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Parametric Objective Function (Part 1)

Author

Listed:
  • Thomas Saaty

    (Melpar, Inc., Alexandria, Virginia)

  • Saul Gass

    (Computation Division, Department of the Air Force, Washington 25, D. C.)

Abstract

In the linear programming problem where there is a linear function to be optimized (called the objective function), it is desirable to study the behaviour of solutions when the (cost) coefficients in the objective function are parametrized. The problem is then to find the set of x j ( j = 1, 2, ..., n ) which minimizes the linear form (sum)( c j + (lambda) c j ') x j and satisfy the constraints x j (ge) 0 and (sum) a ıj x j = a ı 0 ( i = 1, 2, ..., m ) where c j , c j ', a ıj , and a ı 0 are constants, with at least one c j (ne) 0, and (lambda) a parameter. Using the simplex method, a computational procedure is described which enables one, given a feasible solution, to determine the values of the parameter (if any) for which the solution minimizes the objective function; and given a minimum feasible solution how one can generate by the simplex algorithm all minimum feasible solutions and the corresponding values of the parameter. To a minimum solution there corresponds one interval of values of (lambda) (nondegeneracy assumed). The process indicates how to obtain a new solution and the corresponding interval of values of (lambda) which is contiguous to the previous one both having only one point in common. Operations Research , ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.

Suggested Citation

  • Thomas Saaty & Saul Gass, 1954. "Parametric Objective Function (Part 1)," Operations Research, INFORMS, vol. 2(3), pages 316-319, August.
    • Handle: RePEc:inm:oropre:v:2:y:1954:i:3:p:316-319
    DOI: 10.1287/opre.2.3.316
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    Cited by:

    1. Goberna, M.A. & Gomez, S. & Guerra, F. & Todorov, M.I., 2007. "Sensitivity analysis in linear semi-infinite programming: Perturbing cost and right-hand-side coefficients," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1069-1085, September.
    2. Aadhityaa Mohanavelu & Bankaru-Swamy Soundharajan & Ozgur Kisi, 2022. "Modeling Multi-objective Pareto-optimal Reservoir Operation Policies Using State-of-the-art Modeling Techniques," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 36(9), pages 3107-3128, July.
    3. José Niño-Mora, 2007. "A (2/3) n 3 Fast-Pivoting Algorithm for the Gittins Index and Optimal Stopping of a Markov Chain," INFORMS Journal on Computing, INFORMS, vol. 19(4), pages 596-606, November.
    4. Efstratios Pistikopoulos & Luis Dominguez & Christos Panos & Konstantinos Kouramas & Altannar Chinchuluun, 2012. "Theoretical and algorithmic advances in multi-parametric programming and control," Computational Management Science, Springer, vol. 9(2), pages 183-203, May.
    5. Walter Gutjahr & Alois Pichler, 2016. "Stochastic multi-objective optimization: a survey on non-scalarizing methods," Annals of Operations Research, Springer, vol. 236(2), pages 475-499, January.
    6. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.
    7. Walter J. Gutjahr & Alois Pichler, 2016. "Stochastic multi-objective optimization: a survey on non-scalarizing methods," Annals of Operations Research, Springer, vol. 236(2), pages 475-499, January.
    8. María Jesús Gisbert & María Josefa Cánovas & Juan Parra & Fco. Javier Toledo, 2019. "Lipschitz Modulus of the Optimal Value in Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 133-152, July.

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