Problem 1. Consider the following LP:
Max 4x1+3x2+x3
st
2x1+4x2+x3 < 20
10x1+3x3 < 30
2x1+10x2+4x3 < 25
Add slack variables x4, x5 and x6.
In the final tableau variables x1, x2 and x4 are basic.
1) Using matrix methods compute the elements of the final tableau.
2) Using matrix methods compute the range for the cost coefficient of variable x1 within which the solution remains optimal.
Problem 2. A sculpture producing company manufactures two art products with seasonal demands: A Cup and a Brickspoon. The monthly demand (1000's) for these two items over the next year is tabled below:
| Product |
Jan |
Feb |
March |
April |
May |
June |
July |
Aug |
Sept |
Oct |
Nov |
Dec |
| Cup |
10 |
10 |
10 |
10 |
30 |
30 |
30 |
30 |
30 |
80 |
80 |
80 |
| Brickspoon |
50 |
50 |
15 |
15 |
15 |
15 |
15 |
15 |
15 |
50 |
50 |
50 |
The unit cost of the Cup and Brickspoon is $5 and $8 respectively when manufactured prior to June. From June to the end of the year the costs fall to $4.50 and $ 7.00 respectively because of the installation of an improved manufacturing system. The combined total of the two products that can be manufactured in any one month cannot exceed 120,000. Further, each unit of a Cup occupies 2 cubic feet and each unit of Brickspoon, 4 cubic feet of inventory space. Suppose that the maximum inventory space allocated to these items is 150,000 cubic feet and that the holding cost per cubic foot is $0.10/month.
a. Formulate and solve the LP to determine the minimum cost production plan that meets the demand and satisfies the constraints.