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# Analysis of Linear Programming

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Problems must be completed in QM or Lingo 10.

Please see the attached file.

1.0 Given the following linear programming problem:
Min Z = 2x + 8y
Subject to (1) 8x + 4y 64
(2) 2x + 4y 32
(3) y 2

What is the minimal solution?
Use Lingo10 or QM to solve. Include your solution screen shots

2.0 Max Z = \$0.30x + \$0.90y
Subject to : 2x + 3.2y 160
4x + 2y 240
y 40
x, y

Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?
Use Lingo10 or QM to solve. Include your solution screen shots

3.0 The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are \$2 per bottle, and profits for dark beer are \$1 per bottle.

What is the objective function?
What is the constraint for malt?
What is the constraint for wheat?
Use Lingo10 or QM to solve. Include your solution screen shots

4.0 The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are \$3.00 per case and profits for diet soft drink are \$2.00 per case. How many cases of regular and how many cases of diet soft drink should Whoppy produce to maximize daily profit?

What is the objective function?
What is the constraint for Time?
What is the constraint for Syrup?
Use Lingo10 or QM to solve. Include your solution screen shots

5.0 Consider the following transportation problem:
1 2 Supply
1 5 6 100
2 4 2 200
3 3 6 150
4 9 7 50
Demand 250 250

What is the objective function?
What are the Supply side constraints?
What are the Demand side constraints?
Use Lingo10 or QM to solve. Include your solution screen shots

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#### Solution Summary

This solution provides explanation and answers for 5 linear programming problems in the attached Word document.

\$2.19

## optimum solutions in a linear programming

2. A company produces two products that are processed on two assembly lines. Assembly line 1 has 100 available hours, and assembly line 2 has 42 available hours. Each product requires 10 hours of processing time on line 1, while on line 2 products 1 requires 7 hours and product 2 requires 3 hours. The profit for product 1 is \$6 per unit, and the profit for product 2 is \$4 per unit.
a) Solve this model by using graphical analysis
b. Show mathematically that the optimum solutions in a linear programming will always happen at an extreme point.

12. The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics, in different proportions. One gram of ingredient 1 contributes 3 units, and 1 gram of ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is \$80, and the cost for a gram of ingredient 2 is \$50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the in order to meet the antibiotic requirements at the minimum cost.
a) Solve this model by using graphical analysis
b. Show mathematically that the optimum solutions in a linear programming will always happen at an extreme point.

18. Solve the following linear programming model graphically:
Maximize Z= 1.5x1 + x 2
Subject to
x1 ? 4
x2 ? 6
x1+ x2 ? 5
x1, x2 ? 0

Show mathematically that the optimum solutions in a linear programming will always happen at an extreme point.

22. Solve the following linear programming model graphically:
Maximize Z= 6.5x1 +10 x 2
Subject to
2x1 + 4x2 ? 40
x1 + x2 ? 15
x1 ? 8
x1, x2 ? 0
Show mathematically that the optimum solutions in a linear programming will always happen at an extreme point.

38. A manufacturing firm produces two products. Each product must undergo an assembly process and a finishing process. It is then transferred to the warehouse, which has space for only a limited number of items. The firm has 80 hours available for assembly and 112 hours for finishing, and it can store a maximum of 10 units in the warehouse. Each unit of product 1 has a profit or \$30 and requires 4 hours to assemble and 14 hours to finish. Each unit of product 2 has a profit of \$70 and requires 10 hour to assemble and 8 hours to finish. The firm wants to determine the quantity of each product to produce in order to maximize profit.
a. Solve this model by using graphical analysis
b. Show mathematically that the optimum solutions in a linear programming will always happen at an extreme point.

42. Kroeger sells its own brand of canned peas as well as several national brands. The store makes a profit of \$0.28 per can for its own peas and a profit of \$0.19 for the national brands. The store has 6 sq ft of shelf space available for canned peas, and each can of peas takes up 9 sq inches of that space. Point-of-sale records show that each week the store never sells more than one half as many cans of its own brand as it does of the national brands. The store wants to know how many cans of its own brand of peas and how many cans of the national brands to stock each week on the allocated shelf space in order to maximize profit.

Show mathematically that the optimum solutions in a linear programming will always happen at an extreme point.

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