决策理论与方法——决策建模与严格不确定决策问题

1、1 决策建模与严格不确定决策问题n学习目标：（1）掌握两种常用的决策建模方法：n决策表n决策树（2）掌握五种严格不确定决策问题的决策准则：n1 .乐观准则n2. 悲观准则n3. Hurwitz准则n4. 等概率准则(Laplace)n5. 后悔值极小化极大(Savage)1 决策建模与严格不确定决策问题n1.1 决策建模方法n1.2 严格不确定决策问题1.1 决策建模方法n1.1.1 Example: a decision problemn1.1.2 Modeling the decision with decision treen1.1.3 Modeling the decision wit

2、h pay-off table (or decision matrix)n1.1.4 Decision making under uncertaintyn1.1.5 Decision making using probability information1.1.1 Example: a decision problemnThe Hewlett-Packard (HP) company specializes in assembling and selling PC systems for use by family doctor practices throughout the U.S.A.

3、 The company is developing a new PC-based system. At present the company is trying to decide on the manufacturing and assembly process to be used. One aspect of this relates to the keyboard that will be used in the system, which will have specially labeled function keys. nThe company has decided tha

4、t it faces three alternatives:nIt can manufacture/assemble the keyboard itself.nIt can buy the keyboards from a domestic manufacturer.nIt can buy the keyboards from a manufacturer in the Far East.1.1.1 Example: a decision problem (continued)nThe problem is that each of these options has different co

5、sts and benefits associated with it. nTo manufacture/assemble the keyboard itself, the company would require major investment in new production equipment as well as extensive training of the work force. It is felt that such an investment is only likely to be cost-effective if sales of the new produc

6、t are particularly good.nBuying the keyboard from a domestic supplier will involve the company in less up-front expense and will be safer if large sales do not materialize in the future.nBuying from an overseas supplier offers better quality but with the risk of disruptions in supply if there are pr

7、oblems in the delivery system.1.1.1 Example: a decision problem (cont.)nUncertainty in the decision problemnThere is uncertainty as to which decision to take because there is uncertainty over future sale.nTo help simplify the situation, the company is planning for one three possible sales levels in

8、the future:nLownMediumnhigh 1.1.1 Example: a decision problem (cont.)nWhat is under control and what is outside of controlnThe company is faced with a range of alternative decisions over which it has control (i.e. it can choose among them) nAnd it also faces an uncertain future in terms of sales. Th

9、is future position is generally referred to as the possible states of nature: future sale levels, are outside the direct control of the company but they do include all the possibilities and only one of them can actually occur. 1.1.1 Example: a decision problem (cont.)nStatement of the decisionnSo, t

10、he basic decision to be made is: which of the three supply options shall we choose, given the uncertainty we face as to the state of nature that will actually occur? 1.1.2 Modeling the decision with decision treenIt is also possible - and frequently useful - to represent the type of problem we have

11、been examining in graphical form by constructing what is known as a decision tree. nThe tree diagram shows the logical progression that occurs over time in terms of decisions and outcomes and is particularly useful in sequential decision problems - where a series of decisions need to be made with ea

12、ch, in part, depending on earlier decisions and outcomes. 1.1.2 Modeling the decision with decision tree (cont.)HPLowMBDBAMediumHighLowMediumHighLowMediumHighFig.2-1 Decision tree InThe decision tree starts from the left-hand side and gradually moves across to the right. nA box is used to indicate t

13、hat at this point we must take a decision (the box is technically known as a decision node)nThe three alternatives branch out from this node: nto manufacture (M)nto buy domestically (BD) nto buy abroad (BA) nEach of these branches leads to a outcome node (indicated by a circle), which presents the p

14、ossible states of nature:nLow (L)nMedium (M)nHigh (H)1.1.2 Modeling the decision with decision tree (cont.)nInformation for Decision making:nIn order to make decision we clearly need additional information to help us assess the alternatives. nThe key pieces of information would relate tonthe financi

15、al consequences of each combination of decision (called pay-off)nstate of nature (call probability)Decision tree with pay-offLowMBDBAMediumHighLowMediumHighLowMediumHighFig.2-2 Decision tree with pay-off (value in $000s)-15105510302552040nWith the pay-off information, which of the three alternative

16、decisions would you recommend? nThe answer has to be: it depends.nHow reliable you think the information is?nHow risky the various options are?nHow critical the decision is to the companys future?nThe key factor will depend on your own attitude to these future sates of nature.Complete decision treeL

17、ow 0.2MBDBAMedium 0.5High 0.3Fig.2-3 Completed decision tree (pay-off and probability)-15105510302552040Low 0.2Medium 0.5High 0.3Low 0.2Medium 0.5High 0.3nLet us assume that the company has been able to quantify the likelihood of each of the states of nature occurring by attaching a probability to e

18、ach.nSuch probabilities may have been derived from market research, from sales forecasting, or may simple be guesstimate based on some hard evidence and the experience of the decision maker.nThe probability of low sale is assessed at 0.2, medium at 0.5 and high at 0.3. Note that, they must sum to 1

19、to include all possible outcomes.1.1.3 Modeling the decision with pay-off table (or decision matrix)nIt is also possible to represent the type of problem we have been examining in a table called pay-off table or decision matrix.nThe pay-off table shows the financial consequences (or pay-off) in term

20、s of the alternative decisions that can be made and the alternative states of nature that might result. Pay-off table I: without probabilityFuture sales levelDecisionLowMediumHighManufacture (M)-151055Buy domestic (BD)103025Buy abroad (BA)52040Table 2-1 Pay-off table: profit contribution ($000s)Pay-

21、off table II: without probability (transposed)Table 2-2 Transposed Pay-off table: profit contribution ($000s)DecisionFuture sales levelManufacture(M)Buy domestic (BD)Buy abroad (BA)Low-15105Medium103020High552540Pay-off table III: with probability and pay-offTable 2-3 Pay-off table: probability and

22、pay-off ($000s)DecisionFuture sales levelProbabilityManufacture(M)Buy domestic (BD)Buy abroad (BA)Low0.2-15105Medium0.5103020High0.35525401.1.4 Decision making under uncertaintynA pure uncertainty situation is that there is no information available about the future states of the world to help a deci

23、sion maker. The decision maker is in a position of complete ignorance. nIn the absence of any other information on the likelihood of each state of nature actually occurring, we can consider a number of common attitudes.1.1.4 Decision making under uncertainty (cont.)nThere are five criteria for the d

24、ecision making under uncertainty (or pure uncertainty):n(1) The maximax criterionn(2) The maximin criterionn(3) The Hurwicz criterionn(4) The Laplace criterion n(5) The minimax Regret criterion1.1.5 Decision making using probability informationnRisk vs. UncertaintynUncertainty refers to outcomes whe

25、re estimates have been made but no probabilities can be attached to the expected comes. nTherefore, no objective probabilities can be assigned to outcomes, though subjective likelihood or confidence levels can be ascribed on statistically unverifiable grounds. nThe source of expected probabilities a

26、re the decision makers guesses and hunches about future patterns of events.1.1.5 Decision making using probability Information (cont.)nRisk vs. UncertaintynRisk refers to outcomes where the range of potential future outcomes is known from past experience. nFuture values and objective probabilities c

27、an therefore be attached to all possible outcomes. nThe values of possible alternative outcomes are known and so too are the likelihoods of the given outcomes occurring.1.1.5 Decision making using probability Information (cont.)nThe expected monetary value (EMV)nThe expected monetary value (EMV) is

28、the outcome anticipated when the range of pay-offs have attached to them some estimate of objective probability or subjective likelihood of potential outcome.nSuch probabilities may have been derived from market research, from sales forecasting, or may simply be a guesstimate based on some hard evid

29、ence and experience of the decision maker.The expected monetary value (EMV) criterionnAccording to the expected monetary value (EMV) criterion, a decision maker always chooses the action which will lead to the highest expected profit / value.1.1.5 Decision making using probability Information (cont.

30、)Low 0.2MBDBAMedium 0.5High 0.3Fig.2-3 Completed decision tree (pay-off and probability)-15105510302552040Low 0.2Medium 0.5High 0.3Low 0.2Medium 0.5High 0.3nThe EMV for the decision of M is: -15nThe EMV for the decision of BD is: 100.2+ 300.5+ 250.3=nThe EMV for the decision of BA is: 50.2+ 200.5+ 4

31、00.3=231.1.5 Decision making using probability Information (cont.)DecisionFuture sales levelProbabilityManufacture(M)Buy domestic (BD)Buy abroad (BA)Low0.2-15105Medium0.5103020High0.3552540EMV18.524.523Limitations of the EMV criterionnThe EMV criterion may have been appropriate for the decision make

32、r because he was only concerned with monetary rewards, and his decision was repeated a large number of times so that a long-run average result would have been of relevance to him.Sensitivity analysisnOf course, the probabilities and profits used in this problem may only be rough estimates or, if the

33、y are based on reliable past data, they may be subject to change. nWe should therefore carry out sensitivity analysis to determine how large a change there would need to be in these values before the alternative course of action would be preferred. Sensitivity analysisnSuppose the probabilities of N

34、atures states are PL, PM, and PH. We have: PL+PM+ PH =1 Thus: PH =1 - PL- PMnThe EMV for the decision of M is: -15PL+ 10PM+ 55(1 - PL-PM)=55-70PL-40PMnThe EMV for the decision of BD is: 10 PL + 30 PM + 25 (1 - PL-PM)=25-15PL+5PMnThe EMV for the decision of BA is: 5 PL + 20 PM + 40 (1 - PL-PM)=40-35P

35、L-20PMSensitivity analysisnFor every possible PL and PM, the decision of BD is still the best action. we should:25-15PL+5PM 55-70PL-40PM25-15PL+5PM 40-35PL-20PMnSolve the above two inequations, we have: PL 3/19 PM 9/19 PL+PM 12/19 (1-PL+PM) 7/19 PH 7/19 nIt means that all the probabilities of Nature

36、s states (PL, PM, and PH) satisfy the above conditions. The BD is still the best action.The value of perfect informationnWhat is the decision with perfect information?DecisionFuture sales levelProbabilityManufacture(M)Buy domestic (BD)Buy abroad (BA)Low0.2-15105Medium0.5103020High0.3552540nThe EMV w

37、ith perfect information is : 100.2+ 300.5+ 550.3=1.2 严格不确定型决策问题的决策准则n1.2.1 决策问题的表格表示决策问题的表格表示n1.2.2 决策问题的分类决策问题的分类n1.2.3 严格不确定型决策问题的决策准严格不确定型决策问题的决策准则则n1.2.4 理想的决策规则理想的决策规则1.2.1 决策问题的表格表示决策问题的表格表示nj,21决策问题可以用表格表示，这种表格叫做决策表，也叫决策矩阵。如果决策问题的后果是用损失表示的，亦可称作损失矩阵。 假设1:自然界只有有限种互不相容的可能状态,记为: = ; 假设2:决策人只有有限种可

38、行的行动方案,记为: A= . miaaaa,21则: 在自然状态j采取行动ai的后果记为: xij.决策表的一般形式决策表的一般形式 无论决策问题的后果xij是什么形式，我们都假设决策人能够用实值效用函数u（或价值函数v）来评价xij。决策矩阵、损失矩阵与转置1.2.2 决策问题的分类决策问题的分类 确定型决策问题 确定型决策问题的特点是决策人在进行选择之前了解真实的自然状态，即他可以确切地知道各种行动的后果。事实上，这就相当于假设n=1，这时决策表形如表。 1.2.2 决策问题的分类决策问题的分类 严格不确定型问题 决策人只能知道有哪些自然状态可能出现，他无法以任何方式量化这种不确定性，即

39、他只能给出各种可能状态 的列表，而对各种状态出现的可能性的大小一无所知，各种自然状态出现的概率无法估计。n,211.2.2 决策问题的分类决策问题的分类n风险型决策（重点）（重点） 这一类决策问题中，决策人虽然无法确知将来的真实自然状态，但他不仅能给出各种可能出现的自然状态 ，还可以给出各种状态出现的概率，通过设定概率分布 来量化不确定性。n,21)(,),(1n1.2.3 严格不确定型决策问题严格不确定型决策问题的决策准则的决策准则n无论是不确定型问题还是风险型问题，都需要根据某种准则来选择决策规则，使结果最优(或满意)，这种准则就叫决策准则，或称决策原则。n严格不确定型决策问题的五大决策准

40、则：n(1)乐观准则(The maximax criterion)n(2)悲观准则(The maximin criterion)n(3)Hurwitz准则(The Hurwicz criterion)n(4)等概率准则(The Laplace criterion )n(5)后悔值极小化极大(The minimax Regret criterion or the Savage criterion)(1) The maximax criterionnLet us assume for the moment that you have a very optimistic view of the fu

41、ture. If this were the case you would tend to choose the decision which could generate the highest possible pay-off. Such an approach is known as the maximax criterion, since we are searching for the maximum of the maximum pay-offs. nSuch an approach follows two steps:nFor each possible decision, id

42、entify the maximum possible pay-off.nComparing these pay-offs, select the decision that will give the maximum pay-off.(1) The maximax criterion (cont.)LowMBDBAMediumHighLowMediumHighLowMediumHighFig.2-4 The maximax criterion -15105510302552040(1) The maximax criterion (cont.)DecisionFuture sales lev

43、elManufacture(M)Buy domestic (BD)Buy abroad (BA)Low-15105Medium103020High552540(1) The maximax criterion (cont.)minminmin111i jnjmiimikloomaxmaxmax111i jnjmiimikuoo Mathematic definition of the maximax criterion with benefit Mathematic definition of the minimin criterion with lose(2) The maximin cri

44、terionnHowever, we are not all optimistic about the future. There is something to be said for examining the worst-case scenario - being pessimistic about future outcomes. In such a situation we can apply the maximin criterion, since we search for the maximum of the minimum pay-offs. For each possibl

45、e decision we determine the minimum (worst) possible pay-off and then choose the largest of these. nThe summary of the approach is:nFor each possible decision, identify the minimum possible pay-off.nComparing these pay-offs, select the decision that will give the maximum pay-off.(2) The maximin crit

46、erion(cont.)LowMBDBAMediumHighLowMediumHighLowMediumHighFig.2-5 The maximin criterion -15105510302552040(2) The maximin criterion(cont.)DecisionFuture sales levelManufacture(M)Buy domestic (BD)Buy abroad (BA)Low-15105Medium103020High552540(2) The maximin criterion(cont.)maxminmin111i jnjmiimiklss Ma

47、thematic definition of the maximin criterion with benefit Mathematic definition of the minimax criterion with lossminmaxmax111i jnjmiimikuss(3) The Hurwicz criterionnIn an attempt to achieve a compromise between the optimism of maximax criterion and the pessimism of the maximin criterion, Hurwicz pr

48、oposed a criterion which is a weighted average of the two extremes. Ok + (1- ) Sk where 0,1, named the optimism parameter; then (1- ) is named the pessimism parameter.(3) The Hurwicz criterion(cont.)minmax1min1min)1 (1111jinjjinjmiiimikkllososmaxmin1max1max)1 (1111jinjjinjmiiimikkuuosos Mathematic d

49、efinition of the Hurwicz criterion with benefit Mathematic definition of the Hurwicz criterion with lose(3) The Hurwicz criterion(cont.)DecisionDecision criterionManufacture(M)Buy domestic (BD)Buy abroad (BA)Maximax(Ok)553040Maximin(Sk)-15105Hurwicz()55 -(1- ) 1530 +(1- ) 1040 +(1- ) 5(4) The Laplac

50、e criterion nThe Laplace criterion is based on the assumption that the decision-maker is in a state of ignorance about the probabilities of Natures states.nEach of the Natures state has equal probability and is uniformly distributed.nThe decision-maker will choose the action that will get the maximum expected monetary value (EMV).(4) The Laplace criterion(cont.)DecisionFuture sales levelProbabilityManufac