Cutting Stock Problem¶
An optimization problem that chooses feasible cutting patterns for available stock and their use counts to meet item demands while minimizing a declared stock-use, cost, or waste objective.
Core Idea¶
The cutting stock problem chooses how to obtain specified quantities of smaller pieces from larger stock at minimum declared cost, stock use, or waste. A feasible cutting pattern describes what one stock unit can yield; whole-number pattern counts then form a plan meeting each item demand. Gilmore and Gomory's original one-dimensional statement concerns ordered lengths cut from stock lengths of stated cost. The named problem also has higher-dimensional, multistage variants, so the uniform-roll model is not its entire scope.[ref-11847f6a0378][ref-2654a24de072]
Scope of Application¶
In paper-roll slitting, master-roll widths are stock, narrower customer rolls are items, and compatible width combinations are patterns. Kallrath and colleagues distinguish stock-count minimization from variants involving multiple master-roll sizes and pattern-setup costs. In corrugated-box production, sheets, rectangular blanks and permitted stages of cutting fill the same roles but have different geometric and sequencing constraints.[ref-baf881eac896][ref-2654a24de072]
Guillotine cutting is a possible pattern restriction, not a universal parent. The proposal awaits independent graph review.
Clarity¶
The problem is the stock, orders, feasible cuts and objective; column generation is one solution method. Its LP relaxation can use fractional pattern counts, which do not themselves tell a mill how many whole rolls or sheets to cut. Likewise, minimum stock count, minimum trim and minimum number of machine setups need not select the same plan. State the objective, overproduction rule and stock availability before comparing results.[ref-11847f6a0378][ref-2654a24de072][^ref-baf881eac896]
Manages Complexity¶
Pattern variables represent repeated cutting layouts instead of every individual cut. Demand constraints then compare output counts across all chosen patterns. Gilmore and Gomory addressed the huge number of possible patterns by keeping a small active LP matrix; later work shows that higher-dimensional pattern generation can be harder and sometimes requires practical stage restrictions.[ref-11847f6a0378][ref-2654a24de072]
This simplification is valid only when its feasible patterns reflect the real instance. A width-fitting arrangement may fail a staged sheet-cut rule; a geometrically feasible plan can fail an availability or sequencing condition.[ref-2654a24de072][ref-baf881eac896]
Abstract Reasoning¶
List stock units, item dimensions and required counts; specify admissible patterns and an objective. Select integral uses of patterns, verify every order quantity, then compare total stock or cost. For identical one-dimensional stock of length \(L\), total demanded length divided by \(L\), rounded up, is a necessary lower bound on the number of units, not a guarantee that pieces combine without leftover gaps. Mixed-stock and two-dimensional cases require their own feasibility and cost checks.[ref-11847f6a0378][ref-2654a24de072]
Knowledge Transfer¶
The roles transfer from paper rolls to corrugated-box sheets: available stock, demanded output, feasible cutting patterns, repetitions and objective. What changes is the pattern test—width combinations for roll slitting versus two-dimensional staged subdivisions and possible cut sequencing for sheets. The common optimization skeleton does not make the two production settings mechanically identical.[ref-baf881eac896][ref-2654a24de072]
[^ref-11847f6a0378]: P. C. Gilmore and R. E. Gomory, “A Linear Programming Approach to the Cutting-Stock Problem,” Operations Research 9(6) (1961), 849–859, publisher abstract. https://doi.org/10.1287/opre.9.6.849 [^ref-2654a24de072]: P. C. Gilmore and R. E. Gomory, “Multistage Cutting Stock Problems of Two and More Dimensions,” Operations Research 13(1) (1965), 94–120, publisher abstract. https://doi.org/10.1287/opre.13.1.94 [^ref-baf881eac896]: Julia Kallrath, Steffen Rebennack, Josef Kallrath and Rüdiger Kusche, “Solving real-world cutting stock-problems in the paper industry,” European Journal of Operational Research 238(1) (2014), 374–389, publisher abstract and visible Introduction. https://doi.org/10.1016/j.ejor.2014.03.027
Relationships to Other Abstractions¶
Current abstraction Cutting Stock Problem Domain-specific
Parents (1) — more general patterns this builds on
-
Cutting Stock Problem is a kind of Packing Problem Domain-specific
A cutting-stock plan packs demanded items without overlap into stock units under a declared optimization objective.
Hierarchy path (1) — routes to 1 parentless root
- Cutting Stock Problem → Packing Problem → Optimization
Neighborhood in Abstraction Space¶
Cutting Stock Problem sits in a sparse region of the domain-specific corpus (73rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Supply Chain & Inventory Management (28 abstractions)
Nearest neighbors
- Mass Customization — 0.85
- Assemble-to-order system — 0.84
- Economic Order Quantity — 0.83
- Portfolio Optimization — 0.83
- Quadratic Assignment Problem — 0.83
Computed from structural-signature embeddings · 2026-10-08