# Decision Analysis and Probability Tree

Problems: 9-13, 15, 16, 18, 19, 20

Scenario 15-1

An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.

A B C D

1 Payoff Matrix

2

3 Economy

4 Investment Decline Expand

5 A 0 85

6 B 25 65

7 C 40 30

8 Bank 10 10

Payoffs

____ 9. Refer to Scenario 15-1. What decision should be made according to the maximin decision rule?

a. A

b. B

c. C

d. Bank

____ 10. Refer to Scenario 15-1. What decision should be made according to the minimax regret decision rule?

a. A

b. B

c. C

d. Bank

Scenario 15-2

An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.

A B C D E F G H

1 Payoff Matrix Regret Matrix

2

3 Economy Economy

4 Investment Decline Expand Investment Decline Expand

5 A 0 85 A

6 B 25 65 B

7 C 40 30 C

8 Bank 10 10 Bank

____ 11. Refer to Scenario 15-2. What formula should go in cell F5 of the Regret Matrix to compute the regret value?

a. =B$5-MAX(B$5:B$8)

b. =MAX(B$5:B$8)-MAX(B5)

c. =MAX(B$5:B$8)-MIN(B$5:B$8)

d. =MAX(B$5:B$8)-B5

____ 12. Expected regret is also called

a. EMV.

b. EOL.

c. EPA.

d. EOQ.

Scenario 15-3

An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 70% and an expanding economy at 30%.

A B C D

1 Payoff Matrix

2

3 Economy

4 Investment Decline Expand EMV

5 A -10 90

6 B 20 50

7 C 40 45

8 Bank 15 20

9

10 Probability 0.7 0.3

Payoffs

____ 13. Refer to Scenario 15-3. What decision should be made according to the expected monetary value decision rule?

a. A

b. B

c. C

d. Bank

Scenario 15-5

An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following decision tree has been developed for the problem. The investor has estimated the probability of a declining economy at 40% and an expanding economy at 60%.

____ 15. Refer to Scenario 15-5. What is the correct decision for this investor based on an expected monetary value criteria?

a. A

b. B

c. C

d. D

____ 16. Refer to Scenario 15-5. What is the expected monetary value for the investor's problem?

a. 32

b. 36

c. 38

d. 42

Scenario 15-6

A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 69% chance that demand for their products will be high and a 31% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 63% chance that the customers will like the products and a 37% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is a 92% chance that demand will be high for the products. If the survey is unfavorable there is only a 30% chance that the demand will be high. The following decision tree has been built for this problem. The company has computed that the expected monetary value of the best decision without sample information is 154.35 million. The company has developed the following conditional probability table for their decision problem.

A B C D

1

2 Joint Probabilities

3 High Demand Low Demand Total

4 Favorable Response 0.58 0.05 0.63

5 Unfavorable

Response 0.11 0.26 0.37

6 Total 0.69 0.31 1.00

7

8

9 Conditional Probability

10 For A Given Survey Response

11 High Demand Low Demand

12 Favorable Response 0.92 0.08

13 Unfavorable Response 0.30 0.70

14

15 Conditional Probability

16 For A Given Demand Level

17 High Demand Low Demand

18 Favorable Response 0.84 0.16

19 Unfavorable Response 0.16 0.84

____ 18. Refer to Scenario 15-6. What formula should go in cell C13 of the probability table?

a. =C5/$D4

b. =C5/C$6

c. =C5/$D5

d. =C4/$D4

Scenario 15-7

A decision maker is faced with two alternatives. The decision maker has determined that she is indifferent between the two alternatives when p=0.45.

Alternative 1: Receive $82,000 with certainty

Alternative 2: Receive $143,000 with probability p and lose $15,000 with probability (1-p).

____ 19. Refer to Scenario 15-7. What is the decision maker's certainty equivalent for this problem?

a. -$15,000

b. $82,000

c. $56,100

d. $82,000

____ 20. What is the formula for the weighted average score for alternative j when using a multi-criteria scoring model?

a.

b.

c.

d.

https://brainmass.com/math/probability/decision-analysis-and-probability-tree-308293

#### Solution Summary

The solution is comprised of a detailed explanation of the various aspects of Decision Analysis. This step-by-step calculation of these complicated topics provides students with a clear perspective of Decision Tree, Maximin Criterion, Minimax Regret Criterion, Regret Matrix, Expected Regret, Expected Value Criterion, Certainity Equivalent, etc.

Pepsi: probabilities, sensitivity analysis, decision tree

Please construct the decision tree for the problem attached, insert the probabilities and values as given in the scenario (make sure to include in the tree the possibility that the one-month forecast is favourable or not), roll back the tree, and determine the course of action that PEPSI should take.

Perform a sensitivity analysis on the probabilities of there being a significant increase in demand or not.

Describe each step of the analysis (including the sensitivity analysis), and the conclusions you have reached as a result of this analysis: include graphs and tables.

Identify advantages and limitations of this Decision Tree as a method for making decisions. Suggest potential means for overcoming any weaknesses.

PEPSI , a manufacturer of soft drinks, sells directly to restaurants, bars and public houses with sales representing about 20% of the local soft-drinks market. The company sells its products in bottles and cans, but chiefly in kegs to be served in draught form.

In the financial year 2009/10 the company achieved excellent results. The value of sales increased by 28% to £60 million and pre-tax profits of £6 million represented an increase of 55% on the previous year. Much of the company's success was based on sales of draught drinks. The number of outlets selling PEPSI?s product had increased substantially over recent years and the upward trend in sales had been accelerating.

The growth in draught soft-drink sales had, however, created some problems for the company's managers. In particular, there was concern that in the coming August, when sales reached a seasonal peak, there might not be enough kegs available to meet demand. Kegs are 45 litres, stainless steel containers in which the drinks are supplied to customers. After use, empty kegs are returned to the factory where they are inspected for damage, cleaned and refilled. In 2010, PEPSI owned about 100,000 kegs, but in view of the expansion of sales, it was felt by some managers that this stock should be increased.

In January 2011, the recently formed Keg Steering Committee met to consider the position for the coming summer. This committee consisted of managers from the Operations, Accounting, Sales and Marketing departments. John Fleetwood, the Operations Manager, put forward a proposal that 4000 new kegs should be ordered immediately from the manufacturer in Birmingham so that they would be available in time for the peak summer demand (the manufacturer would only supply in batches of 1000 kegs).

Dave Mitchell, the accountant was less sure. At £50 per keg, this would amount to an expenditure of £200,000 and there was no certainty that all these kegs would be needed. However, Fleetwood pointed out that, even if they were not needed this year, the kegs might be required in the following year.

"In that case," retorted Mitchell, "We'll have tied up £200,000 for a year unnecessarily. Given a 5% rate of interest, that would cost us £10,000."

At this point, Sally Martin, the Marketing Manager, intervened. "Surely there are intermediate positions. We could presumably order any number of batches from 0 to 4.

What we need is some estimate of the likely levels of demand in August and the consequences of having a given number of kegs available to meet these levels of demand."

Richard Skills, the Sales Manager, pointed out that the sales forecasts were really only reliable one month ahead, partly because this was linked to the long-range weather forecast. In hot, dry summers, sales rocketed. "Perhaps we should delay our decision until July when we'll get the August forecast," he said.

"If we do that," replied Fleetwood, "it may be too late. Other brewers might be placing orders with the keg manufacturers and I reckon there would be a 10% chance that they could not get the kegs produced for us in time for August. Surely you must have some idea of what sales are likely to do. After all, your reps are constantly touring the country, talking to people in the trade."

Skills were silent for a few moments. "Well, if you really twist my arm, the best I can say is this. I reckon there's about a 70% chance that sales in August this year will be significantly up on August last year, in other words, by at least 5% and a 30% chance that we will see no significant increase. But remember, this is only a very rough estimate."

Martin turned to Mitchell. "OK, can we do some calculations on what the consequences are of having different numbers of kegs available for August assuming first no significant increase in sales, and secondly an increase of at least 5%?"

Mitchell was sceptical, but under pressure, he agreed to do some rough calculations. For simplicity, he assumed that:

i) either 0, 2000 or 4000 kegs would be ordered;

ii) if it was needed, each new keg would make only one journey in August thereby supplying 45 litres to customers and earning a contribution of £8;

iii) if sales increase by at least 5% then at least 4000 kegs could be used in August;

iv) the only costs are those resulting from tying-up the capital expenditure on the kegs (at 5% interest rate) until they will be needed the following year, for example, to replace worn-out kegs.

[Calculate Mitchell?s figures for change in profit on the basis of his assumptions given above]

"I still think it might be worth delaying our decision until we get the August sales forecast," said Skills, "even if that does mean taking a risk that the kegs will not be available."

"How reliable are these one-month-ahead forecasts?" asked Martin.

"They're not bad. If you can give me a couple of days I'll let you have a summary of their recent performance."

After some further discussion, it was agreed to reconvene the committee at the end of the week when Skills' figures could be looked at. A decision would then be made on whether to go ahead and order a specific number of kegs immediately (and if so how many, 2000 or 4000) or to delay the purchasing decision until the sales forecast was available.

On the basis of Skills? data about the accuracy of the one-month forecasts it was possible to recompute some of the probabilities of the outcomes. It was most likely that the one-month forecast would be that sales would increase significantly in August (representing both seasonal characteristics and the long-term trend) ? Skills estimated this probability to be around 0.8 .

Having more accurate forecasts of demand means that you are better able to estimate the actual amount of demand ? using Bayes? Theorem we can now make a new computation of the conditional probability of there actually being a significant increase in sales given a one-month forecast of a significant increase in sales (which is p = 0.9), and the probability of significant sales increase given an unfavourable forecast (which is p = 0.3).

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