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

Received: 18 May 2012 / Accepted: 31 July 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

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*Corresponding author; e-mail: asr@ukm.my

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