1. 10 points] A farmer in Kentucky has 1,500 acres of land on which she intends to plant corn, wheat, and soybeans. Each acre of corn costs $150 for preparation, requires 6 worker-days of labor, and yields a profit of $35. An acre of wheat costs $130 to prepare, requires 9 worker-days of labor, and yields a profit of $45. An acre of soybeans costs $80 to prepare, requires 7 worker-days of labor, and yields a profit of $25. The farmer has taken out a loan of $95,000 for crop preparation and has contracted with a union for 6,500 worker-days of labor. A granary has agreed to purchase 400 acres of corn, 650 acres of wheat, and 450 acres of soybeans. The farmer has established the following goals, in order of their importance:
   1. To maintain good relations with the union, the labor contract must be honored; that is, the full 6,500 worker-days of labor contracted for must be used.
   2. Preparation costs should not exceed the loan amount so that additional loans will not have to be secured.
   3. The farmer desires a profit of at least $100,000 to remain in good financial condition.
   4. Contracting for excess labor should be avoided.
   5. The farmer would like to use as much of the available acreage as possible.
   6. The farmer would like to meet the sales agreement with the granary. However, the goal should be weighted according to the profit returned by each crop.

Formulate a goal programming model to determine the number of acres of each crop the farmer should plant to satisfy the goals in the best possible way.

Solve this model using the computer.

Decision Analysis [20 points]

1. [10 points] 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 different 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 Tackle	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
Probability	.40	.10	.20	.20	.10

1. 1. If the coaches employ an offensive game plan, they will use the maximax criterion. What will be their best play?
   2. If the coaches employ a defensive plan, they will use the maximin criterion. What will be their best play?
   3. What will be their best offensive play if State is equally likely to use any of its five defenses?
   4. Using expected value, rank Techs plays from best to worst.
   5. During the actual game, a situation arises in which Tech has a third down and 10 yards to go, and the coaches are 60% certain that State will run a blitz, with a 10% chance of any of the other four defenses. What play should Tech run? Is it likely the team will make the first down?
2. [10 points] The management of First American Bank was concerned about the potential loss that might occur in the event of a physical catastrophe such as a power failure or a fire. The bank estimated that the loss from one of these incidents could be as much as $100 million, including losses due to interrupted service and customer relations. One project the bank is considering is the installation of an emergency power generator at its operations headquarters. The cost of the emergency generator is $800,000, and if it is installed, no losses from this type of incident will be incurred. However, if the generator is not installed, there is a 10% chance that a power outage will occur during the next year. If there is an outage, there is a 0.05 probability that the resulting losses will be very large, or approximately $80 million in lost earnings, alternatively, it is estimated that there is a 0.95 probability of only slight losses of around $1 million. Using decision tree analysis, determine whether the bank should install the new power generator.