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BU.520.601 BU.520.601 Decision Models DecisionAnalysis1 Summer 2013

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BU.520.601 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.520.601 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.520.601 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.520.601 DecisionAnalysis5 Decision analysis without probabilities Alternatives Economic Condition RecessionNormalBoom Project A 407550006100 Project B 0525012080 Project C 2500700010375 Project D 150060009500 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.520.601 DecisionAnalysis6 MaximaxMaximaxAlternatives Economic Condition RecessionNormalBoom Project A 407550006100 Project B 0525012080 Project C 2500700010375 If you are an optimist, you will decide on the basis of Maximax. Step 1: Pick the max value for each alternative. 6100 12080 10375 Step 2:Then pick the alternative with max payoff.

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BU.520.601 DecisionAnalysis7 MaximinMaximinAlternatives Economic Condition RecessionNormalBoom Project A 407550006100 Project B 0525012080 Project C 2500700010375 1: Pick the min value for each alternative. 4075 0 2500 If you are a conservative you will use Maximin. 2: Then pick the alternative with max payoff.

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BU.520.601Alternatives Regret Table RecessionNormalBoom Project A 020005980 Project B 407517500 Project C 157501705 DecisionAnalysis8 Minimax Regret Alternatives Economic Condition RecessionNormalBoom Project A 407550006100 Project B 0525012080 Project C 2500700010375 You are neither optimist nor conservative. Step 1: Calculate the maximum for each outcome. 4075| 7000| 12080 4075| 7000| 12080 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.520.601Alternatives Regret Table RecessionNormalBoom Project A 020005980 Project B 407517500 Project C 157501705 DecisionAnalysis9 Minimax Regret.. Alternatives Economic Condition RecessionNormalBoom Project A 407550006100 Project B 0525012080 Project C 2500700010375 Step 3: Pick the max value for each alternative. 5980 4075 1705 Step 4: Pick the alternative with minimum regret. 4075| 7000| 12080 4075| 7000| 12080

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BU.520.601 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 407550006100 Project B 0525012080 Project C 2500700010375 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.520.601 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.520.601 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.520.601 DecisionAnalysis13 A larger tree diagram

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BU.520.601 DecisionAnalysis14 You bought 500 units of X @$10 each. Demand: X Demand: X300400500600Pr(X)0.300.450.200.05 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 you @$14 each ( you can make $4/unit profit). What’s your decision? Example 1

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BU.520.601 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 300400500600Pr(X)0.300.450.200.05 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.520.601 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 300400500600Pr(X)0.300.450.200.05 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*1200 + 0.45*2100 + 0.2* 3000 + 0.05*3000 = 2055 Your decision?

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BU.520.601 DecisionAnalysis17 Risk Profile Payoff = 1200 Payoff = 2100 Payoff = 3000 Self Sale 300400 500 600 20% 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.520.601 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.520.601 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 0115120125 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 $125 0000015-515151520-5-5202025-5-5-525

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BU.520.601 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 0115120125 1. No bid 0152025 2a. Under $115 0-5-5-5 2b. $115 to under $120 015-5-5 2c. $120 to under $125 01520-5 2d. Over $125 0152025 Actual Prob. 30% 14% 28% 21% 7%

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BU.520.601 DecisionAnalysis21 Example 3: Profit for our bid Competitor 0115120125 1. No bid 0152025 2a. < $115 0-5-5-5 2b. $115 to < $120 015-5-5 2c. $120 to < $125 01520-5 2d. > $125 0152025 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.520.601 Profit for our bid Competitor 0115120125 1. No bid 0152025 2a. < $115 0-5-5-5 2b. $115 to < $120 015-5-5 2c. $120 to < $125 01520-5 2d. > $125 0152025 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.520.601 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.520.601 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.520.601 Profit for our bid Competitor’s bid 0115120125 1. No bid 0152025 2a. Under $115 0-5-5-5 2b. $115 to under $120 015-5-5 2c. $120 to under $125 01520-5 2d. Over $125 0152025 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*25+0.14*0+0.28*15+0.21*20+0.07*25 = 17.65 25 0.30 0.7 * 0.2 = 0.14 0.7 * 0.4 = 0.28 0.7 * 0.3 = 0.21 0.7 * 0.1 = 0.07 Sometimes we may have partial information. DecisionAnalysis

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BU.520.601 $0 No bid OB=$115 $15 -$5 $15 $15 $15OB=$120 OB=$125 CB=0CB 30% 70% 15(.3)+11(.7) = $12.2 30% 70%$20$5 $9.5 30% 70% $25 -$2 $6.1 <115 115 to <120 120 to < 125 >125 30% 10% 40% 20% EMVPayoff 26DecisionAnalysis -5(.2)+15(.4+.3+.1) = $11 bid $12.2 Our decision Example 3: Alternate method

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BU.520.601 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.520.601 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.520.601 DecisionAnalysis29 0 100 0 100 0 100 U(100)U(0)U(100)U(0)U(100) U(0 ) 0 100 0 100 0 100 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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