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A General Framework for Designing Approximation Schemes for Combinatorial Optimization Problems with Many Objectives Combined into One

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  • Shashi Mittal

    (Amazon.com, Seattle, Washington 98109)

  • Andreas S. Schulz

    (Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

In this paper, we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function. Examples of such problems include those in which the objective function is a product or ratio of two linear functions, parallel machine scheduling problems with the makespan objective, robust versions of weighted multiobjective optimization problems, and assortment optimization problems with logit choice models. The main idea behind our approximation schemes is the construction of an approximate Pareto-optimal frontier of the functions that constitute the given objective. Using this idea, we give the first fully polynomial-time approximation schemes for the max-min resource allocation problem with a fixed number of agents, combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions, or the product of a fixed number of linear functions, and assortment optimization problems with logit choice model.

Suggested Citation

  • Shashi Mittal & Andreas S. Schulz, 2013. "A General Framework for Designing Approximation Schemes for Combinatorial Optimization Problems with Many Objectives Combined into One," Operations Research, INFORMS, vol. 61(2), pages 386-397, April.
    • Handle: RePEc:inm:oropre:v:61:y:2013:i:2:p:386-397
    DOI: 10.1287/opre.1120.1093
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    References listed on IDEAS

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    Cited by:

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    3. H. Edwin Romeijn & Dolores Romero Morales & Wilco Van den Heuvel, 2014. "Computational complexity of finding Pareto efficient outcomes for biobjective lot‐sizing models," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(5), pages 386-402, August.
    4. Nathan Adelgren & Akshay Gupte, 2022. "Branch-and-Bound for Biobjective Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 909-933, March.
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    6. Büsing, Christina & Goetzmann, Kai-Simon & Matuschke, Jannik & Stiller, Sebastian, 2017. "Reference points and approximation algorithms in multicriteria discrete optimization," European Journal of Operational Research, Elsevier, vol. 260(3), pages 829-840.
    7. Meng Qi & Ho‐Yin Mak & Zuo‐Jun Max Shen, 2020. "Data‐driven research in retail operations—A review," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(8), pages 595-616, December.
    8. Brad D. Woods & Abraham P. Punnen, 2020. "A class of exponential neighbourhoods for the quadratic travelling salesman problem," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 303-332, August.
    9. Nan Liu & Yuhang Ma & Huseyin Topaloglu, 2020. "Assortment Optimization Under the Multinomial Logit Model with Sequential Offerings," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 835-853, July.
    10. Gallego, Guillermo & Li, Anran & Truong, Van-Anh & Wang, Xinshang, 2020. "Approximation algorithms for product framing and pricing," LSE Research Online Documents on Economics 101983, London School of Economics and Political Science, LSE Library.
    11. Rui Chen & Hai Jiang, 2020. "Capacitated assortment and price optimization under the nested logit model," Journal of Global Optimization, Springer, vol. 77(4), pages 895-918, August.
    12. Rui Chen & Hai Jiang, 2020. "Assortment optimization with position effects under the nested logit model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(1), pages 21-33, February.
    13. Jacob B. Feldman & Huseyin Topaloglu, 2015. "Capacity Constraints Across Nests in Assortment Optimization Under the Nested Logit Model," Operations Research, INFORMS, vol. 63(4), pages 812-822, August.
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    15. Guillermo Gallego & Anran Li & Van-Anh Truong & Xinshang Wang, 2020. "Approximation Algorithms for Product Framing and Pricing," Operations Research, INFORMS, vol. 68(1), pages 134-160, January.

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