Question

A cardboard cutting company received three orders to cut rolls to the measurements shown below:

Order	Width (meters)	Length (meters)
A	0.50	1000
B	0.70	3000
C	0.90	2000

This company buys the cardboard to be cut into two standard widths: 1 and 2 meters and then cuts it according to what each order requires. The standard rolls do not have a defined length, since for practical purposes the cardboard can be glued to comply with the required length.

Formulate the problem as a PL model to determine the optimal cut patterns that minimize waste as a PL model (all leftovers less than 0.50 meters wide are considered waste).

Answer 1

Following patterns of rolls for orders A, B, C can be cut from Standard rolls of 1 meter width

Width (m)		0.5	0.7	0.9
Order		A	B	C	Waste (m)
1 m roll	X1	2			0
	X2		1		0.3
	X3			1	0.1

Following patterns of rolls for orders A, B, C can be cut from Standard rolls of 2 meter width  

Width (m)		0.5	0.7	0.9
Order		A	B	C	Waste (m)
2 mtr roll	Y1	1	2		0.1
	Y2	2	1		0.3
	Y3	2		1	0.1
	Y4	4			0
	Y5		1	1	0.4
	Y6			2	0.2

where, X1, X2, X3 represent the length (in meters) of patterns to be cut from 1 meter standard rolls and Y1, Y2, ... Y6 represent the length (in meters) of patterns to be cut from 2 meter standard rolls

LP model to minimize the total waste is following:

Minimize 0X1+0.3X2+0.1X3+0.1Y1+0.3Y2+0.1Y3+0Y4+0.4Y5+02Y6

s.t.

2X1+1Y1+2Y2+2Y3+4Y4 >= 1000

1X2+2Y1+1Y2+1Y5 >= 3000

1X3+1Y3+1Y5+2Y6 >= 2000

Xi, Yi >= 0