Is bases the same as basis ? (Simplex Algorithm)

[flyingpig]

Homework Statement

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

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

 

[Fredrik]

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.

 

[flyingpig]

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.

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

 

[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

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

 

[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?

 

[Pyrrhus]

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?

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