Solving Minimization with Integar programming

Question

I am asked to solve the following problem:

The problem:

You were asked to repair a farm house with sheets of plywood.

• You were given thirty sheets of plywood. (each size = 10ft x 10ft)

• The house requires 20 circles (radius = 2.5ft ) and 15 rectangles (size = 6ft x 4ft)

• It costs 20 dollars to cut a circle and 15 dollars to cut a rectange. And there are three ways to cut the sheets as shown below:

Here is the image: https://i.stack.imgur.com/mOomz.png

Basically,

1st way: cutting 4 circles out of 1 sheet

2nd way: cutting 4 rectangles out of 1 sheet

3rd way: cutting 2cicles and 1 rectangle out of 1 sheet

(I set these into x,y,z:numbers of sheets cut in the way)

• You can also buy 1 single circle with 45 dollars and 1 rectangle with 30 dollars.

• You CANNOT waste more than 30% of the material. (Assuming total unused area <= 30%)

Here is how I solve the problem:

x,y,z = number of sheets cut in the ways shown above

Objective function:

M = 80x+60y+55z

Constrants:

1. 4x+2z<=20

2. 4y+z<=15

3. x+y+z<=30

4. 0.215x+0.04y+0.367z<=0.3(x+y+z)

It seems that I am getting all zero but I cannot figure out how to set the constraints.

I am asked to solve this with ORtools python.

But it doesn't make any sense to do it with the incorrect equations.

Answer 1

I think you got the first two constraints wrong, should be greater than.

Here a model where also the possibility to buy separate circles (c) and rectangles (r) have been included:

M = 80x+60y+55z+45c+30r

Constraints:

1. 4x+2z+c >= 20

2. 4y+z+r >= 15

3. x+y+z <= 30

4. 0.215x+0.04y+0.367z <= 0.3(x+y+z)

5. x >= 0, y >= 0, z >= 0, c >= 0, r >= 0

gives:

M=640, x=5, y=4, z=0, c=0, r=0