{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "gilmore-gomory-pattern-column-generation-1961", "title": "Gilmore and Gomory: generate useful cutting patterns instead of listing them all", "summary": "Gilmore and Gomory solve the linear relaxation of a cutting-stock model by generating profitable cutting patterns through a knapsack pricing problem, keeping only a small working basis rather than enumerating every possible pattern.\n\nA factory cutting standard lengths into ordered pieces can have an enormous number of possible cutting patterns. Listing every pattern before optimizing may be impractical. Gilmore and Gomory let the current linear solution assign a value to each ordered piece, then ask a smaller knapsack problem to construct a pattern worth more than its stock cost. A profitable pattern enters the working basis; if exact pricing finds none for every stock length, the linear solution is optimal. Fast ad hoc searches can find improvements, but their failure must be checked by the more complete pricing calculation. The paper solves a worked example costing 170 and notes that its whole-number answer is fortuitous. In general the master solution can be fractional: rounding produces a feasible or repaired integer plan, but does not establish the cheapest integer plan.", "boundary": "matrix/formula glyphs are damaged, not certified transcriptions. Unlimited stock of specified lengths and costs; one-dimensional cutting patterns and order demands. Page 851 explicitly restricts this report to the continuous LP relaxation; integer pattern counts can remain fractional and rounding may be suboptimal. An integer knapsack pricing subproblem does not make the LP master integer. Heuristic pricing can find useful columns; failure requires the complete dynamic-programming check across stock types before claiming LP optimality. Degeneracy/anti-cycling treatment is discussed, not ignored. The worked example has stock lengths 5,6,9 costs 6,7,10 and demands 20,10,20 for lengths 2,3,4; its integer cost 170 is explicitly called fortuitous. This example is not a general benchmark. Ford/Fulkerson, Dantzig/Wolfe and knapsack/dynamic-programming predecessors retain credit. Part II 1963 and later integer/branch-and-price/multidimensional methods are separate. Original reprint says November 1961; Crossref deposits December 1961; only year and issue are used, without silently resolving the discrepancy. The editorial account uses the publication year and does not assert a manuscript receipt date. The complete implementation and cost-model results have not been independently certified here.", "area": "Optimization and operations research", "topic": "Mathematics and computing", "year": 1961, "lane": "HUMAN", "result": "ALGORITHM", "package_id": "gilmore-gomory-pattern-column-generation-1961", "target": { "record_id": "urn:uuid:68147a11-7780-4ff8-a75d-14230038e22e", "version": 1, "content_hash": "sha256:7dd818bf2e28b071a5e2116c9ac463c9bf71cfad408cf545ed9763906dbd96cd" }, "humans": [ "P. C. Gilmore", "R. E. Gomory" ], "authors_complete": true, "human_role_note": "Source-listed contributors to this scoped work. 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Unlimited stock of specified lengths and costs; one-dimensional cutting patterns and order demands. Page 851 explicitly restricts this report to the continuous LP relaxation; integer pattern counts can remain fractional and rounding may be suboptimal. An integer knapsack pricing subproblem does not make the LP master integer. Heuristic pricing can find useful columns; failure requires the complete dynamic-programming check across stock types before claiming LP optimality. Degeneracy/anti-cycling treatment is discussed, not ignored. The worked example has stock lengths 5,6,9 costs 6,7,10 and demands 20,10,20 for lengths 2,3,4; its integer cost 170 is explicitly called fortuitous. This example is not a general benchmark. Ford/Fulkerson, Dantzig/Wolfe and knapsack/dynamic-programming predecessors retain credit. Part II 1963 and later integer/branch-and-price/multidimensional methods are separate. Original reprint says November 1961; Crossref deposits December 1961; only year and issue are used, without silently resolving the discrepancy. The editorial account uses the publication year and does not assert a manuscript receipt date. 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Unlimited stock of specified lengths and costs; one-dimensional cutting patterns and order demands. Page 851 explicitly restricts this report to the continuous LP relaxation; integer pattern counts can remain fractional and rounding may be suboptimal. An integer knapsack pricing subproblem does not make the LP master integer. Heuristic pricing can find useful columns; failure requires the complete dynamic-programming check across stock types before claiming LP optimality. Degeneracy/anti-cycling treatment is discussed, not ignored. The worked example has stock lengths 5,6,9 costs 6,7,10 and demands 20,10,20 for lengths 2,3,4; its integer cost 170 is explicitly called fortuitous. This example is not a general benchmark. Ford/Fulkerson, Dantzig/Wolfe and knapsack/dynamic-programming predecessors retain credit. Part II 1963 and later integer/branch-and-price/multidimensional methods are separate. Original reprint says November 1961; Crossref deposits December 1961; only year and issue are used, without silently resolving the discrepancy. The editorial account uses the publication year and does not assert a manuscript receipt date. The complete implementation and cost-model results have not been independently certified here.", "domain": "Optimization and operations research" }, "sources": [ { "citation": "Gilmore and Gomory: generate useful cutting patterns instead of listing them all; exact original technical paper", "url": "https://ralphgomory.org/wp-content/uploads/2023/02/10_cutting_stock.pdf" }, { "citation": "Publisher-deposited identity for gilmore-gomory-pattern-column-generation-1961", "url": "https://api.crossref.org/works/10.1287%2Fopre.9.6.849" } ], "statement": "# Gilmore and Gomory: generate useful cutting patterns instead of listing them all\n\nGilmore and Gomory solve the linear relaxation of a cutting-stock model by generating profitable cutting patterns through a knapsack pricing problem, keeping only a small working basis rather than enumerating every possible pattern.\n\nA factory cutting standard lengths into ordered pieces can have an enormous number of possible cutting patterns. Listing every pattern before optimizing may be impractical. Gilmore and Gomory let the current linear solution assign a value to each ordered piece, then ask a smaller knapsack problem to construct a pattern worth more than its stock cost. A profitable pattern enters the working basis; if exact pricing finds none for every stock length, the linear solution is optimal. Fast ad hoc searches can find improvements, but their failure must be checked by the more complete pricing calculation. The paper solves a worked example costing 170 and notes that its whole-number answer is fortuitous. In general the master solution can be fractional: rounding produces a feasible or repaired integer plan, but does not establish the cheapest integer plan.\n\n## Scientific scope\n\nmatrix/formula glyphs are damaged, not certified transcriptions. Unlimited stock of specified lengths and costs; one-dimensional cutting patterns and order demands. Page 851 explicitly restricts this report to the continuous LP relaxation; integer pattern counts can remain fractional and rounding may be suboptimal. An integer knapsack pricing subproblem does not make the LP master integer. Heuristic pricing can find useful columns; failure requires the complete dynamic-programming check across stock types before claiming LP optimality. Degeneracy/anti-cycling treatment is discussed, not ignored. The worked example has stock lengths 5,6,9 costs 6,7,10 and demands 20,10,20 for lengths 2,3,4; its integer cost 170 is explicitly called fortuitous. This example is not a general benchmark. Ford/Fulkerson, Dantzig/Wolfe and knapsack/dynamic-programming predecessors retain credit. Part II 1963 and later integer/branch-and-price/multidimensional methods are separate. Original reprint says November 1961; Crossref deposits December 1961; only year and issue are used, without silently resolving the discrepancy. The editorial account uses the publication year and does not assert a manuscript receipt date. The complete implementation and cost-model results have not been independently certified here.\n\n## People and systems\n\nP. C. Gilmore; R. E. Gomory\n\n\n\n## Original sources\n\n- [Gilmore and Gomory: generate useful cutting patterns instead of listing them all; exact original technical paper](https://ralphgomory.org/wp-content/uploads/2023/02/10_cutting_stock.pdf)\n- [Publisher-deposited identity for gilmore-gomory-pattern-column-generation-1961](https://api.crossref.org/works/10.1287%2Fopre.9.6.849)\n", "summary": "Gilmore and Gomory solve the linear relaxation of a cutting-stock model by generating profitable cutting patterns through a knapsack pricing problem, keeping only a small working basis rather than enumerating every possible pattern.\n\nA factory cutting standard lengths into ordered pieces can have an enormous number of possible cutting patterns. Listing every pattern before optimizing may be impractical. Gilmore and Gomory let the current linear solution assign a value to each ordered piece, then ask a smaller knapsack problem to construct a pattern worth more than its stock cost. A profitable pattern enters the working basis; if exact pricing finds none for every stock length, the linear solution is optimal. Fast ad hoc searches can find improvements, but their failure must be checked by the more complete pricing calculation. The paper solves a worked example costing 170 and notes that its whole-number answer is fortuitous. In general the master solution can be fractional: rounding produces a feasible or repaired integer plan, but does not establish the cheapest integer plan.", "title": "Gilmore and Gomory: generate useful cutting patterns instead of listing them all" }, "record_id": "urn:uuid:68147a11-7780-4ff8-a75d-14230038e22e", "version": 1, "source_ref": { "record_id": "urn:uuid:68147a11-7780-4ff8-a75d-14230038e22e", "version": 1, "content_hash": "sha256:7dd818bf2e28b071a5e2116c9ac463c9bf71cfad408cf545ed9763906dbd96cd" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:852660380b17b85b20a23e4b2d896b06bd30af3704c5bef17b327159d711eb3e" }, "target": { "record_id": "urn:uuid:68147a11-7780-4ff8-a75d-14230038e22e", "version": 1, "content_hash": "sha256:7dd818bf2e28b071a5e2116c9ac463c9bf71cfad408cf545ed9763906dbd96cd" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "gilmore-gomory-pattern-column-generation-1961", "entry_id": "gilmore-gomory-pattern-column-generation-1961", "revision": 1, "title": "Gilmore and Gomory: generate useful cutting patterns instead of listing them all", "context": "Optimization and operations research", "track": "Optimization and operations research", "band": "A", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-04", "evidence_cutoff": "2026-10-04", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "An implicit-pattern LP formulation made computationally usable by knapsack pricing, with a small working basis and an explicit distinction between finding a column heuristically and certifying no improving column.", "scope": "An implicit-pattern LP formulation made computationally usable by knapsack pricing, with a small working basis and an explicit distinction between finding a column heuristically and certifying no improving column. matrix/formula glyphs are damaged, not certified transcriptions. Unlimited stock of specified lengths and costs; one-dimensional cutting patterns and order demands. Page 851 explicitly restricts this report to the continuous LP relaxation; integer pattern counts can remain fractional and rounding may be suboptimal. An integer knapsack pricing subproblem does not make the LP master integer. Heuristic pricing can find useful columns; failure requires the complete dynamic-programming check across stock types before claiming LP optimality. Degeneracy/anti-cycling treatment is discussed, not ignored. The worked example has stock lengths 5,6,9 costs 6,7,10 and demands 20,10,20 for lengths 2,3,4; its integer cost 170 is explicitly called fortuitous. This example is not a general benchmark. Ford/Fulkerson, Dantzig/Wolfe and knapsack/dynamic-programming predecessors retain credit. Part II 1963 and later integer/branch-and-price/multidimensional methods are separate. Original reprint says November 1961; Crossref deposits December 1961; only year and issue are used, without silently resolving the discrepancy. The editorial account uses the publication year and does not assert a manuscript receipt date. The complete implementation and cost-model results have not been independently certified here.", "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.", "references": [ { "title": "Gilmore and Gomory: generate useful cutting patterns instead of listing them all; exact original technical paper", "url": "https://ralphgomory.org/wp-content/uploads/2023/02/10_cutting_stock.pdf", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." }, { "title": "Publisher-deposited identity for gilmore-gomory-pattern-column-generation-1961", "url": "https://api.crossref.org/works/10.1287%2Fopre.9.6.849", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." } ], "roles": [ { "who": "P. C. Gilmore", "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.", "credit": "SCIENCE" }, { "who": "R. E. 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