the Run Preferences
icon and
change the maximum number of trials to 1000.
Also, select _________________as the sampling
method and Use Same Sequence as the random
5. Click on the Define Decision icon and define the
decision __________as cells C17:F17 (each as a
continuous variable with lower bound of 0% and
upper bound of 50%. Select the variable type as
_______________.
6. Click on the OptQuest icon to start the program.
7. In OptQuest, select File – New – Yes to optimize
all ________variables. Check that the Type
Column indicates that these are __________
values and click OK.
8. Define the constraints as follows:
a. Intel% + Microsoft% + P&G% + MM% = 1
b. Intel% <= 0.5
c. Microsoft% <= 0.5
d. P&G% <= 0.5
e. MM% <= 0.5
f. 0.466*Intel% + 0.621*Microsoft% +
0.208*P&G% + 0.052*MM% >=0.25
9. From the Select menu, choose ____________
Objective and click OK.
10. In the Options window, click OK to accept the
default process. To start the optimization
under_____________, click Yes.
After 10 minutes, you should get results similar to
the following:
11. To interpret the results, select Edit – Copy to
Excel which will copy the resulting ________for
the decision variables back into your
spreadsheet and automatically give a
___________diagram for the chosen statistic.
In the Forecast window, you can also select
View – Statistics to view the summary
____________.
The OptQuest engine basically gave the same answer as
Solver (in a much longer period of time).
The result from OptQuest confirmed that after running
1,000 random________, the average portfolio variance
was indeed closely represented by the mean values
used in our original spreadsheet, which Solver used to
_____________.
PROJECT SELECTION
Consider the R&D group at a major public utility that has
identified eight possible projects for the coming year.
Each project has an initial ___________required and the
resulting NPV from a __________cash flow analysis has
also been tabulated.
The CFO of the company has only authorized $2 million
to be spent on R&D projects for the coming year. If
implemented, these 8 projects would require an
_____________of $2.8 million, so we much choose those
projects which will return the largest NPV and still
remain within the___________.
Set up the model for optimization using __________
decision variables (yes/no) for each of the eight potential
projects.
The resulting solution from Solver shows that projects 1,
2, 3, 4, and 7 should be chosen with the __________NPV
obtained of $3.55 million and using all $2 million of the
budget.
This is an
application of
___________
programming
where the NPVs
are assumed to
be certain. Now
add some
____________to
the success of
each project.
Now, due to the uncertainty of each project___________,
re-optimize with Solver (maximize cell G15, changing
cells are E5:E12, constraint: C15 <= C14).
This gives a new solution that ______________projects
1, 2, 4, 6, 7, with an expected NPV of $2.184 million and
spending only $1.85 million of the budgeted $2 million.
How do we know if this is the best___________, given all
the uncertainty?
With OptQuest we can specify that it maximize the 25th
_____________of the NPV distribution that results from a
simulation of 1000 trials.
Then, for each possible combination of decision
variables (28 = 256 combinations), the software will make
“___________” choices as it searches for the best
combination.
To set this up in OptQuest, follow the steps outlined
below:
1. Click on the Define Assumption
icon and
define the assumption cells as F5:F12 (each as a
_____________with success rate as given in the
original spreadsheet, and 1 trial).
2. Define the decision variables as cells E5:E12
(each as a discrete variable with _____________
of 0, upper bound of 1).
3. Click on the Define Forecast
icon and
specify cell G15 with “NPV” as ______________
name and “Dollars” as the forecast units.
4. Click on the Run Preferences
icon and
specify a maximum number of trials equal to
1,000. Also, specify Latin Hypercube and Use
same sequence with an __________value of 999.
5. Now, start OptQuest by clicking on its icon or by
going to the Tools – OptQuest pull-down menu.
6. In OptQuest, click on File – New and select Yes
to optimize all ________variables. Check that
the Type column indicates that these are
discrete values.
7. Define the single _________as:
250*Project1 + 650*Project2 + 250*Project3 +
500*Project4 + 700*Project5 + 30*Project6 +
350*Project7 + 70*Project 8 <= 2000
8. From the Select drop-down menu, select
Maximize Objective and change the Forecast
Statistic from the ________to the Percentile and
enter the number 25.
9. In the Options window, click OK to accept the
_________process. To begin the optimization
under uncertainty, click Yes.
10. To interpret the results, select Edit – Copy to
Excel to copy the resulting values for the
decision variables to the ________________and
automatically provide frequency diagrams of
the chosen __________(NPV).
Under View switch to Cumulative Chart and
enter the right-hand side value as $1.6 million.
In the Forecast window, you can also select View –
Statistics to view the summary statistics.
The results from OptQuest showed that the best projects
to choose are 1, 2, 5, 6, 7 which spend $1.98 million of
the budget and generate a maximum 25th percentile
value of $1.6 million.