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# Linear Programing: The Simplex Method: Homework

Module by: UniqU, LLC. E-mail the author

Summary: This chapter covers principles of the simplex method to Linear Programming. After completing this chapter students should be able to: solve linear programming maximization problems using the simplex method and solve the minimization problems using the simplex method.

## MAXIMIZATION BY THE SIMPLEX METHOD

Solve the following linear programming problems using the simplex method.

### Exercise 1

Maximize z=x1+2x2+3x3z=x1+2x2+3x3 size 12{z=x rSub { size 8{1} } +2x rSub { size 8{2} } +3x rSub { size 8{3} } } {}

subject to x1+x2+x3122x1+x2+3x318x1+x2+x3122x1+x2+3x318 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} # <= {} {} # "12" {} ## 2x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # +{} {} # 3x rSub { size 8{3} } {} # <= {} {} # "18"{} } } {}

x 1 , x 2 , x 3 0 x 1 , x 2 , x 3 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } ,x rSub { size 8{3} } >= 0} {}

### Exercise 2

Maximize z=x1+2x2+x3z=x1+2x2+x3 size 12{z=x rSub { size 8{1} } +2x rSub { size 8{2} } +x rSub { size 8{3} } } {}

subject to x1+x23x2+x34x1+x35x1+x23x2+x34x1+x35 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # <= {} {} # 3 {} ## x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} # <= {} {} # 4 {} ## x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{3} } {} # <= {} {} # 5{} } } {}

x 1 , x 2 , x 3 0 x 1 , x 2 , x 3 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } ,x rSub { size 8{3} } >= 0} {}
(1)

### Exercise 5

The Acme Apple company sells its Pippin, Macintosh, and Fuji apples in mixes. Box I contains 4 apples of each kind; Box II contains 6 Pippin, 3 Macintosh, and 3 Fuji; and Box III contains no Pippin, 8 Macintosh and 4 Fuji apples. At the end of the season, the company has altogether 2800 Pippin, 2200 Macintosh, and 2300 Fuji apples left. Determine the maximum number of boxes that the company can make.

## MINIMIZATION BY THE SIMPLEX METHOD

In problems 1-2, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

### Exercise 6

Minimize z=6x1+8x2z=6x1+8x2 size 12{z=6x rSub { size 8{1} } +8x rSub { size 8{2} } } {}

subject to 2x1+3x274x1+5x292x1+3x274x1+5x29 size 12{ matrix { 2x rSub { size 8{1} } {} # +{} {} # 3x rSub { size 8{2} } {} # >= {} {} # 7 {} ## 4x rSub { size 8{1} } {} # +{} {} # 5x rSub { size 8{2} } {} # >= {} {} # 9{} } } {}

x 1 , x 2 0 x 1 , x 2 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } >= 0} {}

### Exercise 7

Minimize z=5x1+6x2+7x3z=5x1+6x2+7x3 size 12{z=5x rSub { size 8{1} } +6x rSub { size 8{2} } +7x rSub { size 8{3} } } {}

subject to 3x1+2x2+3x3104x1+3x2+5x3123x1+2x2+3x3104x1+3x2+5x312 size 12{ matrix { 3x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # 3x rSub { size 8{3} } {} # >= {} {} # "10" {} ## 4x rSub { size 8{1} } {} # +{} {} # 3x rSub { size 8{2} } {} # +{} {} # 5x rSub { size 8{3} } {} # >= {} {} # "12"{} } } {}

x 1 , x 2 , x 3 0 x 1 , x 2 , x 3 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } ,x rSub { size 8{3} } >= 0} {}

In the next two problems, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

### Exercise 8

Minimize z=4x1+3x2z=4x1+3x2 size 12{z=4x rSub { size 8{1} } +3x rSub { size 8{2} } } {}

subject to x1+x2103x1+2x224x1+x2103x1+2x224 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # >= {} {} # "10" {} ## 3x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # >= {} {} # "24"{} } } {}

x , x 2 0 x , x 2 0 size 12{x,x rSub { size 8{2} } >= 0} {}

### Exercise 19

A factory manufactures three products, A, B, and C. Each product requires the use of two machines, Machine I and Machine II. The total hours available, respectively, on Machine I and Machine II per month are 180 and 300. The time requirements and profit per unit for each product are listed below.

 A B C Machine I 1 2 2 Machine II 2 2 4 Profit 20 30 40

How many units of each product should be manufactured to maximize profit, and what is the maximum profit?

### Exercise 20

A company produces three products, A, B, and C, at its two factories, Factory I and Factory II. Daily production of each factory for each product is listed below.

 Factory I Factory II Product A 10 20 Product B 20 20 Product C 20 10

The company must produce at least 1000 units of product A, 1600 units of B, and 700 units of C. If the cost of operating Factory I is $4,000 per day and the cost of operating Factory II is$5000, how many days should each factory operate to complete the order at a minimum cost, and what is the minimum cost?

### Exercise 21

For his classes, Professor Wright gives three types of quizzes, objective, recall, and recall-plus. To keep his students on their toes, he has decided to give at least 20 quizzes next quarter. The three types, objective, recall, and recall-plus quizzes, require the students to spend, respectively, 10 minutes, 30 minutes, and 60 minutes for preparation, and Professor Wright would like them to spend at least 12 hours(720 minutes) preparing for these quizzes above and beyond the normal study time. An average score on an objective quiz is 5, on a recall type 6, and on a recall-plus 7, and Dr. Wright would like the students to score at least 130 points on all quizzes. It takes the professor one minute to grade an objective quiz, 2 minutes to grade a recall type quiz, and 3 minutes to grade a recall-plus quiz. How many of each type should he give in order to minimize his grading time?

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