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An observation on the structure of production sets with indivisibilities

In: Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research

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  • Herbert E. Scarf

    (Yale University)

Abstract

A subset of the constraints of an integer programming problem is said to be binding if, when the remaining constraints are eliminated, the smaller problem has the same optimal solution as the original problem. It is shown that an integer programming problem with n variables has a set of binding constraints of cardinality less than or equal to 2n−1. The bound is sharp.

Suggested Citation

  • Herbert E. Scarf, 2008. "An observation on the structure of production sets with indivisibilities," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 1, pages 1-5, Palgrave Macmillan.
    • Handle: RePEc:pal:palchp:978-1-137-02441-1_1
    DOI: 10.1057/9781137024411_1
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    Cited by:

    1. I. Bárány & H. E. Scarf & D. Shallcross, 2008. "The topological structure of maximal lattice free convex bodies: The general case," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 11, pages 191-205, Palgrave Macmillan.
    2. Queyranne, M. & Tardella, F., 2015. "Carathéodory, Helly and Radon Numbers for Sublattice Convexities," LIDAM Discussion Papers CORE 2015010, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Walter Briec & Kristiaan Kerstens & Ignace Van de Woestyne, 2022. "Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review," Springer Books, in: Subhash C. Ray & Robert G. Chambers & Subal C. Kumbhakar (ed.), Handbook of Production Economics, chapter 18, pages 721-754, Springer.
    4. Herbert E. Scarf, 2008. "Integral Polyhedra in Three Space," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 4, pages 69-104, Palgrave Macmillan.
    5. Wiesława Obuchowska, 2010. "Minimal infeasible constraint sets in convex integer programs," Journal of Global Optimization, Springer, vol. 46(3), pages 423-433, March.
    6. Friedrich Eisenbrand & Gennady Shmonin, 2008. "Parametric Integer Programming in Fixed Dimension," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 839-850, November.
    7. Valentin Borozan & Gérard Cornuéjols, 2009. "Minimal Valid Inequalities for Integer Constraints," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 538-546, August.
    8. Maurice Queyranne & Fabio Tardella, 2017. "Carathéodory, Helly, and Radon Numbers for Sublattice and Related Convexities," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 495-516, May.
    9. Obuchowska, Wiesława T., 2012. "Feasibility in reverse convex mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 218(1), pages 58-67.

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