Decision analysis 量化決策分析方法專論 2011/5/26 1
Problem formulation- states of nature In the decision analysis, decision alternatives are referred to as chance events. The possible outcomes for a chance event are referred to as states of nature. The states of nature are defined so that one and only one of the possible states of nature will occur. 2011/5/26 2
Problem formulation- states of nature s s 1 2 strong demand for the units weak demand for the units 2011/5/26 3
Problem formulation- alternative plans for the complex d d d 1 2 3 to buid a small complex of 30 business units to buid a medium sized complex of 60 business to buid a large complex of 90 business units units 2011/5/26 4
Decision trees 1 d 1 d 2 d 3 s 1 2 3 4 s 1 s 2 s 1 s 2 8 7 14 5 20 s 2-9 2011/5/26 5
Decision making without probabilitiesoptimistic approaches The optimistic approach evaluates each decision alternative in terms of the best payoff that can occur. The decision alternative that is recommended is the one that provides the best possible payoff. For a problem in which maximum return is desired, the optimistic approach would lead the decision maker to choose the alternative corresponding to the largest return. For problems involving minimization, this approach leads to choose the alternative with the smallest payoff. 2011/5/26 6
Maximum payoff for each PDC Decision alternative decision alternative Maximum payoff Small complex d1 8 Medium complex d2 14 Large complex d3 20 Maximum of the maximum payoff values 2011/5/26 7
Decision making without probabilitiesconservative approaches The conservative approach evaluates each decision alternative in terms of worst payoff that can occur. The decision recommended is the one that provides the best of the worst possible payoffs. For a problem in which the output measure is return, the conservative would lead the decision maker to choose the alternative that maximizes the minimum possible return. For problems involving minimization, this approach identifies the alternative that will minimize the maximum payoff. 2011/5/26 8
Minimum payoff for each PDC Decision alternative decision alternative Minimum payoff Small complex d1 7 Medium complex d2 5 Large complex d3-9 Maximum of the minimum payoff values 2011/5/26 9
Decision making without probabilitiesminimax regret approach R ij where R V V V ij * j ij * j ij the regret (opportunity loss) associated with decision alternative s the payoff value corresponding to the best V decision for the state of nature s the payoff and the state of nature s j corresponding to decision alternative j j d i 2011/5/26 10
Opportunity loss (regret) table for the Decision alternative PDC project (million) State of nature Strong demand s 1 Weak demand s 2 Small complex d 1 12 0 Medium 6 2 complex d 2 Large complex d 3 0 16 2011/5/26 11
Maximum regret for each PDC Decision alternative decision alternative Maximum regret Small complex d 1 12 Medium complex d 2 6 Large complex d 3 16 Minimum of the maximum regret values 2011/5/26 12
Decision making with probabilitiesexpect value approach N i1 P s P ij s where ij 0 N d j Psij N P V ij s 1 i1 ij d j V the number of ij the probability of the state of nature s for decision alternative d the expected value of alternative d alternative d states of j j j nature decision the payoff of the state of nature s for decision i i 2011/5/26 13
Expect value approach for the PDC project d 0.88 0.27 1 d 0.814 0.25 2 7.8 12.2 d 0.820 0.2 914. 2 3 2011/5/26 14
Expected value of perfect information PI PI expected value of perfect information W Wo W PI Wo PI PI expected value with perfect information about the states of nature PI expected value without perfect information about the states of nature 2011/5/26 15
Sensitivity analysis Sensitivity analysis also helps the decision maker by describing how changes in the state-of-nature probabilities and/or changes in the payoffs affect the recommended decision alternative. 2011/5/26 16
Sensitivity analysis d 0.28 0.87 1 d 0.214 0.85 2 7.2 6.8 d 0.220 0.8 9 3. 2 3 2011/5/26 17
Sensitivity analysis with p variations for the PDC project If P s p, then Ps d Ps 8 Ps 7 d 1 1 2 9 7 5 d 29 p 9 3 p p 1 2 1 2 p 2011/5/26 18
Sensitivity analysis with p variations for the PDC project 2011/5/26 19
Sensitivity analysis with the payoff values for the PDC project 3 3 12.2 0.8S If W remains constant then d d d 0.8S 0.2 9 3 0.2W 12.2 S 17.5 2011/5/26 20
Sensitivity analysis with the payoff values for the PDC project 3 3 12.2 0.8S If S remains constant then d d d 0.820 3 0.2W 0.2W 12.2 W 19 2011/5/26 21
Decision analysis with sample information Prior probability for the states of nature Posterior probability 2011/5/26 22
Decision analysis with sample information- the PDC decision tree 2011/5/26 23
Decision analysis with sample information- branch probabilities If P P If P P If P P If the market research study is undertaken, then favourable report unfavourable report strong demand weak demand strong demand weak demand the prior probabilities are as follows, P P strong demand weak demand 0. 20 0.77 the market research project is favourable, then 0.94 0.06 the market research report is unfavourable, then 0.35 0.65 the market research report is not undertaken, 0.80 0.23 2011/5/26 24
The PDC tree with branch probabilities 2011/5/26 25
Decision analysis with sample information- steps of decision strategy At chance nodes, calculate the expected value by multiplying the payoff at the end of each branch by the corresponding branch probabilities. At decision nodes, select the decision branch that leads to the best expected value. This expected value becomes the expected value at the decision node. 2011/5/26 26
Calculation of the expected values at chance nodes 6-14 CN6 0.948 0.067 7.94 CN7 0.9414 0.065 13.46 CN8 0.9420 0.06 918.26 CN9 0.358 0.657 7.35 CN10 0.3514 0.655 8.15 CN11 0.3520 0.65 91.15 CN12 0.808 0.207 7.80 CN13 0.8014 0.205 12.20 CN14 0.8020 0.20 914. 20 2011/5/26 27
The PDC decision tree after calculating expected values at chance nodes 6-14 2011/5/26 28
The PDC decision tree after calculating expected values at decision nodes 3-5 2011/5/26 29
The PDC decision tree reduced to two decision branches 2011/5/26 30
The PDC decision tree showing the branches associated with optimal decision strategy 2011/5/26 31
The probability distribution for the payoffs for the PDC optimal decision strategy Payoff (million) Probability -9 0.05 5 0.15 14 0.08 20 0.72 Total 100 2011/5/26 32
Utility and decision making Utility is a measure of the total worth of a particular outcome. It reflects the attitude of decision makers towards a collection of factors such as profit, loss and risk. 2011/5/26 33
Expected values for the three decision alternatives in the Swofford case study d1 0.330,000 0.520,000 0.2 50,000 d 2 0.350,000 0.5 20,000 0.2 30,000 d 0.30 0.50 0.20 0 3 9,000 1,000 2011/5/26 34
Developing utilities for payoffs Utility of Utility of -50,000 50,000 U Preference between a guaranteed 30,000 payoff and lottery (Swofford obtains a payoff of 50,000 with probability of p and a payoff of - 50,000 with probability of (1-p)) U 50,000 50,00010 0 2011/5/26 35
Developing utilities for payoffs U 30,000 p U 50,000 1 pu 50,000 0.9510 0.050 9.5 lottery p50,000 1 p 50,000 0.95 45,000 50,000 0.05 50,000 2011/5/26 36
Developing utilities for payoffs Lottery: Swofford obtains a payoff of 50,000 with probability of p and a payoff of - 50,000 with probability of (1-p) U 20,000 p U 50,000 1 pu 50,000 0.5510 0.450 5.5 lottery p 50,000 1 p 50,000 27,500-22,500 5,000 2011/5/26 37
Developing utilities for payoffs U M p U 50,000 1 p 50,000 p 10 1 p0 10 p where M is the specific monetary payoff p is the probability in which the decision maker was indifferent between a guaranteed payoff M a lottery with of 50,000 with that probability and 2011/5/26 38
Utility of monetary payoffs for the Swofford problem Monetary value Indifference value of p Utility value 50,000 Does not apply 10.0 30,000 0.95 9.5 20,000 0.90 9.0 0 0.75 7.5-20,000 0.55 5.5-30,000 0.40 4.0-50,000 Does not apply 0.0 2011/5/26 39
Utility table for Swofford problem States of nature Decision Prices up Prices stable Prices down alternative Investment A 9.5 9.0 0.0 Investment B 10.0 5.5 4.0 Do not invest 7.5 7.5 7.5 2011/5/26 40
Expected utility approach EU where d Ps i N d EU U i d U the decision alternative i ij N j1 is is i is for the state of j ij the possible states of is the decision alternative i, and the expected utility of the utility with regard to the decision alternative nature j nature the i 2011/5/26 41
Expected utility for each of the decision alternatives in the Swfford problem EU EU EU d 0.39.5 0.59.0 0.20 1 d 0.310 0.59.0 0.24.0 2 7.35 6.55 d 0.37.5 0.57.5 0.27.5 7. 50 3 2011/5/26 42
Ranking of the alternatives and the Ranking of the alternatives associated monetary value Expected utility Expected monetary value Do not invest 7.5 0 Investment A 7.35 9,000 Investment B 6.55-1,000 2011/5/26 43
Summary Decision analysis is used to evaluate alternative decisions where the outcomes are uncertain. For decision problems without probability information, three approaches can be used, namely, the optimistic, the conservative and the minimax regret approach. Decision analysis can be used to assess the value of additional information about the decision problem. The expected utility approach can be used in situations where monetary value is not the only measure of performance. 2011/5/26 44