MgtOp 470—Business Modeling with Spreadsheets

Professor Munson

Topic 6

Monte Carlo Simulation

Set 2: @RISK Notes

“This software alone is worth the price of your textbook.”Chuck Munson

Example Problem

Selling T-shirts at a rock concertDecisions: purchase quantity Q, selling price P, and

advertising level ANo salvage value for unsold shirtsDemand D = a – bP + c(A/100)Profit = P×MIN(Q, D) – VQ − A

Baseline parameters: Variable Cost V = $5.00Demand Intercept a = 5000Demand Price Slope per Dollar b = 180Demand Advertising Slope per 100 Dollars c = 6

This problem could be optimized to maximize expected profit. However, some inputs are uncertain, so we want to run a Monte Carlo simulation to compare several different decision strategies.

UncertaintiesVariable Cost: There’s a 30% chance that the supplier

will be unable to deliver on time, in which case an expedited supplier must be used, costing $12 per shirt.

Intercept: The upper limit of 5000 is based on the assumption of 20,000 tickets being sold for the concert. Expected concert sales are normally distributed with a mean of 20,000 and a standard deviation of 2,000, so the intercept is normally distributed with a mean of 5,000 and a standard deviation of 500.

Advertising Slope: Historical data suggests that this is exponentially distributed with a mean of 6 (β = 6).

Price Slope: The effect of price on demand depends upon a variety of factors. A list of historical slopes is provided below:182, 162, 202, 184, 178, 172, 180, 192, 177, 176, 175

The Model123456789

1011121314151617181920

A B C D E F G HT-Shirt SalesProfit/Loss Analysis

PARAMETERSUnit Variable Cost $5.00

Price Slope per Dollar 180 Advertising Slope per $100 6

Demand Intercept 5,000

DECISIONSPrice $25.00

Advertising $200.00Quantity 500

OUTPUTDemand 512

Profit (Loss) $9,800C17: =C11*MIN(C13,C16)-C5*C13-C12

=MAX(0,C8-C6*C11+C7*(C12/ 100)

@RISK Implementation

After loading @RISK, click on the @RISK tab to display the following ribbon:

Output VariablesTo define output variables, put the cursor on the output cell of interest and press the “Add Output” button.

After selecting <OK>, the formula in cell C16 changes to: =RiskOutput()+MAX(0,C8-C6*C11+C7*(C12/100))

For our example, we’ll select Profit (Loss) as an output as well.

Input VariablesTo define distributions for input variables, put the cursor on the input cell of interest and press the “Define Distributions” button. Then select your distribution and plug in the parameters (if known).

For the Advertising Slope:

For the Variable Cost:For a discrete distribution, click on the arrow in the

“X-Table” box to input the values and associated probabilities.

Fitting a Distribution to a Data Set@RISK can automatically attempt to find the best-fitting distribution for your data. Highlight your data and then click on the “Distribution Fitting” button on the @RISK ribbon. Then click:

Fit...

If you know that there are or are not bounds on either end of the data, click the appropriate buttons on the left-hand side of the dialog box. Then press Fit.

You can then write the formula for the best fitting distribution directly into a cell. Click on:

Write To Cell.

You can even have the distribution parameters updated and refitted before each simulation run as original inputs change.

Reviewing Your SettingsYou can verify all of your inputs and outputs by clicking on the “Model Window” button of the @RISK ribbon.

Starting Seed

The default starting seed is random. To set a specific starting seed, click on “Simulation Settings,” then select:

Sampling: Initial Seed

Type your desired seed number to the right of the “Fixed” setting (the seed is 2 in this example).

Running the Simulation

Set the number of iterations under the “Simulation” tab of the @RISK ribbon.

Automatically show the output graph after the simulation.

Press the “Start Simulation” button to run it. Various output reports can be generated.

To display graphs

To display summaries

To display simulation statistics

To display each iteration’s data

Example of Computing the Run LengthChoose desired confidence interval ± A

$400Choose desired confidence level

90%z =NORMSINV(1−(.10/2)) = 1.645

Find S from the run of 100 trials:

n = (zS/A)2 = [(1.645)(5424.02)/400]2

= 497.6 or 498 runs

New S = 5465.32, so new n = 505.

New S = 5463.346, so run length is large enough.

Decision ComparisonsOur policy now has an expected profit of $4599; however, it is highly risky. There’s a 27.13% chance of making no money at all:

Let’s compare to a “Low Risk” strategy that lowers the price from $25.00 to $20.00 and lowers the order quantity from 500 to 300 units.

To compare different decisions on the same set of random inputs, set “Number of Simulations” to more than one. Then use the “RiskSimtable” command on the decision variables that you want to change.

Price of $20 and $25Enter into Cell C11:

=RiskSimtable({20,25})

Quantity of 300 and 500Enter into Cell C13:

=RiskSimtable({300,500})

Set the number of simulations to 2.

To name each scenario, click on “Simulation Settings,” then select: General: Simulation Names…

Type in the simulation names.

Press the “Start Simulation” button.

To switch display between either or both