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Decision Analysis ADM2302 ~ Rim Jaber

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Introduction LP models were all formulated under the assumption that certainty existed. Several decision making techniques are available to aid the decision maker in dealing with the type of decision situation in which there is uncertainty. ADM2302 ~ Rim Jaber

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**Learning objectives List steps of decision making process.**

Describe different types of decision making environments. Make decisions under uncertainty when probabilities are not known. Make decisions under risk when probabilities are known. Develop accurate and useful decision trees. Revise probability estimates using Bayesian analysis (may not be discussed) Understand the importance and use of utility theory in decision making. ADM2302 ~ Rim Jaber

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**Components of Decision Making**

Decisions themselves States of Nature: the uncontrollable events that may occur in the future Example: Payoff Tables: ADM2302 ~ Rim Jaber

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Payoff Tables A mean of organizing and illustrating the payoffs from the different decision (alternative), given the various states of nature in a decision problem. Payoff Table States of Nature a b Decisions (Alternatives) Payoff1a Payoff2a Payoff1b Payoff2b 1 2 ADM2302 ~ Rim Jaber

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**Types of Decision Making Environments**

Decision Making under Certainty Example Decision Making under Risk (Decision making with probability) Decision Making Under Uncertainty (Decision making without probability) ADM2302 ~ Rim Jaber

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**The Six Steps in Decision Theory**

1. Clearly define the problem at hand 2. List all the possible alternatives (decisions to be made) 3. Identify the possible outcomes (state of nature) of each alternative 4. List the payoff or the profit of each combination of alternatives and outcomes 5. Select one of the mathematical decision theory models (e.g. Decision Making under Risk) 6. Apply the model and make your decision

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**The Six Steps in Decision Theory**

Example An investor is going to purchase one of three types of real estate. The investor must decide among an apartment building, office building and a warehouse. The future states of nature that will determine how much profit the investor will make are good economic conditions and poor economic conditions. The profits that will result from each decision in the event of each state of nature are shown in the following table. ADM2302 ~ Rim Jaber

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**Payoff Table for the real Estate Investments**

States of Nature Good Economic Conditions Decisions (Purchase) Bad Economic Conditions Apartment Building Office Building Warehouse $ 50,000 100,000 30,000 $ 30,000 - 40,000 10,000 ADM2302 ~ Rim Jaber

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**Decision Making Under Uncertainty**

Four Criteria 1. MAXIMAX - find the alternative that maximizes the maximum outcome for every alternative 2. MAXIMIN - find the alternative that maximizes the minimum outcomes for every alternative 3. EQUALLY LIKELY- find the alternative with the highest average outcome 4. MINIMAX REGRET- minimizes the maximum regret (regret is the difference between the payoff from the best decision and all the other decision payoffs) ADM2302 ~ Rim Jaber

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**The Optimistic Approach**

Maximax Criterion: The Optimistic Approach States of Nature Maximum Payoff ($) Decisions (Purchase) Good Economic Conditions Bad Economic Conditions Apartment Building Office Building Warehouse $ 30,000 - 40,000 10,000 $ 50,000 100,000 30,000 50,000 100,000 30,000 Max ADM2302 ~ Rim Jaber

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**The Conservative Approach**

Maximin Criterion: The Conservative Approach States of Nature Minimum Payoff ($) Decisions (Purchase) Good Economic Conditions Bad Economic Conditions Apartment Building Office Building Warehouse $ 30,000 - 40,000 10,000 $ 50,000 100,000 30,000 30,000 -40,000 10,000 Max ADM2302 ~ Rim Jaber

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**Equally Likely Criterion**

States of Nature Row Average ($) Decisions (Purchase) Good Economic Conditions Bad Economic Conditions Apartment Building Office Building Warehouse $ 30,000 - 40,000 10,000 $ 50,000 100,000 30,000 40,000 30,000 20,000 equally likely ADM2302 ~ Rim Jaber

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**The Minimax Regret Criterion**

Construct a regret table by calculating for each state of nature the difference between each payoff and the best payoff for that state of nature. Find the maximum regret for each alternative. Select the alternative with the minimum of these values ADM2302 ~ Rim Jaber

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**The Minimax Regret Criterion**

States of Nature Maximum Regret ($) Decisions (Purchase) Good Economic Conditions Bad Economic Conditions Apartment Building Office Building Warehouse $ 0 70,000 20,000 $ 50,000 70,000 50,000 70,000 Min ADM2302 ~ Rim Jaber

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**Decision Making Under Risk (probabilities assigned to the states of nature)**

Decision Criteria The Maximum Likehood Criterion Expected Monetary Value (EMV) = Expected Payoff (EP) The Expected Opportunity loss Expected value of Perfect Information ADM2302 ~ Rim Jaber

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**Decision Making Under Risk- Cont’d**

Decision Maker must first estimate the probability of occurrence of each state of nature (prior probabilities) Once these estimates have been made, then the decision criterion mentioned can be applied ADM2302 ~ Rim Jaber

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**Real Estate Investment Example**

Let us suppose that based on several economic forecasts, the investor is able to estimate 0.6 probability that good economic conditions will prevail and 0.4 probability that poor economic conditions will prevail in the future. ADM2302 ~ Rim Jaber

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**The Maximum Likehood Criterion**

Identify the state of nature with the largest Probability. Choose the decisions alternative that has the largest Payoff ADM2302 ~ Rim Jaber

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**Payoff Table for the real Estate Investments**

States of Nature Good Economic Conditions Decisions (Purchase) Bad Economic Conditions 60% 40% Apartment Building Office Building Warehouse $ 30,000 - 40,000 10,000 $ 50,000 100,000 30,000 ADM2302 ~ Rim Jaber

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**Expected payoff (EP) Criterion**

EP (alternative i/decision i) = (outcome of first state of nature)*(its prob.) + (outcome of second state of nature)*(its prob.)+…+ (outcome of last state of nature) * (its prob.) The Best decision is the one with the greatest EP ADM2302 ~ Rim Jaber

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**Payoff Table for the real Estate Investments**

States of Nature Good Economic Conditions Decisions (Purchase) Bad Economic Conditions 60% 40% Apartment Building Office Building Warehouse $ 30,000 - 40,000 10,000 $ 50,000 100,000 30,000 ADM2302 ~ Rim Jaber

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**Expected Monetary Value (EP)--Cont’d**

The EP means that if the decision situation of purchasing an office building occurred a large number of times, an average payoff of $44,000 would result If the payoffs were in terms of costs, the best decision would be the one with the lowest EP ADM2302 ~ Rim Jaber

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**Expected Opportunity Loss(EOL)**

Alternative approach in decision making under risk is to minimize expected opportunity loss (EOL). Opportunity loss, also called regret, refers to difference between optimal profit or payoff and actual payoff received. EOL for an alternative is sum of all possible regrets of alternative, each weighted by probability of state of nature for that regret occurring. EOL (alternative i) = (regret of first state of nature) x (probability of first state of nature) + (regret of second state of nature) x (probability of second state of nature) (regret of last state of nature) x (probability of last state of nature) ADM2302 ~ Rim Jaber

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**EOL: Opportunity loss table (=regret table)**

States of Nature EOL ($) Decisions (Purchase) Good Economic Conditions Bad Economic Conditions 0.6 0.4 Apartment Building Office Building Warehouse $ 0 70,000 20,000 $ 50,000 70,000 30,000 28,000 50,000 Min ADM2302 ~ Rim Jaber

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EOL Using minimum EOL as decision criterion, best decision would be second alternative, “purchase an office building" with an EOL of $28,000. Minimum EOL will always result in same decision alternative as maximum EMV. ADM2302 ~ Rim Jaber

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**Expected Value of Perfect Information (EVPI)**

Is used to place an upper limit on what you should pay for information that will aid in making a better decision. Is the increase in the EP that could be obtained if it were possible to learn the true state of nature before making the decision. Is the difference between the expected value under certainty and the expected value under risk ADM2302 ~ Rim Jaber

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**Example: Payoff Table for the Real Estate Investments**

States of Nature Good Economic Conditions Decisions (Purchase) Bad Economic Conditions 60% 40% Apartment Building Office Building Warehouse $ 30,000 - 40,000 10,000 $ 50,000 100,000 30,000 ADM2302 ~ Rim Jaber

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**Expected value under certainty= Expected value with perfect information**

If market is booming Invest in the office building ($100,000). If market is not so booming invest in an apartment building ($30,000). There is a 60% chance that a payoff will be $100,000 and a 40% chance that it will be $30,000 for an expected value under certainty of 0.6x100, x30,000 = 72,000 ADM2302 ~ Rim Jaber

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**Expected value under certainty= Expected value with perfect information**

Expected value with perfect information= (best outcome for first state of nature)x(its prob.) + (best outcome for second state of nature)x(its prob.) +…+(best outcome for last state of nature)x(its prob.) 0.6x100, x30,000 = 72,000 $72,000 is the expected on average return, in the long run, if we have perfect information before a decision is to be made (we are certain which state of nature is going to occur before a decision has to be made). ADM2302 ~ Rim Jaber

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Computing EVPI Expected Value of Perfect Information=EVPI = Expected value with perfect information - maximum EP = (expected value under certainty) – (the expected value under risk) = 72,000 – 44,000 = $28,000 $28,000 is the maximum amount that would be paid to gain information that would result in a decision better that the one made without perfect information ADM2302 ~ Rim Jaber

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Example ADM2302 ~ Rim Jaber

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**Solution p(SL) = p(BD) = 2 p(RB) p(SL) = 0.4, p(BD) = 0.4, p(RB) = 0.2**

EP(GasPro)=0.4* * *(-100)=20 EP(Moscow)=0.4* *(-50) + 0.2*(-500) =-40 EP (TzarBank)=0.4* * *(-1000) = 40 Choose TzarBank ADM2302 ~ Rim Jaber

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Decision Trees A Graphical diagram used for making decisions. It represents the sequence of events in a decision situation. What are the benefits and advantages of decision trees? ADM2302 ~ Rim Jaber

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**Decision Trees Symbols used in decision tree:**

:A decision node from which one of several alternatives may be selected. The branches emanating from them reflect the alternative decisions possible at that point. :A state of nature node out of which one state of nature will occur. The branches emanating from them indicate the state of nature that can occur. ADM2302 ~ Rim Jaber

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**Structure or draw the decision tree **

Analyzing Problems with Decision Trees The Five Steps Define the problem Structure or draw the decision tree Assign probabilities to the states of nature Estimate the payoffs for each possible combination of alternative and state of nature Solve the problem by computing expected payoff (EP) for each state of nature node Make your decision ADM2302 ~ Rim Jaber

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**A State of Nature Node (Probability Node)**

Investor’s Decision Tree A State of Nature Node (Probability Node) Good Economic Condition (GEC) A Decision Node Poor Economic Condition (PEC) Purchase Apart. Building 2 GEC Office Building 1 3 PEC GEC Warehouse 4 PEC ADM2302 ~ Rim Jaber

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How to compute the EP Start with the final outcomes(payoffs) and work backward through the decision tree towards node 1 EP of the outcomes is computed at each probability node ADM2302 ~ Rim Jaber

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Decision Tree Example 1 This is just a beginning of ADM2302 course and Andrew does not know if he should attend all classes. He consulted some other students and came to the following conclusions: chances of passing a course while attending all classes are 80%; chances of passing a course while attending randomly are 50%. It is well known that professor who is teaching that course is giving second chance to the students who failed. They have to solve a pretty nasty case study. Again, Andrew estimates that chances of solving this case if he would go to all the classes are 60%, while they drop to just 10% if he would attend classes randomly. Andrew would be very happy if he passes the course (5 on a happiness scale of 0 - 5). Clearly, he would be very disappointed if he fails (0 on a happiness scale). Going to a classroom requires an effort and diminished happiness associated with passing the course. It goes down by 3 points (happiness scale) for attending all classes and 1 point for random attendance. ADM2302 ~ Rim Jaber

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**Solution Pass 5 - 3 = 2 0.8 Pass the case study Fail 5 - 3 = 2 0.6**

0.8x2+0.2x0=1.6 0.8 0.6x2+0.4x(-3)=0 Pass the case study Fail 5 - 3 = 2 0.6 Attend all 0.2 Fail the course 0.4 0 - 3 = -3 Randomly Pass 5 - 1 = 4 0.5 Fail Pass the case study 0.5x4+0.5x(-0.5)=1.75 5 - 1 = 4 0.5 0.1 Fail the course 0 - 1 = -1 0.9 0.9x(-1)+0.1x4= -0.5 ADM2302 ~ Rim Jaber

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**Decision Tree Example 2: House Insurance**

You are currently considering to insure the contents of your house against theft for one year. You estimate that the contents of your house would cost $20,000 to replace. Ottawa crime statistics indicate that there is a probability of 3% that your house will be broken into. In that event your loss would be 10%, 20%, or 40% of the contents with probabilities 0.5, 0.35 and 0.15 respectively. An insurance policy from General Accident (GA) costs $200/year but guarantees to replace any losses due to theft. An insurance policy from the Federation Insurance (FI) is cheaper and costs $100/year but you have to pay first $x (if x=200 then it is first $200) of any loss. An insurance policy from Prudential Insurance (PI) is even cheaper at $75/year, but it replaces only a fraction of y% (if y=60 then it is 60%) of any loss suffered. Of course, you could alternatively choose not to insure your house at all. Assume that there can be at most one theft a year.

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**House Insurance—Cont’d**

Decide what to do if x=50 and y=40 and your goal is to minimize your losses. Assuming that y=40, what would be the highest value for “x” so the insurance company selected in answer to “1” is still the best choice? ADM2302 ~ Rim Jaber

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**Solution for part 1 No theft 0.97 Theft -2000 (= 0.1x20,000) 0.03**

-108= 0x x0.03 -3600=-2000x x x0.15 No theft 0.97 Theft 10% loss with 0.5 -2000 (= 0.1x20,000) 0.03 20% loss with 0.35 -4000 (= 0.2x20,000) 40% loss with 0.15 -8000 (=0.4x20,000) No policy -200 No theft 0.97 -200 Theft -200 -200 GA 10% loss with 0.5 0.03 20% loss with 0.35 -200 40% loss with 0.15 FI -101.5=-100x x0.03 -200 No theft 0.97 -100 -150 Theft -150 (= ) or in general x 10% loss with 0.5 PI 0.03 -150 (= ) or in general x 20% loss with 0.35 40% loss with 0.15 -139.8=-75x x0.03 -150 (= ) or in general x No theft 0.97 -75 -2235 Theft 10% loss with 0.5 -1275 (= x(0.6)) or in general x(1-y/100) 0.03 20% loss with 0.35 -2475 (= x(0.6)) or in general x(1-y/100) 40% loss with 0.15 -4875 (= x(0.6)) or in general x(1-y/100)

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**Solution for part 2 EMV (no insurance) = -108 EMV(GA) = -200**

EMV (FI) = EMV (PI) = What should be “deductible” x? Compare FI with the next best option (no insurance) ( x)x x0.97 = -108 Approximately $267 ADM2302 ~ Rim Jaber

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**Sequential Decision Trees**

Introduction Example: We will alter our real estate investment example to encompass a 10-year period during which several decisions must be made. ADM2302 ~ Rim Jaber

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**2 6 4 1 3 7 5 Population Growth No Population Growth Population Growth**

Build Apart. Purchase an Apart. Build. 6 Population Growth (for 3 years) No Population Growth 4 1 Sell Land Purchase Land 3 Population Growth Develop Commercially No Population Growth (for 3 years) 7 5 No Population Growth Sell Land ADM2302 ~ Rim Jaber

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Module 5 Part 2: Decision Theory

Module 5 Part 2: Decision Theory

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