# Is bases the same as basis ? (Simplex Algorithm)

1. Sep 24, 2011

### flyingpig

1. The problem statement, all variables and given/known data

[PLAIN]http://img193.imageshack.us/img193/3662/unledmcg.png [Broken]

3. The attempt at a solution

I rewrote the whole thing in dictionary

$$x_3 = 15 - 8x_1 - 4x_2$$
$$x_4 = 7 - 2x_1 - 6x_2$$
$$z = 0 + 22x_1 - 12x_2$$

$$x_i \geq 0$$
$$1\leq i \leq 4$$

a) So my basis/bases is x = (x_3,x_4)

b) Do I have to solve the above using the Simplex Algorithm first?

c) I have to set x_1 or x_2 = 0 and the other one not 0.

So doing so I get (I chose x_1 and let x_2 = t)

$$x_3 = 15 - 4x_2$$
$$x_4 = 7 - 6x_2$$
$$z = -12x_2$$

So I get a line of

x = (0,t|15-t,7-6t)^t

So my answer is a line with $$t \in (-\infty, \frac{7}{6})$$

I have infinity in my interval...something does not feel right

Last edited by a moderator: May 5, 2017
2. Sep 24, 2011

### Fredrik

Staff Emeritus
Bases is the plural of basis. One basis, two bases. I have to go and do something else now, so I don't have time to help you with the actual problem, but I'm sure someone else will.

3. Sep 24, 2011

### flyingpig

How could there be more than one basis for any LOP?

4. Sep 28, 2011

### flyingpig

I need to make a correction

When they mean bases, they just mean all combination of the basic variables.

I happen to have 4 variables and I could only only choose 2 slacks, so that's 6 combination namely

$$\beta = \left \{ 1,2 \right \}, \left \{ 1,3 \right \}, \left \{ 1,4 \right \}, \left \{ 2,3 \right \}, \left \{ 2,4 \right \}, \left \{ 3,4 \right \}$$

5. Oct 2, 2011

### flyingpig

I just have one question, it's given that at least one basic solution HAS to be an optimal right? There can't be two optimal?

6. Oct 2, 2011

### Pyrrhus

There may be multiple optimal solutions, but yes at least one (if the Weierstrass theorem is satisfied).