Linear Cutting Optimization: The 1D Cutting Stock Problem

What is 1D cutting stock?

In the one-dimensional cutting stock problem, you have stock of a fixed length — say 6 m steel bars — and a list of required pieces, such as ten 1.2 m and eight 2.1 m lengths. The goal is to decide how to cut the bars so the total number used is minimized and the leftover drop is as small as possible.

It is the same family of problem as 2D sheet cutting and bin packing, just along a single dimension. Because the part list can be combined in many ways, software finds far better combinations than cutting by eye.

Where linear cutting applies

• Metal: steel bars, aluminium extrusions, tube, and angle iron.
• Construction: rebar cut to a bar bending schedule.
• Woodworking: lumber, trim, and moulding cut to length.
• Profiles: PVC and aluminium window and door profiles.
• Piping: cutting pipe runs from standard stock lengths.

Accounting for the saw kerf

Every cut removes a few millimetres of material as kerf. In linear cutting this matters because many short cuts from one bar accumulate kerf loss. A good optimizer subtracts the kerf for each cut so the plan reflects the real usable length.

Reducing linear waste

• Combine part lengths that sum close to the stock length.
• Reuse usable drops (offcuts) on later jobs instead of scrapping them.
• Order stock in lengths that divide cleanly into your common parts.
• Group identical cuts to reduce saw setup changes.