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1 Decision Analysis Decision-Making Under Uncertainty updated 10.24.01 Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow NTU SY-521-N SMU EMIS 5300/7300

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2 Decision Making Under Uncertainty The decision-maker knows for sure which state of nature will occur, and he or she makes the decision based on the optimal payoff available under that state. It is unknown which states of nature will occur and the probability of the likelihood of a state occurring is also unknown The decision-maker has virtually no information regarding which state of nature will occur, and he or she attempts to develop a strategy based on payoffs

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3 Approaches to Decision Making Under Uncertainty Maximax criterion Maximin criterion Hurwicz criterion Minimax regret

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4 Maximax Criterion An optimistic approach in which the decision-maker acts based on a notion that the best things will happen The decision-maker isolates the maximum payoff under each decision alternative and then selects the decision alternative that produces the highest of these maximum payoffs The name maximax means selecting the maximum overall payoff from the maximum payoffs of each decision alternative

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5 Decision Table State of the Economy Stagnant Slow Rapid GrowthGrowth Maximum Stocks -$500 $700$2200 $2200 InvestmentBonds -$100 $600$900 $900 Decision AlternativeCD’s $300 $500$750 $750 Mixture -$200 $650$1300 $1300

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6 Decision Table The maximax criterion approach requires that the decision-maker select the maximum payoff of these four Maximum of {$2200, $900, $750, $1300} = $2200 Since maximax criterion results in $2200 as the optimal payoff, the decision alternative selected is the stock alternative, which is associated with the $2200

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7 Maximin Criterion A pessimistic approach to decision-making under uncertainty This approach assumes that the worst will happen and attempts to minimize the damage Using the maximin criterion approach, the decision-maker starts by examining the payoffs under each decision alternative and selects the worst, or minimum, payoff that can occur under that decision

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8 Decision Table State of the Economy Stagnant Slow Rapid GrowthGrowth Minimum Stocks -$500 $700$2200 -$500 InvestmentBonds -$100 $600$900 -$100 Decision AlternativeCD’s $300 $500$750 $300 Mixture -$200 $650$1300 -$200

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9 Decision Table With the maximin criterion, the decision-maker examines the minimum payoffs for each decision alternative given in the last column and selects the maximum of these values Maximum of {-$500, -$100, $300, -$200} = $300 The decision is to invest in CD’s because this investment alternative yields the highest, or maximum, payoff under the worst-case scenario

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10 Hurwicz Criterion An approach that lies somewhere in between the Maximax and the Maximin approaches Selects the maximum and the minimum payoff from each decision alternative A value called (not the same as the probability of a Type I error), which lies between 0 and 1, is selected as a weight of optimism The nearer is to 1, the more optimistic is the decision-maker

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11 Hurwicz Criterion The maximum payoff under each decision alternative is multiplied by and the minimum payoff under each decision alternative is multiplied by 1 - These weighted products are summed for each decision alternative, resulting in a weighted value for each decision alternative The maximum weighted value is selected, and the corresponding decision alternative is chosen

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12 Decision Table State of the Economy Stagnant SlowRapid GrowthGrowth MaximumMinimum Stocks -$500 $700$2200 $2200-$500 Bonds -$100 $600$900 $900-$100 CD’s $300 $500$750 $750$300 Mixture -$200 $650$1300 $1300-$200

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13 Decision Table Suppose we are more optimistic than pessimistic and select = 0.7 for the weight of optimism. The calculations of weighted values for each decision alternative are as follows: stocks($2200)(.7) + (-$500)(.3) = $1390 bonds($900)(.7) + (-$100)(.3) = $600 CD’s($750)(.7) + ($300)(.3) = $615 mixture($1300)(.7) + (-$200)(.3) = $850

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14 Decision Table The Hurwicz criterion leads the decision-maker to choose the maximum of these values, $1390 The result under the Hurwicz criterion with = 0.7 is to choose stocks as the decision alternative An advantage of the Hurwicz criterion is that it allows the decision-maker the latitude to explore various weights of optimism A decision-maker’s outlook might change from scenario to scenario and from day to day

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15 Decision Table In this case, if we had been fairly pessimistic and chosen an of 0.2, the result would have been stocks($2200)(.2) + (-$500)(.8) = $40 bonds($900)(.2) + (-$100)(.8) = $100 CD’s($750)(.2) + ($300)(.8) = $390 mixture($1300)(.2) + (-$200)(.8) = $100 Under this scenario, the decision-maker would choose the CD option because it yielded the highest weighted payoff ($390) with = 0.2

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16 Decision Alternatives for Various Values of stocksbondsCD’smixture 1- max 2200max 900max 750max 1300 min -500min -100min 300min -200.01-500-100300-200.1.9-2300345-50.2.840100390100.3.7310200435250.4.6580300480400.5.5850400525550.6.41120500570700.7.31390600615850.8.216607006601000.9.119308007051150 1022009007501300 Bold indicates the choice given for the value

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17 Graph of Hurwicz Criterion for Selected Values of $2300 2100 1900 1700 1500 1300 1100 900 700 500 300 100 -100 -300 -500 0.1.2.3.4.5.6.7.8.9 1

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18 Decision Table stocks weighted payoff = CD’s weighted payoff $2200() + (-$500)(1 - ) = 750() + 300(1 - ) 2200 - 500 + 500 = 750 + 300 - 300 2250 = 800 = 0.3555 At = 0.3555, both stocks and CD’s yield the same payoff under the Hurwicz criterion. For values less than = 0.3555, CD’s are the chosen investment. Neither bonds nor mixture produce the optimum payoff under the Hurwicz criterion for any value of .

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19 First State of Economy Suppose the state of economy turns out to be stagnant. The optimal decision choice would be CDs, which pay off $300. Any other decision would lead to an opportunity loss. The opportunity loss for each decision alternative other than CDs can be calculated by subtracting he decision alternative payoff from $300. Stocks$300-(-$500)= $800 Bonds$300-(-$100)= $400 CDs$300-($300) = $0 Mixture$300-(-$200)= $500

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20 Second State of Economy The opportunity losses for the slow-growth state of economy are calculated by subtracting each payoff from $700, because $700 is the maximum payoff that can be obtained under this state; any other payoff is an opportunity loss. These opportunity losses are: Stocks$700-($700)= $0 Bonds$700-($600)= $100 CDs$700-($500) = $200 Mixture$700-($650)= $50

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21 Third State of Economy The opportunity losses for a rapid-growth state of economy are Stocks$2200-($2200)= $0 Bonds$2200-($900)= $1300 CDs$2200-($750) = $1450 Mixture$2200-($1300)= $900

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22 Minimax Regret The strategy of minimax regret is based on lost opportunity Lost opportunity occurs when a decision-maker loses out on some payoff or portion of a payoff because he or she chose the wrong decision alternative. In analyzing decision-making situations under uncertainty, an analyst can transform a decision table (payoff table) into an opportunity loss table, which can be used to apply the minimax regret criterion.

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23 Opportunity Loss Table State of the Economy Stagnant Slow Rapid GrowthGrowth Stocks $800 $0 $0 InvestmentBonds $400$100$1300 Decision AlternativeCD’s $0$200$1450 Mixture $500 $50 $900

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24 Opportunity Loss Table In summary, the maximum regrets under each decision alternative are stocks$800 bonds$1300 CD’s$1450 mixture$900 In making a decision based on a minimax regret decision-maker examines the maximum regret under each decision alternative given and selects the minimum of these

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25 Opportunity Loss Table The result is the stocks option, which has the minimum regret of $800 An investor who wants to minimize the maximum regret under the various states of the economy will choose to invest in stocks under the minimax regret strategy

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26 Summary of Example Decision - Analysis Under Uncertainty Decision Criteria Maximin Hurqicz ( = 0.7) Minimax Regret Investment Decision Alternative Stocks CDs Stocks Payoff $2200 $300 $1390 $800

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