1. Introduction to Operations Research
Revised Simplex Method

2. Linear programming How many basic variables are there in this problem?
If we know that x2 and x3 are non-zero, how can we find the optimal solution directly?
How can this help us use the simplex method more efficiently, assuming we start from the usual initial solution?
What if I told you that x3 was the only non-zero decision variable in a solution. How could you find that solution?
Maximize
5x1 + 3x2 + 4x3
Subject to:
2x1 + x2 + x3 ≤ 20
3x1 + x2 + 2x3 ≤ 30
x1, x2, x3 ≥ 0

3. Linear Programming Description
Maximize
Z = c1x1 + … + cnxn
Subject to.
a11x1 + … + a1nxn ≤ b1
am1x1 + … + amnxn ≤ bm
x1 ≤ 0, …, xn ≤ 0

4. Revised Simplex Method
Re-write problem in matrix form
Max Z = cx
s.t.
Ax ≤ b and x ≥ 0
c = [c1,c2,…,cn] contribution of each xi to Z

5. Revised simplex method

6. Revised Simplex method
Initial Tableau

7. Revised Simplex Method
Put Wyndor Glass problem in Matrix form
Fill in values for each Matrix or vector: c, A, x, b, I. xs
Convince yourself that this really represents the initial tableau.
Maximize Z = 3x1 + 5x2
Subject to:
x ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
x1, x2 ≥ 0

8. Revised Simplex method
Initial Tableau

9. Revised Simplex method

10. Revised simplex method
Recall that in any basic solution m variables are basic (positive) and n variables are non-basic (equal to zero)
Ignore all the non-basic variables.
Let xB be the vector of basic variables
Let B be the Matrix of coefficients of basic variables.
Then it is true that BxB=b

11. Initial Basis
BxB =b

12. Revised Simplex method
Consider the following constraint set, in augmented form.
(x3 and x4 are slacks)
x1 + 2x2 + x = 5
2 x1 + 3x x4 = 8
If x1 and x3 are basic, then we know that x2 and x4 each equal 0, so we can rewrite the constraint set as
x1 + x = 5
2 x = 8
In matrix form: xB = [x1, x3] and
You can see that BxB = b

13. Revised Simplex method
In any iteration it is true that BxB = b
We can solve for xB
B-1BxB = B-1b
xB = B-1b
So, if I tell you the basis, you can tell me the value of the basic variables.

14. Wyndor example Maximize Z = 3x1 + 5x2 Subject to: x1 ≤ 4 2x2 ≤ 12
The basic variables are x1, x3, x4. What are their values?
xB = B-1b
Maximize Z = 3x1 + 5x2
Subject to:
x ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
x1, x2 ≥ 0