How rebar cutting plans are optimised: the cutting stock problem

Updated: 2026-08-04 · ISKRIT

Cutting a bar schedule out of stock lengths is a classic one-dimensional cutting stock problem. It has a well developed theory, and the gap between a naive plan and an optimised one is usually a few percent of the total steel tonnage.

What is being minimised

Not the number of bars, and not the number of cuts. The objective is total stock length consumed, because that is what is bought and weighed. A plan that uses fewer bars but leaves long unusable ends is worse than one that uses more bars and returns reusable offcuts to the rack.

Greedy against column generation

The straightforward approach fills one bar at a time, always taking the best pattern for the pieces still outstanding. It is fast and it is often within a percent of optimal, but it can strand awkward lengths at the end of the run, which is exactly when there is nothing left to pair them with.

The Gilmore and Gomory method treats the problem as a linear program over cutting patterns. Since there are far too many patterns to enumerate, patterns are generated on demand: the linear relaxation produces dual prices for each required length, and a knapsack solve against those prices returns the pattern worth adding. The integer part of the resulting mix is applied and a greedy pass finishes the remainder. Taking whichever of the two solutions consumes less stock is a safe rule, because both are legitimate plans.

Kerf and discretisation

Every cut removes material. In the arithmetic the remainder of a bar is stock length minus the sum of pieces minus one kerf per piece, and ignoring that term makes a plan that does not fit on the saw. Lengths are handled on a fixed step, one millimetre being the practical choice, so that piece lengths and kerfs stay exact integers and no rounding drift accumulates across a bar.

Offcuts and splices

A real yard has a rack of previous offcuts. Consuming those first before touching a new bar changes the answer, so they belong in the input rather than in a separate manual pass. Pieces longer than a stock bar need splices, and every splice adds a lap length to the total demand, which the plan must account for before it decides how many bars to open.

Questions

How much is there to gain? Against a manual plan, commonly three to six percent of stock length; against a decent greedy plan, often under one.

Is the optimum worth chasing? Only where the run is large. On a small schedule the difference is one bar, and the time spent is worth more than the steel.

The rebar cutting tool runs both the greedy and the column generation passes and keeps the better plan.