Solved Problem 3-1 Maria Rojas is considering the possibility of opening a small dress shop on Fairbanks Avenue, a few blocks from the university. She has located a good mall that attracts students. Her options are to open a small shop, a medium-sized shop, or no shop at all. The market for a dress shop can be good, average, or bad. The probabilities for these three possibilities are 0.2 for a good market, 0.5 for an average market, and 0.3 for a bad market. The net profit or loss figures for the medium-sized and small shops for the various market conditions are given in the following table. Building no shop at all yields no loss and no gain.
What do you recommend?
Calculate the EVPI.
Develop the opportunity loss table for this situation. What decisions would be made using the minimax regret criterion and the minimum EOL criterion?
ALTERNATIVE | GOOD MARKET ($) | AVERAGE MARKET ($) | BAD MARKET ($) |
---|---|---|---|
Small shop | 75,000 | 25,000 | –40,000 |
Medium-sized shop | 100,000 | 35,000 | –60,000 |
No shop | 0 | 0 | 0 |
Since the decision-making environment is risk (probabilities are known), it is appropriate to use the EMV criterion. The problem can be solved by developing a payoff table that contains all alternatives, states of nature, and probability values. The EMV for each alternative is also computed, as in the following table:
STATE OF NATURE | ||||
---|---|---|---|---|
ALTERNATIVE | GOOD MARKET ($) | AVERAGE MARKET ($) | BAD MARKET ($) | EMV ($) |
Small shop | 75,000 | 25,000 | –40,000 | 15,500 |
Medium-sized shop | 100,000 | 35,000 | –60,000 | 19,500 |
No shop | 0 | 0 | 0 | 0 |
Probabilities | 0.20 | 0.50 | 0.30 |
As can be seen, the best decision is to build the medium-sized shop. The EMV for this alternative is $19,500.
The opportunity loss table is shown here.
STATE OF NATURE | |||||
---|---|---|---|---|---|
ALTERNATIVE | GOOD MARKET ($) | AVERAGE MARKET ($) | BAD MARKET ($) | MAXIMUM ($) | EOL ($) |
Small shop | 25,000 | 10,000 | 40,000 | 40,000 | 22,000 |
Medium-sized shop | 0 | 0 | 60,000 | 60,000 | 18,000 |
No shop | 100,000 | 35,000 | 0 | 100,000 | 37,500 |
Probabilities | 0.20 | 0.50 | 0.30 |
The best payoff in a good market is 100,000, so the opportunity losses in the first column indicate how much worse each payoff is than 100,000. The best payoff in an average market is 35,000, so the opportunity losses in the second column indicate how much worse each payoff is than 35,000. The best payoff in a bad market is 0, so the opportunity losses in the third column indicate how much worse each payoff is than 0.
The minimax regret criterion considers the maximum regret for each decision, and the decision corresponding to the minimum of these is selected. The decision would be to build a small shop, since the maximum regret for this is 40,000, while the maximum regret for each of the other two alternatives is higher, as shown in the opportunity loss table.
The decision based on the EOL criterion would be to build the medium shop. Note that the minimum EOL ($18,000) is the same as the EVPI computed in part (b). The calculations are
Solved Problem 3-2 Cal Bender and Becky Addison have known each other since high school. Two years ago they entered the same university and today they are taking undergraduate courses in the business school. Both hope to graduate with degrees in finance. In an attempt to make extra money and to use some of the knowledge gained from their business courses, Cal and Becky have decided to look into the possibility of starting a small company that would provide word processing services to students who needed term papers or other reports prepared in a professional manner. Using a systems approach, Cal and Becky have identified three strategies. Strategy 1 is to invest in a fairly expensive microcomputer system with a high-quality laser printer. In a favorable market, they should be able to obtain a net profit of $10,000 over the next 2 years. If the market is unfavorable, they can lose $8,000. Strategy 2 is to purchase a less expensive system. With a favorable market, they could get a return during the next 2 years of $8,000. With an unfavorable market, they would incur a loss of $4,000. Their final strategy, strategy 3, is to do nothing. Cal is basically a risk taker, whereas Becky tries to avoid risk.
What type of decision procedure should Cal use? What would Cal’s decision be?
What type of decision maker is Becky? What decision would Becky make?
If Cal and Becky were indifferent to risk, what type of decision approach should they use? What would you recommend if this were the case?
The problem is one of decision making under uncertainty. Before answering the specific questions, a decision table should be developed showing the alternatives, states of nature, and related consequences.
ALTERNATIVE | FAVORABLE MARKET ($) | UNFAVORABLE MARKET ($) |
---|---|---|
Strategy 1 | 10,000 | –8,000 |
Strategy 2 | 8,000 | –4,000 |
Strategy 3 | 0 | 0 |
Since Cal is a risk taker, he should use the maximax decision criteria. This approach selects the row that has the highest or maximum value. The $10,000 value, which is the maximum value from the table, is in row 1. Thus, Cal’s decision is to select strategy 1, which is an optimistic decision approach.
Becky should use the maximin decision criteria because she wants to avoid risk. The minimum or worst outcome for each row, or strategy, is identified. These outcomes are for strategy 1, for strategy 2, and $0 for strategy 3. The maximum of these values is selected. Thus, Becky would select strategy 3, which reflects a pessimistic decision approach.
If Cal and Becky are indifferent to risk, they could use the equally likely approach. This approach selects the alternative that maximizes the row averages. The row average for strategy 1 is The row average for strategy 2 is $2,000, and the row average for strategy 3 is $0. Thus, using the equally likely approach, the decision is to select strategy 2, which maximizes the row averages.
Solved Problem 3-3 Monica Britt has enjoyed sailing small boats since she was 7 years old, when her mother started sailing with her. Today, Monica is considering the possibility of starting a company to produce small sailboats for the recreational market. Unlike other mass-produced sailboats, however, these boats will be made specifically for children between the ages of 10 and 15. The boats will be of the highest quality and extremely stable, and the sail size will be reduced to prevent problems of capsizing.
Her basic decision is whether to build a large manufacturing facility, a small manufacturing facility, or no facility at all. With a favorable market, Monica can expect to make $90,000 from the large facility or $60,000 from the smaller facility. If the market is unfavorable, however, Monica estimates that she would lose $30,000 with a large facility and she would lose only $20,000 with the small facility. Because of the expense involved in developing the initial molds and acquiring the necessary equipment to produce fiberglass sailboats for young children, Monica has decided to conduct a pilot study to make sure that the market for the sailboats will be adequate. She estimates that the pilot study will cost her $10,000. Furthermore, the pilot study can be either favorable or unfavorable. Monica estimates that the probability of a favorable market, given a favorable pilot study, is 0.8. The probability of an unfavorable market, given an unfavorable pilot study, is estimated to be 0.9. Monica feels that there is a 0.65 chance that the pilot study will be favorable. Of course, Monica could bypass the pilot study and simply make the decision as to whether to build a large plant, small plant, or no facility at all. Without doing any testing in a pilot study, she estimates that the probability of a favorable market is 0.6. What do you recommend? Compute the EVSI.
Before Monica starts to solve this problem, she should develop a decision tree that shows all alternatives, states of nature, probability values, and economic consequences. This decision tree is shown in Figure 3.14.
The EMV at each of the numbered nodes is calculated as follows:
At each of the square nodes with letters, the decisions would be:
Based on the EMV criterion, Monica would select Do Not Conduct Study and then select Large Facility. The EMV of this decision is $42,000. Choosing to conduct the study would result in an EMV of only $32,900. Thus, the expected value of sample information is
Solved Problem 3-4 Developing a small driving range for golfers of all abilities has long been a desire of John Jenkins. John, however, believes that the chance of a successful driving range is only about 40%. A friend of John’s has suggested that he conduct a survey in the community to get a better feeling of the demand for such a facility. There is a 0.9 probability that the research will be favorable if the driving range facility will be successful. Furthermore, it is estimated that there is a 0.8 probability that the marketing research will be unfavorable if indeed the facility will be unsuccessful. John would like to determine the chances of a successful driving range given a favorable result from the marketing survey.
This problem requires the use of Bayes’ Theorem. Before we start to solve the problem, we will define the following terms:
Now, we can summarize what we know:
From this information we can compute the three additional probabilities that we need to solve the problem:
Now we can put these values into Bayes’ Theorem to compute the desired probability:
In addition to using formulas to solve John’s problem, it is possible to perform all calculations in a table:
STATE OF NATURE | CONDITIONAL PROBABILITY | PRIOR PROBABILITY | JOINT PROBABILITY | POSTERIOR PROBABILITY | ||
---|---|---|---|---|---|---|
Favorable market | 0.9 | 0.4 | 0.36 | 0.36 0.48 0.75 | ||
Unfavorable market | 0.2 | 0.6 | 0.12 | 0.12 0.48 0.25 | ||
0.48 |
As you can see from the table, the results are the same. The probability of a successful driving range, given a favorable research result, is , or 0.75.