1. Cutting Stock Problem
Problem of cutting an unlimited number of pieces of material (paper rolls, for instance) of length l
to produce
ni pieces of length li , i = 1, 2, …, I.
The objective is to minimize the number of pieces of material to meet the demands.
Note:
minimize number of pieces ≡ minimize total waste

2. Cutting pattern Pj (j = 1, 2, …, NJ) corresponds to a specific way of cutting a piece of material:
aij = number of pieces of length li produced with cutting pattern Pj
where aij ≥ 0 and integer, i = 1, 2, …, I

3. Mathematical Model
where xj represents the number of pieces of material cut with the pattern j

4. Integer programming problem difficult to solve.
Drop the constraints that the xj are required to be integer
When the demands ni are large, little is lost with respect to optimality by rounding the solution obtained by solving the relaxed problem

5. The number NJ of patterns may be very large.
For problems where the piece of material have length l = 800 cm., and where there are demands for 40 different lengths li from 20 cm. to 80 cm., the number of admissible patterns can easily reach 10 to 100 millions.
Interest to solve this problem using column generation.
At each iteration, a subset of patterns are available to specify a restricted problem of

7. Solve the restricted problem.