# Quantitative Methods, Maximax and Maximin Conditions

(See attached file for full problem description)

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5. A farmer in Georgia must decide which crop to plant next year on his land: corn, peanuts, or soybeans. The return from each crop will be determined by whether a new trade bill with Russia passes the Senate. The profit the farmer will realize from each crop given the two possible results on the _ trade bill is shown in the following payoff table.

Trade Bill

Pass

Fail

Crop Corn Peanuts Soybean

$35,000 18,000 22,000

$ 8,000 12,000 20,000

Determine the best crop to plant using the following decision criteria.

a. Maximax

b. Maximin

c. Minimax regret

d. Hurwicz (α = .3)

e. Equal likelihood

7. The owner of the Columbia Construction Company must decide between building a housing devel¬opment, constructing a shopping center, or leasing all the company's equipment to another company.

The profit that will result from each alternative will be determined by whether material costs remain stable or increase. The profit from each alternative given the two possibilities for material costs is shown in the following payoff table.

Material Costs

Stable

Increase

Decision

Houses Shopping center Leasing

$ 70,000 105,000 40,000

$30,000 20,000 40,000

Determine the best decision using the following decision criteria.

a. Maximax

b. Maximin

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I

I ,I

f.

11. The Tech football coaching staff has six basic offensive plays it runs every game. Tech has an upcoming game against State on Saturday, and the Tech coaches know that State employs five dif¬ferent defenses. The coaches have estimated the number of yards Tech will gain with each play against each defense as shown in the following payoff table.

Defense

Play 54 63 Wide Tack1e Nickel Blitz

Off tackle 3 -2 9 7 -1

Option -1 8 -2 9 12

Toss sweep 6 16 -5 3 14

Draw -2 4 3 10 -3

Pass 8 20 12 -7 -8

Screen -5 -2 8 3 16

a. If the coaches employ an offensive game plan, they will use the maximax criterion. What will their best play be?

b. If the coaches employ a defensive plan, they will use the maximin criterion. What will their best play be?

c. What will their best offensive play be if State is equally likely to use any of its five defenses?

Allen Abbott has a wide-curving, uphill driveway leading to his garage. When there is a heavy snow, Allen hires a local carpenter, who shovels snow on the side in the winter, to shovel his driveway. The snow shoveler charges $30 to shovel the driveway. Following is a probability distribution of the number of heavy snows each winter.

Heavy Snows Probability

1 .13

2 .18

3 .26

4 .23

5 .10

6 .07

7 .03

1.00

Allen is considering the purchase of a new self-propelled snowblower for $625 that would allow him, his wife, or his children to clear the driveway after a snow. Discuss what you think Allen's deci¬sion should be and why.

Probability

PROBLEMS 111

29. The Palm Garden Greenhouse specializes in raising carnations that are sold to florists. Carnations are sold for $3.00 per dozen; the cost of growing the carnations and distributing them to the florists is $2.00 per dozen. Any carnations left at the end of the day are sold to local restaurants and hotels for $0.75 per dozen. The estimated cost of customer ill will if demand is not met is $1.00 per dozen. The expected daily demand for the carnations is shown as follows.

Daily Demand

20 .05

22 .10

24 .25

26 .30

28 .20

30 .10

1.00

a. Develop the payoff table for this decision situation.

b. Compute the expected value of each alternative number of carnations that could be stocked, and select the best decision.

c. Construct the opportunity loss table and determine the best decision.

d. Compute the expected value of perfect information.

47. Blue Ridge Power and Light is an electric utility company with a large fleet of vehicles including automobiles, light trucks, and construction equipment. The company is evaluating four alterna1 strategies for maintaining its vehicles at the lowest cost including: (I) no preventive maintenance all and repair vehicle components when they fail; (2) take oil samples at regular intervals and t form whatever preventive maintenance is indicated by the oil analysis; (3) change the vehicle oil a regular basis and perform repairs when needed; (4) change the oil at regular intervals, take samples regularly, and perform maintenance repairs as indicated by the sample analysis.

For autos and light trucks, strategy 1 (no preventive maintenance) costs nothing to implement and results in two possible outcomes: There is a .10 probability that a defective component' occur requiring emergency maintenance at a cost of $1,200, or there is a .90 probability that defects will occur and no maintenance will be necessary.

Strategy 2 (take oil samples) costs $20 to implement (i.e., take a sample) and there is a .10 pr ability that there will be a defective part and .90 probability that there will not be a defect. If the] actually a defective part, there is a .70 probability the sample will correctly identify it, resulting,' preventive maintenance at a cost of $500. However, there is a .30 probability that the sample will not identify the defect and indicate everything is okay, resulting in emergency maintenance late a cost of $1,200. On the other hand, if there are actually no defects, there is a .20 probability that sample will erroneously indicate that there is a defect, resulting in unnecessary maintenance cost of $250. There is a .80 probability that the sample will correctly indicate there are no def resulting in no maintenance and no costs.

Strategy 3 (changing the oil regularly) costs $14.80 to implement and has two outcomes: a probability of a defective component, which will require emergency maintenance at a cost of $1,200, and a .96 probability that no defects will occur resulting in no maintenance and no cost

Strategy 4 (changing the oil and sampling) costs $34.80 to implement and results in the s probabilities of defects and no defects as strategy 3. If there is a defective component, there is ~ probability that the sample will detect it and $500 in preventive maintenance costs will be incur Alternatively, there is a .30 probability that the sample will not detect the defect, resulting in emergency maintenance at a cost of $1,200. If there is no defect, there is a .20 probability the sample indicate there is a defect, resulting in an unnecessary maintenance cost of $250, and a .80 probability that the sample will correctly indicate no defects, resulting in no cost.

Develop a decision strategy for Blue Ridge Power and Light and indicate the expected value of strategy.

#### Solution Summary

This handwritten solution provides decision tree analysis on all the scenarios and calculates the cost and expected value as well as minimum cost cost. All steps are shown with brief explanations.