t2 decision theory

Ruzanita Mat RaniQMT 337 - FSKM

Topic 2: Decision Theory

Introduction

What is Decision Theory? is an analytical techniques to identify the optimal decision with several alternatives

and risk or uncertain pattern of future events.

What makes difference between good and bad decision?

i. A good decision is one that is based on logic, considers all available data and possible alternatives, and applied quantitative approach.

ii. A bad decision is one of that is not based on logic, does not use all available information, does not consider all alternatives and does not employ appropriate quantitative techniques.

The Six Steps in Decision Theory

1. Define the problem.

2. List the possible alternatives. Alternative is a course of action or a strategy that may be chosen by a decision

maker (DM).

3. Identify the possible outcomes or state of nature. State of nature (SON) is an outcome or occurrence over which the DM has little or no

control.

4. List the payoff or profit of each combination of alternatives and outcomes.

Payoff or profit is called conditional value. The easiest way to present these values is by constructing a decision table (payoff

table). States of nature

Alternatives

Payoffs

5. Select one of the mathematical decision theory models.

6. Apply the model and make your decision.

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Types of Decision Making Environment

The types of decisions people make depend on how much knowledge or information they have about the situation. Three decision making environments are:

1. Decision making under certainty DM knows with certainty the consequence of every alternatives or decision choice.

2. Decision making under risk There are several possible outcomes for each alternative and DM knows the probability of

occurrence of each outcome. The most popular methods of making decision under risk are EMV, EVPI and EOL.

3. Decision making under uncertainty There are several possible outcomes for each alternative and DM does not know the probability of the various outcomes.

Decision Making Under Risk

1. Expected monetary value (EMV) EMV is the weighted sum of possible payoffs for each alternative.

This is determined by multiplying monetary values by their respective probabilities. The results are then added to arrive at the EMV.

EMV (alternative I)= (payoff of the first SON) x (prob. of first SON) +(payoff of the second SON) x (prob. of second SON)+…..+(payoff of the last SON) x (prob. of last SON)

Example 2.1:

The manager’s problem is whether to expand the product line by manufacturing and marketing a new product, backyard storage sheds, the alternatives is either to construct a large plant, a small plant or no plant at all. The manager determines that there are only two possible outcomes, which are the market storage sheds could be favorable (high demand) or the market storage sheds could be unfavorable (low demand). The manager thinks that the payoff for each alternatives and state of nature is as table below.

AlternativesState of nature (RM)

Favorable market Unfavorable marketConstruct a large plant 200,000 -180,000Construct a small plant 100,000 - 20,000Do nothing 0 0

The manager believes that the probability of a favorable market is exactly the same as the probability of an unfavorable market (0.5, 0.5). Which alternatives would be the maximum EMV?

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Solution:


Favorable market Unfavorable marketConstruct a large plant 200,000 -180,000Construct a small plant 100,000 - 20,000Do nothing 0 0Probability 0.5 0.5

EMV (large plant) = (200,000)0.5 + (-180,000)0.5= 10,000EMV (small plant) = (100,000)0.5 + (-20,000)0.5 = 40,000EMV (do nothing) = (0)0.5 + (0)0.5 = 0

*The decision that manager should take is construct a small plant. 2. Expected Opportunity loss (EOL)

EOL is the cost of not picking the best solution sometimes called regret.

The amount you would lose by not picking the best alternative. For any SON, this is the difference between the consequences of any alternative and the best possible alternative.

The minimum expected opportunity loss is found by constructing an EOL table and computing for each alternative.

Step 1- Create the EOL table.- Subtracting each outcome in the column from the best outcome in the same

column.

Step 2- EOL is computed by multiplying the probability of each SON times the

appropriate EOL table.

*EVPI = Minimum EOL

Example 2.2:

From our previous example (2.1) which alternatives give minimum EOL?

Solution:Step 1 – Create a loss table


Favorable market Unfavorable marketConstruct a large plant 200,000 – 200,000 0 – (-180,000)Construct a small plant 200,000 – 100,000 0 – 20,000Do nothing 200,000 – 0 0 – 0

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Favorable market Unfavorable marketConstruct a large plant 0 180,000Construct a small plant 100,000 20,000Do nothing 200,000 0Probability 0.5 0.5

Step 2 – Compute EOL

EOL (large plant) = (0)0.5 + (180,000)0.5 = RM 90,000EOL (small plant) = (100,000)0.5 + (20,000)0.5 = RM 60,000EOL (do nothing) = (200,000)0.5 + ((0)0.5 = RM 100,000

*The decision that manager should take is construct a small plant.

Exercise 2.1

An investor is to purchase one of three types of real estate. The investor must decide among an apartment building, an 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 table below.

Decision State of natureGood economic Poor economic

Apartment 50,000 30,000Office 100,000 -40,000

Warehouse 30,000 10,000

Based on several economic forecasts, the investor is able to estimate a 0.60 probability that good economic conditions will prevail and a 0.40 probability that poor economic conditions will prevail. Calculate the expected value for each decision and select the best one.

Exercise 2.2

Farah Sufiah, president of FS Manufacturing is considering whether to build more manufacturing plants in Klang Valley. Her decision is summarized in the following table:

Alternatives FavorableMarket (RM)

UnfavorableMarket (RM)

Build a large plant 400,000 -300,000Build a small plant 80,000 -10,000Don’t build 0 0Probabilities 0.4 0.6

Construct an opportunity loss table. What is the best strategy based on EOL?

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3. Expected value of perfect information (EVPI)

EPVI is the expected outcome with perfect information minus the expected outcome without perfect information (maximum EMV).

The average or expected value of information if it were completely accurate. The increase in EMV that results from having perfect information.

Expected value with perfect information (EVwPI) is the average or expected value of the decision if you knew what would happen ahead of time. You have perfect knowledge.

EVwPI= (best outcome or consequence for first SON) x (prob. of first SON) + (best outcome or consequence for second SON)x (prob. of second SON)+

…..+ (best outcome or consequence for last SON) x (prob. of last SON)

EVPI= EVwPI – Maximum EMV

Example 2.3:

From our previous example (2.2), assume that the manager has been approached by the Scientific Marketing Inc. (SMI), a firm that proposes to help the manager make the decision about whether to build the plant to produce storage sheds. SMI claims that its technical analysis will tell the manager with certainty whether the market is favorable for his proposed product. SMI would charge the manager RM 65,000 for the information. Should the manager hire the firm to make the marketing study, is it worth RM 65,000?

Solution:

i. EVwPI



EVwPI = (200,000) 0.5 + (0) 0.5 = RM100,000

ii. Maximum EMV = RM40,000

iii. EVPI = EVwPI – Maximum EMV = RM100,000 – RM40,000 = RM60,000

*EVPI = minimum EOL = RM60,000

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* The most the manager would be willing to pay for perfect information is RM60,000 so the manager should not hire SMI to make the marketing study.

Exercise 2.3

The manager of a store’s purchasing department must decide on the orders for clothes for the coming festival month. Two new styles were introduced at the recent show. The payoff for the various state have been estimated as follows:

AlternativesMarket Acceptance

Style A Style B BothStyle A 75 25 80Style B 35 84 53Both 54 62 42

The probability for market acceptance of style A, style B and both styles are 0.35, 0.25 and 0.4 respectively.

What is the most the manager should pay for this perfect information?

Problems:

2.1 The manager of Excel Overseas Sdn. Bhd. is considering the possibility of opening a stationery shop at section 11, Shah Alam. His options are to open a large shop, small shop or no shop at all. The market for a stationery shop can be good, average or bad. The net profit or loss for the large or small shops for the various market conditions are given below. Building no shop at all yields no loss or no gain.

Decision State of nature

Good market Average market Bad marketLarge shop 250,000 90,000 -120,000Small shop 150,000 60,000 -80,000

No shop 0 0 0

The manager estimates that the probability of a good market is 0.4, an average market is 0.4 and a bad market is 0.2.

a) Determine the best alternative using EMV.b) Construct an EOL table. Determine the best alternative using EOL.c) Calculate the EVPI.

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2.2 Osman is considering several alternatives with regard to opening satay restaurants. His expected payoffs (RM) are shown below:

Alternative State of Nature (RM)Good Market Fair Market Poor Market

Open 1 restaurant 380,000 70,000 -400,000Open 2 restaurants 200,000 80,000 -200,000Do not open any restaurants

0 0 0

Osman believe that the probability the market will be good, fair or poor are 40%, 30% and 30% respectively. A market research firm will analyze the market conditions and will provide a perfect forecast of the future. What is the most that Osman should pay for this information?

2.3 Dr Hassan and wife are both experienced doctors. They are considering the construction of private clinic. If there is a favorable market for the clinic, they could realize a net profit of RM100,000. If the market is just able to sustain the clinic’s operations, they could be earning RM60,000. If the market is not favorable, they could lose RM40,000. Of course they have the alternative not to proceed at all, in which case there is no cost. Without any market data, the best that Dr Hassan and wife could guess is that there is a 50% chance that the clinic will be successful and a 25% chance that the clinic will be sustainable.

Using the minimum Expected Opportunity loss criteria, advice Dr Hassan and wife on whether they should proceed with the construction of the private clinic.

2.4 The manager of an electric shop has to decide how many units of EQUIP STAR have to be ordered at the beginning of each month. The demand for EQUIP Star is projected to be either 10, 15 or 20 units per month with the probability of 0.35, 0.40 and 0.25 respectively. Each unit costs RM200 and selling price is RM350 per unit.

a) Construct a payoff table for this problem.b) Determine the best alternative using Expected Monetary Value criterion?

2.5 Style Sdn. Bhd. Realized that orders for clothing from a particular manufacturer for the next year’s festive season must be placed now. The cost per unit for a particular dress is RM20 while the anticipated selling price is RM50. Any goods not sold during the season can be sold for RM15 to a local discount store. Demand is projected to be either 50, 60 or 70 units. There is 40% chance that demand will be 50 units, a 50% chance that the demand will be 60 units and a 10% chance that demand will be 70 units.

a) Draw the payoff table for Style Sdn. Bhd.b) How many units should be ordered under Expected Monetary value criterion?c) What is the Expected Value of Perfect Information.

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Decision Tree

Any problem that can be presented in a decision table can also be graphically illustrated in a decision tree.

All decision trees are similar in that they contain decision points or nodes and state of nature points or nodes:

A decision node from which one of several alternatives may be chosen

A state of nature node out which one state of nature will occur

Five steps of decision tree analysis

1. Define the problem.2. Draw the decision tree.3. Assign probabilities to the states of nature.4. Estimate payoffs for each possible combination of alternatives and the state of nature.5. Solve the problem by computing expected monetary value (EMV).

A simple decision tree for previous example:

*The decision that manager should take is construct a small plant.

When a decision situation requires only a single decision, an expected value payoff table will yield the same result as a decision tree. However, a decision table is usually limited to a single decision situation. A sequential decision tree illustrates a situation requiring a series of decisions.

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1

2

Constructlarge plant

Construct small plant

Do nothing

Favorable market (0.5)

Unfavorable market (0.5)



Payoffs

200,000

-180,000

100,000

-20,000

0

10,000

40,00040,000


Example 2.4:

From previous example (2.1), let say the manager has two decisions to make, with the second decision dependent on the outcome of the first. Before deciding a new plant, Ahmad has the option of conducting marketing research survey at a cost of RM10,000. The information from this survey could help the manager decide whether to construct a large plant, a small plant or not to build at all. The manager recognizes that such a market survey will not provide with perfect information, but it may help quite a big nevertheless. There is a 45% chance that the survey results will indicate positive market for storage sheds and 55% that the survey results will negative. 0.78 is the probability of a favorable market for the sheds given a positive result from the market survey. 0.22 probability that the market for sheds will be unfavorable given that the survey results are positive. 0.27 is the probability of a favorable market for the sheds given a negative result from the market survey. 0.73 probability that the market will be unfavorable given that the survey results are negative.



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First Decision Point Second Decision Point

*59,200-10,000 = 49,200

15

49, 2

00

7

6

5

4

3

2

1

40 ,0

0012

,400

116

,400

Payoffs

200,000

-180,000

100,000

-20,000

0

200,000

-180,000

100,000

-20,000

0

200,000

-180,000

100,000

-20, 000

0

Large plant

Small plant

Large plant

Small plant

Large plant

Small plant

Survey (0.45)Result positive

Survey (0.55)Result negative

Conduct market survey

Do not conduct

116,400

73,600

-77,400

12,400

10,000

40,000





No plant


Unfavorable (0.73)

Favorable (0.27)

Unfavorable (0.73)

No plant



Favorable (0.50)

Unfavorable (0.50)

No plant

*The best decision is to conduct market survey. If the survey results favorable construct a large plant, but if the survey

results unfavorable construct a small plant.

1)


2)

First Decision Point Second Decision Point

49,200

16

49, 2

00

7

6

5

4

3

2

1

40 ,0

002,

400

106

,400





No plant


Unfavorable (0.73)

Favorable (0.27)

Unfavorable (0.73)

No plant



Favorable (0.50)

Unfavorable (0.50)

No plant

Payoffs

190,000

-190,000

90,000

-30,000

-10,000

190,000

-190,000

90,000

-30,000

-10,000

200,000

-180,000

100,000

-20,000

0

Large plant

Small plant

Large plant

Small plant

Large plant

Small plant

Survey (0.45)Result positive

Survey (0.55)Result negative

Conduct market survey

Do not conduct

106,400

63,600

-87,400

2,400

10,000

40,000

*The best decision is to conduct market survey. If the survey results favorable construct a large plant, but if the survey

results unfavorable construct a small plant.


Exercise 2.4

A housing developer is considering three housing projects: Anggerik, Mawar and Dahlia. The developer has to decide which housing project to launch. The profit from each project depends on the demand for houses. The payoffs (million ringgit) from each project under high demand and low demand are given below,

House project High demand Low demandAnggerik 12 7Mawar 15 6Dahlia 10 8

The developer estimates that there is a 50-50 chance of a high demand.

For a fee of RM15000, an economist can be engaged to give an economic outlook. The economist claims that from past studies, the probability of a high demand for houses given a good economy is 0.8, and the probability of a low demand given a poor economy is 0.9. The probability of a good economy is 0.70.

Use a decision tree to determine the best strategy for the developer.

Bayesian Theorem

There are many ways of getting probability data for a problem. The numbers can be assessed by a manager based on experienced or intuition.

They can be derived from historical data or they can be computed from other available data using Bayes’ theorem.

Bayes’ theorem allows decision makers to revise probability values. The revised probabilities are called posterior probabilities (the altered prior probability of an event based on additional information).

Example 2.5:

From previous example (2.4) we made the assumption that the following four posterior probabilities (the revised probability of past events based on new information) were known:

P(favorable market (FM) survey results positive) = 0.78P(unfavorable market (UM) survey results positive) = 0.22P(favorable market (FM) survey results negative) = 0.27P(unfavorable market (UM) survey results negative) = 0.73

Recall that without any market survey information, decision maker estimates of a favorable and unfavorable market are:

P(FM) = 0.50

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P (UF) = 0.50

These are referred to as the prior probabilities (probability of single event occurring).

Joint probability is the probability both events will occur.

The general form of Bayes’ theorem:

P(AB)=

Where A.B = any two events complement of A

Additional information comes from the experts.

The experts have told the manager that, statistically of all new products with a FM, market surveys were positive and predicted success correctly 70% of the time. 30% of the time the surveys incorrectly predicted negative results.The experts have told the manager that, statistically of all new products with a UFM, market surveys were negative and predicted success correctly 80% of the time. 20% of the time the surveys incorrectly predicted positive results.

Market survey reliability in predicting states of nature

Result of survey Actual state of natureFavorable market Unfavorable market

Positive P(survey positive FM)=0.70 P(survey positive UM)=0.20Negative P(survey negative FM)=0.30 P(survey negative UM)=0.80

Probability revisions given a positive survey

State of

nature(SON)

Conditional probabilityP(survey positive SON)

Priorprobability

Jointprobability

Posterior probabilityP (SON survey positive)

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Prior Probability

Additional information (Conditional Probability)

Bayes’ Process

Posterior probability


FMUM

0.700.20

x 0.50x 0.50

= 0.35= 0.10= 0.45

0.35 / 0.45 = 0.780.10 / 0.45 = 0.22 = 1.00

P (survey results positive)

Probability revisions given a negative survey

State of

nature(SON)

Conditional probabilityP(survey negative SON)

Priorprobability

Jointprobabili

ty

Posterior probabilityP (SON survey negative)

FMUM

0.300.80

x 0.50x 0.50

= 0.15= 0.40= 0.55

0.15 / 0.55 = 0.270.40 / 0.55 = 0.73 = 1.00

P (survey results negative)

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Exercise 2.5

A company faces a decision problem with respect to a new product, M337, developed by one of its research laboratories. A plant is needed to facilitate the production of M337. The company is considering either to construct large plant or a small plant. The production of M337 might be responded with similar product competitors. The company’s marketing department projected that profits (in RM) for M337, based on the plant size and the presence competition, will be as follows:

Alternatives States of natureNo competition Competition

Large plant 120,000 -25,000Small plant 95,000 5,000Probability 0.60 0.40

The marketing department suggested that M337 be test marketed. It is estimated that test marketing will cost RM10,000. The test market will indicate whether the new product is successful or unsuccessful. The reliability of the test marketing data is given by the conditional probabilities shown below.

Test market result

Actual state of natureNo competition Competition

Successful 0.7 0.2Unsuccessful 0.3 0.8

Example:P(test market indicates product is successful given no competition) = 0.7P(test market indicates product is unsuccessful given there is competition) = 0.8

a) Find the following probabilities:i) Probability that there is no competition given the test market indicates the

product is successful.ii) Probability that there is competition given the test market indicates the product is

unsuccessful.iii) Probability that the test market indicates the product is successful.

b) Draw the decision tree for the above situation.c) Analyze the tree and determine the best course of action for the company.

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Expected Value of Sample Information (EVSI)

EVSI is the maximum amount to pay to access sample information.

From example 2.4, the manager realizes that conducting the market research is not free. The manager would like to know what the actual value of doing a survey is. One way of measuring the value of market information is to compute the EVSI.

EVSI =

__

= 59,200 – 40,000 (see solution ex 2.4 (1))

Or

= (EV with sample information + cost) – (EV without sample information) = (49,200 + 10,000) – (40,000) (see solution ex 2.4(2)) = 19 200

The maximum amount is RM19,200 for a market study. Since the cost to conduct market survey is RM10,000 so the survey is worthwhile.

Problems:

2.6 Ali is considering opening a competing bookstore near the campus and he has begun an analysis of the situation. There are two possible sites under consideration. One is relatively small while the other is large. If he opens at site A and demand is high, he will generate a profit of RM55,000. If demand is low he will lose RM10,000, If he opens at site B and demand is high, he will generates profit of RM85,000, but he will lose RM30,000 if the demand is low. He also has the option of not opening either. He believes that there is 50% chance that demand will be high. Ali can hire an expert to conduct a market research study. The probability of a high demand given a favorable study is 0.85. The probability of a high demand given an unfavorable study is 0.1. There is a 60% chance that the study will be favorable. Should Ali conduct the study? What is the maximum amount Ali will be willing to pay for this study?

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expected value of best decision with

sample information, assuming no cost to

gather it

expected value of best decision without sample

information


2.7 The director of telecommunication company is going to introduce one of the two models of mobile phone: Persona Elegant or Persona Compact. The company’s profit is determined by the market conditions and is presented in the payoff table below:

Market Condition (RM)Model Favorable Stable UnfavorablePersona Elegant 100,000 80,000 -34,000Persona Compact 95,000 82,000 -29,000

From his records the probability of a favorable market is estimated at 0.4 and that of unfavorable market 0.1. At the same time he is considering hiring a market research firm to do a survey to determine the future market conditions. The result of the survey will indicate either positive or negative market conditions. It is estimated that there is 0.6 probability that the survey will be positive. If the survey is positive the probability that the market will be favorable, stable or unfavorable are 0.72, 0.26 and 0.02, respectively. On the other hand, if the survey is negative, the probability that the market will be favorable, stable or unfavorable are 0.20, 0.66 and 0.14, respectively.

Using decision tree analysis, determine:

a) The decision strategy the company should follow.b) The maximum amount the company should pay the market research firm.

2.8 A firm that manufactures bicycles needs to expand its production facilities. The firm must decide whether to construct a large plant or a small plant. The market is either good or fair. The payoff (in RM) for each combination of alternatives and states of nature are given below:

Alternative Market condition

Good Fair

Small plant 600,000 200,000

Large plant 900,000 -100,000

Probability 0.60 0.40

The firm is considering hiring a market research team to conduct a survey on future market condition. The result of the survey would indicate either positive or negative market condition. There is 80% chance of a good market given a positive survey, and 70% chance of a fair market given a negative survey. The chance of a positive survey result is 56%.

a) Construct a decision tree to help the firm choose the best decision.b) Advise the firm on the optimal decision strategy.c) How much should the firm pay for the market research?

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2.9 Abdul Rahman is thinking about producing a new product for his company. If the market were favorable, he would get a return of RM100,000, but if the market were unfavorable he would lose RM40,000. He estimates that the probability of a favorable market is 0.5. He is also considering doing a survey to gather additional information about the market. The cost of the survey is RM4,000. Furthermore, the revised probability for the favorable market given that the survey result is positive is 0.73. The probability of the favorable market given that the survey is negative is 0.22. The probability that the survey will result in a positive market is 0.55.

a) Draw a decision tree for this problem. b) Give the best decision that Abdul Rahman should take.c) Compute the Expected Value of Sample Information for the problem and explain the

value obtained.

2.10 Ramzi & Co, a house developer, is trying to decide whether to build double-storey terrace houses or single-storey terrace houses or he could also choose not to proceed with the project. Given a favorable market, he will earn a profit of RM30,000 if he builds double storey houses and RM10,000 if he builds single storey houses. However, with an unfavorable market, Ramzi could lose RM40,000 with the double-storey houses and RM20,000 with single-storey houses. He believes that the probability of a favorable market is 0.7. Prior to this decision, Ramzi can get additional information from market research analyst at the cost of RM6,000 and the probability that the result will be positive is 0.5. A positive result from the study will increase the probability of a favorable market to 0.9. Furthermore, a negative result from the study will decrease the probability of a favorable market to 0.4.

a) Construct a decision tree for this problem.b) Analyze the decision tree and advise Ramzi for the best decision.c) If the cost to gather additional information is reduced to only RM2,000, what will be your

advised to Ramzi.

2.11 Borneo Manufacturing Company must decide whether it should manufacture component A or component B. The profit (RM) depends on the demand of the components and is given in the following table:

AlternativesState of natureHigh demand (H) Low demand (L)

Component A 100,000 -20,000Component B 70,000 10,000Probability 0.6 0.4

A market research firm offers to perform a study of the market for a fee of RM4,000 and will report either a favorable market (F) or unfavorable market (U).

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The relevant conditional probabilities are as follows:

P(FH) = 0.88 P(UH) = 0.12 P(FL) = 0.3 P(UL) = 0.7

The company has calculated the following posterior probabilities:

P(HF) = 0.82 P(LF) = 0.18

a) Show that P(HU) = 0.2 and P(LU) = 0.8b) Draw a decision tree for the above problem.c) Determine whether the study should be performed.d) Calculate the Expected Value of Sample Information (EVSI).

2.12 Jones is a farmer in New Zealand. He must determine whether to plant corn or wheat. If he plants corn and the weather is warm, he earns RM8,000. If he plants corn and the weather is cold, he earns RM5,000. If he plants wheat and the weather is warm, he earns RM7,000. If he plants wheat and the weather is cold, he earns RM4,500. In the past, 40% of the years have been cold and 60% have been warm. Before planting, Jones can pay RM600 for an expert weather forecast.

The reliability of the forecast is given by the conditional probabilities below:

Given the actual weather

Weather forecast predictCold Warm

Cold 0.7 0.3Warm 0.2 0.8

Example:P(weather forecast will predict a cold year given the actual weather is warm) = 0.2P(weather forecast will predict a warm year given the actual weather is cold) = 0.3

a) Draw a decision tree for the above situation.b) Analyze the tree and determine the best course of action for Jones.c) State the maximum amount Jones will be willing to pay to the weather forecaster.

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t2 decision theory

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