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Lexicographically Minimum and Maximum Load Linear Programming Problems

Author

Listed:
  • Dritan Nace

    (Laboratoire Heudiasyc UMR CNRS 6599, Université de Technologie de Compiègne, 60205 Compiègne Cedex, France)

  • James B. Orlin

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

In this paper, we introduce the lexicographically minimum load linear programming problem, and we provide a polynomial approach followed by the proof of correctness. This problem has applications in numerous areas where it is desirable to achieve an equitable distribution or sharing of resources. We consider the application of our technique to the problem of lexicographically minimum load in capacitated multicommodity networks and discuss a special nonlinear case, the so-called Kleinrock load function. We next define the lexicographically maximum load linear programming problem and deduce a similar approach. An application in the lexicographically maximum concurrent flow problem is depicted followed by a discussion on the minimum balance problem as a special case of the lexicographically maximum load problem.

Suggested Citation

  • Dritan Nace & James B. Orlin, 2007. "Lexicographically Minimum and Maximum Load Linear Programming Problems," Operations Research, INFORMS, vol. 55(1), pages 182-187, February.
    • Handle: RePEc:inm:oropre:v:55:y:2007:i:1:p:182-187
    DOI: 10.1287/opre.1060.0341
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    References listed on IDEAS

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    1. Freund, Robert Michael. & Roundy, Robin. & Todd, Michael J., 1947-, 1985. "Identifying the set of always-active constraints in a system of linear inequalities by a single linear program," Working papers 1674-85., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    2. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    4. Hans Schneider & Michael H. Schneider, 1991. "Max-Balancing Weighted Directed Graphs and Matrix Scaling," Mathematics of Operations Research, INFORMS, vol. 16(1), pages 208-222, February.
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    2. Dukkanci, Okan & Karsu, Özlem & Kara, Bahar Y., 2022. "Planning sustainable routes: Economic, environmental and welfare concerns," European Journal of Operational Research, Elsevier, vol. 301(1), pages 110-123.
    3. Tang, Ming & Liao, Huchang, 2024. "Group efficiency and individual fairness tradeoff in making wise decisions," Omega, Elsevier, vol. 124(C).
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    5. Karsu, Özlem & Morton, Alec, 2015. "Inequity averse optimization in operational research," European Journal of Operational Research, Elsevier, vol. 245(2), pages 343-359.
    6. Christ, Quentin & Dauzère-Pérès, Stéphane & Lepelletier, Guillaume, 2019. "An Iterated Min–Max procedure for practical workload balancing on non-identical parallel machines in manufacturing systems," European Journal of Operational Research, Elsevier, vol. 279(2), pages 419-428.
    7. Mohammadmehdi Hakimifar & Vera C. Hemmelmayr & Fabien Tricoire, 2022. "A Bi-Objective Field-Visit Planning Problem for Rapid Needs Assessment under Travel-Time Uncertainty," Sustainability, MDPI, vol. 14(5), pages 1-16, March.
    8. Mohammadmehdi Hakimifar & Vera C. Hemmelmayr & Fabien Tricoire, 2023. "A lexicographic maximin approach to the selective assessment routing problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 45(1), pages 205-249, March.

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