Jérémie GallienClass 2 Outline1.Monte-Carlo:• Framework/Definition• Algorithm• Examples2.Random Number GenerationJérémie GallienMonte-Carlo FrameworkEstimate q = E[ h(X) ]where X = {X1,…,Xm} is a random vector in Rm,h(.) is a function Rm-> R,and E[ | h(X) | ] < ¥Jérémie Gallien1.Generate n samples of X: X1, … , Xn2.Compute h(X1), h(X2), …, h(Xn)3.Estimate q = E[ h(X) ] withq = [h(X1)+h(X2)+…+h(Xn)]/nMonte-Carlo Algorithm•Why is q a good estimator?Jérémie GallienA Sales Incentive PlanA company has a sales incentive plan with the following structure: In the months when sales are lower than $10,000, a salesrepget a base salary of $2000. If the monthly sales achieved are between $10,000 and $15,000, then he/she is paid 20% of the sales. Finally, if the monthly sales figure is larger than $15,000, the salesrepget 30% of the sales.Assuming the monthly sales achieved by a salesman follows an exponential distribution with mean $15,000, what is the mean and standard deviation of the salesman’s monthly salary under this compensation plan?Jérémie GallienProduct Reliability• Consider a product reliability model given by:• Component life lengths:XA~ exp(5.1); XB~ N(4.5,1); XC~ exp(6.5)XD~ exp(5.4); XE~ exp(6.4); XF~ N(5.5,0.8)• What is the probability that the product will function for at least 3 years (warranty period)?ABCEDFJérémie GallienDreamCast Launch ProjectTask Description Duration (weeks) PrecedenceAMarket Study14BExternal Developer Focus Group10CFeature Selection4A,BDHardware Engineering25CEOperating System Devlpt.16D,BFAdvertising Campaign20D,M,NGSupplier Selection & Negotiation20DHComponent Inventory Buildup45GIAssembly Facility Setup18DJFinished Good Inventory Buildup7I,HKLibrary & Programmer Toolkit Devlpt.12ELExternal Development Support Setup5KMInternal Game Devlpt.30KNExternal Game Devlpt.32L,OOPlatform Promotion6KPPublisher Selection & Negotiation5MQWebsite Setup20MRRelease Promotion Material Design3D,M,NSDistribution Channels Devlpt. & Negotiation9RTCarrier Selection & Negotiation4SULaunch Event Organization & PR5RVHardware & Software Shipment1J,T,P,NWLaunch!0V,U,Q,FJérémie GallienB10C4E16D25G20I18H45J7K12M30P5N32L5O6R3S9T4U5F20Q20V1W0A14Ext DevFocus GrpMktStudyFeaturesHW EngO/S DevAdvert PlanSupp Sel & NegCompsInvSetup FactoryFG Inv BuildSW Dev KitDev SuppIntDevExt DevPlatformMktgPublisherSel& NegSetupWebsiteReleasePromoDistr NegCarrier NegPlan LaunchEventShip HW & SWGO!Jérémie GallienCritical Path Algorithm1.A task Early Start Date is the largest of all Early Finish Dates of its predecessors2.A task Late Finish Date is the earliest of all Late Start Dates of its successorsA14D3C5B10ES, EFLS, LFA142 , 16LS, LF5 , 10LS, LF5 , 15LS, LFA14D3C5B10ES, EF20, 34ES, EF14, 19ES, EF15, 25LS, LFES, EFJérémie GallienB10C4E16D25G20I18H45J7K12M30P5N32L5O6R3S9T4U5F20Q20V1W0A140, 104,14ES, EFLS, LF0, 140,1414, 1814,1843, 5943, 5918, 4318, 4359, 7159, 7143, 61103,12143, 6356,76108, 115121,12863, 10876, 12171, 7771, 7771, 10179, 10971, 7672, 77101, 121109,129101, 106123,12877, 10977, 109109, 129109, 129108, 111112,115111, 116124,129124, 125128, 129111, 120115,124120, 124124,128129, 129129, 129AdJérémie Gallien71, 7672, 77120, 124124,128111, 120115,124111, 116124,129108, 111112,115101, 106123,12871, 10179, 109124, 125128, 129101, 121109,12943, 61103,12143, 6356,76108, 115121,12863, 10876, 1210, 104,14B10C4E16D25G20I18H45J7K12M30P5N32L5O6R3S9T4U5F20Q20V1W0A14ES, EFLS, LF0, 140,1414, 1814,1843, 5943, 5918, 4318, 4359, 7159, 7171, 7771, 7777, 10977, 109109, 129109, 129129, 129129, 129AdJérémie GallienTaskDescriptionDuration (weeks)PrecedenceLSLFESEFSlackAMarket Study14-129-115-129-1150BExternal Developer Focus Group10-125-115-129-1194CFeature Selection4A,B-115-111-115-1110DHardware Engineering25C-111-86-111-860EOperating System Devlpt.16D,B-86-70-86-700FAdvertising Campaign20D,M,N-200-2000GSupplier Selection & Negotiation20D-73-53-86-6613HComponent Engineering & Inventory Buildup45G-53-8-66-2113IAssembly Facility Setup18D-26-8-86-6860JFinished Good Inventory Buildup7I,H-8-1-21-1413KLibrary & Programmer Toolkit Devlpt.12E-70-58-70-580LExternal Development Support Setup5K-57-52-58-531MInternal Game Devlpt.30K-50-20-58-288NExternal Game Devlpt.32L,O-52-20-52-200OPlatform Promotion6K-58-52-58-520PPublisher Selection & Negotiation5M-6-1-28-2322QWebsite Setup20M-200-28-88RRelease Promotion Material Design3D,M,N-17-14-20-173SDistribution Channels Devlpt. & Negotiation9R-14-5-17-83TCarrier Selection & Negotiation4S-5-1-8-43ULaunch Event Organization & PR5R-50-17-1212VHardware & Software Shipment1J,T,P,N-10-4-33WLaunch!0V,U,Q,F00000Critical Path SolutionJérémie GallienTaskDescriptionDuration (weeks)PrecedenceLSLFESEFSlackAMarket Study14-129-115-129-1150BExternal Developer Focus Group10-125-115-129-1194CFeature Selection4A,B-115-111-115-1110DHardware Engineering25C-111-86-111-860EOperating System Devlpt.16 D,B-86-70-86-700FAdvertising Campaign20D,M,N-200-2000GSupplier Selection & Negotiation20D-73-53-86-6613HComponent Engineering & Inventory Buildup45G-53-8-66-2113IAssembly Facility Setup18D-26-8-86-6860JFinished Good Inventory Buildup7I,H-8-1-21-1413KLibrary & Programmer Toolkit Devlpt.12E-70-58-70-580LExternal Development Support Setup5K-57-52-58-531MInternal Game Devlpt.30 K-50-20-58-288NExternal Game Devlpt.32 L,O-52-20-52-200OPlatform Promotion6K-58-52-58-520PPublisher Selection & Negotiation5M-6-1-28-2322QWebsite Setup20M-200-28-88RRelease Promotion Material Design3D,M,N-17-14-20-173SDistribution Channels Devlpt. & Negotiation9R-14-5-17-83TCarrier Selection & Negotiation4S-5-1-8-43ULaunch Event Organization & PR5R-50-17-1212VHardware & Software Shipment1J,T,P,N-10-4-33WLaunch!0V,U,Q,F00000Stochastic Task DurationsDuration of development tasks E, M and N now follow exponential distributions with means 16, 30 and 32 respectively.What is the probability that the total project duration will be more than 135 weeks?Jérémie GallienStochastic ProgrammingX1~N(1500,350)Stochastic Demand X2~N(2000,600)X3~N(1500,450) Plants (i)DistributionCenters (j)X1X2X3Cap1Cap2CapacityDemandTransportationUnit CostsCijJérémie GallienRandom Number Generation• The key in a Monte-Carlo simulation is to generate sample values drawn from a known probability distribution. How is this done?Generate Sample Value from Uniform[0,1]Discrete Distribution• Inverse transform method• Rejection methodContinuous Distribution• Inverse transform method• Rejection method•