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BU BU Decision Models DecisionAnalysis1 Summer 2013

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BU DecisionAnalysis2 Let us flip a fair coin 1000 (there is a fee). If you win a toss, I give you $102. If I win, you give me $100. How much fee will you pay for the playing the entire game? Let us flip a fair coin once (there is a fee). If you win I give you $102. If I win, you give me $100. How much fee will you pay me for playing the game: $5, $2, $1, $0? You can select any other amount. Suppose you are getting ready to go the office in a crowded metro. Carrying an umbrella is a hassle; you will carry it only when you feel necessary. Forecast for today is 70% chance of rain and the sky is overcast. Should you carry an umbrella - Yes or no? My decision would be “Yes” and it is a good decision. However, there are two possible outcomes - it will rain or not. If it does not rain, it does not mean I have made a bad decision.

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BU DecisionAnalysis3 Decision Analysis (DA) DA is a methodology applicable to analyze a wide variety of problems. DA is a methodology applicable to analyze a wide variety of problems. Although DA was used in the 1950s (at Du Pont) and early 1960s (at Pillsbury), major DA development took place in mid sixties. One of the earliest application (at GE) was to analyze whether a super heater should be added to the current power reactor. Although DA was used in the 1950s (at Du Pont) and early 1960s (at Pillsbury), major DA development took place in mid sixties. One of the earliest application (at GE) was to analyze whether a super heater should be added to the current power reactor. DA has been considered as a technology to assist (individuals and) organizations in decision making by quantifying the considerations (even though they may be subjective) to deduce logical actions. DA has been considered as a technology to assist (individuals and) organizations in decision making by quantifying the considerations (even though they may be subjective) to deduce logical actions.

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BU DecisionAnalysis4 Decision Analysis (DA) One can discuss many topics listed below; we will look at a few. Problem Formulation.Problem Formulation. Decision Making with / without Probabilities.Decision Making with / without Probabilities. Risk Analysis and Sensitivity Analysis.Risk Analysis and Sensitivity Analysis. Decision Analysis with Sample / Perfect Information.Decision Analysis with Sample / Perfect Information. Multistage decision making.Multistage decision making. Tools and terminology Basic statistics and probabilityBasic statistics and probability Influence diagram / payoff table / decision treeInfluence diagram / payoff table / decision tree EMV: Expected Monetary ValueEMV: Expected Monetary Value EVSI / EVPI : Expected Value of Sample / Perfect InformationEVSI / EVPI : Expected Value of Sample / Perfect Information Bayes’ ruleBayes’ rule Decision vs. outcomeDecision vs. outcome Risk managementRisk management Minimax / maximin /Minimax / maximin / Utility theoryUtility theory

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BU DecisionAnalysis5 Decision analysis without probabilities Alternatives Economic Condition RecessionNormalBoom Project A Project B Project C Project D Example: There are four projects; I can select only one. The payoff table shows potential “payoff” depending upon likely economic conditions. Concepts covered: Payoff table. Different approaches: Maximax, maximin, minimax re gret Since the payoff in project C is higher than the payoff for D for every economic condition, we say that project C is dominant. We can eliminate project D from consideration.

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BU DecisionAnalysis6 MaximaxMaximaxAlternatives Economic Condition RecessionNormalBoom Project A Project B Project C If you are an optimist, you will decide on the basis of Maximax. Step 1: Pick the max value for each alternative Step 2:Then pick the alternative with max payoff.

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BU DecisionAnalysis7 MaximinMaximinAlternatives Economic Condition RecessionNormalBoom Project A Project B Project C : Pick the min value for each alternative If you are a conservative you will use Maximin. 2: Then pick the alternative with max payoff.

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BU Alternatives Regret Table RecessionNormalBoom Project A Project B Project C DecisionAnalysis8 Minimax Regret Alternatives Economic Condition RecessionNormalBoom Project A Project B Project C You are neither optimist nor conservative. Step 1: Calculate the maximum for each outcome. 4075| 7000| | 7000| Stet 2: Prepare “Regret Table” by subtracting each outcome cell value from its maximum. At least one number for each regret table outcome is zero and there are no negative numbers. Why?

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BU Alternatives Regret Table RecessionNormalBoom Project A Project B Project C DecisionAnalysis9 Minimax Regret.. Alternatives Economic Condition RecessionNormalBoom Project A Project B Project C Step 3: Pick the max value for each alternative Step 4: Pick the alternative with minimum regret. 4075| 7000| | 7000| 12080

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BU DecisionAnalysis10 General comments The above three approaches we used involved Decision Making without Probabilities. Table columns show outcomes (also called state of nature ). Payoff table Alternatives Economic Condition RecessionNormalBoom Project A Project B Project C The maximax payoff criterion seeks the largest of the maximum payoffs among the actions.The maximax payoff criterion seeks the largest of the maximum payoffs among the actions. The maximin payoff criterion seeks the largest of the minimum payoffs among the actionsThe maximin payoff criterion seeks the largest of the minimum payoffs among the actions. The minimax regret criterion seeks the smallest of the maximum regrets among the actions.The minimax regret criterion seeks the smallest of the maximum regrets among the actions.

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BU DecisionAnalysis11 Decisionpoint Decision analysis with probabilities Typically, we use a tree diagram for the decision analysis. 1. A decision point is shown by a rectangle 20%55% 25% Chance events must be mutually exclusive and exhaustive (total probability = 1). 4. At the end of each branch is an endpoint shown as a triangle where a payoff will be identified. 2. Alternatives available at a decision point are shown as decision branches (DB). 3. At the end of each DB, there can be two or more chance events shown by a node and chance branches (CB). CB DB

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BU DecisionAnalysis12 Decision analysis with probabilities At the chance node, we calculate the average (i.e. expected) payoff. The terminology used is Expected Monetary Value (EMV) Decision point: Chance event : End point: DB: Decision Branch CB: Chance Branch If there is no chance event for a particular decision branch, it’s EMV is equal to the payoff. 20% 55% 25% DB CB We select the decision with the highest EMV. What if we are dealing with costs?

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BU DecisionAnalysis13 A larger tree diagram

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BU DecisionAnalysis14 You bought 500 units of each. Demand: X Demand: X Pr(X) Obviously, if demand exceeds 500, you will sell all 500. On the other hand, if demand is under 500, you will have leftover units. These leftover items can disposed off for $7 each ($3 loss, the dealer will no longer buy these leftover units from you). You can sell these yourself for $16 each ($6/unit profit) but the demand is uncertain. The demand distribution is shown in the table. A dealer has offered to buy these from each ( you can make $4/unit profit). What’s your decision? Example 1

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BU DecisionAnalysis15 Suppose you have 500 units of X in stock, purchased for $10 each. Dealer sales price:$14, self sale price:$16 with salvage value:$7. Demand: X Pr(X) DealerSale Self sale Example 1.. Start with the tree having 2 branches (DB) at the decision point. There are no chance events in the dealer sale branch, 500, 20% 600, 5% 400, 45% 300, 30% For the self sale, there are 4 mutually exclusive possibilities.

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BU DecisionAnalysis16 Suppose you have 500 units of X in stock, purchased for $10 each. Dealer sales price:$14, self sale price:$16 with salvage value:$7. Demand: X Pr(X) Example 1... DealerSale Self sale 500, 20% 600, 5% 400, 45% 300, 30% Payoff = 500*4 = 2000 EMV = 2000 Payoff = 300*6 – 200*3 = 1200 Payoff = 400*6 – 100*3 = 2100 Payoff = 500*6 = 3000 EMV = 0.3* * * *3000 = 2055 Your decision?

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BU DecisionAnalysis17 Risk Profile Payoff = 1200 Payoff = 2100 Payoff = 3000 Self Sale % 5% 45% 30% Risk profile is the probability distribution for the payoff associated with a particular action. The risk profile shows all the possible economic outcomes and provides the probability of each: it is a probability distribution for the principal output of the model.

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BU DecisionAnalysis18 Example 3 We have received RFP (Request For Proposal). We may not want to bid at all (our cost: 0) We may not want to bid at all (our cost: 0) If we bid, we will have to spend $5k for proposal preparation. Based on the information provided in the RFP, a quick decision is to bid either $115k or $120k or $125k. If we bid, we will have to spend $5k for proposal preparation. Based on the information provided in the RFP, a quick decision is to bid either $115k or $120k or $125k. We must select among 4 alternatives (including no bid). A quick estimate of the cost of the project (in addition to the preparation cost) is $95k. A quick estimate of the cost of the project (in addition to the preparation cost) is $95k. Looks like we may have a competitor. Looks like we may have a competitor. If we bid the same amount as the competitor, we will get the project because of our reputation with the client. If we bid the same amount as the competitor, we will get the project because of our reputation with the client. We have gathered some probabilities based on past experience. We have gathered some probabilities based on past experience.

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BU DecisionAnalysis19 Our bid (OB) must be 0 (no bid), 115, 120 or 125. Competitor’s bid (CB): 0, under 115, 115 to under 120, 120 to under 125, 125 and over. Assumption: If bids are equal, we get the contract. Information : Preparation cost: $5 + Cost of work : $95 = $100 total Profit for our bid All numbers in thousand dollars Use mini-max, maxi-max, etc? There are probabilities involved. Example 3.. Competitor’s bid 1. No bid 2a. Under $115 2b. $115 to under $120 2c. $120 to under $125 2d. Over $

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BU DecisionAnalysis20 1. There is a 30% probability that the competitor will not bid. 2. If the competitor does bid, there is (a) 20% probability of bid under $115. (b) 40% probability of bid $115 to under $120. (c) 30% probability of bid under $120 to under $125. (d) 10% probability of bid over $125. Example 3… Prob. - 20% 40% 30% 10% Prob. 30% 70% Profit for our bid Competitor’s bid No bid a. Under $ b. $115 to under $ c. $120 to under $ d. Over $ Actual Prob. 30% 14% 28% 21% 7%

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BU DecisionAnalysis21 Example 3: Profit for our bid Competitor No bid a. < $ b. $115 to < $ c. $120 to < $ d. > $ Actual Prob. 30% 14% 28% 21% 7% $0 No bid bid $115 WinLose Payoff = (-5), Probability 14% Payoff = 15, Probability 86% (-5)*(0.14) + 15 * (0.86) = $12.2

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BU Profit for our bid Competitor No bid a. < $ b. $115 to < $ c. $120 to < $ d. > $ DecisionAnalysis22 Example 3: Actual Prob. 30% 14% 28% 21% 7% 20, 58% -5, 42% 25, 37% -5, 63% 15, 86% -5, 14% Bid$120 Bid$115 Bid=$125LW L W L W No bid $0 $9.5 $6.1 $12.2 Our decision? We will now use Excel to solve the problem.

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BU DecisionAnalysis23 Ex. 3: Excel =SUMPRODUCT(Profit_bid_115,Probabilities) =MAX(D9:G9) INDEX+MATCH HLOOKUP ? Value we are looking (12.2) is not in the ascending order in the table.

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BU Example 3: Sensitivity analysis What if 30% probability of no bid from competitor is incorrect? 24DecisionAnalysis We can build a one variable data table. Variable: Competitor’s no bid probability. We select two outputs: bid and (corresponding maximum) profit.

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BU Profit for our bid Competitor’s bid No bid a. Under $ b. $115 to under $ c. $120 to under $ d. Over $ Ex. 3: DA and value of information Our decision was to bid $115 and EMV was $12.2. Suppose we get competitor’s bid information. Can we improve our profit? What is the probability? Earned Value of Perfect Information (EVPI) = $17.65 – $12.2 = $5.45 EMV = 0.3* *0+0.28* * *25 = * 0.2 = * 0.4 = * 0.3 = * 0.1 = 0.07 Sometimes we may have partial information. DecisionAnalysis

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BU $0 No bid OB=$115 $15 -$5 $15 $15 $15OB=$120 OB=$125 CB=0CB 30% 70% 15(.3)+11(.7) = $ % 70%$20$5 $9.5 30% 70% $25 -$2 $6.1 < to < to < 125 >125 30% 10% 40% 20% EMVPayoff 26DecisionAnalysis -5(.2)+15( ) = $11 bid $12.2 Our decision Example 3: Alternate method

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BU Values 12.2, 9.5 and 6.1 represent Expected Monetary Values (EMV). This line indicates the decision made. This line indicates the decision made. This is called folding back the decision tree. 27DecisionAnalysis $0 OB=$115 OB=$125 $12.2 $9.5 $6.1 bid $12.2 OB=$120 No bid Example 3…..

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BU DecisionAnalysis28 Utility theory Consider the gambling problems again. –Let us flip a fair coin once. –If you win I give you $102 –If I win, you give me $100 –How much will you pay me to play this game: $5, $2, $1, $0 ? Consider another gamble –Let us flip the same coin (500 times) with the same payoffs –How much will you pay me to play this game? Different people will pay different amounts to play the first gameDifferent people will pay different amounts to play the first game Expected payoff in the first game is $1 but most people do not want to play the game at all. Why? Losing $100 is a bigger event than winning $102 Most people will play the second game.Most people will play the second game. Still differ in how much they will pay. For most people a gain that is twice as big is not twice as good.For most people a gain that is twice as big is not twice as good. A loss of twice as much is more than twice as bad.A loss of twice as much is more than twice as bad. People’s attitude towards risk can be categorized as: risk averse, risk seeker and risk neutral.People’s attitude towards risk can be categorized as: risk averse, risk seeker and risk neutral. A common way to express it is through the decision-maker’s utility function.A common way to express it is through the decision-maker’s utility function.

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BU DecisionAnalysis U(100)U(0)U(100)U(0)U(100) U(0 ) U(100)U(0)U(100)U(0)U(100) U(0 ) Risk seeker Risk averse Risk neutral Utility is a measure of relative satisfaction. We can plot a graph of amount of money spent vs. “utility” on a 0 to 100 scale. Typical shapes for different types of risk takers generally follow the patterns shown below. Graphs above show that to achieve 50% utility, risk seekers will pay maximum, risk averse will pay minimum and risk neutral will pay an average amount.

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