1D Cutting Stock Optimizer

How 1D Cutting Stock Optimization Works

The one-dimensional cutting stock problem asks how to cut required pieces from standard-length stock (bars, pipes, profiles, extrusions) using as few stocks as possible. The problem is NP-hard, so practical tools use fast heuristics that come very close to the optimum for typical workshop order lists.

First Fit Decreasing (FFD) sorts all pieces from longest to shortest, then places each piece into the first stock that still has room; if none fits, a new stock is opened. Best Fit Decreasing (BFD) also sorts pieces descending but places each piece into the stock that would have the least remaining space afterwards, which can pack tight combinations slightly better on some order mixes.

Each cut consumes material equal to the saw kerf. This calculator applies the kerf between consecutive pieces on the same stock (n pieces need n − 1 cuts), so the reported waste separates true end-of-bar offcut from kerf loss. For FFD, theory guarantees the result never exceeds 11/9 × optimum + 6/9 stocks (Dósa, 2007) — in practice it is usually optimal or within one stock of optimal.

When to Use This Tool

• Cutting rebar, structural steel, pipe, or aluminum extrusion order lists from standard 6 m or 12 m stock
• Estimating how many stock bars to purchase for a fabrication job before ordering material
• Comparing waste between two stock lengths offered by a supplier to pick the cheaper option
• Generating a cut sequence sheet for the saw operator with labeled pieces per bar
• Quantifying kerf loss separately from offcut waste when justifying a thinner blade

Common Mistakes to Avoid

• Ignoring kerf — on a 6 m bar cut into 20 pieces, a 3 mm kerf consumes ~57 mm, which can push the last piece onto a new bar
• Mixing units between the stock length and the piece list (entering stock in meters but pieces in millimeters)
• Entering gross piece counts for multiple assemblies instead of using the quantity column, which makes the order list hard to verify
• Treating the heuristic result as a guaranteed optimum for safety-critical material budgets — for large industrial runs, column-generation LP methods can save a few more stocks
• Forgetting that usable remnants (long offcuts) count as waste here — if you stock remnants for reuse, subtract them from the reported waste

How to Interpret Results

• Stocks Used is the headline number — compare it to the lower bound ⌈total piece length ÷ stock length⌉ to judge how tight the layout is
• Utilization above ~90% is good for mixed order lists; above 95% is excellent and usually means piece lengths combine well with the stock length
• Waste per pattern shows which bars carry the offcut — a single bar with most of the waste often yields a usable remnant
• Kerf Loss grows with piece count, not piece length — many short pieces lose proportionally more material to the blade
• If FFD and BFD give the same stock count, the layout is almost certainly optimal for that stock length; if they differ, take the better one

Related Standards & References

• Gilmore & Gomory (1961) — A Linear Programming Approach to the Cutting-Stock Problem (Operations Research 9:849-859), the foundational exact method
• Dósa (2007) — tight worst-case bound for First Fit Decreasing: FFD(I) ≤ 11/9·OPT(I) + 6/9
• Johnson (1973) — Near-optimal bin packing algorithms (MIT), the classical analysis of FFD/BFD family heuristics
• Commercial stock lengths commonly follow 6000 mm and 12000 mm mill bars for rebar and structural sections; verify your supplier list before optimizing

Frequently Asked Questions

Should I choose FFD or BFD?

Both sort pieces longest-first and differ only in bin choice: FFD takes the first stock with room, BFD the tightest fit. They usually produce the same stock count; when they differ, the gap is one stock. Run both (one click) and keep the better layout.

How is the kerf applied?

The kerf is added between consecutive pieces on the same stock — n pieces need n − 1 cuts, the trailing offcut needs no extra cut. Set the kerf to your actual blade or torch width (typically 2-4 mm for cold saws, 1-2 mm for band saws).

Can the result be worse than the true optimum?

Yes, by a provably small margin: FFD never needs more than 11/9 of the optimal stock count plus a constant. For workshop-scale order lists the result is almost always optimal; for mill-scale runs with thousands of pieces, exact column-generation solvers can sometimes save additional stocks.