Introduction to @Risk
Background Information
 Recall that Walton Bookstore buys calendars
for $7.50, sells them at a regular price of $10,
and gets a refund for all calendars that
cannot be sold.
 The company does not know exactly how
many calendars its customers will demand,
but it does have historical data on demands
for similar calendars in previous years.
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WALTON4.XLS
 The Data sheet of this file contains the historical data.
 Walton wants to use these historical data to
determine a reasonable probability distribution for
next year’s demand for calendars.
 Then it wants to use this probability distribution,
together with @Risk, to simulate the profit for any
particular order quantity.
 It eventually wants to find the “best” order quantity.
3
Solution
 We will use this example to illustrate many of
@Risk’s features.
 We first see how it helps use to choose an
appropriate “input” distribution for demand.
 Then we will us it to build a simulation model for a
specific order quantity and generate outputs from this
model.
 Finally we will see how the RISKSIMTABLE function
enables us to simultaneously generate outputs from
several order quantities so that we can chose a “best”
order quantity.
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Loading @Risk
 The first step, if you have not already done it, is to
install Palisade Decision tools suite.
 Once @Risk is loaded, you will see two new toolbars,
the Decision Tools toolbar shown here and the @Risk
toolbar shown on the next slide.
5
Fitting a Probability Distribution
 Some of the historical
demand data appears on the
next slide.
 As the text box indicates,
Walton believes the
probability distribution of
demand for next year’s
calendars should closely
match the histogram for the
historical data.
 To see which probability
distributions match the
histogram well, we can use
@Risk’s fitting ability, using
the following steps.
6
Fitting a Probability Distribution -continued



Model window. Click on the Show @Risk-Model
Window toolbar button. @Risk has two windows that
get you outside of Excel: The Model and Results
windows. The former helps in setting up the model; the
latter shows results from running a simulation. For now,
we require the Model window.
Insert a Fit Tab. Once the Model window is showing,
select the Insert/Fit Tab menu item. This brings up a
one-column “spreadsheet” on the left.
Copy and paste data. We want to copy the historical
data to this mini-spreadsheet. To do so, go back to the
Excel windows, copy the historical data, go back to the
@Risk Model window, and paste the data – copy and
paste work in the usual way.
7
Fitting a Probability Distribution -continued

Select candidate distributions. @Risk has
many probability distributions from which to
select. To see the candidates, select the
Fitting/Specify Distributions to Fit menu item.
This brings up the dialog box shown on the
next slide. You can check as many of the
candidates as you like. Some are undoubtedly
unfamiliar to you so you might want to stick
with familiar distributions such as the normal
and triangular. However, we clicked on the OK
to accept the defaults shown in the figure.
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Fitting a Probability Distribution -continued
9
Fitting a Probability Distribution -continued


Do the fitting. Select the Fitting/Run Fit Now menu item
to see which of the candidate distributions most closely
match the historical data. @Risk evaluates the fits in
several different ways, and it also allows you to check
the fits visually. After it runs, you will see a screen as
shown on the next slide. This screen shows one of the
candidate distributions superimposed on the histogram
of the data.
Examine the selected distribution. To do so, select the
Insert/Distribution Window menu item, and fill it out as
shown on the slide after the next. Specifically, select Fit
Results in the Source box, select By Name in the
Choose box and click on Normal. @Risk provides a
very friendly interface for examining the resulting
normal distribution.
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Fitting a Probability Distribution -continued
11
Fitting a Probability Distribution -continued


It has two “sliders” that you can drag in either direction
to see probabilities of various areas under the curve.
Also you can enter X values” or P values” directly into
the boxes in the right column to obtain equivalent
information.
A caution about negative values. We should point out
that there is a potential drawback to using this normal
distribution. Although the mean demand in this
example is approximately three standard deviations to
the right of 0, so that a negative demand is very
unlikely there is still some chance that one can occur –
which would not make physical sense in our model. To
ensure that negative demand do not occur, there are
two possibilities.
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Fitting a Probability Distribution -continued

First, we could use a truncated normal
distribution of the form
=RISKNORMAL(Meandem,StDev,0,1000).
The function disallows values below the third
argument or above the fourth argument. The
other possibility is to choose a probability
distribution that, by its very definition, does not
allow negative values. On such distribution is
the Weibull distribution, which provides one of
the best fits to the historical data.
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Developing The Simulation Model
 Now that we have chosen a probability distribution for
demand, the spreadsheet model for profit is essentially
the same as we developed earlier without @Risk. It
appears on the next slide. The only new things to be
aware of are as follows.

Input distribution. We want to use the normal distribution
for demand found from @Risk’s fitting procedure. To do
this, enter the fitted mean and standard deviation in cells
E4 and E5. Then enter the formula
=ROUND(RISKNORMAL(MeanDem,StdevDem),0) in
cell A13 for the random demand. This uses the
RISKNORMAL function to generate a normally
distributed demand with the fitted mean and standard
deviation. Because demands should be integers, we use
Excel’s ROUND function, with second argument 0, to 14
round this value to 0 decimals.
Developing The Simulation Model
-- continued

Output cell. When we run the simulation, we want
@Risk to keep track of profit. In @Risk’s terminology,
we need to designate the Profit cell, E13, as an output
cell. There are two ways to designate a cell as an
output cell. One way is to highlight it and then click on
the Add Output Cell button on the @Risk toolbar. An
equivalent way is to add RISKOUTPUT( )+ to the cell’s
formula. Either way, the formula in cell E13 changes
from =B13+D13-C13 to =RISKOUTPUT( )+B13+D13C13. The plus sign following RISKOUTPUT ( ) does
not indicate addition. It is simply @Risk’s way of
saying: Keep track of the value in this cell as the
simulation progresses. Any number of cells can be
designated in this way as output cells. They are
typically “bottom line values of primary interest.”
15
Developing The Simulation Model
-- continued

Inputs and outputs. @Risk keeps a list of all
input cells and output cells. If you want to
check the list at any time, click on the Display
Inputs, Outputs button on the @Risk toolbar. It
provides an Explorer-like list as shown here.
16
Developing The Simulation Model
-- continued

Summary functions. @Risk provides several
functions for summarizing output values. We
illustrate these in the range B16:B19. They
contain the formulas =RISKMIN(Profit),
=RISKMAX(Profit), RISKMEAN(Profit), and
RISKSTDDEV(Profit). The values in these
cells are not of any use until we run the
simulation. However, once the simulation runs,
these formulas capture summary statistics of
profit.
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Running the Simulation
 Now that we have developed the model for Walton,
the rest if straightforward.
 The procedure is always the same. We specify the
simulation settings and the report setting and then
run the simulation.

Simulation settings. We must first tell @Risk how we
want the simulation to be run. To do so, click on the
Simulation Settings button on the @Risk toolbar. Click
on the Iterations tab and fill out the dialog box as
shown on the next slide. This says that we want to
replicate the simulation 1000 times, each with a new
random demand.
18
Running the Simulation -continued
19
Running the Simulation -continued

Then click on the Sampling tab and fill out the dialog
box as shown here. For technical reasons it is always
best to use Latin Hypercube sampling, it is more
efficient.
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Running the Simulation -continued


We also recommend checking the Monte Carlo button
on the Standard Recalc group. Although this has no
effect on the ultimate results, it means that you will see
random numbers in the spreadsheet.
Report settings. @Risk has many options for
displaying the outputs from a simulation. The outputs
can be placed in an @Risk Results window or on new
sheets of your Excel workbook. They can also be
shown in more or less detail. Click on the Report
settings button on the @Risk toolbar to select some of
these options. In the dialog box on the next slide we
have requested a summary of the simulation and
detailed statistics, and we have asked that they be
shown both in the @Risk Results window and on new
sheets in the current workbook.
21
Running the Simulation -continued
22
Running the Simulation -continued

Run the simulation. We are finally ready to run
the simulation! To do so, simply click on the
Start Simulation button on the @Risk toolbar.
At this point, @Risk repeatedly generates a
random number for each random input cell,
recalculates the worksheet, keeps track of all
output cell values. You can watch the progress
at the bottom left of the screen.
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Analyzing the Output
 @Risk generates a large number of output
measures. We discuss the most important of
these now.

Summary Report. Assuming that the top box
was checked in the @Risk Reports dialog box,
we are immediately transferred to the @Risk
Results window. This window contains the
summary results shown here.
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Analyzing the Output -- continued


Detailed Statistics. We can also request more
detailed statistics within the @Risk Results
window with the Insert/Detailed Statistic menu
item. Some of these detailed statistics appear
on the next slide. All of the information in the
Summary Report is here, plus some.
Target values. By scrolling to the bottom of the
detailed statistics list, as shown on the slide
after next, you can enter any target value or
target percentile. If you enter a target value,
@Risk calculates the corresponding
percentile, and vice versa.
25
Analyzing the Output -- continued
26
Analyzing the Output -- continued

Simulation data. The results to this point summarize
the simulation. It is also possible to see the full results
– the data, demands and profits, from all 1000
replications. To do this select the Insert/Data menu
item. A portion of the data appears on the next slide.
27
Analyzing the Output -- continued

Charts. To see the results graphically, click on the
Profit item in the left pane of the Results window and
then select the Insert/Graph/Histogram menu item.
This creates a histogram of the 1000 profits from the
simulation.
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Analyzing the Output -- continued


The same interface is available that we saw earlier –
namely, we can move the “sliders” at the top of the
chart to the left or right to see various probabilities.
Outputs in Excel. Often we will want the simulation
outputs, including charts, in an Excel workbook. The
easiest way to get the numerical information shown
earlier is to fill out the Report Settings dialog box as we
did. Then separate sheets are created to hold the
reports.
 This has been a quick tour through @Risk’s report
capabilities.
 The best way to become more familiar with @Risk is
to experiment with the user-friendly interface.
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Using RISKSIMTABLE
 Walton’s ultimate goal is to choose an order quantity
that provides a large average profit.
 We could rerun the simulation model several times,
each time with a different order quantity in the
OrderQuan cell, and compare the results.
 However, this has two drawbacks.


First, it takes a lot of time and work.
Second, each time we run the simulation, we get a
different set of random demands. Therefore, one of the
order quantities could win the contest just by luck. For
a fairer comparison, it would be better to test each
order quantity on the same set of random demands.
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WALTON5.XLS
 The RISKSIMTABLE function in @Risk enables us to
obtain a fair comparison quickly and easily.
 This file includes the setup for this model.
 The next slide shows the comparison model.
 There are two modifications to the previous model.


First, we have listed order quantities we want to test in
the range names OrderQuanList.
Second, instead of entering a number in cell B9, we
enter the formula =RISKSIMTABLE(OrderQuanList).
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The Spreadsheet
32
Using RISKSIMTABLE -continued
 Note that the list does not need to be entered
in the spreadsheet.
 However the model is now set up to run the
simulation for all order quantities in the list.
 To do this, click on the Simulation Settings
button on the @Risk toolbar and fill out the
Iterations dialog box as shown on the next
slide.
 Specifically, enter 1000 for the number of
iterations and 5 for the number of simulations.
33
Using RISKSIMTABLE -continued
34
Using RISKSIMTABLE -continued
 After running the simulations, the Report window
shows the results for all five simulations.
 For example, the basic summary report appears on
the next slide.
 The first five lines show summary statistics of profit.
 Although we do not show them here, the same
information can be seen graphically. A separate
histogram of profit for each simulation is easy to
obtain.
35
Using RISKSIMTABLE -continued
36
Using RISKSIMTABLE -continued
 Indeed, much of the appeal of @Risk is that
we can see all of these characteristics –
averages, minimums, maximums, percentiles,
charts – and use them to make informed
decisions.
37
Background Information
 As in the previous example, Walton needs to
place an order for next year’s calendar.
 We continue to assume that the calendars will
sell for $10 and customer demand for the
calendars at this price is normally distributed
with mean 168.1 and standard deviation 57.6.
 However, there are now two other sources of
uncertainty.
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Background Information -continued
 First, the maximum number of calendars Walton’s
supplier can supply is uncertain and is modeled with
a triangular distribution.
 It’s parameters are 125, 250, and 200. Once Walton
places and order, the supplier will charge $7.50 per
calendar if he can supply the entire Walton order.
Otherwise, he will charge only $7.25 per calendar.
 Second, unsold calendars can no longer be returned
to the supplier for a refund. Instead, Walton will put
them on sale for $5 a piece after February 1.
39
Background Information -continued
 At that price, Walton believes the demand for
leftover calendars is normally distributed with
mean 50 and standard deviation 10.
 Any calendars still left over, say after March
1, will be thrown away.
 Walton plans to order 200 calendars and
wants to use simulation to analyze the
resulting profit.
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Solution
 As before, we first need to develop the model.
 Then we can run the simulation with @Risk
and examine the results.
 The completed model appears on the next
slide.
 The model itself requires a bit more logic than
the previous Walton model.
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Solution
42
Developing The Simulation Model
 The model can be developed with the following steps.
 Random inputs. There are three random inputs in this
model: the most the supplier can supply Walton, the
customer demand when the selling price is $10, and
the customer demand for sale-price calendars.
Generate these cells in A16, D16 and G16 with the
formulas =ROUND(RiskTrian(E9,E10,E11),0),
=ROUND(RiskNormal(E5,E6),0) and
=ROUND(RiskNormal(F5,F6),0). Note that we
generate the random potential demand for calendars at
the sale price even though there might not be any
calendars left to put on sale.
43
Developing The Simulation Model
-- continued


Actual supply. The number of calendars supplied to
Walton is the smaller of the number ordered and the
maximum the supplier is able to supply. Calculate this
value in cell B16 with the formula
=MIN(MaxSupply,OrderQuan).
Order cost. Walton gets the reduced price, $7.25, if
the supplier cannot supply the entire order. Otherwise,
Walton must pay $7.50 per calendar. Therefore
calculate the total order cost in cell C16 with the
formula
=IF(MaxSupply>=OrderQuan,UnitCost1,UntiCost2)*
Supply
44
Developing The Simulation Model
-- continued

Other quantities. The rest of the model is
straightforward. Calculate the revenue from
regular-price sales in cell E16 with the formula
=UnitPrice1*MIN(Supply,Demand1).
Calculate the number left over after regularprice sales in cell H16 with the formula
=UnitPrice2*MIN(Leftover, Demand2).
Finally, calculate profit and designate it as an
output cell for @Risk in cell I16 with the
formula =RISKOUTPUT( )+E16+H16-C16.
45
Using @Risk
 As always, the next steps are to specify the
simulation settings, specify the report settings and
run the simulation.
 When there are several input cells, @Risk generates
a value from each of them independently and
calculates the corresponding profit on each iteration.
 Selected results appear on the next slide.
 They indicate an average profit of $255.66, a 5th
percentile of - $410.50, a 95th percentile of $514.25,
and a distribution of profits that is again skewed to
the left.
46
Using @Risk
47
Sensitivity Analysis
 We now demonstrate a feature of @Risk that
is particularly useful when there are several
random input cells.
 This feature lets us see which of these inputs
is most related to, or correlated with, an
output cell.
 To perform this analysis, select the
Insert/Graph/Tornado Graph menu item from
the @Risk Results window.
48
Sensitivity Analysis -- continued
 In the resulting dialog box, select Profit as the
output variable and click on the Correlation
Sensitivity button.
 This produces the results shown here.
49
Sensitivity Analysis -- continued
 The “regression” option produces similar results, but
we believe the correlation option is easier to
understand.
 This figure shows graphically and numerically how
each of the random inputs correlates with profit – the
higher the correlation, the stronger the relationship
between that input and profit.
 In this sense, we see that the regular-price demand
has by far the strongest effect on profit.
50
Sensitivity Analysis -- continued
 The other two inputs, maximum supply and
sale-price demand, are not nearly as
important because they are nearly unrelated
to profit.
 Identifying important input variables can be
important for real applications.
 If a random input is highly correlated with an
important output, then it might be worth the
time and cost to learn more about this input
and possibly reduce the amount of
uncertainty involving it.
51