Illustrative Examples of Planning Under UncertaintyMS&E348Winter 2011/2012Professor Gerd InfangerWinter 2011/2012 MS&E348/Infanger 2Multi-period investmentStocksBondsCash(Example by Birge, Network formulation based on Mulvey)Winter 2011/2012 MS&E348/Infanger 3Returns processWinter 2011/2012 MS&E348/Infanger 4Defining a goalWinter 2011/2012 MS&E348/Infanger 5Deterministic problem formulationWinter 2011/2012 MS&E348/Infanger 6Deterministic model, spreadsheet representationWinter 2011/2012 MS&E348/Infanger 7Decision treet =1 t =2 t =3 t =4Stage 1 Stage 2 Stage 3Winter 2011/2012 MS&E348/Infanger 8Stochastic ProblemWinter 2011/2012 MS&E348/Infanger 9Stochastic model, spreadsheet representationWinter 2011/2012 MS&E348/Infanger 10Optimal expected utility = -7.6Optimal solution of the 3-period programWinter 2011/2012 MS&E348/Infanger 11Evaluation of the deterministic solutionExpected utility of the deterministic solution = -19.2Winter 2011/2012 MS&E348/Infanger 12The “wait and see” solutionInvest in each period wherever is the higher return!Perfect Information!Winter 2011/2012 MS&E348/Infanger 13Evaluation of the “wait and see” solutionExpected utility of the “wait and see” solution = 52.5Winter 2011/2012 MS&E348/Infanger 14Value of the stochastic solution and Expected value of perfect informationWinter 2011/2012 MS&E348/Infanger 15Telecommunications planningBCADEProblem: Place capacities on the above links to minimize expected unserved demand given a budget constraint.Demand forecasts are uncertain, some of the links may fail(Example by Sen, simplification of a much larger real-life model)Winter 2011/2012 MS&E348/Infanger 16DemandH: mean = 2.0L: mean = 1.5Winter 2011/2012 MS&E348/Infanger 17Capacity, cost and likelihood of failureBCADE2.5112.5 2.511Failure probability of links:BA, BE, BD, BC 0.3AE, ED, DC 0.0Budget Constraint:Winter 2011/2012 MS&E348/Infanger 18Model formulationWinter 2011/2012 MS&E348/Infanger 19Model formulation (cont.)Winter 2011/2012 MS&E348/Infanger 20Winter 2011/2012 MS&E348/Infanger 21Deterministic solutionBCADE553.5 6.5Tree (not reliable in case of failures)Winter 2011/2012 MS&E348/Infanger 22Stochastic solutionBCADE1.12.6 2.71.1 0.853.5 3.5Full network (reliable in case of failures)Note there are 14 random parametersThe total number of scenarios is 245446 = 41 Million!Winter 2011/2012 MS&E348/Infanger 231 2 3 4 5A BLogistics planningwarehousesshopsshippinginventorydemandsWinter 2011/2012 MS&E348/Infanger 241 2 3 4 5CA B20020200 20030 501.01.251.080 20Shipping Cost12345A 2.00 1.00B 1.00 1.50 2.00C 2.20 1.10 1.10 1.65 2.20Cost of unserved demand: 2000.110 0.220 0.430 0.240 0.160 0.170 0.280 0.490 0.2100 0.100.110 0.220 0.430 0.240 0.110 0.120 0.230 0.440 0.250 0.130 0.140 0.250 0.460 0.270 0.1Winter 2011/2012 MS&E348/Infanger 25Logistics Planning Example: SolutionsDeterministic StochasticOpt. Objective 485.0 543.5Exp. Cost 726.9 543.5A 100 100B 100 110C030Winter 2011/2012 MS&E348/Infanger 26Safety stock considerations• Assumptions– Lead, order cycle and forecast time within same period– Normally distributed demand– Forecast equal to the mean demand • Safety Stock– Safety stock = (standard deviation)*(service factor)Stores Warehouse Warehouse1, 2, 3, 4, 5 A BMean 20,80,20,30,50 100 100Stdev 11 15.5 19Winter 2011/2012 MS&E348/Infanger 27Safety stock calculationService Service Warehouse WarehouseFactor Level A B15.5 19.01.64 95.0% 125.4 131.21.75 96.0% 127.1 133.31.88 97.0% 129.1 135.72.05 98.0% 131.8 139.02.33 99.0% 136.1 144.32.58 99.5% 140.0 149.02.65 99.6% 141.1 150.42.75 99.7% 142.6 152.32.88 99.8% 144.6 154.73.09 99.9% 147.9 158.7Winter 2011/2012 MS&E348/Infanger 28Comparison of the different solutionsDeterministic Det w/ Safety StochasticOpt. Objective 485.0 580.0 543.5Exp. Cost 726.9 580.0 543.5Cost (%) 133.7% 106.7% 100.0%Pred. Service Level 99.7%Act. Service Level 100.0% 99.7%A 100 142.6 100B 100 152.3
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