灵敏度分析学习.pptx

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1、Session TopicsqSimultaneous Change in OFC(coefficient)同时改变目标函数系数q Shadow Price Analysis for RHS 右端项的影子价格分析q Range-of-Feasibility What-If Analysis 可行域的敏感性分析q Simultaneous Change in RHS 同时改变右端项第1页/共121页4.1 The Importance of What-If Analysis to ManagersWhat-If 分析对管理者的重要性第2页/共121页What is Sensitivity Ana

2、lysis 什么是敏感性分析qA particular mathematical model provides only a rough representation of the real problem.数学模型只是实际问题的一个粗略的抽象。qAn optimal solution is only optimal with respect to a particular mathematical model.最优解一般只是针对某一特定的数学模型。第3页/共121页What is Sensitivity Analysis 什么是敏感性分析qThe purpose of a linear pr

3、ogramming study is to help guide managements final decision by providing insights into the likely consequences of pursuing various managerial options under a variety of assumptions about future conditions.管理者要对未来做各种假设，在这些假设下，测试可能产生的结果，通过对各种结果深入分析来指导决策。第4页/共121页What is Sensitivity Analysis 什么是敏感性分析qM

4、ost of the important insights are gained while conducting analysis after finding an optional solution for the original version of the basic model.通常，在取得最初版本模型的最优解之后，进行分析才能取得对问题深入的认识。第5页/共121页What is Sensitivity Analysis 什么是敏感性分析qThis analysis is commonly referred to as what-if analysis because it in

5、volves addressing some questions about what would happen to the optimal solution if different assumptions are made about future conditions.这种分析称为what-if分析或敏感性分析（Sensitivity Analysis），因为其中所涉及的问题主要是:如果未来情况的假设变动的话，最优解将会有怎样的变化？第6页/共121页Parameters of the model模型的参数（1）Parameters of the model:the problems

6、provide the data needed to determine precisely that should go into the data cells to determine the LP optimal solution.The parameters of the model are rough estimates.Substantial time and efforts are needed to track down the needed data.For example,in the Wyndor case:the unit profits$300 for doors a

7、nd$500 for the windows.第7页/共121页Parameters of the model模型的参数（2）Some factors cannot be estimated with real accuracy until long after the linear programming study has been completed and the new products have been on the market for some time.For example,the profits depend on many factors:the costs of r

8、aw materials,production,shipping,advertising,and so on,as well as such things as the market reception to the new products and the amount of competition encountered.第8页/共121页Parameters of the model模型的参数（3）Wyndors managers want to know what the effect would be if the unit profits turn out to differ si

9、gnificantly from the estimates.管理者想要知道如果这些单位利润的估计值与实际情况相差很多，会有怎样的结果？For example,would the optimal solution change if the profit of doors:$300$200?第9页/共121页Parameters of the model模型的参数（4）How inaccurate can the estimate be in either direction before the optimal solution would change?估计值可以在什么范围内变动，而不会改

10、变最优解？第10页/共121页Parameters of the model模型的参数（5）If the optimal solution will remain the same over a wide range of values for a particular coefficient in the objective function,then management will be content with a fairly rough estimate for this coefficient.目标函数中某一系数在一定的范围内变动不会改变最优解，那么，管理者是能够接受该系数的估计值

11、。第11页/共121页Parameters of the model模型的参数（6）On the other hand,if even a small error in the estimate would change the optimal solution,then management will want to take special care to refine（精确）this estimate.第12页/共121页First benefit of what-if analysis1.Typically,many of the parameters of a LP model ar

12、e only estimates of quantities(e.g.,unit profits)that cannot be determined precisely at this time.线性规划模型的许多参数在建模时是很难精确确定的，只能是对一些数据的估计（如；单位利润）。第13页/共121页First benefit of what-if analysis What-if analysis reveals how close each of these estimates needs to be to avoid obtaining an erroneous optimal sol

13、ution,and therefore pinpoints the sensitive parameters where extra care is needed to refine their estimates.通过What-if 分析可以表明，系数估计值必须精确到怎样的程度，才能避免得出错误的最优解，而且，可以找出哪些系数是需要重新精确定义的灵敏度参数。第14页/共121页Second benefit of what-if analysis2.If conditions change after the study has been completed(a common occurren

14、ce),what-if analysis leaves signposts that indicate(without solving the model again)whether a resulting change in a parameter of the model changes the optimal solution.如果在研究结束后，问题的条件发生了变化，通过what-if分析，即使不求解，也可以表明模型参数的变化是否会改变最优解。第15页/共121页Sensitivity Analysis灵敏度分析To study how changes in the parameters

15、 of a LP model affect the optimal solution.研究线性规划模型的参数如何变化会影响最优解。It involves checking how sensitive the optimal solution is to the value of each parameters.最优解对每一参数变化的灵敏度。第16页/共121页Third benefit of what-if analysis3.When certain parameters of the model represent managerial policy decisions,what-if a

16、nalysis provides valuable guidance to management about what the impact would be of altering these policy decision.当模型特定的参数反映管理政策决策时，what-if分析可以表明改变这些决策对结果的影响，从而有效指导管理者作出最终的决策。第17页/共121页4.2 Continuing the Wyndor Case Study继续研究伟恩德公司案例第18页/共121页Wyndor Case1.Two new products:An 8-foot glass door with al

17、uminum framing 一种8英尺的铝框玻璃门A 4-foot6-foot double-hung wood-framed window 一种4X6英尺的双把木框窗2.Three plants：vPlant 1 produces aluminum frames and hardware 工厂1：生产铝框和硬制件vPlant 2 produces wood frames 工厂2：生产木框vPlant 3 produces the glass and assembles the windows and doors 工厂3：生产玻璃、组装窗和门第19页/共121页Fig 4.1 Spreads

18、heet for Wyndor Case第20页/共121页Optimal solution for Wyndor CaseOptimal solution:D=2(produce 2 doors per week)W=6(produce 6 windows per week)Profit=3,600(The estimate total weekly profit is$3,600)第21页/共121页Managements Discussion of the Recommended Product Mix（1）1.the unit profit？What happens if the es

19、timate of the unit profit of one of Wyndors new products is inaccurate?How far off can one of these estimates be without changing the optimal product mix？Range-of-optimality sensitivity analysis第22页/共121页Managements Discussion of the Recommended Product Mix（2）2.What happens if the estimates of the u

20、nit profits of both of Wyndors new products are inaccurate?第23页/共121页Managements Discussion of the Recommended Product Mix（3）3.available production timeWhat happens if a change is made in the number of hours of production time per week being made available to Wyndors new products in one of the plant

21、s?How much more profit could we get from the new products for each additional hour of production time？We would compare this additional profit with the profit we could lose by cutting back on the production of current products.第24页/共121页Managements Discussion of the Recommended Product Mix（4）4.What h

22、appens if simultaneous changes are made in the number of hours of production time per week being made available to Wyndors new products in all the plants?第25页/共121页4.3 Range-of-Optimality Sensitivity Analysis最优域灵敏度分析第26页/共121页Question 1What happens if the estimate of the unit profit of one of Wyndor

23、s new products is inaccurate?如果伟恩德公司一种新产品单位利润的估计值是不精确的，结果会怎么样？第27页/共121页Example（1）For example,(PD,PW)=($300,$500),how much larger or how smaller PD can be and still have(D,W)=(2,6)as the optimal solution?在保持最优解(D,W)=(2,6)不变的条件下，PD可能的最大值与最小值。第28页/共121页Example（2）In other words,how much can the estimat

24、e of$300 for the unit profit for these doors be off before the model will give an erroneous optimal solution?门的单位利润$300 的这一估计可以在多大程度上偏离实际值，而不会产生错误的最优解。第29页/共121页Wyndor Case Study 伟恩德公司案例研究ThespreadsheetmodelanditsoptimalsolutionfortheoriginalWyndorproblembeforebeginningwhat-ifanalysis.第30页/共121页Fig

25、4.2PD=$300$200No change at all in the optimal solution.第31页/共121页WyndorCase Study 伟恩德公司案例研究TherevisedWyndorproblemwheretheestimateoftheunitprofitfordoorshasbeendecreasedfromPD=$300toPD=$200,butnochangeoccursintheoptimalsolution.第32页/共121页Fig 4.3PD=$300$500No change at all in the optimal solution.PD

26、is said to be not a sensitive parameter.Allowable range to stay optimal for PD,or just its range of optimality for short.第33页/共121页Wyndor Case Study 伟恩德公司案例研究TherevisedWyndorproblemwheretheestimateoftheunitprofitfordoorshasbeenincreasedfromPD=$300toPD=$500,butnochangeoccursintheoptimalsolution.第34页/

27、共121页Wyndor Case Study 伟恩德公司案例研究TherevisedWyndorproblemwheretheestimateoftheunitprofitfordoorshasbeenincreasedfromPD=$300toPD=$1000,butchangeoccursintheoptimalsolution.第35页/共121页Wyndor Case Study 伟恩德公司案例研究AdatatablethatshowstheeffectofsystematicallyvaryingtheestimateoftheunitprofitfordoorsintheWyndo

28、rproblem.第36页/共121页Wyndor Case Study 伟恩德公司案例研究PartofthesensitivityreportgeneratedbytheExcelSolverfortheoriginalWyndorproblem,wherethelast three columns enable identifying for ranges ofoptimalityfortheunitprofitsfordoorsandwindows.第37页/共121页Range of optimality of PDCurrent value of PD:300Allowable in

29、crease in PD:450 So PD300+450=750Allowable decrease in PD:300 So PD300-300=0Range of optimality for PD:0PD750第38页/共121页ConclusionThe range of optimality for PD:0PD750,because(D,W)=(2,6)remains optimal over this range but not beyond.(when PD=0 or PD=750,there are multiple optimal solution,but(D,W)=(2

30、,6)still is one of them.)With the range this wide around the original estimate of$300 for the unit profit for doors,we can be quite confident of obtaining the correct optimal solution for the true unit profit even though the discussion in Section 4.2 indicates that estimate is fairly rough.第39页/共121

31、页Range of optimality of PwCurrent value of Pw:500Allowable increase in Pw:unlimited.So Pw has no upper limit.Allowable decrease in Pw:300 So PD500-300=200Range of optimality for Pw:Pw 200第40页/共121页Sensitive ParametersA small change in the value of certain coefficients in the objective function can c

32、hange the optimal solution.目标函数系数微小的变动都可能会影响最优解，这样的系数称为敏感参数。第41页/共121页4.4 Simultaneous Change in Objective Function Coefficients目标函数系数同时变动第42页/共121页QuestionThe range of optimality for a particular coefficient assumes that the original estimates for all the other coefficients are completely accurate

33、so that this coefficient is the only one whose true value may differ from its original estimate.在求解一个变动系数的最优域时，假设其他所有系数都是正确的，研究的系数是惟一可能与实际值不符的变动系数。第43页/共121页Purposing This section focuses on how to determine,without solving the problem again,whether the optimal solution might change if certain chang

34、es occur simultaneously in the coefficients of the objective function.本节将着重介绍如何在不重新求解模型的条件下，确定如果目标函数的几个系数同时变化，可能造成的对最优解的影响。第44页/共121页Question 2What happens if the estimates of the unit profits of both of Wyndors new products are inaccurate?如果伟恩德公司两种新产品单位利润的估计值都是不精确的，将会对结果产生怎样的影响？第45页/共121页Wyndor Cas

35、e Study 伟恩德公司案例研究TherevisedWyndorproblemwheretheestimatesoftheunitprofitsfordoorsandwindowshavebeenchangedtoPD=$450andPW$400,respectively,butnochangeoccursintheoptimalsolution.第46页/共121页Wyndor Case Study 伟恩德公司案例研究TherevisedWyndorproblemwheretheestimatesoftheunitprofitsfordoorsandwindowshavebeenchang

36、edtoPD=$600andPW$300,respectively,whichresultinachangeintheoptimalsolution.第47页/共121页Wyndor Case Study 伟恩德公司案例研究AdatatablethatshowstheeffectofsystematicallyvaryingtheestimatesoftheunitprofitsforbothdoorsandwindowsintheWyndorproblem.第48页/共121页100 Percent Rule for Simultaneous Changes in Objective Fun

37、ction CoefficientsIf Simultaneous Changes are made in the objective function,calculate for each change the percentage of the allowable change(increase or decrease)for that coefficient to remain within its range of optimality.If the sum of the percentage changes does not exceed 100 percent,the origin

38、al optimal solution definitely will still be optimal.(if the sum does exceed 100 percent,then we cannot be sure.)第49页/共121页The 100 percent rule百分之百法则目标函数系数同时变动的百分之百法则(The100percentruleforsimultaneouschangesinobjectivefunctioncoefficients)：如果目标函数的系数同时变动，计算出每一系数变动量占该系数最优域允许变动量的百分比，而后，将各个系数的变动百分比相加，如果所

39、得的和不超过百分之一百，最优解不会改变，如果超过百分之一百，则不能确定最优解是否改变。第50页/共121页Example 1PD:$300$450 Percentage of allowable increase =(450-300)/450=33.3%Pw:$500$400 Percentage of allowable decrease =(500-400)/300=33.3%Sum=66.6%100%,Changed in optimal solution.第52页/共121页Example 3PD:$300$150 Percentage of allowable decrease =(

40、300-150)/300=50%Pw:$500$250 Percentage of allowable decrease =(500-250)/300=83%Sum=133%100%,No change in optimal solution.第53页/共121页Wyndor Case Study 伟恩德公司案例研究当门，窗的单位利润的估计值分别改为PD$525，PW$350，刚好处在百分百法则所允许的临界点上，（D,W）（2，6）还是最优解，但同时，目标函数直线上在（2，6）（4，3）之间的所有点均为最优解第54页/共121页WyndorCase Study 伟恩德公司案例研究当门，窗的单位

41、利润的估计值改为PD$150，PW$250，（最初解的一半）作图法可知（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第55页/共121页Role of the 100 percent rule百分之百法则的作用qThe100percentrulecanbeusedtodeterminejusthowlargethechangesintheobjectivefunctioncoefficientsneedtobebeforetheoriginaloptimalsolutionmaynolongerbeoptimal.q可用于确定在保持最优解不变的条件下，目标函数系数的变动范

42、围；第56页/共121页Roleofthe100percentrule百分之百法则的作用qBydividingeachcoefficientsallowableincreaseorallowabledecreasebythenumberofdecisionvariables,the100percentruleimmediatelyindicateshowmucheachcoefficientcanbesafelychangedwithoutinvalidatingthecurrentoptimalsolution.q百分百法则通过将允许的增加或减少值在各个系数之间分摊，从而可以直接显示出每个系

43、数的允许变动值；第57页/共121页Roleofthe100percentrule百分之百法则的作用qAfter completing the study,if conditions change in the future that cause some or all of the coefficients in the objective function to change,the 100 percent rule quickly indicates whether the original optimal solution must remain optimal.q线性规划研究结束以后

44、，如果将来条件变化，致使目标函数中一部分或所有系数都发生变动，百分百法则可以直接表明最初最优解是否保持不变。第58页/共121页4.5 Shadow Price Analysis for Right-Hand Sides右端值影子价格分析第59页/共121页ShadowPriceAnalysisforRHS右端项的影子价格分析qWe might be interested in the effect of changes in the right-hand for these parameters of the model are only estimates of quantities th

45、at cannot be determined precisely at this time so we want to determine in the effect if these estimates are inaccurate.q分析函数约束右端值变动的原因，是因为在建模时，还不能得到模型的参数的精确值，只能对其作大略的估计。因此要知道万一这些估计不准确产生的后果；第60页/共121页ShadowPriceAnalysisforRHS右端项的影子价格分析qA more common reason for this interest is the one that the right-

46、hand sides may well represent managerial policy decisions rather than quantities that are largely outside the control of management.q更主要的理由是因为这些常数往往不是由外界决定的而是管理层的政策决策。在建模并求解后，管理者想要知道如果改变这些决策是否会提高最终收益。第61页/共121页ShadowPriceAnalysisforRHS右端项的影子价格分析qWhat shadow price analysis does is provide valuable gu

47、idance to management about what the effect would be of altering these policy decisions.q影子价格分析就是为管理者提供这方面的信息第62页/共121页TheProfit&GambitCo.Advertising-MixProblem利博公司广告组合问题The Profit&Gambit Co.produces cleaning products for home use.利博公司生产家用的清洁产品：1.A spray prewash stain remover一种喷雾去污剂2.A new liquid lau

48、ndry detergent一种新液体洗涤剂3.A well-established powder laundry detergent一种成熟的洗衣粉A new advertising campaign新的广告运动。第63页/共121页TheProfit&GambitCo.Advertising-MixProblem利博公司广告组合问题Management has set minimum goals for the campaign管理者设定广告组合的最低目标：1.The stain remover should capture an additional 3 percent of its m

49、arket.喷雾去污剂必须再增加3%的市场份额。2.The new liquid detergent should gain 18 percent of the laundry market.新液体洗涤剂必须在洗涤剂市场中获得18%的份额。3.A 4 percent increase of this same market should be captured by the powder detergent.洗衣粉占洗涤剂市场的份额必须增加4%。第64页/共121页DatafortheProfit&GambitCo.Advertising-MixProblem利博公司广告组合问题的数据 产品

50、电视 印刷媒体 需要最小的增加量 去污剂 0%1%3%液体洗涤剂 3%2%18%洗衣粉 1%4%4%单位成本 100万 200万第65页/共121页Formulating a spreadsheet Model for this problem构建电子表格模型 Decision variables决策变量：TV=number of units of advertising on the television 电视广告的单位数量PM=number of units of advertising in the print media印刷媒体广告的单位数量第66页/共121页TheMathemati