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Application of Programs with Maximin Objective Functions to Problems of Optimal Resource Allocation

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  • Seymour Kaplan

    (New York University, New York, New York)

Abstract

A mathematical program with a maximin objective function is defined as an optimization problem of the following type: Maxz = min i c i x i , subject to AX = b, X ≧ 0. Although the c i can be in the interval (− ∞, ∞), the paper discusses the more common practical case where all c i ≧ 0. It shows that problems of this type arise in a variety of applications where it is required to maximize a production function of the “fixed proportion” type subject to a set of linear constraints. Although it is well known that the solution to this type of problem can be found by linear programming, this paper shows that, if the existence of a certain condition can be demonstrated, then a simplified method can be used to determine the optimum solution. Many problems of practical interest can be solved by this simplified method; an example involving the readiness of a ship is presented.

Suggested Citation

  • Seymour Kaplan, 1974. "Application of Programs with Maximin Objective Functions to Problems of Optimal Resource Allocation," Operations Research, INFORMS, vol. 22(4), pages 802-807, August.
    • Handle: RePEc:inm:oropre:v:22:y:1974:i:4:p:802-807
    DOI: 10.1287/opre.22.4.802
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    Cited by:

    1. G. Yu, 1998. "Min-Max Optimization of Several Classical Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 98(1), pages 221-242, July.
    2. Yamada, Takeo & Futakawa, Mayumi & Kataoka, Seiji, 1998. "Some exact algorithms for the knapsack sharing problem," European Journal of Operational Research, Elsevier, vol. 106(1), pages 177-183, April.
    3. Takahito Kuno & Kouji Mori & Hiroshi Konno, 1989. "A mofified gub algorithm for solving linear minimax problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 36(3), pages 311-320, June.
    4. Kasin Ransikarbum & Scott J. Mason, 2016. "Multiple-objective analysis of integrated relief supply and network restoration in humanitarian logistics operations," International Journal of Production Research, Taylor & Francis Journals, vol. 54(1), pages 49-68, January.
    5. Hanan Luss, 1999. "On Equitable Resource Allocation Problems: A Lexicographic Minimax Approach," Operations Research, INFORMS, vol. 47(3), pages 361-378, June.
    6. Buchanan, John & Gardiner, Lorraine, 2003. "A comparison of two reference point methods in multiple objective mathematical programming," European Journal of Operational Research, Elsevier, vol. 149(1), pages 17-34, August.
    7. K. Mathur & M. Puri & S. Bansal, 1995. "On ranking of feasible solutions of a bottleneck linear programming problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 3(2), pages 265-283, December.
    8. Fujimoto, Masako & Yamada, Takeo, 2006. "An exact algorithm for the knapsack sharing problem with common items," European Journal of Operational Research, Elsevier, vol. 171(2), pages 693-707, June.
    9. Hanan Luss & Donald R. Smith, 1988. "Multiperiod allocation of limited resources: A minimax approach," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(4), pages 493-501, August.
    10. M Kumral, 2011. "Incorporating geo-metallurgical information into mine production scheduling," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(1), pages 60-68, January.
    11. I. Stancu-Minasian & R. Caballero & E. Cerdá & M. Muñoz, 1999. "The stochastic bottleneck linear programming problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(1), pages 123-143, June.

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