Zohra Sabrina DELHOUM, Sonia RADJEF, Fatima BOUDAOUD

Generation of efficient and ϵ-efficient solutions in multiple objective linear programming

We develop an algorithm to solve a multiple objective linear programming problem with bounded variables.It is based on the scalarization theorem of optimal solutions of multiobjective linear programs and the single objectiveadaptive method. We suggest a process for the search for the first efficient solution without having to calculate a feasiblesolution, and we elaborate a method to generate efficient solutions, weakly efficient solutions, and ϵ -efficient solutions.Supporting theoretical results are established and the method is demonstrated on a numerical example.

PDF

___

[1] Benayoun R, de Montgolfier J, Tergeny J, Larichev O. Linear programming with multiple objective functions: step method (STEM). Math Program 1971; 1: 366-375.
[2] Benson HP. An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J Glob Optim 1998; 13: 1-24.
[3] Dimkov MP. Research on the multicriteria linear programming problems. In: Gabasov R, Kirillova FM, editors. Optimal Control Problems. Moscrow, USSR: Nauka i Technika, 1981, pp. 25-42.
[4] Ecker JG, Hegner NS. On computing an initial efficient extreme point. J Oper Res Soc 1978; 29: 1005-1007.
[5] Ecker JG, Kouada IA. Finding efficient points for linear multiple objective programs. Math Program 1975; 8: 375-377.
[6] Ecker JG, Kouada IA. Finding all efficient extreme points for multiple objective linear programs. Math Program 1978; 14: 249-261.
[7] Ehrgott M, Wiecek MM. Multiobjective programming. In: Figueira J, Greco S, Ehrgott M, editors. Multiple Criteria Decision Analysis: State of the Art Surveys. New York, NY, USA: Springer, 2005, pp. 708.
[8] Evans JP, Steuer RE. A revised simplex method for linear multiple objective programs. Math Program 1973; 5: 54-72.
[9] Evans JP, Steuer RE. Generating efficient extreme points in linear multiple objective programming: two algorithms and computing experience. In: Cochrane JL, Zeleny M, editors. Multiple Criteria Decision Making. Columbia, SC, USA: University of South Carolina Press, 1973, pp. 349-365.
[10] Gabasov R. Adaptive Method of Linear Programming. Preprint. Karlsruhe, Germany: University of Karlsruhe, 1993.
[11] Gabasov R, Kirillova FM, Tyatyushkin AI. Constructive methods of optimization. Minsk, Belarus: P.I.-University Press, 1984.
[12] Loridan P. ϵ -Solutions in vector minimization problems. J Optimiz Theory App 1984; 43: 265-276.
[13] Radjef S. Application de la m´ethode adapt´ee aux probl`emes multicrit`ere. PhD, A Mira University, Bejaia, Algeria, 2011 (in French).
[14] Radjef S, Bibi MO. An effective generalization of the direct support method. Math Probl Eng 2011; 2011: 374390.
[15] Radjef S, Bibi MO. A new algorithm for linear multiobjective programming problems with bounded variables. Arab J Math 2014; 3: 79-92.
[16] White DJ. Epsilon efficiency. J Optimiz Theory App 1986; 49: 319-337.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv