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Practice Problems: Linear Programming

Problem 1:

Chad’s Pottery Barn has enough clay to make 24 small vases or 6 large vases. He has only enough of a special glazing compound to glaze 16 of thesmall vases or 8 of the large vases. Let X1 = the number of small vases and X2 = the number of large vases.

The smaller vases sell for $3 each, and the larger vases would bring $9 each.

(a)Formulate the problem

(b) Solve the problem graphically

Problem 2:

A fabric firm has received an order for cloth specified to contain at least 45 pounds of cotton and 25 pounds of silk. Thecloth can be woven out of any suitable mix of two yarns A and B. They contain the proportions of cotton and silk (by weight) as shown in the following table:

| Cotton | Silk |
A | 30% | 50% |
B| 60% | 10% |

Material A costs $3 per pound, and B costs $2 per pound. What quantities (pounds) of A and B yarns should be used to minimize the cost of this order?

ANSWERS

Problem 1:

(a)Formulation:
Objective function: Maximize 3X1 + 9X2
Subject to: Clay constraint: 1X1 + 4X2 24
Glaze constraint: 1X1 + 2X2 16

(b) Graphical Solution

| X1 @ $3.00 | X2 @ $9.00 |Income |
A | 0 | 0 | 0 |
B | 0 | 6 | $54 |
C | 8 | 4 | $60* |
D | 16 | 0 | $48 |

Evaluating all possible corner points that might be the optimal solution, the optimum income of $60 will occurby making and selling 8 small vases and 4 large vases. An iso-profit line on the graph from (20,0) to (0,6.67) shows the point that returns value of $60.
Problem 2:

Formulation:
Objectivefunction: min C = 3A + 2B
Constraints: Cotton .30A + .60B 45
Silk .50A + .10B 25

We can learn the values of A and B at intersection of the Silk and Cotton constraints by simultaneouslysolving the equations that determine the point. To solve for A we first multiply the Silk equation by 6 then subtract the Cotton equation.



Following the same basic procedure for the value of...
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