Unformatted text preview: QMT437 Operations Research Pn Paezah TOPIC: DECISION THEORY – Part 2: Decision Tree Analysis LEARNING OUTCOMES At the end of this lesson, students should be able to: • Draw decision trees • Determine posterior (revised) probabilities using probability tree when conditional probabilities and prior probabilities are given. • Make decisions using decision tree technique. • Calculate Expected Value of Sample Information (EVSI). Decision Trees In decision tree analysis, a problem is depicted as a diagram which displays all possible acts, events, and payoffs (outcomes) needed to make choices at different points over a period of time. A decision tree contains four elements: 1. Decision nodes, decision maker; which indicate all possible courses of actions/alternatives open to the 2. Chance nodes, , which shows the uncertain events and all their possible outcomes; 3. Probabilities for each possible state of nature or outcome of a chance event, and 4. Payoffs, which summarize the consequences of each possible combination of choice and chance. Drawing a decision tree You start a decision tree with a decision that needs to be made. This decision is represented by a small square (box) node. From this box draw out lines towards the right for each possible alternative and write that alternative along the line. Keep the lines apart as far as possible so that you can expand your thoughts. Shown below is a decision node with three alternatives. Alternative 1 Alternative 2 Alternative 3 At the end of each alternative line, consider the results. 1 • • If the result of taking that decision is uncertain, draw a small circle. For example, if the payoff of Alternative 2 depends on whether State of Nature 1 or State of Nature 2, then draw a circle (state of nature) node and draw lines from this node and list each state of nature on these lines. If the result is another decision that needs to be made (e.g. Alternative 3), draw another square. Alternative 1 State of nature 1
Alternative 2 State of nature 2 Alternative A
Alternative 3 Alternative B Keep on doing this until you have drawn as many of the possible states of nature (or outcomes) and decisions as you can see leading on from your original decision. Once the tree is completed, include the fine details of the probabilities of outcomes and the cash flows that may occur at different times.
Alternative 1 Payoff 1
State of Nature 1
State of Nature 2
(Probability 2) Payoff 2
Alternative 2 Payoff 3 Alternative A
… Alternative 3 Alternative B
… Decision Tree Analysis Solving problems using decision trees involve five steps: 1. Define the problem. 2. Draw the decision tree. 3. Assign probabilities to states of nature. 4. Estimate payoffs for each possible combination of alternatives and states of nature. 5. Solve the problem by computing EMVs at each node. The calculation is (probability X payoff) = EMV. This is accomplished by working from the end points (right hand side) of the decision tree and folding it back towards the start (left hand side) choosing at each decision point the course of action with the highest expected monetary value (EMV). 2 Example 1: ABC Sdn. Bhd. needs to decide whether to switch to a new product or not. The product that the company is currently making provides a fixed payoff of RM150,000. If the company switches to the new product, its payoff depends on the level of sales. It is estimated that there are about 30% chance of high‐level sales (RM300,000 payoff), 50% chance of medium‐level sales (RM100,000 payoff), and 20% chance of low‐level sales (losing RM100,000). a) Develop a payoff table for this problem. Determine the best decision using Expected Monetary Value (EMV). b) Draw a decision tree and determine the best strategy. Note: All decision problems that can be represented by a payoff table can also be solved using decision trees. Exercise: Last year, Puan Mona was involved in an accident due to a faulty part in her newly purchased car. The accident has caused her to be out of job for 6 months. She is now suing the car manufacturer for all the hardships she has to endure. She can settle out of court for RM40,000, or go to court. If Puan Mona goes to court, there is a 40% chance that she will win the case. If she wins, a small and large settlement are equally likely (a small settlement nets RM28,000, and a large settlement nets RM225,000). If she loses she will incur a cost of RM6,000 for lawyer and court fee. Draw a decision tree for Puan Mona and recommend the best strategy for her. 3 Example 2: ABC Sdn. Bhd. needs to decide whether to switch to a new product or not. The product that the company is currently making provides a fixed payoff of RM150,000. If the company switches to the new product, its payoff depends on the level of sales. It is estimated that there are about 30% chance of high‐level sales (RM300,000 payoff), 50% chance of medium‐level sales (RM100,000 payoff), and 20% chance of low‐level sales (losing RM100,000). A survey which costs RM20,000 can be performed by a marketing research firm to provide information regarding the sales to be expected. Based on past records, if the survey shows high‐level sales, then there are about 60% chance of high‐level sales and 40% chance of medium‐level sales when the company sells the product. On the other hand, if the survey shows low‐level sales, then there are about 60% chance of medium‐level sales and 40% chance of low‐level sales when the company sells the product. The probability that the firm will predict a high‐level sales is 0.7, and a low‐level sales is 0.3. a) Determine the optimal strategy for ABC. b) Calculate the Expected Value of Sample Information. Is it worthwhile to pay RM20,000 to the marketing research firm to obtain information on expected future sales? Example 3: The Faculty of Computer Science (FCS) is developing a software to combat plagiarism which is a serious problem in many colleges. Based on its experience with other software projects, FCS estimates that the total cost to develop a prototype is RM700,000. If the software is a major success, FCS can sell the rights to the software to a software house for RM1.7 million. If it is a moderate success, then it can sell the right to the software house for RM0.6 million. However, if it is a failure, it will not be able to sell the software, and hence will lose all development costs incurred. FCS estimates the probability of a major success, a moderate success and a failure at 0.20, 0. 45, and 0.35, respectively. FCS can hire a consultant to review the idea for the new software and make recommendation on whether FCS should develop the prototype. This will cost FCS RM10,000. The reliability of the consultant’s information is given by the conditional probabilities shown in the following table: Given Actual Outcome Major Moderate Failure Success Success Consultant’s recommendation Recommended 0.7 0.8 0.1 on prototype Not Recommended 0.3 0.2 0.9 development For example, P(consultant recommends⎥ major success)= 0.7. Draw a decision tree for FCS and determine the faculty’s best strategy. 4 Comments: In FCS’s problem, without the consultant’s review, the faculty’s own estimates on the success rate of the prototype are: P (major success) = 0.20, P (moderate success) = 0.45, and P (failure) = 0.35. These are called prior probabilities. The probabilities given in the above table are referred to as conditional probabilities. These are: P(recommend⎥ major success) = 0.7 P(not recommend⎥ major success) = 0.3 P(recommend⎥ moderate success) = 0.8 P(not recommend⎥ moderate success) = 0.2 P(recommend⎥ failure) = 0.1 P(not recommend⎥ failure) = 0.9 To analyze the decision tree, however, we need to have posterior (revised) probabilities such as P(major success⎥ consultant recommends), P(moderate success⎥ consultant recommends), and etc. The revised probabilities can be obtained by constructing probability trees, using the definition of conditional probability P(A⎥B)= P( A ∩ B)
; where P( B) > 0 P( B) and applying Bayes theorem. According to Bayes Theorem, P(A⎥B) = P( B A) P( A)
P ( B A).P( A) + P( B A).P( A) 5 Exercises: 1. Your corporation has been presented with a new product development proposal. The cost of the development project is RM500,000. The probability of successful development is projected to be 70%. If the development is unsuccessful, the project will be terminated. If it is successful, the manufacturer must then decide whether to begin manufacturing the product on a new production line or a modified production line. If the demand for the new product is high, the incremental revenue for a new production line is RM1,200,000, and the incremental revenue for the modified production line is RM850,000. If the demand is low, the incremental revenue for the new production line is RM700,000, and the incremental revenue for the modified production line is RM150,000. All of these incremental revenue values are gross figures, i.e., before subtracting the RM500,000 development cost, RM300,000 for the new production line and RM100,000 for the modified production line. The probability of high demand is estimated as 40%, and of low demand as 60%. Draw a decision tree and determine the optimal strategy for your corporation. 2. Pantas Manufacturing Company must decide whether it should purchase or manufacture a component at its plant in Kota Bharu. The demand for the component varies from high, moderate, and low. The company also has a choice to do nothing if it finds that it is not profitable to sell the components. The table below summarizes the estimated profits (in thousand of ringgit) under different circumstances. Option Profit (RM’000) High Demand Manufacture Purchase Do nothing Moderate Demand Low Demand 80 60 0 50 30 0 ‐100 ‐70 0 a) The states of nature have the following probabilities: P(Low demand) = 0.30 P(Moderate demand) = 0.35 P(High demand) i) = 0.35 Decide the best course of action for Pantas. ii) How much should Pantas be willing to pay for perfect information? b) A marketing research to test the potential demand for the product is expected to report either favorable (F) or unfavorable (U) conditions. The relevant revised probabilities are as follows: P(High | F) = 0.55 P(High | U) = 0.23 6 P(Moderate | F) = 0.37 P(Moderate | U) = 0.34 P(Low | F) = 0.08 P(Low | U) = 0.43 P(F) P(U) = 0.62 = 0.38 Required: i.
iv. 3. Construct a decision tree to help Pantas choose the best decision. Calculate the expected monetary value required for making the decision. What is the expected value of sample information for the market research? What is the optimal decision strategy for Pantas? A developer intends to build double‐storey and bungalow houses in Melaka. He also has the option of not to proceed with his project. Given a favorable market, he will earn a profit of RM50,000 if he builds double‐storey houses and RM80,000 if he builds bungalow houses in that area. However, with an unfavorable market, he would lose RM30,000 with the double storey houses and RM 45,000 with bungalow houses. The probability of a favorable market is 0.6. He has the choice to obtain additional information from market research analyst at the cost of RM 10,000 and the probability that the result is positive is 0.7. A positive result from the research will increase the probability of a favorable market to 0.8. Furthermore, a negative result from the research will decrease the probability of a favorable market to 0.3. a) Construct a decision tree for the above situation. b) Analyze the decision tree and advice the developer for the best decision. c) If the cost to gather additional information is reduced to RM5,000, what is your advice? 4. [Final Exam, April 2007] The Semporna Manufacturing Company must decide whether to purchase a component from a supplier or manufacture the component at its plant in Melaka. If demand is high, Semporna Manufacturing could profitably manufacture the component. However, if demand is low, its unit manufacturing cost would be high due to underutilization of equipment. The following table shows the projected profit (RM’000) for Semporna Manufacturing’s make‐or‐
buy decision. Decision Alternative Demand Low (L) High (H) Manufacture component ‐20 100 Purchase component 20 70 7 The company estimates the probabilities of low and high demand at 0.45 and 0.55, respectively. Semporna Manufacturing is considering conducting a market research to study the potential demand for the component. The study is expected to report either favourable (F) or unfavourable (U) conditions. The reliability of the study is given below: P(F⎟ L) = 0.10 P(F⎟ H) = 0.60 a) Use a decision tree to recommend the optimal strategy for Semporna Manufacturing. b) How much is the market research information worth to Semporna Manufacturing? 5. Dina Cosmetics is considering the introduction of a new line of beauty products. In order to produce the new line, the company is considering either renovating the current plant or leasing another plant. The following payoff table has been developed by the company: Alternative Profit (RM000)
Favorable Market Unfavorable Market Renovate Plant 100 ‐40 Lease Plant 50 10 Do nothing 0 0 The company believes the probability of a favorable market to be 0.55 and the probability of an unfavorable market to be 0.45. Dina Cosmetics is considering contracting with a market research firm to do a survey to determine future market conditions. The market research firm wants a fee of RM12,000. The survey will indicate either positive or negative market conditions. There is a 60‐40 chance of a positive report. Furthermore, there is a 0.70 probability of a favorable market given a positive report, and a 0.90 probability of an unfavorable market given a negative report. a) Develop a decision tree to reflect the decision‐making problem for Dina Cosmetics. Determine the decision strategy the company should take, and the expected value of this strategy. b) Calculate the maximum amount Dina Cosmetics should pay for the survey. Is it worthwhile to pay RM12,000 for the survey? 8 ...
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- Spring '12
- consultant, FCS