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# Decision tree

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### Decision tree

1. 1. DECISION TREE Bayesian Approach DR. KALPNA SHARMA,D E PA R T M E N T O F M AT H E M AT I C S M A N I PA L U N I V E R S I T Y J A I P U R 1
2. 2. DECISION TREES A decision tree is a chronological representation of the decision problem. Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives.  The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives. At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb. 2 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
3. 3. FIVE STEPS TO DECISION TREE ANALYSIS1. Define the problem.2. Structure or draw the decision tree.3. Assign probabilities to the states of nature.4. Estimate payoffs for each possible combination of alternatives and states of nature.5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node. 3 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
4. 4. EXAMPLE A developer must decide how large a luxurycondominium complex to build – small, medium, orlarge. The profitability of this complex depends uponthe future level of demand for the complex’scondominiums. 4 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
5. 5. ELEMENTS OF DECISION THEORYStates of nature: The states of nature could be defined as lowdemand and high demand.Alternatives: Developer could decide to build a small, medium,or large condominium complex.Payoffs: The profit for each alternative under each potentialstate of nature is going to be determined.We develop different models for this problem on the followingslides. 5 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
6. 6. PAYOFF TABLE THIS IS A PROFIT PAYOFF TABLE States of NatureAlternatives Low HighSmall 8 8Medium 5 15Large -11 22 (payoffs in millions) DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR 6
7. 7. DECISION TREE 8 8 5Medium Complex 15 -11 7 22DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
8. 8. EXAMPLE: BURGER PRINCE Burger Prince Restaurant is contemplatingopening a new restaurant on Main Street. It has threedifferent models, each with a different seating capacity.Burger Prince estimates that the average number ofcustomers per hour will be 80, 100, or 120. The payofftable (profits) for the three models is on the next slide. 8 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
9. 9. EXAMPLE: BURGER PRINCEPayoff Table Average Number of Customers Per Hour s1 = 80 s2 = 100 s3 = 120 Model A \$10,000 \$15,000 \$14,000 Model B \$ 8,000 \$18,000 \$12,000 Model C \$ 6,000 \$16,000 \$21,000 9 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
10. 10. EXAMPLE: BURGER PRINCEExpected Value Approach Calculate the expected value for each decision.The decision tree on the next slide can assist in thiscalculation. Here d1, d2, d3 represent the decision alternativesof models A, B, C, and s1, s2, s3 represent the states of natureof 80, 100, and 120. 10 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
11. 11. EXAMPLE: BURGER PRINCE Payoffs s1 .4 10,000 s2 .2 2 s3 15,000 .4 d1 14,000 s1 .4 d2 8,0001 s2 .2 3 18,000 d3 s3 .4 12,000 s1 .4 6,000 s2 .2 4 16,000 s3 .4 21,000 11 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
12. 12. EXAMPLE: BURGER PRINCEExpected Value For Each Decision EMV = .4(10,000) + .2(15,000) + .4(14,000) = \$12,600 d1 2 Model A EMV = .4(8,000) + .2(18,000) + .4(12,000) Model B d2 = \$11,6001 3 d3 EMV = .4(6,000) + .2(16,000) + .4(21,000) Model C = \$14,000 4 Choose the model with largest EV, Model C. DR. KALPNA SHARMA, 12 DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
13. 13. EXAMPLE PROBLEM: THOMPSON LUMBER COMPANY Thompson Lumber Company is trying to decide whether toexpand its product line by manufacturing and marketing a new product which is “backyard storage sheds.” The courses of action that may be chosen include: (1) large plant to manufacture storage sheds, (2) small plant to manufacture storage sheds, or (3) build no plant at all. 13 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
14. 14. THOMPSON LUMBER COMPANYProbability 0.5 0.5 14 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
15. 15. EXPECTED MONETARY VALUEThompson Lumber CompanyProbability of favorable market is same as probability of unfavorablemarket.Each state of nature has a 0.50 probability. 15 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
16. 16. CALCULATING THE EVPIBest outcome for state of nature "favorable market" is"build a large plant" with a payoff of \$200,000.Best outcome for state of nature "unfavorable market"is "do nothing," with payoff of \$0.Therefore, Expected profit with perfect informationEPPI = (\$200,000)(0.50) + (\$0)(0.50) = \$ 100,000If one had perfect information, an average payoff of\$100,000 could be achieved in the long run.However, the maximum EMV (EV BEST) or expectedvalue without perfect information, is \$40,000. 16Therefore, EVPI = \$100,000 - \$40,000 = \$60,000. DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, UNIVERSITY JAIPUR MANIPAL
17. 17. TO TEST OR NOT TO TEST Often, companies have the option to perform market tests/surveys, usually at a price, to get additional information prior to making decisions. However, some interesting questions need to be answered before this decision is made: How will the test results be combined with prior information? How much should you be willing to pay to test?The good news is that Bayes’ Theorem can be used tocombine the information, and we can use our decision tree to find EVSI, the Expected Value of Sample Information.In order to perform these calculations, we first need to know how reliable the potential test may be. 17 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
18. 18. MARKET SURVEY RELIABILITY IN PREDICTING ACTUAL STATES OF NATUREAssuming that the above information is available, wecan combine these conditional probabilities with ourprior probabilities using Bayes’ Theorem. 18 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
19. 19. MARKET SURVEY RELIABILITY IN PREDICTING ACTUAL STATES OF NATURE 19 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
20. 20. PROBABILITY REVISIONS GIVEN POSITIVE SURVEYAlternatively, the following table will produce the same results: 20 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
21. 21. PROBABILITY REVISIONS GIVEN NEGATIVE SURVEY 21DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
22. 22. PLACING POSTERIORPROBABILITIES ON THE DECISION TREEThe bottom of the tree is the “no test” part of the analysis;therefore, the prior probabilities are assigned to these events. P(favorable market) = P(FM) = 0.5P(unfavorable market) = P(UM) = 0.5 The calculations here will be identical to the EMVcalculations performed without a decision tree.The top of the tree is the “test” part of the analysis; therefore,the posterior probabilities are assigned to these events. DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR 22
23. 23. DECISION TREES FOR TEST/NO TEST MULTI-STAGE DECISION PROBLEMS 23 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
24. 24. DECISION TREE SOLUTIONThompson Lumber Company 24 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
25. 25. IN-CLASS PROBLEM 3Leo can purchase a historic home for \$200,000 or land in a growing area for\$50,000. There is a 60% chance the economy will grow and a 40% change it willnot. If it grows, the historic home will appreciate in value by 15% yielding a\$30,00 profit. If it does not grow, the profit is only \$10,000. If Leo purchases theland he will hold it for 1 year to assess the economic growth. If the economy grewduring the first year, there is an 80% chance it will continue to grow. If it did notgrow during the first year, there is a 30% chance it will grow in the next 4 years.After a year, if the economy grew, Leo will decide either to build and sell a houseor simply sell the land. It will cost Leo \$75,000 to build a house that will sell for aprofit of \$55,000 if the economy grows, or \$15,000 if it does not grow. Leo cansell the land for a profit of \$15,000. If, after a year, the economy does not grow,Leo will either develop the land, which will cost \$75,000, or sell the land for aprofit of \$5,000. If he develops the land and the economy begins to grow, he willmake \$45,000. If he develops the land and the economy does not grow, he willmake \$5,000. 25 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
26. 26. IN-CLASS PROBLEM 3: 2 SOLUTION Economy grows (.6) No growth (.4)Purchase Economy growshistoric home (.8) Build house 6 No growth (.2)1 4 Sell Economy grows landPurchase land (.6) 3 Economy grows (.3) No growth (.4) Develop land 7 No growth (.7) 5 Sell land 26 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
27. 27. IN-CLASS PROBLEM 3: SOLUTION \$22,000 Economy grows (.6) \$30,000 2 No growth (.4) \$10,000Purchase Economy growshistoric home \$55,000 \$47,000 (.8) Build house 6 \$35,000 No growth (.2) \$15,0001 4 Sell \$15,000 Economy grows \$47,000 landPurchase land (.6) 3 Economy grows \$45,000 \$17,000 (.3) \$35,000 No growth (.4) Develop land 7 No growth (.7) \$5,000 5 Sell land \$5,000 \$17,000 27 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
28. 28. SIMPLE EXAMPLE: UTILITY THEORYLet’s say you were offered \$2,000,000 right now on a chance to win\$5,000,000. The \$5,000,000 is won only if you flip a coin and gettails. If you get heads you lose and get \$0. What should you do? \$2,000,000 \$0 Heads (0.5) Tails (0.5) \$5,000,000 28 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
29. 29. Decision Trees 29DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
30. 30. Planning Tool 30DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
31. 31. DECISION TREES Enable a business to quantify decision making Useful when the outcomes are uncertain Places a numerical value on likely or potential outcomesAllows comparison of different possible decisions to be made 31 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
32. 32. DECISION TREES Limitations:How accurate is the data used in the construction of the tree? How reliable are the estimates of the probabilities? Data may be historical – does this data relate to real time? Necessity of factoring in the qualitative factors – humanresources, motivation, reaction, relations with suppliers and other stakeholders 32 DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
33. 33. Process 33DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
34. 34. THE PROCESS Economic growth rises Expected outcome 0.7 £300,000Expand by opening new outlet Economic growth declines Expected outcome -£500,000 0.3 Maintain current status £0 The circle denotes the point where different outcomes could occur. The estimates of the probability and the knowledge of the expected outcome allow theadenotes the pointuncertainty is maintain thethe economy – quo! This wouldcontinuesoutcome of is: A square firm to make a calculation of the likely return. In this example it where a decision is made, In this example, a business is contemplating There is also the outlet. The nothing and the state of current status if the economy have an to grow opening new option to do £0. healthily the option is estimated to yield profits of £300,000. However, if the economy fails to grow as Economicthe potentialrises:estimated£300,000 = £210,000 expected, growth loss is 0.7 x at £500,000. Economic growth declines: 0.3 x £500,000 = -£150,000 34 The calculation would suggest it is wise to go ahead with the decision ( a net ‘benefit’ figure. ofA +£60,000)A , U N IPVAERRTSMI E Y TJ A IFP U RA T H E M A T I C S , M A N I P A L DR K LPNA SHARM DE T N O M
35. 35. The Process Economic growth rises Expected outcome 0.5 £300,000Expand by opening new outlet Economic growth declines Expected outcome -£500,000 0.5 Maintain current status £0Look what happens however if the probabilities change. If the firm is unsure of thepotential for growth, it might estimate it at 50:50. In this case the outcomes will be:Economic growth rises: 0.5 x £300,000 = £150,000Economic growth declines: 0.5 x -£500,000 = -£250,000 35In this instance,D the A L P N benefit A , D E P A R T M E N T – Fthe TdecisionS ,looksPless favourable! R. K net A S H A R M is -£100,000 O M A H E M A T I C M A N I A L UNIVERSITY JAIPUR
36. 36. Advantages 36DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
37. 37. Disadvantages 37DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR
38. 38. 38DR. KALPNA SHARMA, DEPARTMENT OF MATHEMATICS, MANIPAL UNIVERSITY JAIPUR