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Tolerance Sensitivity and Optimality Bounds in Linear Programming

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  • Richard E. Wendell

    (Katz Graduate School of Business, University of Pittsburgh, Pittsburgh, Pennsylvania 15260)

Abstract

Traditional sensitivity analysis in linear programming usually focuses on variations of one coefficient or term at a time. The tolerance approach was proposed to provide a decision maker with an effective and easy-to-use method to summarize the effects of simultaneous and independent changes in selected parameters. In particular, for variations of the objective function coefficients, the approach gives a maximum-tolerance percentage within which selected coefficients may vary from their estimated values (within a priori limits) while still retaining the same optimal basic feasible solution. Although an optimal solution may cease being optimal for variations beyond the maximum-tolerance percentage, it may still be close to optimal. Herein we characterize the potential loss of optimality for variations beyond the maximum-tolerance percentage as a maximum-regret function. We consider theoretical properties of this function and propose a method to compute a relevant portion of it.

Suggested Citation

  • Richard E. Wendell, 2004. "Tolerance Sensitivity and Optimality Bounds in Linear Programming," Management Science, INFORMS, vol. 50(6), pages 797-803, June.
    • Handle: RePEc:inm:ormnsc:v:50:y:2004:i:6:p:797-803
    DOI: 10.1287/mnsc.1030.0221
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    References listed on IDEAS

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    1. Wang, Hsiao-Fan & Huang, Chen-Sheng, 1993. "Multi-parametric analysis of the maximum tolerance in a linear programming problem," European Journal of Operational Research, Elsevier, vol. 67(1), pages 75-81, May.
    2. Richard E. Wendell, 1985. "The Tolerance Approach to Sensitivity Analysis in Linear Programming," Management Science, INFORMS, vol. 31(5), pages 564-578, May.
    3. Harvey M. Wagner, 1995. "Global Sensitivity Analysis," Operations Research, INFORMS, vol. 43(6), pages 948-969, December.
    4. Harvey J. Greenberg, 1993. "How to Analyze the Results of Linear Programs—Part 2: Price Interpretation," Interfaces, INFORMS, vol. 23(5), pages 97-114, October.
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    Cited by:

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    2. Filippi, Carlo, 2005. "A fresh view on the tolerance approach to sensitivity analysis in linear programming," European Journal of Operational Research, Elsevier, vol. 167(1), pages 1-19, November.
    3. Hinojosa, M.A. & Mármol, A.M., 2011. "Axial solutions for multiple objective linear problems. An application to target setting in DEA models with preferences," Omega, Elsevier, vol. 39(2), pages 159-167, April.
    4. M. A. Hinojosa & A. M. Mármol, 2011. "Egalitarianism and Utilitarianism in Multiple Criteria Decision Problems with Partial Information," Group Decision and Negotiation, Springer, vol. 20(6), pages 707-724, November.
    5. Scott E. Sampson, 2008. "OR PRACTICE---Optimization of Vacation Timeshare Scheduling," Operations Research, INFORMS, vol. 56(5), pages 1079-1088, October.
    6. Curry, Stewart & Lee, Ilbin & Ma, Simin & Serban, Nicoleta, 2022. "Global sensitivity analysis via a statistical tolerance approach," European Journal of Operational Research, Elsevier, vol. 296(1), pages 44-59.
    7. Henriques, C.O. & Inuiguchi, M. & Luque, M. & Figueira, J.R., 2020. "New conditions for testing necessarily/possibly efficiency of non-degenerate basic solutions based on the tolerance approach," European Journal of Operational Research, Elsevier, vol. 283(1), pages 341-355.
    8. Neralić, Luka & Wendell, Richard E., 2019. "Enlarging the radius of stability and stability regions in Data Envelopment Analysis," European Journal of Operational Research, Elsevier, vol. 278(2), pages 430-441.
    9. Xuefei Lu & Alessandro Rudi & Emanuele Borgonovo & Lorenzo Rosasco, 2020. "Faster Kriging: Facing High-Dimensional Simulators," Operations Research, INFORMS, vol. 68(1), pages 233-249, January.
    10. Hladík, Milan, 2010. "Multiparametric linear programming: Support set and optimal partition invariancy," European Journal of Operational Research, Elsevier, vol. 202(1), pages 25-31, April.
    11. Ma, Kang-Ting & Lin, Chi-Jen & Wen, Ue-Pyng, 2013. "Type II sensitivity analysis of cost coefficients in the degenerate transportation problem," European Journal of Operational Research, Elsevier, vol. 227(2), pages 293-300.

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