Simplex Method: Maximization with Equalities

[langhost]

Argh! I'm so confused now, I have no idea where to begin! We're supposed to use the Simplex Method to solve it but our textbook only deals with mixed constraints, wherein we use the Big M method (ie. add in surplus, artificial variables, etc). I'm not sure whether you would use the Big M method with this one because the contraints are all equalities. I need a walkthrough the steps you would take to solve it, because I don't even know where to begin and I've been trying everything I can think of.

Anyway, the specific question is:

MAX P= 28x₁ + 30x₂ + 20x₃ + 1x₄ + 3x₃ + 2x₆

Constraints:
2x₁ + 3x₂ + 5x₃ = 65
4x₁ + 1x₂ + 3x₃ = 60
3x₁ + 2x₂ + 1x₃ = 60
x₁, ..., x₆ < 0

 

[Zone Ranger]

Argh! I'm so confused now, I have no idea where to begin! We're supposed to use the Simplex Method to solve it but our textbook only deals with mixed constraints, wherein we use the Big M method (ie. add in surplus, artificial variables, etc). I'm not sure whether you would use the Big M method with this one because the contraints are all equalities. I need a walkthrough the steps you would take to solve it, because I don't even know where to begin and I've been trying everything I can think of.

Anyway, the specific question is:

MAX P= 28x₁ + 30x₂ + 20x₃ + 1x₄ + 3x₃ + 2x₆

Constraints:
2x₁ + 3x₂ + 5x₃ = 65
4x₁ + 1x₂ + 3x₃ = 60
3x₁ + 2x₂ + 1x₃ = 60
x₁, ..., x₆ < 0

Are you sure the problem is stated as above?

if \(\displaystyle X_i\leq0\) then none of your other constraints are possible.

 

[langhost]

Oops. You're right. That last part should be x1, ..., x6 >= 0

 

[tkhunny]

Why is x3 listed twice in the the maximization function?

Since there are no constraints on x4, x5, or x6, you had better do something unusual. What's your plan?

 

[Zone Ranger]

Constraints:
2x₁ + 3x₂ + 5x₃ = 65
4x₁ + 1x₂ + 3x₃ = 60
3x₁ + 2x₂ + 1x₃ = 60

With these constraints you are forced to have


\(\displaystyle x_1=\frac{67}{6}\)

\(\displaystyle x_2=\frac{77}{6}\)

\(\displaystyle x_3=\frac{5}{6}\)

and as tkhunny pointed out

Why is x3 listed twice in the the maximization function?

Since there are no constraints on x4, x5, or x6, you had better do something unusual. What's your plan?

this makes me wonder if you have the problem stated correctly.

 

[langhost]

Gah....so embarrassed at my carelessness (I had the problem constraints wrong too). Restating the problem correctly (quadruple-checked I promise):

MAX P= 28x1 + 30x2 + 20x3 + 1x4 + 3x5 + 2x6

Constraints:
2x1 + 3x2 + 5x3 + 1x4 = 65
4x1 + 1x2 + 3x3 + 1x5 = 60
3x1 + 2x2 + 1x3 + 1x6= 60
x1, ..., x6 >= 0