Business analytics methods, models and decisions evans analytics2e ppt 12

 Monte Carlo simulation is the process of generating random values for uncertain inputs in a model, computing the output variables of interest, and repeating this process for many trial

Chapter 12

Monte Carlo Simulation and Risk Analysis

 Many situations dictate that randomness be explicitly incorporated into our models This is usually

done by specifying probability distributions for the appropriate uncontrollable inputs

◦ Such models are called stochastic, or probabilistic.

 Risk is the likelihood of an undesirable outcome It can be assessed by evaluating the probability

that the outcome will occur along with the severity of the outcome.

 Risk analysis seeks to examine the impact of uncertain inputs on various outputs.

Model Uncertainty and Risk Analysis

 Production volume is uncertain; assume normal with a mean of 1000 and standard deviation of 100.

 Replace cell B12 with =ROUND(NORM.INV(RAND(), 1000, 100, true), 0)

 Whenever F9 key or Formula > Calculation > Calculate Now is clicked, the value of demand will change randomly

Example 12.1: Incorporating Uncertainty in the Outsourcing Decision Model

 Monte Carlo simulation is the process of generating random values for uncertain inputs in a

model, computing the output variables of interest, and repeating this process for many trials to understand the distribution of the output results.

 Monte Carlo simulation can easily be accomplished on a spreadsheet using a data table.

Monte Carlo Simulation

 Excel file Outsourcing Decision Monte Carlo Simulation Model.

 Enter the trial number (1 to 20) in column D

 Reference the cells associated with model outputs in row 3: (E3, F3, G3)  (=B12, =B19, =B20)

 Select the range for the data table (D3:G23)

 In the Data Table dialog, enter any blank cell for the Column Input Cell

Example 12.2: Using Data Tables for Monte Carlo Spreadsheet Simulation

1 Develop a spreadsheet model.

2 Determine probability distributions for uncertain

input variables.

3 Identify output variables you want to predict.

4 Choose the number of trials for the simulation.

5 Run the simulation.

6 Interpret the results.

Monte Carlo Simulation Using Analytic Solver Platform

 For many decision models, empirical data may be available, either in historical records or collected through

special efforts.

probability distributions to help choose a representative distribution that has the shape that would most reasonably represent the analyst’s understanding about the uncertain variable.

Defining Uncertain Model Inputs

 Outsourcing Decision Model

 Demand (production volume) is normally distributed with a mean of

1000 and standard deviation of 100 units

◦ =PsiNormal(1000, 100) in cell B12

◦ Use ROUND function to ensure that the result is a whole number:

=ROUND(PsiNormal(1000,100),0)

 Unit cost has a triangular distribution with a minimum of $160, most

likely value of $175, and a maximum of $200

◦ =PsiTriangular(160, 175, 200) in cell B10.

Example 12.3: Using Analytic Solver Platform Probability Distribution

Functions

 For demand, select cell B12.

 Click the Distributions button in the

Analytic Solver Platform ribbon and

select the normal distribution from

the Common category.

Example 12.4: Using the Distributions Button in Analytic Solver Platform

 Normal distribution dialog

 Change the parameters to mean = 1000, stdev = 100

Example 12.4 Continued

 For unit cost, select cell B10 and select the triangular distribution

 Change the parameters in the dialog

Example 12.4 Continued

 To define a cell you wish to predict and create a distribution of output values from your model, first

select it, and then click on the Results button in the Simulation Model group in the Analytic Solver

Platform ribbon Choose the Output option and then In Cell.

◦ Analytic Solver Platform calls output cells uncertain function cells.

◦ Uncertain function cells must be numeric

 Analytic Solver Platform adds the function PsiOutput( ) to uncertain function cell formulas.

◦ You may also add +PsiOutput( ) to any output cells manually

Defining Output Cells

 Select cell B19

 After defining the cell as an uncertain

function, the formula should read:

=B16 – B17 + PsiOutput( )

Example 12.5: Using the Results Button in Analytic

Solver Platform

 First click on the Options button in the Options group in

the Analytic Solver Platform ribbon This displays a dialog

in which you can specify the number of trials and other

options to run the simulation (make sure the Simulation

tab is selected)

 Trials per Simulation allows you to choose the number of

times that the simulation will generate random values for the uncertain cells in the model and recalculate the entire spreadsheet

Running a Simulation

 Analytic Solver Platform generates a stream of random

numbers from which the values of the uncertain inputs are selected from their probability distributions

◦ Every time you run the model, you will get slightly different results because of sampling error.

 Setting a value for Sim Random Seed will guarantee that

the same sequence of random numbers will be used for generating the random values for the uncertain inputs every time the simulation is run

Random Number Seed

 Monte Carlo sampling selects random variates

independently over the entire range of possible values of the distribution

 With Latin Hypercube sampling, the uncertain variable’s

probability distribution is divided into intervals of equal probability and generates a value randomly within each interval

◦ Monte Carlo sampling is more representative of reality and should be used if you are interested in evaluating the model performance under various what-if scenarios.

Sampling Methods

 Click the Simulate button in the Solve Action group

 When the simulation finishes, you will see a message “Simulation finished successfully” in the

lower-left corner of the Excel window.

Running a Simulation

 You may specify whether you want output charts to automatically appear after a simulation is run

by clicking the Options button in the Analytic Solver Platform ribbon, and either checking or

unchecking the box Show charts after simulation in the Charts tab

 An easy way to view results for any uncertain function is to double-click an uncertain function cell.

Viewing and Analyzing Results

 Frequency distribution of cost difference

Example 12.6: Analyzing Simulation Results for the Outsourcing

Decision Model

Select other options: Percentiles, Chart Type, Chart Options, Axis Options,

and Markers

 Set Upper Cutoff = 0 to find the probability of a negative cost difference.

Example 12.6 Continued

 Moore Pharmaceuticals spreadsheet With

uncertain data:

1 What is the risk that the net present value

over the 5 years will not be positive?

2 What are the chances that the product will

show a cumulative net profit in the third year?

3 What cumulative profit in the fifth year are we

likely to realize with a probability of at least

0.90?

New Product Development Model

 Market size: normal with mean of 2,000,000 units and standard deviation of 400,000 units

 R&D costs: uniform between $600,000,000 and $800,000,000

 Clinical trial costs: lognormal with mean of $150,000,000 and standard deviation $30,000,000

 Annual market growth factor: triangular with minimum = 2%, maximum = 6%, and most likely =

3%

 Annual market share growth rate: triangular with minimum = 15%, maximum = 25%, and most

likely = 20%

Model Assumptions

◦ Cumulative net profit each year and net present value

Example 12.7: Setting Up the Simulation Model for Moore Pharmaceuticals

 Summary of output functions and uncertain variables

◦ Customize this by checking or unchecking the boxes in the Filters pane.

Simulation Results: Variables Chart

1 What is the risk that the NPV over the 5 years will not be positive?

Example 12.8: Risk Analysis for Moore Pharmaceuticals

2 What are the chances the product will show a cumulative net profit in the third year?

Example 12.8 Continued

3 What cumulative profit in the 5th year are we likely to realize with a probability of at least 0.90 (that is, the 10th percentile)?

Example 12.8 Continued

 Each time you run a simulation, you will obtain slightly different results.

 Confidence interval:

the standard normal z-value instead of the t-distribution in the confidence interval formula.

Confidence Interval for the Mean

 Moore Pharmaceuticals

 95% Confidence interval

Example 12.9: A Confidence Interval for the Mean Net Present Value

 A sensitivity chart allows you to determine the influence that each uncertain model input has individually on

an output variable based on its correlation with the output variable.

◦ Displays rankings of uncertain variables according to their impact on an output cell.

◦ It tells which uncertain variables influence output variables the most and which would benefit from better estimates.

◦ It tells which uncertain variables influence output variables the least and can be ignored or discarded altogether.

◦ By providing understanding of how the uncertain variables affect your model, it allows you to develop more realistic spreadsheet models and improve the accuracy of your results.

Sensitivity Chart

 If a simulation has multiple related forecasts, an overlay chart superimposes the frequency distributions from selected forecasts on one chart in order to compare differences and similarities that might not be apparent.

Overlay Charts

 Moore Pharmaceuticals

 Click the Charts button in the Analysis group

 Click Multiple Simulation Results (do not choose

Multiple Simulations!) and then choose Overlay.

 In the Reports dialog that appears, select the output

variable cells you wish to include in the chart and

move them to the right side of the dialog

Example 12.11: Creating an Overlay Chart

 Result for year 1 and year 5

cumulative profit

Example 12.11 Continued

 If a simulation has multiple output variables that are related to one another (such as over time), you can view the distributions of all output variables on a single chart, called

a trend chart.

the mean.

Trend Charts

 Moore Pharmaceuticals

 Click the Charts button in the Analysis group

 Click Multiple Simulation Results and then

choose Trend.

 In the Reports dialog that appears, select the

output variable cells you wish to include in the

chart and move them to the right side of the

dialog

Example 12.12: Creating a Trend Chart

 A box-whisker chart shows the

minimum, first quartile, median, third quartile, and maximum values in a data set graphically

 The first and third quartiles form a box

around the median, showing the middle

50 percent of the data, and the

whiskers extend to the minimum and maximum values

Box-Whisker Charts

 Analytic Solver Platform creates reports in the form of Excel worksheets that summarize a

simulation

 Click the Reports button in the Analysis group in the ribbon, and choose Simulation from the

options that appear

 The report summarizes basic statistical information about the model, simulation options,

uncertain variables, and output variables,

Simulation Reports

 Apply Monte Carlo simulation to forecast the profitability of different purchase quantities when the future demand is

uncertain

 Suppose that the store owner kept records for the past 20 years on the number of boxes sold

Newsvendor Model

 Historical candy sales average 44.05

 Using 44 for demand and purchase quantity,

the model predicts a profit of $264.00

 However, if we construct a data table to

evaluate the profit for each of the historical

values, the average profit is only $255.00

Example 12.13: Using Average Values in the Newsvendor Model

 The evaluation of a model output using the average value of the input is not necessarily equal to the average value of the outputs when evaluated with each of the input values.

◦ In the newsvendor example, the quantity sold is limited to the smaller of the demand and purchase quantity, so even when demand exceeds the purchase quantity, the profit is limited.

 Using average values in models can conceal risk.

The Flaw of Averages

 We can perform a Monte Carlo simulation by resampling from the historical sales distribution— that is, by selecting a value randomly from the historical data.

Monte Carlo Simulation Using Historical Data

 Generate candy sales by resampling from the 20 historical values.

 Enter the formula =PsiDisUniform(D2:D21) into cell B11.

 Set profit in B17 as an uncertain function.

Example 12.14: Simulating the Newsvendor Model Using Resampling

 Simulation results for purchase quantity = 44

Example 12.14 Continued

 Sampling from empirical data has some drawbacks

sampling error

 It is usually advisable to fit a distribution using the techniques described in Chapter 5 and use it for the uncertain variable.

Monte Carlo Simulation Using a Fitted Distribution

 Newsvendor Model with Historical Data

 The best-fitting distribution is a negative binomial distribution

Example 12.15: Using a Fitted Distribution for Monte Carlo Simulation

 After fitting the distribution, when you

attempt to close the dialog, Analytic Solver

Platform will ask if you wish to accept the

fitted distribution

first cell of the data (cell D2)

Example 12.15 Continued

 Results

Example 12.15 Continued

 Whenever the Simulate button is clicked, you will notice that the lightbulb in the icon turns bright If you change any

number in the model, Analytic Solver Platform will automatically run the simulation for that quantity; this makes it

easy to conduct what-if analyses

◦ Example: change the purchase quantity to 50; mean profit is less than if purchase quantity is 44

Interactive Simulation

 Historical demand data shown in columns D and

E

 Assume that each reservation has a constant

probability p = 0.04 of being canceled; therefore,

the number of cancellations (cell B14) can be

modeled using a binomial distribution with n =

number of reservations made and p =

probability of cancellation

Overbooking Model with Custom Demand

 Select cell B12 and then click on the Distributions button

in the ribbon and choose Discrete from the Custom

category.

 Edit the range for “values” and “weights” in the

Parameters section

◦ Values correspond to the range of demand in cells D2:D13,

and weights are the relative frequencies or probabilities in

 To model the number of cancellations in cell B14, choose the binomial distribution from the Discrete category in the

Distributions list The number of trials is the value in cell B13 and is referenced in the Parameters section.

 Or, use the function =PsiBinomial(B13, 0.04) in cell B14.

Example 12.16 Continued

 Frequency charts for number of

overbooked customers and net

revenue if 310 reservations are

accepted

 You can use Interactive

Simulation to quickly change the

number of reservations to find

the best solution

Overbooking Model Results

 Cash Budgeting is the process of projecting and summarizing a company’s cash

inflows and outflows expected during a planning horizon.

 Because of the inherent uncertainty in sales forecasts, Monte Carlo simulation is an

appropriate tool for modeling cash budgets.

Cash Budget Model