Solve questions using excel appropriate functions (Optimization)
- Perform the primal simplex method to solve the following LPs. The slack variables are the initial basic variables. Write the basis and the simplex tableau at each iteration. Fill in the blanks of the tableaux and the under-lined blanks.
1-1. (2 points)
Maximize 4 x1 + 6 x2,
Subject to – x1 + x2 ≤ 11,
x1 + x2 ≤ 27,
2 x1 + 5 x2 ≤ 90,
x1 , x2 ≥ 0.
The Simplex Tableau with respect to B0 = {s1, s2, s3}:
| x1 | x2 | s1 | s2 | s3 | Max | ||
| 0th row | 4 | 6 | 0 | 0 | 0 | Z – 0 | ratio |
| 1st row | -1 | 1 | 1 | 0 | 0 | 11 | 11 |
| 2nd row | 1 | 1 | 0 | 1 | 0 | 27 | 27 |
| 3rd row | 2 | 5 | 0 | 0 | 1 | 90 | 18 |
What is the objective value? What is the basic feasible solution? Is the basis optimal?
The simplex Tableau with respect to B1 = {x2, s2, s3}:
| x1 | x2 | s1 | s2 | s3 | Max | ||
| 0th row | 10 | 0 | -6 | 0 | 0 | Z – 66 | ratio |
| 1st row | -1 | 1 | 1 | 0 | 0 | 11 | |
| 2nd row | 2 | 0 | -1 | 1 | 0 | 16 | |
| 3rd row | 7 | 0 | -5 | 0 | 1 | 35 |
What is the objective value? What is the basic feasible solution? Is the basis optimal?
- (Continued)
The simplex Tableau with respect to B2 = { }:
| x1 | x2 | s1 | s2 | s3 | Max | ||
| 0th row | ratio | ||||||
| 1st row | |||||||
| 2nd row | |||||||
| 3rd row |
What is the objective value? What is the basic feasible solution? Is the basis optimal?
2-1. (2 points) Compute the simplex tableau corresponding to initial basis = { x1, x4, x3} and then perform the simplex iteration; i.e., the column of the most negative reduced cost enters into basis.
min x1 + 3 x2 + 3 x3 +2 x4 – 2 x5,
s t – 4 x2 + 3 x3 – x4 = 1,
4 x1 + 3 x2 + x4 + x5 = 2,
– 3 x1 + 2 x2 + x4 + x5 = 2,
x1, x2, x3, x4, x5 ≥ 0.
| x1 | x2 | x3 | x4 | x5 | Min | ||
| 0th row | 1 | 3 | 3 | 2 | -2 | 0 | Ratio |
| 1st row | 0 | -4 | 3 | -1 | 0 | 1 | |
| 2nd row | 4 | 3 | 0 | 1 | 1 | 2 | |
| 3rd row | -3 | 2 | 0 | 1 | 1 | 2 |
2.1 (Continued)
The Simplex Tableau with respect to initial basis B0 = { x1, x4, x3}
| x1 | x2 | x3 | x4 | x5 | Min | ||
| 0th row | ratio | ||||||
| 1st row | |||||||
| 2nd row | |||||||
| 3rd row |
What is the objective value? What is the basic feasible solution? Is the basis optimal?
DETAILED ASSIGNMENT
