6-Simulation.pdf

DSC1007 Lecture 6Simulation

War Simulation

Graf Helmuth von Moltke

• Regarded as the grandfather of modern military simulation.

• Although not the inventor of Kriegspiel, he was greatly impressed by it as a young officer

• As Chief of Staff of the Prussian Army promoted its use as a training aid.

• Kriegspiel is sometimes credited with the Prussian victory in the Franco-Prussian War.

What is Simulation?

• A simulation model is a computer model that imitates a real-life situation.

• The fundamental advantage of a simulation model is that it provides an entire distribution of results, not simply a single bottom-line result.

• Each different set of values for the uncertain quantities can be considered a scenario. – Simulation models allow the company to generate many

scenarios, each leading to a particular outcome.

Introduction Continued

• Simulation models are also useful for determining how sensitive a system is to changes in operating conditions.

• Another benefit of a computer simulation is that it enables managers to answer what-if question without actually changing (or building) a physical system.

• Simulations are used in a variety of business settings.

SimulationinBusinessSimulation models are widely used in many management settings:

•Modeling of manufacturing operations•Modeling of service operations where queues form•Modeling of investment alternatives•Analyzing and pricing of sophisticated financial instruments

Aircraft Boarding Strategy

How to board all passengers in the shortest possible time?

SimulationModeling

ProbabilisticSimulation

MonteCarlosimulationisatechniquethatallowspeopletoaccountforuncertainty inquantitativeanalysisanddecisionmaking.

SimulationModeling

WhousesMonteCarlosimulation?ManycompaniesuseMonteCarlosimulationasanimportantpartoftheirdecision‐makingprocess.

• GM, ProctorandGamble,Pfizer,Bristol‐MyersSquibb,andEliLilly:toestimateboththeaveragereturnandtheriskfactorofnewproducts.

• EliLilly : todeterminetheoptimalplantcapacityforeachdrug.

• ProctorandGamble: tomodelandoptimallyhedgeforexrisk.

• Sears :todeterminehowmanyunitsofeachproductlineshouldbeorderedfromsuppliers.

• Oilanddrugcompanies:tovalue"realoptions,"suchasthevalueofanoptiontoexpand,contract,orpostponeaproject.

Simulating a Random Variable

• The fundamental technique in simulation modeling is to simulate a random variable following certain probability distribution.

UniformRandomNumbers

Uniformrandomnumbersrefertoasequenceofnumbersthatareindependent andobeytheuniformdistributionU[0,1]

EXCELrandomnumbergenerator:RAND()

Properties of RAND():•Uniform property: All numbers between 0 and 1 have the same chance of occurring.•Independence property: Different random numbers are probabilistically independent. A number generated previously has no effect on the values of the following random numbers.

UniformDistributionU[a,b]

Q : HowtogenerateU[a,b]randomnumbers?

IfX U[0,1]

thenY =a +(ba)X U[a,b]

GeneratingU[0,1]randomnumbersiseasy– useRAND()

A:

GeneratingU[a,b]randomnumbers– usea +(ba)RAND()

OtherDistributions

GeneratingU[0,1]randomnumbers – RAND()

GeneratingU[a,b]randomnumbers – a +(ba)RAND()

Next:howtogeneraterandomnumbersthatobey– adiscrete probabilitydistribution

– acontinuous probabilitydistribution

Discrete Distribution

• Example: Let X be a random variable representing race of a randomly selected Singaporean.

X ProbabilityChinese 74.2%Malay 13.3%Indian 9.2%Others 3.3%

* Data from Department of Statistics, Singapore

RouletteWheel

Using RAND() to Generate X[0,1]uniformrandomnumber assigned X

0.00―0.742 Chinese0.742―0.875 Malay0.875―0.967 Indian0.967―1.00 Others

Trial RandomNumber X1 .6622 .9233 .3004 .8125 .999

Chinese

IndianChineseMalayOthers

andsoon...

GentleLentilCaseLOOKUPfunction– generatingvaluesofX

• Mostsimulationsoftwarepackages(e.g.,CrystalBall)cangeneraterandomnumbersfromdiscrete andavarietyofcontinuous distributions,suchastheNormal distribution,theuniform distribution,etc.

• Theuserneedtospecifythetypeofdistributionandtheparameters( and fortheNormal,a andb fortheuniform)

• However,itisworthwhiletopointouthowthecomputeraccomplishesthistask.

• WillfocusonusingEXCELformulatogeneraterandomnumbers

GeneratingRandomNumberswithagivenContinuous ProbabilityDistribution

GeneratingRandomNumberswithagivenContinuous ProbabilityDistribution

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PDFf(y) oftheRandomVariable

Example

GeneratingRandomNumberswithagivenContinuous ProbabilityDistribution

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Example

StepstogenerateaRNthatfollowsagivenCDFF(y)

GeneratingRandomNumberswithagivenContinuous ProbabilityDistribution

1. Usearandomnumbergeneratortogenerateanumberu thatobeysauniform distributionbetween0.0and1.0.

2. Placethenumberu ontheverticalaxisofthegraphoftheCDFF(y)ofthegivendistribution.Thenfindthepointy onthehorizontalaxiswhoseCDFvalueF(y) isequaltou.

3. Thenumbery generatedthiswayhasthedesiredCDFF(y).

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F(y)

Supposethe[0,1]uniformRNwegethappenstobeu =0.826

y = 6.851

StepstogenerateaRNthatfollowsagivenCDFF(y)

y

F(y) u = 0.826

F(y) =u

StepstogenerateaRNthatfollowsagivenCDFF(y)

1. Usearandomnumbergeneratortogenerateanumberu thatobeysauniform distributionbetween0.0and1.0.

2. Placethenumberu ontheverticalaxisofthegraphoftheCDFF(y)ofthegivendistribution.Thenfindthepointy onthehorizontalaxiswhoseCDFvalueF(y) isequaltou.

3. Thenumbery generatedthiswayhasthedesiredCDFF(y).

GeneratingRandomNumberswithagivenContinuous ProbabilityDistribution

Example: SupposewewanttogenerateRNsthatfollowtheNormaldistributionN(,)

y =NORMINV(u,,)F(y) =u

Case – Ordering Calendars at Walton Bookstore

• In August, Walton Bookstore must decide how many of next year’s nature calendars to order.

• Each calendar costs the bookstore $7.50 and sells for $10. After January 1, all unsold calendars will be returned to the publisher for a refund of $2.50 per calendar.

• Walton believes that the number of calendars it can sell by January 1 follows some probability distribution with mean 200.

• How many calendars should Walton order in order to maximize the expected profit?

Decision by Common SenseWalton's bookstore - deterministic model

Cost dataUnit cost $7.50Unit price $10.00Unit refund $2.50

Uncertain quantityDemand (average shown) 200

Decision variableOrder quantity 200

Profit modelDemand Revenue Cost Refund Profit

200 $2,000.00 $1,500.00 $0.00 $500.00

Is it correct?

Simulation Model

Simulation with Excel

Histogram

• Step 1. Initiate “Analysis ToolPak” in Excel.

Histogram

• Step 2. Define bins in Excel worksheet.

Histogram

• Step 3. Launch Analysis ToolPak and select “Histogram”.

Histogram

• Step 4. Define inputs to create the histogram.

Histogram

• Step 5. Create histogram chart with the result.

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Frequency

Frequency

Find Optimal Order with “Goal-Seek”

Configure “Goal-Seek”

What-if with “Data Table”Step 1. Build a list of possible order quantities

What-if with “Data Table”Step 2. Add formula of “Expected Profit” to the top of the table

What-if with “Data Table”Step 3. Highlight the table and choose “Data Table” button

What-if with “Data Table”Step 4. Specify B13 as the cell to be replaced by the list of options.

Result

Press F9 if the result doesn’t show.

Simulation with @Risk

Task

• Please try to do the problem by yourself for the cases where demand follows different probability distributions.