数据模型与决策-第三次课3what-if分析举例线性要确定变量x1,x2xn使得

1线性规划举例：线性规划要确定决策变量x1x2xnMaximizeZc1x1cnsubject

a11x1a1nxna21x1a2nxn

am1x1amnxnbmx10,,xn 已知c1,…,cn;a11,…,amn;b1,…,2线性规划的what-if最优解只是针对某一特定的数学模型而数学模型中的参数经常是估计值最优解或影子价格保持不变的参数变化范围是如果最优解对参数的变化很敏感，就需要对参准确估以上所说的分析称为what-if3使用EXCEL进行what-if改变目标函数单个多个改变约束函数右单个多个分析单个系统利用敏感性分析4伟恩德公司生产能力、产品所需资源、工厂工厂门

工厂工厂工厂窗

利润5伟恩德公司案代数模maxP300D500WD2W3D2WD,W6伟恩德公司案what-if分析之78单个目标函数门的单位利润PD=$300降到PD=$200，而最优解不变9单个目标函数门的单位利润PD=$300增加到PD=$500，而最优单个目标函数门的单位利润从PD=$300增加到PD=$1000，最优解改单个目标函BCDEFG34 567 2489 thesecellstheSolverfor CDE单个目标函数BCDEfor2626262626262626434343系统地改变门单位利润得单个目标函数用Excel求解伟恩德公司门窗生产问题输出的敏感可变元终递目标允许允许单元名值成系增减UnitsProduced20Units60约终阴约允许允许单元名值价限制增减Plant12042Plant266Plant366单个目标函数 伟恩德模型用ExcelSolver产生的灵敏度分析报告的一目标函数

3P1+5 P1<2P2<3P1+2P2<18P1,P2>0

(Plant(Plant(Plant3)(非负约束

目标函数系数C1在多大范围内变化最优解

最优解对应的3P1+50/23P1+50/2<C1/5<0<C1<现在C1最多减少3，最多增加，在此范围内最优解不变3P1+2P2<18(Plant

目标函数系数C2在多大范围内变化最优解不

最优解对应的起作用的约2P2<12(Plant3P1+2P2<18(Plant 3P1+50/2<3/C2<2<C2<现在C2 目标函数系数WProductionrateforwindows8 Line

(2,6)isoptimalfor0<PD< PD=0(Profit=0D+50042

Line

PD=300(Profit=300D+500PD=750(Profit=750D+500Line Productionratefor通过约束边界线的虚线B和虚线C是单位门的利润PD等于最优（0PD750）两个端点时的目标函数多个目标函数门、窗的单位利润分别被改为45，＝$00，多个目标函数门、窗的单位利润分别被改为最优解改多个目标函数BCDEFGHI345671024802thesecellstheSolver93226TotalforC多个目标函BCDEFGHTotalfor系统地改变门、窗单位利润得到的结果（总利润）多个目标函BCDEFGH(2,(4,(4,(2,(2,(2,(4,(4,(2,(2,(2,for(4,(4,(4,(2,(2,(4,(4,(4,(4,(2,C25C25="("&DoorsProduced&","&WindowsProduced&系统地改变门、窗单位利润得到的结果（生产率）百分之百法则（The100percentrule目标函数系数同时变动的百分之百法则如果目标函数的系数同时变动，计算出每一系数的变化占该系数允许变化量（增加或减少）的百分比，然后将各个系数的变动百分比相加，如果百分比之和不超过100%，则最初的最优解仍然是最优的，如果超过百分之百法则的,线性规划求解结束以后，如果将来条件变化，致使目多个目标函数百分之百 伟恩德模型用ExcelSolver产生的灵敏度分析报告的一WProductionrateforwindows

多个目标函数百分之百Objectivefunctionlinenow Profit=$3150=525D+350sinceD=$525,PW=$350. (2,Entirelinesegmentis4

Max300PD当门、窗的单位利润估计值分别改为PD＝$525，分之百法则所允许的临界点上，（D,W）＝（2,6）还是最优解，2

(4,

时目标函数直线上（2,6）（4,3）的所有点均为最 ProductionrateforProductionrateforwindows

多个目标函数百分之百W8Profit=$1800=150D+250642

Optimalsolution(2,6)

Max当门、窗单位利润的估计值改为PD＝$150，P半），（D,W）＝（2,6）还是最优解，尽管百分之百法则表示最优解有可能变动。 Productionratefor约束函数右端）变化一个单位，使得目标函数值变化的单个约束函数BCDEFG34567104802932工厂2可用时间从12增加到13，总利润增加了$150单个约束函数BCDEFG34567100480293209工厂2可用时间进一步从13增加到18，总利润增加了$750单个约束函数BCDEFG34567100480293209工厂2可用时间进一步从18增加到20，总利润不再增加了单个约束函数BCDEFG3DoorsWindow4UnitProfit$300$5005HoursHours6HoursUsedPerUnitProducedUsedAvailable7Plant102<=48Plant0212<=129Plant3218<=181011DoorsWindowTotalProfit12UnitsProduced26$3,60013141516TimeAvailableOptimalUnitsProducedTotalIncremental17inPlant2(hours)DoorsWindowProfitProfit1826$3,600442$2,200Selec20$250542.$2,450thesecell643$2,700$250(B18:E35)73.6673.$2,850$150befor2383.3334$3,000$150choosin24934.$3,150$150theSolve25102.6675$3,300$150Table26112.3335.$3,450$150271226$3,600$15028131.6676.$3,750$15029141.3337$3,900$150301517.$4,050$15031160.6678$4,200$15032170.3338.$4,350$150331809$4,500$150341909$4,500$0352009$4,500$0单个约束函数用Excel求解伟恩德公司门窗生产问题输出的敏感可变元终递目标允许允许单元名值成系增减UnitsProduced20Units60约终阴约允许允许单元名值价限制增减Plant12042Plant266Plant366伟恩德公司案例研代数模maxP300DD2W3D2WD,WWProductionrateforwindows

影子价格可行(0, 2W=18

Profit=300(0)+500(9)=$4,5008 Line

(2,

2W=12 Profit=300(2)+500(6)=$3,600 regionfor

(4,3)

2W=6 Profit=300(4)+500(3)=$2,70 LineC(3D+2W=18)

LineA(D=4) Productionratefordoors多个约束函数BCDEFG34567104802932Units 一个可用时间从工厂3转移到工厂2，总利润增加了$50多个约束函数BCDEFGH345671024802932 thesecellstheSolver---百分之百法则（The100percent一个生产问每周的原材料

8SmallProfit=$20/

6LargeProfit=$15/GraphicalSolution(OriginalMaximizeProfit=($20)T+subject 2T+C≤6large2T+2C≤8small42T+2C<8small3

T≥0,C≥2T+C<6large Z=($20)T+($15)C=7LargeMaximizeProfit=($20)T+subject2T+C≤7large 2T+2C≤8small42T+2C<8small

T≥0,C≥22T+C<7large12T+C<6large Z=($20)T+($15)C=9LargeMaximizeProfit=($20)T+subject 2T+C≤9large2T+2C≤8small 2T+C<9large

T≥0,C≥12T+2C<8

2T+C<6large Z=($20)T+($15)C=$25ProfitperMaximizeProfit=($25)T+subject 2T+C≤6large2T+2C≤8small4

T≥0,C≥32T+C<6large Z=($25)T+($15)C=$35ProfitperMaximizeProfit=($35)T+subject2T+C≤6large 2T+2C≤8small T≥0,C≥2T+2C<8Z=($35)T+($15)C=1 GeneratingtheSensitivityBCDEFG34 56Billof7Large 668889 AftersolvingwithSolver,choose“Sensitivity”underTheSensitivityBCDEFG34 56Billof7Large 668889 Adjustable2052055LargeBricksTotal6562285842TheSensitivityAllowableThe (Solutionstaysthe ProductionQuantity: ProductionQuantity:

LargeBricksTotal SmalBricksTotal Usageoftheresource(Left-hand-sideof

Allowablerange(Shadowpriceisvalid)Increaseinobjectivefunctionvalueperunitincreaseinright-hand-∆Z=(shadow$35ProfitperBCDEFG34 56Billof7Large 668869 3050-662660827LargeBCDEFG34 56Billof7Large 778889 Adjustable3051055LargeBricksTotal75713858619LargeBCDEFG34 56Billof7Large 898889 Adjustable4050-5LargeBricksTotal80918818100%RuleforSimultaneousChangesintheObjectiveCoefficients

2052055LargeBricksTotal65622SmalBricksTotal85842Examples:(DoessolutionstaytheProfitperTable=&ProfitperChair=ProfitperTable=&ProfitperChair=ProfitperTable=&ProfitperChair=100%RuleforSimultaneousChangesintheRight-Hand-SidesAdjustable

2052055LargeBricksTotal6562285842Examples:(Aretheshadowpricesvalid?Ifwhat’sthenewtotal(+1LargeBrick)&(+2SmallBricks)(+1LargeBrick)&(–1SmallBrick)SummaryofSensitivityReportforChangesintheObjectiveFunctionCoefficientsFinalThevalueofthedecisionvariables(changingcells) theoptimalsolution.ReducedIncreaseintheobjectivefunctionvalueperunitincreaseinthevalueofazero-valuedvariable(forsmallincreases)—maybeinterpretedastheshadowpriceforthenonnegativityconstraint.ObjectiveThecurrentvalueoftheobjectiveAllowableDefinestherangeofthecoefficientsintheobjectivefunctionforwhichthecurrentsolution(valueofthedecisionvariablesorchangingcellsintheoptimalsolution)willnotchange.SummaryofSensitivityReportforChangesintheRight-Hand-SidesFinal