Thread: Linear Programming Problem!!

Consider the following linear programming problem:

Maximise Z = 100x1 + 200x2

Subject to 3x1 + 2x2 ≤ 120
x1 + 4x2 ≤ 80
2x1 + x2 ≤ 100
x1, x2 ≥ 0

Solution is Z=5600
x1=32
x2=12
s3=24

If a new constraint 2x1+ 2x2 ≤ 100 was included in the original problem, how would this affect the solution?

My answer is that there is no change. But is this correct and is tehre a quicker way of getting the answer rather than the long drawn-out simplex method??


***EDIT I have another question where there are 3 variables X,Y, and Z. How do I plot that/answer the question then??? ***

Hello, DooBeeDoo!

Consider the following linear programming problem:

Maximise: .

Subject to: .

Solution is: .

If a new constraint was included in the original problem,
how would this affect the solution?

My answer is that there is no change. . . . . I agree

Is there a quicker way of getting the answer?
. . You weren't shown how to graph these problems?

Constraint [4] puts us in Quandrant 1.

For , graph the line: .
It has intercepts: (40, 0), (0, 60)
Draw the line and shade the region below it.

For graph the line: .
It has intercepts: (80, 0), (0, 20)
Draw the line and shade the region below it.

The shaded region looks like this . . .

Code:

        |
     60 *
        |*
        | *
        |  *
        |   *
        |    *
     20 o     *
        |:::*  *
        |:::::::o
        |::::::::*  *
        |:::::::::*     *
        |::::::::::*        *
    - - o - - - - - o - - - - - * - -
        |          40          80

Constraint [3] is useless.
The line has intercepts (50, 0), (0, 100)
. . and falls outside of the shaded region.


We are concerned with the vertices of the shaded region.
We can see three of them: .(0, 0), (40, 0), (0, 20)

We find the fourth by locating the intersection of the two slanted lines.

We solve the system: . 32,\:12)" alt="\begin{array}{ccc}3x + 2y &=&120 \\ x + 4y &=& 80\end{array}\;\text{ and get: }\32,\:12)" />


We substitute these coordinates into the Z-function
. . to determine which one maximizes Z.

thanks a lot dude. very helpful.

I have another question where there are 3 variables X,Y, and Z. How do I plot that/answer the question then???

Hello, DooBeeDoo!

I have another question where there are 3 variables X,Y, and Z.
How do I plot that / answer the question then?

With three or more variables, there are mechanical procedures.
. . The most common is the Simplex Method.
Do a Google search.