2.3报童问题模型

报童问题模型1、报童问题的提出2、报童问题所属范畴3、报童模型的建立与求解4、报童模型的推广与应用1、报童问题的提出在日常生活中，经常会碰到一些季节性强、更新快、不易保存等特点的物品，如海产、山货、时装、生鲜食品和报纸等，当商店购进这些商品时，买的数量越多，价格越便宜获利越大。但买得太多也可能卖不出去，需要削价处理，人力物力都受损；如果进货太少，又可能发生缺货现象，失去销售机会而减少利润。这就产生一个问题：订货量过多，出现过剩，会造成损失；订货量少，又可能会失去销售机会，影响利润，那么应该如何确定订货策略呢？将这一现象具体到报童销售报纸上，就引发了报童问题：报童问题：报童每天需订购多少份报纸？问题报童售报：(零售价)a(购进价)b(退回价)c售出一份赚a-b；退回一份赔b-c每天购进多少份使收入最大？分析购进太多卖不完退回赔钱购进太少不够销售赚钱少应根据需求确定购进量每天需求量是随机的优化问题的目标函数应是长期的日平均收入每天收入是随机的存在一个合适的购进量等于每天收入的期望2、报童问题所属范畴单周期随机型存贮模型这种单周期购入—售出（报纸、日历、杂志，各种季节性货物、时装），并且超出该购入—售出周期商品就会严重贬值的存贮问题，存贮论中统称为卖报童问题。这类问题的库存控制策略是以利润期望最大为目标，确定一次购入的经济订货批量。3、模型的建立与求解建模•设每天购进n份，日平均收入为G(n)调查需求量的随机规律——每天需求量为r的概率f(r),r=0,1,2…准备nrnr求n使G(n)最大•已知售出一份赚a-b；退回一份赔b-crnr退回售出,))((,)(rncbrba赔赚nban)(,赚售出nrnrrnfbarfrncbrbanG01)()()()])(()[()(nndrrnpbadrrprncbrbanG0)()()()])(()[()(nnnndrrpbadrrpcbdrrpbannpbadrrpcbnnpbadndG00)()()()()()()()()()()()(求解将r视为连续变量)()()(概率密度rprf0dndGcbbadrrpdrrpnn)()(0cbbadrrpdrrpnn)()(0结果解释nnPdrrpPdrrp201)(,)(cbbaPP21取n使P1~卖不完的概率，a-b~售出一份赚的钱P2~卖超的概率，b-c~退回一份赔的钱ncbnba)(,)(0nrp(r)P1P24、报童问题的推广与应用在科学的管理方法和手段在管理实践中运用越来越多的今天，管理者同样需要考虑，怎样改进粗放的管理模式，才能提高企业的管理水平，从而提高企业的效益。在管理实践中，我们会发现，与报童问题类似的问题非常多，这样我们就可以将报童问题的研究方法运用到实践中，通过科学的调查、计算，把过去经验的管理方法，上升到科学的管理方法。报童问题的推广与应用：多产品报童问题；考虑风险偏好的报童问题；基于需求预测的报童问题；考虑采购提前期的报童问题；12ProductAvailability:TradeoffsHighavailability=responsivetocustomersattractincreasedsaleshigherrevenueHighavailability=largerinventoryhighercostsriskofobsolescence13NewsboyModelsingleperiodmodel(onesellingseason)(one-timeorder,e.g.forquantitydiscount)demanduncertaintyorderplaced(anddelivered)beforedemandisknownunmetdemandislostunsoldinventoryattheendoftheperiodisdiscard(orsalvagedatlowervalue)Howmuchtoorder?NewsvendorModel14FactorsaffectingavailabilityDemanduncertaintyOverstockingcostC0=lossincurredwhenaunitunsoldatendofsellingseasonUnderstockingcostCu=profitmarginlostduetolostsale(becausenoinventoryonhand)Customer/CycleservicelevelCSL=levelofproductavailability=Prob(Demandstocklevel)13-15Copyright©2013PearsonEducation.DeterminingOptimalLevelofProductAvailability•Singleperiod•Possiblescenarios–Seasonalitemswithasingleorderinaseason–One-timeordersinthepresenceofquantitydiscounts–Continuouslystockeditems–Demandduringstockoutisbacklogged–Demandduringstockoutislost16Example:SellingparkasatLLBeanCostperparka=c=$45Salepriceperparka=p=$100Inventoryholding(untilseasonend)andtransportationcost(tooutletstore)perparka=$10Discountpriceperparka(seasonendsales)=$50Salvagevalueperparka=$50-$10=$40=sCostofoverstocking=Co=$45+$10-$50=c-s=$5Marginalprofitfromsellingparka=costofunderstocking=Cu=$100-$45=p-c=$5513-17Copyright©2013PearsonEducation.L.L.BeanExample–DemandDistributionDemandDi(inhundreds)ProbabilitypiCumulativeProbabilityofDemandBeingDiorLess(Pi)ProbabilityofDemandBeingGreaterthanDi40.010.010.9950.020.030.9760.040.070.9370.080.150.8580.090.240.7690.110.350.65100.160.510.49110.200.710.29120.110.820.18130.100.920.08140.040.960.04150.020.980.02160.010.990.01170.011.000.00Table13-118LLBean:ExpectedProfitExpecteddemandExpectedprofitiforder10Expectedprofitiforderk))(10](1[)])(10()([10104cpPscicpipii10174jjpj))(](1[)])(()([4cpkPscikcpipkkii19LLBean:ExpectedprofitDemandProbabilitySum(d(i)xp(i))CumulativeProb.Prob.demandgreaterExpectedprofitd(i)p(i)P(i)=Pr(Dd(i))1-P(i)=Pr(Dd(i))ifstockd(i)40.010.040.010.99220.0050.020.140.030.97274.4060.040.380.070.93327.6070.080.940.150.85378.4080.091.660.240.76424.4090.112.650.350.65465.00100.164.250.510.49499.00110.206.450.710.29523.40120.117.770.820.18535.80130.109.070.920.08541.60140.049.630.960.04541.40150.029.930.980.02538.80160.0110.090.990.01535.0020Newsvendor:MarginalAnalysisStockoneunitif…Stock2units(insteadof1unit)if...Stock1Stock2Stock3D=0D=1D=2D=321Increaseorderfromktok+1ifProb(Demandk)CuCo+CuOrderk+1insteadofkifPr(Dk)CuPr(Dk)(Co)0orPr(Dk)(Co)+[1-Pr(Dk)]Cu0orderk+1keepordersizeatkinsteadofk1moreunsold1fewerlostsale0CuCoAdditionalcontribution22LLBean13917.055555540455545100ordersoCCCratiocriticalscCcpCuououDemandProbabilityCumulativeProb.d(i)p(i)P(i)=Pr(Dd(i))40.010.0150.020.0360.040.0770.080.1580.090.2490.110.35100.160.51110.200.71120.110.82130.100.92140.040.96150.020.98160.010.99L.L.BeanExampleAdditionalHundredsExpectedMarginalBenefitExpectedMarginalCostExpectedMarginalContribution11th5,500x0.49=2,695500x0.51=2552,695–255=2,44012th5,500x0.29=1,595500x0.71=3551,595–355=1,24013th5,500x0.18=990500x0.82=410990–410=58014th5,500x0.08=440500x0.92=460440–460=–2015th5,500x0.04=220500x0.96=480220–480=–26016th5,500x0.02=110500x0.98=490110–490=–38017th5,500x0.01=55500x0.99=49555–495=–440Table13-224NewsvendorModel-DemandDistributionContinuousOrderysuchthatCSL*=Prob(Demandy)=CuCo+CuyCriticalratioCriticalfractileOptimalCycleServicelevel若每份报纸的购进价为0.75元，售出价为1元，退回价为0.6元，需求量服从均值500份，均方差50份的正态分布，报童每天应购进多少份报纸才能使平均收入最高，最高收入是多少？某水果店以每千克1.6元的价格购进每筐重100千克的香蕉，当天以每千克2.4元的价格出售，当天销售余下的香蕉再以平均每千克1.2元的处理价出售，以筐为单位的需求情况由下表列出