Evans Analytics2e Ppt 12

7/24/2019 Evans Analytics2e Ppt 12

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Chapter 12

Monte Carlo Simulationand Risk Analysis


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Many situations dictate that randomness be explicitlyincorporated into our models. This is usually done byspecifying probability distributions for the appropriateuncontrollable inputs. Such models are called stochastic, or probabilistic.

Riskis the likelihood of an undesirable outcome. It canbe assessed by evaluating the probability that theoutcome will occur along with the severity of theoutcome.

Risk analysisseeks to examine the impact of uncertaininputs on various outputs.

Model Uncertainty and Risk Analysis


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Production volume is uncertain assume normal with a mean of!""" and standard deviation of !"". #eplace cell $!% with '()*+)'#M.I)+#-)*+, !""", !"", true, "

/henever F9key or Formula0 Calculation0 Calculate Now isclicked, the value of demand will change randomly

Example 12.1: ncorporatin! Uncertainty in the

Outsourcing Decision Model


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Monte Carlo simulationis the process of generatingrandom values for uncertain inputs in a model,computing the output variables of interest, and repeatingthis process for many trials to understand the distribution

of the output results. Monte 1arlo simulation can easily be accomplished on a

spreadsheet using a data table.

Monte Carlo Simulation


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2xcel file Outsourcing Decision Monte Carlo Simulation Model. 2nter the trial number +! to %" in column *. #eference the cells associated with model outputs in row 34 +23, 53,

63+&$!%, &$!7, &$%" Select the range for the data table +*346%3 In the Data Table dialog, enter any blank cell for the Column Input

Cell.

Example 12.2: Usin! "ata #ables $or Monte

Carlo Spreadsheet Simulation


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!. *evelop a spreadsheet model.%. *etermine probability distributions for uncertain input variables.

3. Identify output variables you want to predict.

8. 1hoose the number of trials for the simulation.9. #un the simulation.

:. Interpret the results.

Monte Carlo Simulation Usin!Analytic Solver Platform


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5or many decision models, empirical data may be available,either in historical records or collected through special efforts.

In other situations, historical data are not available, and wecan draw upon the properties of common probability

distributions to help choose a representative distribution thathas the shape that would most reasonably represent theanalyst;s understanding about the uncertain variable.

(niform or triangular distributions are often used in theabsence of data.

InAnalytic Solver lat!orm, use custom 2xcel functions or theDistributionsbutton in the ribbon.

"e$inin! Uncertain Model nputs


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Outsourcing Decision Model *emand +production volume is

normally distributed with a mean of!""" and standard deviation of !""units.

&Psi)ormal+!""", !"" in cell $!% (se #'()* function to ensure that the

result is a whole number4'()*+Psi)ormal+!""",!"",".

(nit cost has a triangular distributionwith a minimum of


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5or demand, selectcell $!%.

1lick theDistributionsbutton

in theAnalytic Solverlat!ormribbon andselect the normaldistribution from the

Commoncategory.

Example 12.(: Usin! the "istributions )utton in

Analytic Solver Platform


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5or unit cost, select cell $!" and select the triangular distribution 1hange the parameters in the dialog

Example 12.( Continued


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To define a cell you wish to predict and create adistribution of output values from your model, first selectit, and then click on the "esultsbutton in the SimulationModelgroup in theAnalytic Solver lat!orm ribbon.1hoose the Outputoption and then In Cell. Analytic Solver lat!orm calls output cells uncertain $unction

cells#

(ncertain function cells must be numeric.

Analytic Solver lat!orm adds the function Psi'utput+

to uncertain function cell formulas. >ou may also add ?Psi'utput+ to any output cells manually

"e$inin! *utput Cells


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Select cell $!7 -fter defining the cell as

an uncertain function,the formula should read4

&$!: @ $!= ? Psi'utput+

Example 12.+: Usin! the Results

)utton inAnalytic Solver Platform


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5irst click on the Optionsbutton inthe Options group in theAnalyticSolver lat!orm ribbon. Thisdisplays a dialog in which you canspecify the number of trials and

other options to run the simulation+make sure the Simulationtab isselected.

Trials per Simulation allows you tochoose the number of times that

the simulation will generaterandom values for the uncertaincells in the model and recalculatethe entire spreadsheet.

Runnin! a Simulation


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Analytic Solver lat!ormgenerates a stream of randomnumbers from which the values ofthe uncertain inputs are selectedfrom their probability distributions.

2very time you run the model, you willget slightly different results because ofsampling error.

Setting a value for Sim# "andomSeedwill guarantee that the sameseAuence of random numbers willbe used for generating therandom values for the uncertaininputs every time the simulation isrun.

Random ,umber Seed


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Monte Carlo samplin! selectsrandom variates independentlyover the entire range of possiblevalues of the distribution.

/ith -atin ypercube samplin!,

the uncertain variable;s probabilitydistribution is divided intointervals of eAual probability andgenerates a value randomlywithin each interval.

Monte 1arlo sampling is morerepresentative of reality and should beused if you are interested in evaluatingthe model performance under variouswhatBif scenarios.

Samplin! Methods


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1lick the Simulatebutton in the Solve Action group. /hen the simulation finishes, you will see a message

CSimulation finished successfullyD in the lowerBleft cornerof the 2xcel window.

Runnin! a Simulation


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>ou may specify whether you want output charts toautomatically appear after a simulation is run by clickingthe Optionsbutton in theAnalytic Solver lat!orm ribbon,and either checking or unchecking the box S$ow c$arts

a!ter simulationin the C$artstab.

-n easy way to view results for any uncertain function isto doubleBclick an uncertain function cell.

/ie0in! and Analyin! Results


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5reAuency distribution of cost difference

Example 12.: Analyin! Simulation Results

$or the Outsourcing Decision Model

Select otheroptions4Percentiles,

1hart Type,1hart 'ptions,-xis 'ptions,and Markers.


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Set %pper Cuto!! & " to find the probability of a negative

cost difference.

Example 12. Continued


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Moore $armaceuticalsspreadsheet. /ithuncertain data4

!. /hat is the risk that thenet present value over the 9

years will not be positiveE%. /hat are the chancesthat the product will show acumulative net profit in thethird yearE

3. /hat cumulative profit inthe fifth year are we likely torealiFe with a probability ofat least ".7"E

,e0 &roduct "e3elopment Model


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Market siFe4 normal with mean of %,""",""" units andstandard deviation of 8"",""" units

#G* costs4 uniform between

-nnual market share growth rate4 triangular withminimum & !9, maximum & %9, and most likely &%"

Model Assumptions


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Market siFe4 &Psi)ormal+%"""""", 8"""""

#G* costs4

&Psi(niform+:"""""""",H""""""""

1linical trial costs4

&PsiJognormal+!9""""""", 3"""""""

-nnual market growth factor4

&PsiTriangular+%, 3, :

-nnual market share growth rate4

&PsiTriangular+!9, %", %9

(ncertain functions4

1umulative net profit each year and net present value

Example 12.4: Settin! Up the Simulation

Model $or Moore &harmaceuticals


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Summary of output functions and uncertain variables 1ustomiFe this by checking or unchecking the boxes in the Filters

pane.

Simulation Results: /ariables Chart


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!. /hat is the risk that the )P over the 9 years will notbe positiveE

Example 12.5: Risk Analysis $or Moore

Pharmaceuticals


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%. /hat are the chances the product will show acumulative net profit in the third yearE

Example 12.5 Continued


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3. /hat cumulative profit in the 9thyear are we likely torealiFe with a probability of at least ".7" +that is, the !"thpercentileE



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2ach time you run a simulation, you will obtainslightly different results.

1onfidence interval4

$ecause a Monte 1arlo simulation will generally have avery large number of trials, we may use the standardnormal &Bvalue instead of the tBdistribution in theconfidence interval formula.

Con$idence nter3al $or the Mean


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Moore $armaceuticals 79 1onfidence interval

Example 12.6: A Con$idence nter3al $or the

Mean ,et &resent /alue


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- sensiti3ity chart allows you to determine the influence thateach uncertain model input has individually on an outputvariable based on its correlation with the output variable. *isplays rankings of uncertain variables according to their impact on an

output cell.

$enefits4 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 andcan be ignored or discarded altogether.

$y providing understanding of how the uncertain variables affect your

model, it allows you to develop more realistic spreadsheet models andimprove the accuracy of your results.

1lick on the Sensitivitytab in the Simulation "esults window.

Sensiti3ity Chart


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Moore $armaceuticals

Example 12.17: nterpretin! the Sensiti3ity Chart $or

,&/

Market siFe4#G* cost41linical trial cost4


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If a simulation has multiple related forecasts, ano3erlay chart superimposes the freAuencydistributions from selected forecasts on one chartin order to compare differences and similarities

that might not be apparent.

*3erlay Charts


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Moore $armaceuticals

1lick the C$artsbutton in theAnalysisgroup

1lick Multiple Simulation"esults +do not chooseMultiple SimulationsK andthen choose Overlay.

In the "eportsdialog thatappears, select the outputvariable cells you wish toinclude in the chart and move

them to the right side of thedialog

Example 12.11: Creatin! an *3erlay

Chart


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#esult for year! and year 9cumulativeprofit



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If a simulation has multiple output variables thatare related to one another +such as over time,you can view the distributions of all outputvariables on a single chart, called a trend chart. - trend chart shows the mean values as well as =9 and

7" bands +probability intervals around the mean.

#rend Charts


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Moore $armaceuticals 1lick the C$artsbutton in

theAnalysisgroup 1lick Multiple Simulation

"esults and then choose

Trend. In the "eportsdialog that

appears, select the outputvariable cells you wish toinclude in the chart and

move them to the rightside of the dialog.

Example 12.12: Creatin! a #rend Chart


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- box80hisker chartshows the minimum,first Auartile, median,third Auartile, andmaximum values in a

data set graphically. The first and third

Auartiles form a boxaround the median,showing the middle 9"

percent of the data,and the whiskersextend to the minimumand maximum values.

)ox89hisker Charts


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Analytic Solver lat!orm creates reports in the form of2xcel worksheets that summariFe a simulation.

1lick the "eportsbutton in theAnalysisgroup in theribbon, and choose Simulationfrom the options that

appear. The report summariFes basic statistical information

about the model, simulation options, uncertain variables,and output variables,

Simulation Reports


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-pply Monte 1arlo simulation to forecast the profitability of different

purchase Auantities when the future demand is uncertain. Suppose that the store owner kept records for the past %" years on

the number of boxes sold

,e0s3endor Model


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Listorical candy salesaverage 88."9.

(sing 88 for demand andpurchase Auantity, themodel predicts a profit of


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The evaluation of a model output using theaverage value of the input is not necessarily eAualto the average value of the outputs whenevaluated with each of the input values. In the newsvendor example, the Auantity sold is limited to

the smaller of the demand and purchase Auantity, soeven when demand exceeds the purchase Auantity, theprofit is limited.

(sing average values in models can conceal risk.

#he 'la0 o$ A3era!es


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/e can perform a Monte 1arlo simulation byresampling from the historical sales distributionthat is, by selecting a value randomly from the

historical data.

Monte Carlo Simulation Usin!

istorical "ata


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6enerate candy sales by resampling from the %" historical values. 2nter the formula &Psi*is(niform+*%4*%! into cell $!!. Set profit in $!= as an uncertain function.

Example 12.1(: Simulatin! the

,e0s3endor Model Usin! Resamplin!


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Simulation results for purchase Auantity & 88

Example 12.1( Continued


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Sampling from empirical data has somedrawbacks. The empirical data may not adeAuately represent the

true underlying population because of sampling error.

(sing an empirical distribution precludes samplingvalues outside the range of the actual data.

It is usually advisable to fit a distribution using thetechniAues described in 1hapter 9 and use it for

the uncertain variable.

Monte Carlo Simulation Usin! a 'itted

"istribution


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Newsvendor Model wit$ 'istorical Data The bestBfitting distribution is a negative binomial distribution.

Example 12.1+: Usin! a 'itted

"istribution $or Monte Carlo Simulation


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-fter fitting thedistribution, when youattempt to close thedialog,Analytic Solverlat!orm will ask if you

wish to accept the fitteddistribution.

1lick (es, and place thePsi function in the first

cell of the data +cell *%. Then reference cell *% in

cell $!!.

Example 12.1+ Continued


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#esults

Example 12.1+ Continued


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/henever the Simulatebutton is clicked, you will notice that the

lightbulb in the icon turns bright. If you change any number in themodel,Analytic Solver lat!ormwill automatically run the simulationfor that Auantity this makes it easy to conduct whatBif analyses.

2xample4 change the purchase Auantity to 9" mean profit is less than ifpurchase Auantity is 88

nteracti3e Simulation


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Listorical demand data

shown in columns * and 2 -ssume that each

reservation has a constantprobabilityp& "."8 of beingcanceled therefore, the

number of cancellations+cell $!8 can be modeledusing a binomial distributionwith n& number ofreservations made andp&

probability of cancellation.

*3erbookin! Model 0ith Custom "emand


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Select cell $!% and then click onthe Distributionsbutton in theribbon and choose Discretefromthe Customcategory.

2dit the range for CvaluesD andCweightsD in the arameters

section alues correspond to the range of

demand in cells *%4*!3, andweights are the relativefreAuencies or probabilities incells 2%42!3.

'r, use the function in cell $!%4&Psi*iscrete+


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To model the number of cancellations in cell $!8, choose the

binomial distribution from the Discretecategory in the Distributionslist. The number of trials is the value in cell $!3 and is referenced inthe arameterssection.

'r, use the function &Psi$inomial+$!3, "."8 in cell $!8.



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5reAuency chartsfor number ofoverbookedcustomers and netrevenue if 3!"reservations are

accepted. >ou can use

Interactive

Simulation toAuickly change the

number ofreservations to findthe best solution.

*3erbookin! Model Results


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Cash )ud!etin!is the process of proNectingand summariFing a company;s cash inflows andoutflows expected during a planning horiFon.

Most cash budgets are based on sales

forecasts.$ecause of the inherent uncertainty in sales forecasts,

Monte 1arlo simulation is an appropriate tool formodeling cash budgets.

Cash )ud!et Model


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Lighlighted cells are uncertain variables +blue anduncertain functions +green

Example Cash )ud!et Model


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Sales are normally distributed with a standard deviationof !" of the mean.

Sales in adNacent months are correlated with oneanother, with a correlation coefficient of ".:.

'n average, %" of sales are collected in the month ofsale, 9", in the month following the sale, and 3", inthe second month following the sale. These figures are uncertain, so a uniform distribution is used to

model the first two values +!9 to %" and 8" to 9",

respectively, with the assumption that all remaining revenues arecollected in the second month following the sale.

Cash )ud!et Model Assumptions

E l 12 14 Si l ti th C h


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*efine distributions for all uncertain variables. 2xample for -pril sales +cell 294 &Psi)ormal+:""""",:""""

1ell $=4 &Psi(niform+!9, %"

1ell $H4 &Psi(niform+8", 9".

*efine the available balances in row %9 as output variables

Example 12.14: Simulatin! the Cash

Budget Model 9ithout Correlations


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Trend chart



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Jikelihood of not meeting minimum balance in -pril



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(nless you specify otherwise, Monte 1arlosimulation assumes that each of the uncertainvariables is independent of all the others.

Analytic Solver lat!orm allows you to specify

correlations between uncertain variables.

Correlatin! Uncertain /ariables

E l 12 15 ti


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Cas$ )udget MonteCarlo Simulation Model

1lick the Correlationsbutton in the SimulationModel group in the

ribbon. In this example we are

only correlating thevariables in the range294O9. Move these to theright pane.

Example 12.15: ncorporatin!

Correlations in Analytic Sol3er &lat$orm

E l 12 15 C ti d


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Initial correlation matrix The numerical values show

the correlations +initially setto Fero The green distributions are

those used in the uncertaincells.

The blue scatterplots showvisual representations of thecorrelations between thevariables.

#eplace the Feros by thecorrelations you want in themodel.



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Analytic Solver lat!orm will check that the correlationsare mathematically consistent if not, it will ask you toadNust the correlations. -lways choose (es. 1lick the%pdate Matri* button and thenAccept %pdate.

-dNusted correlation matrix4

Analytic Solver lat!orm then adds these to thesimulation model.

Evans Analytics2e Ppt 12

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