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Bounds on the Effect of Aggregating Variables in Linear Programs

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  • Paul H. Zipkin

    (Columbia University, New York, New York)

Abstract

This paper explores the effects of aggregating variables in large linear programs. We define a reasonable criterion for the resulting loss in accuracy, and derive bounds on this quantity. A posteriori bounds may be calculated after solving the aggregated problem, and a priori bounds before. Also, we show that standard iterative methods can be used to improve the accuracy of a given aggregated problem. A numerical example illustrates the results.

Suggested Citation

  • Paul H. Zipkin, 1980. "Bounds on the Effect of Aggregating Variables in Linear Programs," Operations Research, INFORMS, vol. 28(2), pages 403-418, April.
    • Handle: RePEc:inm:oropre:v:28:y:1980:i:2:p:403-418
    DOI: 10.1287/opre.28.2.403
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    Cited by:

    1. Igor Litvinchev & Socorro Rangel, 2006. "Using error bounds to compare aggregated generalized transportation models," Annals of Operations Research, Springer, vol. 146(1), pages 119-134, September.
    2. Bjørndal, Endre & Jörnsten, Kurt, 2009. "Lower and upper bounds for linear production games," European Journal of Operational Research, Elsevier, vol. 196(2), pages 476-486, July.
    3. Fu Lin & Sven Leyffer & Todd Munson, 2016. "A two-level approach to large mixed-integer programs with application to cogeneration in energy-efficient buildings," Computational Optimization and Applications, Springer, vol. 65(1), pages 1-46, September.
    4. Merrick, James H. & Weyant, John P., 2019. "On choosing the resolution of normative models," European Journal of Operational Research, Elsevier, vol. 279(2), pages 511-523.
    5. Merrick, James H., 2016. "On representation of temporal variability in electricity capacity planning models," Energy Economics, Elsevier, vol. 59(C), pages 261-274.
    6. M S Sodhi & C S Tang, 2011. "Determining supply requirement in the sales-and-operations-planning (S&OP) process under demand uncertainty: a stochastic programming formulation and a spreadsheet implementation," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(3), pages 526-536, March.
    7. Fleischmann, Moritz & Kloos, Konstantin & Nouri, Maryam & Pibernik, Richard, 2020. "Single-period stochastic demand fulfillment in customer hierarchies," European Journal of Operational Research, Elsevier, vol. 286(1), pages 250-266.
    8. Haugen, Kjetil K., 1996. "A Stochastic Dynamic Programming model for scheduling of offshore petroleum fields with resource uncertainty," European Journal of Operational Research, Elsevier, vol. 88(1), pages 88-100, January.
    9. Alonso-Ayuso, Antonio & Carvallo, Felipe & Escudero, Laureano F. & Guignard, Monique & Pi, Jiaxing & Puranmalka, Raghav & Weintraub, Andrés, 2014. "Medium range optimization of copper extraction planning under uncertainty in future copper prices," European Journal of Operational Research, Elsevier, vol. 233(3), pages 711-726.
    10. John Turner, 2012. "The Planning of Guaranteed Targeted Display Advertising," Operations Research, INFORMS, vol. 60(1), pages 18-33, February.
    11. Klaassen, Pieter, 1997. "Solving stochastic programming models for asset/liability management using iterative disaggregation," Serie Research Memoranda 0010, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    12. Gonzato, Sebastian & Bruninx, Kenneth & Delarue, Erik, 2021. "Long term storage in generation expansion planning models with a reduced temporal scope," Applied Energy, Elsevier, vol. 298(C).
    13. Knolmayer, Gerhard, 1981. "A simulation study of simplification strategies in the development of optimization models," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 96, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    14. Murwan Siddig & Yongjia Song, 2022. "Adaptive partition-based SDDP algorithms for multistage stochastic linear programming with fixed recourse," Computational Optimization and Applications, Springer, vol. 81(1), pages 201-250, January.
    15. Ali Fattahi & Sriram Dasu & Reza Ahmadi, 2023. "Peak-Load Energy Management by Direct Load Control Contracts," Management Science, INFORMS, vol. 69(5), pages 2788-2813, May.
    16. Beltran-Royo, C., 2017. "Two-stage stochastic mixed-integer linear programming: The conditional scenario approach," Omega, Elsevier, vol. 70(C), pages 31-42.
    17. Sodhi, ManMohan S. & Tang, Christopher S., 2009. "Modeling supply-chain planning under demand uncertainty using stochastic programming: A survey motivated by asset-liability management," International Journal of Production Economics, Elsevier, vol. 121(2), pages 728-738, October.
    18. Jornsten, Kurt & Leisten, Rainer, 1995. "Decomposition and iterative aggregation in hierarchical and decentralised planning structures," European Journal of Operational Research, Elsevier, vol. 86(1), pages 120-141, October.
    19. Alexander H. Gose & Brian T. Denton, 2016. "Sequential Bounding Methods for Two-Stage Stochastic Programs," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 351-369, May.
    20. ManMohan S. Sodhi, 2005. "LP Modeling for Asset-Liability Management: A Survey of Choices and Simplifications," Operations Research, INFORMS, vol. 53(2), pages 181-196, April.
    21. Kremer, Mirko & Schneeweiss, Christoph & Zimmermann, Michael, 2006. "On the validity of aggregate models in designing supply chain contracts," International Journal of Production Economics, Elsevier, vol. 103(2), pages 656-666, October.
    22. Srinivasa, Anand V. & Wilhelm, Wilbert E., 1997. "A procedure for optimizing tactical response in oil spill clean up operations," European Journal of Operational Research, Elsevier, vol. 102(3), pages 554-574, November.

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