Linear Programming Case Study

Week 8 Assignment

MAT 540

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Instructor Name

Date

Problem Introduction

The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4600 ounces of salt, 9400 ounces of flour, and 2200 ounces of herbs. A bag of Lime chips requires 1.5 ounces of salt, 5 ounces of flour, and 2 ounces of herbs to produce; while a bag of Vinegar chips requires 4 ounces of salt, 6 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.48, and for a bag of Vinegar chips $0.59.

a) What is the formulation for this problem?

b) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

c) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

d)  Discuss:  Slack (if any); shadow price, and sensitivity analysis results using the program of your choice. 

Above problem is a maximization problem as one is trying to maximize the profits by making different bags of chips. It takes salt, flour and herbs to make two different types of chips – Lime and Vinegar. There are constrained amounts of salt, flour and herb and the owner want to maximize his profits. The amount of profit per bag is given as well.

The LP problem thus becomes:

Maximize Profits from the sale of bags of both lime and vinegar chips

Constraints:

In mathematical terms, let’s say X1 to be the number of Lime bags and X2 to be the number of Vinegar bags.

• Salt consumed should not exceed 4,600
• Flour consumed should not exceed 9,400
• Herbs consumed should not exceed 2,200

LP is:

Maximize: 0.48 X1 + 0.59 X2

Subject to:

1.5X1 + 4 X2 <=4,600

5X1 + 6X2 <= 9,400

2 X1 + 2X2 <=2,200

When one solves the LP problem above, it turns out that in order to maximize the profits; the owner should not make any bag of lime chips and invest all resources in making vinegar chips. This leads to a maximum profit of $649. This is primarily because the profit per bag of Vinegar chips is a lot higher than profit per bag of lime chips.

b) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

Please note that 800 and 600 is violating the herbs constraints. The amount of herb is now 2,800 but the max was 2,200. To answer this question though flour and salt is left out and not completely used up. 1,000 ounces of salt is left and 1,800 ounces of flour is left.

c) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

Please note that 800 and 600 is violating the herbs constraints. The amount of herb is now 2,800 but the max was 2,200. To answer this question though flour and salt is left out and not completely used up. 1,000 ounces of salt is left and 1,800 ounces of flour is left.

d) Discuss: Slack (if any); shadow price, and sensitivity analysis results using the program of your choice.

In this scenario, there is slack for salt and flour but no slack for herb and that becomes a binding constraint. Said another way, if we had more herbs, then the profit can be increased more. However, adding salt and flour would not increase the profit. Herb also has a shadow price of 0.295 which means that the profit will increase by .295 for every additional herb ounce. However, the shadow price of salt and flour are not provided and so adding additional salt and flour do not impact profits.

Reference:

Gass, S. (1964). Linear programming. 1st ed. New York: McGraw-Hill.