Sains Malaysiana 41(12)(2012): 1651–1656

 

A Stackelberg Solution to a Two-Level Linear Fractional Programming Problem with Interval Coefficients in the Objective Functions

(Penyelesaian Stackelberg bagi Masalah Pengaturcaraan Pecahan Linear Dua-Aras

dengan Pekali Selang dalam Fungsi Objektif)

 

 

M. Borza & A. S. Rambely*

School of Mathematical Sciences, Faculty of Science & Technology

Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

 

M. Saraj

Department of Mathematics, Faculty of Mathematical Sciences & Computer

Shahid Chamran University, Ahvaz-Iran

 

 

Diserahkan: 18 Mei 2012 / Diterima: 31 Julai 2012

 

 

ABSTRACT

In this paper, two approaches were introduced to obtain Stackelberg solutions for two-level linear fractional programming problems with interval coefficients in the objective functions. The approaches were based on the Kth best method and the method for solving linear fractional programming problems with interval coefficients in the objective function. In the first approach, linear fractional programming with interval coefficients in the objective function and linear programming were utilized to obtain Stackelberg solution, but in the second approach only linear programming is used. Since a linear fractional programming with interval coefficients can be equivalently transformed into a linear programming, therefore both of approaches have same results. Numerical examples demonstrate the feasibility and effectiveness of the methods.

 

Keywords: Interval coefficients; linear fractional programming; Stackelberg solution; two-level programming

ABSTRAK

Dalam kajian ini, dua kaedah diperkenalkan untuk mendapatkan penyelesaian Stackelberg bagi masalah pengaturcaraan pecahan linear dua-aras dengan pekali selang dalam fungsi objektif. Kaedah yang digunakan adalah berdasarkan kaedah terbaik peringkat-K dan kaedah penyelesaian masalah pengaturcaraan pecahan linear dengan pekali selang dalam fungsi objektif. Dalam kaedah pertama, pengaturcaraan pecahan linear dengan pekali selang dalam fungsi objektif dan pengaturcaraan linear digunakan untuk mendapatkan penyelesaian Stackelberg, tetapi dalam kaedah kedua hanya pengaturcaraan linear digunakan. Oleh sebab suatu pengaturcaraan pecahan linear dengan pekali selang boleh dijelmakan secara setara kepada pengaturcaraan linear. kedua-dua kaedah menghasilkan keputusan yang sama. Beberapa contoh berangka menunjukkan kesauran dan keberkesanan kaedah-kaedah ini.

 

Kata kunci: Pekali selang; pengaturcaraan dua-aras; pengaturcaraan pecahan linear; penyelesaian Stackelberg

RUJUKAN

Anandalingam, G. & Friesz, T.L. 1992. Hierarichical optimization. Annals of Operation Research 34: 1-11.

Annadalingam, G. & White, D.J. 1990. A solution method for the linear static Stackelberg problem using penalty function. IEEE Transactions on Automatic Control 35: 1170-1173.

Bard, J.F. 1984. An investigation of the linear three-level programming problem. IEEE Transactions on Systems, Man and Cybernetics 14: 711-717.

Bard, J.F. 1998. Practical Bi-level Optimization: Algorithm and Applications. Dordrecht: Kluwer Academic Publishers.

Bard, J.F. & Falk, J.E. 1982. An explicit solution to the multi-level programming problem. Computers and Operations Research 9: 77-100.

Bard, J.F. & Moore, J.T. 1990. A branch and bound algorithm for the bi-level programming problem. SIAM Journal on Scientific and Statistical Computing 11: 281-292.

Bitran, G.R. & Novaes, A.G. 1973. Linear programming with a fractional objective function. Operation Research 21: 22-29.

Bialas, W.F. & Karwan, M.H. 1984. Two-level linear programming. Management Science 30: 1004-1020.

Borza, M., Rambely, A.S. & Saraj, M. 2012. Solving linear fractional programming with interval coefficients in objective function. Applied Mathematical Sciences 69: 3443-3452.

Calvete, H.I. & Gale. C. 1998. On the quasi-concave bi-level programming problem. Journal of Optimization Theory and Applications 98: 613- 622.

Charnes, A. & Cooper, W.W. 1962. Programming with linear fractional functions. Naval Research Logistics Quarterly 9: 181-186.

Hansen, P., Jaumard, B. & Savard, G. 1992. New branch-and-bound rules for linear bi-level programming. SIAM Journal on Scientific and Statistical Computing 13: 1194-1217.

Luo, Z.Q., Pang, J.S. & Ralph, D. 1996. Mathematical Programs with Equilibrium Constraints. Cambridge: Cambridge University Press.

Migdalas, A., Pardalos, P.M. & Varbrand, P. 1998. Multi-level Optimization: Algorithms and Applications. Dordrecht: Kluwer Academic Publishers.

Sakawa, M. & Nishizaki, I. 2001. Interactive fuzzy programming for two-level linear fractional programming problems. Fuzzy Sets and Systems 119: 31-40.

Sakawa, M. & Nishizaki, I. 2009. Cooperative and Noncooperative Multi-Level Programming. New York: Springer.

Sakawa, M., Nishizaki, I. & Uemura, Y. 2000a. Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. Fuzzy Sets and Systems 109: 3-19.

Sakawa, M., Nishizaki, I. & Uemura, Y. 2000b. Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters. Fuzzy Sets and Systems 115: 93-103.

Shih, H.S., Lai, Y.J. & Lee, E.S. 1996. Fuzzy approach for multi-level programming problems. Computers and Operational Research 23: 73-91.

Stancu-Minasian, I.M. 1997. Fractional Programming: Theory, Methods and Applications. Dordrecht: Kluwer Academic Publishers.

Stackelberg, H.V. 1934. Marketform und Gleicgwicht. Berlin: Springer-Verlag.

Wen, U.P. & Bialas, W.F. 1986. The hybrid algorithm for solving the three-level linear programming problem. Computers and Operations Research 13: 367-377.

White, D.J. & Annadalingam, G. 1993. A penalty function approach for solving bi-level linear programs. Journal of Global Optimization 3: 397-419.

 

 

*Pengarang untuk surat-menyurat; e-mail: asr@ukm.my

 

 

sebelumnya