محاسبه مرز کارای مدل دوسطحی خطی چندهدفه

نوع مقاله: مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشگاه شهید چمران اهواز

چکیده

برنامه ریزی دو سطحی، مدلی برای مسایل بهینه سازی سلسله مراتبی است که دو تصمیم گیرنده با توابع هدف، متغیر ها و قید های متفاوتی وجود دارد. آلوز و همکارانش در [1]، روشی برای محاسبه مرز کارای مساله دو‌سطحی خطی با دو تابع هدف در سطح بالا و یک تابع هدف در سطح پایین ارائه دادند. در این مقاله ما روش آنها را برای حالتی که بیش از دو تابع هدف در هر دو سطح وجود دارد، تعمیم داده و با بهره‌گیری از تغییر متغیر مناسب، روش جدیدی برای محاسبه مرز کارای مساله دو‌سطحی خطی با توابع هدف کسری در سطح بالا ارائه می‌دهیم. نهایتاً کارآیی روش های پیشنهادی را با حل چند مثال عددی و مقایسه نتایج آنها با دیگر روش ها نشان می دهیم.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Computing the pareto frontier of a linear Multiobjective bi-level model

نویسندگان [English]

  • Abbas Mehrabani
  • Habibe Sadeghi
چکیده [English]

Bilevel programming is the model for hierarchical optimization problems in which there are two decision makers that have different objective functions, variables and constraints. Alves et al in[1], proposed a method for computing the Pareto frontier of bilevel linear problem with biobjective at the upper level and a single objective function at the lower level. In this paper, we extend their method for the situation in which there exists more than two objective function at both levels, and then by using a suitable exchange variable, we proposed a new method for computing the Pareto frontier of bilevel linear problem with fractional multi-objective at the upper level. Finally we will show the efficiency of the propsed approaches by solving a few numerical examples and comparing the results with other methods.

کلیدواژه‌ها [English]

  • Bilevel programming
  • Multi objective programming
  • Pareto frontier
  • mixed- integer programming
  • Fractional programming
[1] Alves, M.J., Dempe, S. and Júdice, J.J. (2012). Computing the Pareto frontier of a bi-objective bi-level linear problem using a multi-objective mixed-integer programming algorithm. Optimization, 61, 335-358.

 [2] Dempe, S. (2002). Foundations of bi-level programming. Non-convex optimization and its applications, Dordrecht, Kluwer.
 [3] Candler, W., and Norton, R. (1977). Multi-level programming and development policy, Washington, D.C, The World Bank.

 [4] Candler, W., and Townsley, R. (1982). A linear two-level programming problem. Computers and Operations Research, 9, 59-76.

[5] Bialas, W.F. and Karwan, M.H. (1984). Two-level linear programming. Management science, 30, 1004-1020.

[6] Edmunds, T. A., and Bard, J. F. (1991). Algorithms for nonlinear bi-level mathematical programs. Systems, Man and Cybernetics, IEEE Transactions on Systems, 21, 83-89.

 [7] Lai, Y.J. (1996). Hierarchical optimization: a satisfactory solution. Fuzzy Sets and Systems, 77, 321-335.

[8] Calvete, H.I. and Galé, C. (2011). On linear bi-level problems with multiple objectives at the lower level. Omega, 39, 33-40.

 [9] Shi, X., and Xia, H. (1997). Interactive bi-level multi-objective decision-making. Journal of the operational research society, 48, 943-949.

[10] Abo-Sinna, M. A. and Baky, I.A. (2007). Interactive balance space approach for solving multi-level multi-objective programming problems. Information Sciences, 177, 3397-3410.

[11] Farahi, M. H. (2010). A new approach to solve multi-objective linear bi-level programming problems. Journal of Mathematics and Computer Sciences, 1, 313-320.

 [12] Pieume, C.O., Marcotte, P., Fotso, L.P. and Siarry, P. (2011). Solving bi-level linear multi-objective programming problems. American Journal of Operations Research, 1, 214-219.

[13] Lachhwani, K. and Poonia, M.P. (2012). Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach. Journal of Industrial Engineering International, 8, 1-11.

[14] Alves, M.J. and Clımaco, J. (2004). A note on a decision support system for multi-objective integer and mixed-integer programming problems. European Journal of Operational Research, 155, 258-265.

[15] Mersha, A. G. and Dempe, S. (2006). Linear bi-level programming with upper level constraints depending on the lower level solution. Applied Mathematics and Computation, 180, 247-254.

 [16] Ehrgott, M. (2006). Multi-criteria optimization. Berlin, Springer.

[17] Alves, M.J. and Clı́maco, J. (2000). An interactive reference point approach for multi-objective mixed-integer programming using branch-and-bound. European Journal of Operational Research, 124, 478-494.

 [18] Bard, J. F. (1999). Practical Bi-level Optimization: Algorithms and Applications, Berlin, Springer.
[19] Wierzbicki, A.P. (1980). The use of reference objectives in multi-objective optimization. In Multiple criteria decision making theory and application (pp. 468-486). Springer Berlin Heidelberg.

[20] Charnes, A. and Cooper, W. W. (1962). Programming with linear fractional functional. Naval Research logistics quarterly, 9, 181-186.

[21] Chakraborty, M. and Gupta, S. (2002). Fuzzy mathematical programming for multi objective linear fractional programming problem. Fuzzy sets and systems, 125, 335-342.