1線性規(guī)劃舉例：線性規(guī)劃要確定決策變量x1x2xnMaximizeZc1x1cnsubject

a11x1a1nxna21x1a2nxn

am1x1amnxnbmx10,,xn 已知c1,…,cn;a11,…,amn;b1,…,2線性規(guī)劃的what-if最優(yōu)解只是針對某一特定的數(shù)學模型而數(shù)學模型中的參數(shù)經常是估計值最優(yōu)解或影子價格保持不變的參數(shù)變化范圍是如果最優(yōu)解對參數(shù)的變化很敏感，就需要對參準確估以上所說的分析稱為what-if3使用EXCEL進行what-if改變目標函數(shù)單個多個改變約束函數(shù)右單個多個分析單個系統(tǒng)利用敏感性分析4偉恩德公司生產能力、產品所需資源、工廠工廠門

工廠工廠工廠窗

利潤5偉恩德公司案代數(shù)模maxP300D500WD2W3D2WD,W6偉恩德公司案what-if分析之78單個目標函數(shù)門的單位利潤PD=$300降到PD=$200，而最優(yōu)解不變9單個目標函數(shù)門的單位利潤PD=$300增加到PD=$500，而最優(yōu)單個目標函數(shù)門的單位利潤從PD=$300增加到PD=$1000，最優(yōu)解改單個目標函BCDEFG34 567 2489 thesecellstheSolverfor CDE單個目標函數(shù)BCDEfor2626262626262626434343系統(tǒng)地改變門單位利潤得單個目標函數(shù)用Excel求解偉恩德公司門窗生產問題輸出的敏感可變元終遞目標允許允許單元名值成系增減UnitsProduced20Units60約終陰約允許允許單元名值價限制增減Plant12042Plant266Plant366單個目標函數(shù) 偉恩德模型用ExcelSolver產生的靈敏度分析報告的一目標函數(shù)

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

(Plant(Plant(Plant3)(非負約束

目標函數(shù)系數(shù)C1在多大范圍內變化最優(yōu)解

最優(yōu)解對應的3P1+50/23P1+50/2<C1/5<0<C1<現(xiàn)在C1最多減少3，最多增加，在此范圍內最優(yōu)解不變3P1+2P2<18(Plant

目標函數(shù)系數(shù)C2在多大范圍內變化最優(yōu)解不

最優(yōu)解對應的起作用的約2P2<12(Plant3P1+2P2<18(Plant 3P1+50/2<3/C2<2<C2<現(xiàn)在C2 目標函數(shù)系數(shù)WProductionrateforwindows8 Line

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

Line

PD=300(Profit=300D+500PD=750(Profit=750D+500Line Productionratefor通過約束邊界線的虛線B和虛線C是單位門的利潤PD等于最優(yōu)（0PD750）兩個端點時的目標函數(shù)多個目標函數(shù)門、窗的單位利潤分別被改為45，＝$00，多個目標函數(shù)門、窗的單位利潤分別被改為最優(yōu)解改多個目標函數(shù)BCDEFGHI345671024802thesecellstheSolver93226TotalforC多個目標函BCDEFGHTotalfor系統(tǒng)地改變門、窗單位利潤得到的結果（總利潤）多個目標函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&系統(tǒng)地改變門、窗單位利潤得到的結果（生產率）百分之百法則（The100percentrule目標函數(shù)系數(shù)同時變動的百分之百法則如果目標函數(shù)的系數(shù)同時變動，計算出每一系數(shù)的變化占該系數(shù)允許變化量（增加或減少）的百分比，然后將各個系數(shù)的變動百分比相加，如果百分比之和不超過100%，則最初的最優(yōu)解仍然是最優(yōu)的，如果超過百分之百法則的,線性規(guī)劃求解結束以后，如果將來條件變化，致使目多個目標函數(shù)百分之百 偉恩德模型用ExcelSolver產生的靈敏度分析報告的一WProductionrateforwindows

多個目標函數(shù)百分之百Objectivefunctionlinenow Profit=$3150=525D+350sinceD=$525,PW=$350. (2,Entirelinesegmentis4

Max300PD當門、窗的單位利潤估計值分別改為PD＝$525，分之百法則所允許的臨界點上，（D,W）＝（2,6）還是最優(yōu)解，2

(4,

時目標函數(shù)直線上（2,6）（4,3）的所有點均為最 ProductionrateforProductionrateforwindows

多個目標函數(shù)百分之百W8Profit=$1800=150D+250642

Optimalsolution(2,6)

Max當門、窗單位利潤的估計值改為PD＝$150，P半），（D,W）＝（2,6）還是最優(yōu)解，盡管百分之百法則表示最優(yōu)解有可能變動。 Productionratefor約束函數(shù)右端）變化一個單位，使得目標函數(shù)值變化的單個約束函數(shù)BCDEFG34567104802932工廠2可用時間從12增加到13，總利潤增加了$150單個約束函數(shù)BCDEFG34567100480293209工廠2可用時間進一步從13增加到18，總利潤增加了$750單個約束函數(shù)BCDEFG34567100480293209工廠2可用時間進一步從18增加到20，總利潤不再增加了單個約束函數(shù)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單個約束函數(shù)用Excel求解偉恩德公司門窗生產問題輸出的敏感可變元終遞目標允許允許單元名值成系增減UnitsProduced20Units60約終陰約允許允許單元名值價限制增減Plant12042Plant266Plant366偉恩德公司案例研代數(shù)模maxP300DD2W3D2WD,WWProductionrateforwindows

影子價格可行(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多個約束函數(shù)BCDEFG34567104802932Units 一個可用時間從工廠3轉移到工廠2，總利潤增加了$50多個約束函數(shù)BCDEFGH345671024802932 thesecellstheSolver---百分之百法則（The100percent一個生產問每周的原材料

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