1. Solve the linear programming problems by drawing the feasible solution region and using the corner point table.

1.1 Maximize: P = 20x + 10y

subject to: 3x + y 21

x + y 9

x + 3y 21

x, y 0

1.2 Minimize: z = 400x + 100y

Subject to: 3x + y 24

x + y 16

x + 3y 30

x, y 0

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1. A furniture manufacturing company manufacture dining-room tables and chairs. The relevant manufacturing data are given in the table below.

Department	Labor-Hours per Unit	Maximum Labor-Hours Available per Day
Table	Chair
Assembly	8	2	400
Finishing	2	1	120
Profit per Unit	$90	$25

2.1 How many tables and chairs should be manufactured each day to realize a maximum profit? What is the maximum profit? Note: Draw the feasible solution region and solve the problem using the corner point table.

2.2 Discuss the effect on the production schedule and the maximum profit if the marketing department of the company decides that the number of chairs produced should be at least four times the number of tables produced.

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1. Consider the following linear programming problem:

Maximize: P = x₁ + x₂

subject to: 5x₁ + 4x₂ 240

5x₁ + 2x₂ 150

5x₁ + x₂ 120

x₁ 0, x₂ 0

3.1 Using slack variables, convert the system of inequalities (i-system) into a system of equations (e-system).

3.2 Using the table method, solve the linear programming problem. Also, identify which solutions are feasible and which solutions are not feasible.

3.3 What is a maximum value of P? What are the values of x₁ and x₂ that lead to the maximum value of P?

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1. Solve the linear programming problem using the simplex method.

Maximize: P = 10x₁ + 50x₂ + 10x₃

subject to: 3x₁ + 3x₂ + 3x₃ 66

6x₁ - 2x₂ + 4x₃ 48

3x₁ + 6x₂ + 9x₃ 108

x₁, x₂, x₃ 0

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5-

1. Consider the following linear programming problem:

Minimize: C = 4x₁ + 5x₂

subject to: 2x₁ + x₂ 12

x₁ + x₂ 9

x₂ 4

x₁, x₂ 0

5.1 Form the matrix A, using the coefficients and constants in the problem constraints and objective function. Find A^T. State the dual problem.

5.2 Write the initial simplex tableau for the dual problem.

5.3 Solve the linear program problem using the simplex method. Show the pivot operations.

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6-

1. Boston has two heating services that offer annual service contracts for home heating: Apple Heating and Banana Heating. Currently, 25% of homeowners have service contracts with Apple, 30% have service contracts with Banana, and the remainder do not have service contracts. Both companies launch aggressive advertising campaigns to attract new customers, with following effects on service contract purchases for the next several years: each year 35% of homeowners with no current service contract decide to purchase a contract from Apple and 40% decide to purchase one from Banana. In addition, 10% of the previous customers at each company decide to switch to the other company, and 5% decide they do not want a service contract.

6.1 Draw a transition diagram.

6.2 Write a transition matrix.

6.3 What percentage of homes will have service contracts with Apple next year?

6.4 What percentage of homes will have service contracts with Apple next year?

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Question 7

1. Once a year company employees are given the opportunity to join one of three pension plans: A, B, or C. Once an employee decides to join one of these plans, the employee cannot drop the plan or switch to another plan. Past records indicate that each year 4% of employees elect to join plan A, 14% elect to join plan B, 7% elect to join plan C, and the remainder do not join any plan.

7.1 Write a transition matrix in standard form.

7.2 Find the limiting matrix.

7.3 In the long run, what percentage of the employees will elect to join plan A? Plan B? Plan C?

7.4 On average, how many years will it take an employee to decide to join a plan?

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Question 8

8. A person on a high-protein, low-carbohydrate diet requires at least 100 units of protein and at most 24 units of carbohydrates daily. The diet will consist entirely of three special liquid diet foods: A, B, and C. The contents and costs of the diet foods are given in the table. How many bottles of each brand of diet food should be consumed daily in order to meet the protein and carbohydrate requirements at minimal cost? What is the minimum cost? Note: Use the big M method.

	Units per bottle
A	B	C
Protein	10	10	20
Carbohydrates	2	3	4
Cost per bottle ($)	0.60	0.40	1.50