Minimize C=5x+4y given the constraints

Question

0<=x<=6

0<=y<=6

4x+2y<=8

2x+y>=4

 

Answer 1

Minimize C=5x+4y given the constraints
0<=x<=6
0<=y<=6
4x+2y<=8
2x+y>=4

I will do this by Solver, is easy and a modern way of solving this. Anyone knowing Excel knows the tool.

Objective Cell (Min)
Cell Name Original Value Final Value
$B$10 eq 10 10

Variable Cells
Cell Name Original Value Final Value Integer
$B$11 x 2                      2 Contin
$B$12 y 0                      0 Contin
The answer is: the minimum value of the equation is 10

To obtain this the values of the variables are:

x=2

Y=0

Satisfying the following Constraints:
Cell Name Cell Value Formula Status Slack
$B$13 8 $B$13<=8 Binding 0
$B$14 4 $B$14>=4 Binding 0
$B$11 x 2 $B$11<=6 Not Binding 4
$B$11 x 2 $B$11>=0 Not Binding 2
$B$12 y 0 $B$12<=6 Not Binding 6
$B$12 y 0 $B$12>=0 Binding 0

Answer 2

Hi Chalyce,

 

The first thing I think we should do is examine the 4 constraints. #1 and #2 tell us that the solution for the minimum C must come from the 6x6 square in the x-y plane bounded by (0,6) for both x and y. Now let's look at #3: 4x + 2y <= 8. We can divide all terms by 2 to get: 2x + y <= 4. Now let's move the x-term to the right side: y <= 4 - 2x. If we plot y = 4 - 2x on top of our 6x6 square, we have a line cutting off a small triangle near the (0,0) origin. For the "<" part of <=, this means the solution is constrained for (x,y) on the line or inside the triangle.

 

Now we look at the 4th constraint: 2x + y >= 4. Again, let's get the x-term to the right side: y >= 4 - 2x. For the "=" part of >=, this is exactly the same line as constraint #3 !! But the ">" tells us that we are constrained by the line and the trapezoid above the line.

 

To honor constraints #3 & #4, the only part in common is simply the line y = 4 - 2x. I hope you have plotted this line on our 6x6 square for constraints #1 & #2. If you did, you see that x ranges from 0 to 2. We'll need that information soon ...

 

Last step: we know our solution space is the line y = 4 - 2x. We can insert this into our function-to-minimize as follows:

 

     C = 5x + 4y,  with y = 4 - 2x

so, C = 5x + 4(4 - 2x) = 5x + 16 - 8x = 16 - 3x

 

I'll leave the rest for you -- we know C = 16 - 3x, and x ranges from 0 to 2. Please help me by telling what the minimum C is, and the (x,y).

 

Thanks, Pete