Revised Simplex Method

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 ≤ 203x1 + x2 + 2x3 ≤ 30x1, x2, x3 ≥ 0

Maximize Z = c1x1 + … + cnxn

Subject to.

a11x1 + … + a1nxn ≤ b1

am1x1 + … + amnxn ≤ bm

x1 ≤ 0, …, xn ≤ 0

mnmm

n

n

aaa

aaa

aaa

21

21221

11211

A

nx

x

x

2

1

x

mb

b

b

2

1

b

0

0

0

n0

Re-write problem in matrix formMax Z = cxs.t.Ax ≤ b and x ≥ 0c = [c1,c2,…,cn] contribution of each xi to Z

mn

n

n

s

x

x

x

2

1

x

100

010

001

I

bx

xIA

s

, 0

sx

x

b

x

xIA0

0c

s

01Z

Initial Tableau

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:

x1 ≤ 4

2x2 ≤ 12

3x1 + 2x2 ≤ 18

x1, x2 ≥ 0

b

x

xIA0

0c

s

01Z

18

12

40

100

010

001

23

20

01

0

0

0

000531

5

4

3

2

1

x

x

xx

xZ

Initial Tableau

1

2

3

4

5

1 3 5 0 0 0 0

4

0 1 0 1 0 0 12

0 0 2 0 1 0 18

0 3 2 0 0 1

Z

x

x

x

x

x

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

100

010

001

B

5

4

3

x

x

x

xB

18

12

4

b

BxB =b

Consider the following constraint set, in augmented form. (x3 and x4 are slacks)

x1 + 2x2 + x3 = 5

2 x1 + 3x2 + 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 + x3 = 5

2 x1 = 8

In matrix form: xB = [x1, x3] and

You can see that BxB = b

1 1

2 0B

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.

The basic variables are x1, x3, x4. What are their values?

xB = B-1b

003

100

011

B

12

2

6

18

12

4

010

3/101

3/100

Maximize Z = 3x1 + 5x2

Subject to:

x1 ≤ 4

2x2 ≤ 12

3x1 + 2x2 ≤ 18

x1, x2 ≥ 0