Solving Multi Objective Linear Fractional Programming Problem Under Uncertainty via Robust Optimization Approach
In this article, a Multi Objective Linear Fractional Programming (MOLFP) problem with uncertain data in the objective function and the relationship between its Robust Counterpart (RC) formulations is studied. We use box uncertainty set for MOLFP problem and propose an approach to derive its corresponding RC formulation by reducing it into a single objective programming problem. It is shown that the corresponding RC formulation of MOLFP problem under box uncertainty set is a Linear Programming (LP) problem. A numerical example is worked out to illustrate the methodology and proposed approach.
A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations Research Letters 25(1) (1999), 1 – 13, DOI: 10.1016/S0167-6377(99)00016-4.
D. Bertsimas and M. Sim, Robust discrete optimization and network flows, Mathematical Programming 98(1-3) (2003), 49 – 71, DOI: 10.1007/s10107-003-0396-4.
D. Bertsimas, D. B. Brown and Caramanis, Theory and applications of robust optimization, SIAM Review 53(3) (2011), 464 – 501, DOI: 10.1137/080734510.
M. Chakraborty and S. Gupta, Fuzzy mathematical programming for multi objective linear fractional programming problem, Fuzzy Sets and Systems 125(3) (2002), 335 – 342, DOI: 10.1016/S0165-0114(01)00060-4.
A. Charnes and W. W. Cooper, Programming with linear fractional functional, Naval Research Logistics Quarterly 9 (1962), 181 – 186, DOI: 10.1002/nav.3800090303.
T. M. Duran, Taylor series approach to fuzzy multiobjective linear fractional programming, Information Sciences 178(4) (2008), 1189 – 1204, DOI: 10.1016/j.ins.2007.06.010.
L. El Ghaoui, F. Oustry and H. Lebret, Robust solutions to uncertain semi definite programs, SIAM Journal on Optimization 9(1) (1998), 33 – 52, DOI: 10.1137/S1052623496305717.
M. A. Goberna, V. Jeyakumar, G. Li and J. Vicente-Pérez, Robust solutions to multi-objective linear programs with uncertain data, European Journal of Operational Research 242(3) (2015), 730 – 743, DOI: 10.1016/j.ejor.2014.10.027.
N. Güzel and M. Sivri, Proposal of a solution to multiobjective linear fractional programming problem, Sigma journal of Engineering and Natural Sciences 2 (2005), 43 – 50, http://www.ytusigmadergisi.com/scientific/2005-2-5-tam.pdf.
F. Hassanzadeh, H. Nemati and M. Sun, Robust optimization for interactive multiobjective programming with imprecise information applied to R&D project portfolio selection, European Journal of Operational Research 238(1) (2014), 41 – 53, DOI: 10.1016/j.ejor.2014.03.023.
S. L. Janak, X. Lin and C. A. Floudas, A new robust optimization approach for scheduling under uncertainty: II. Uncertainty with known probability distribution, Computers & Chemical Engineering 31(3) (2007), 171 – 195, DOI: 10.1016/j.compchemeng.2006.05.035.
J. S. H. Kornbluth and R. E. Steuer, Goal programming with linear fractional criteria, European Journal of Operational Research 8(1) (1981), 58 – 65, DOI: 10.1016/0377-2217(81)90029-1.
J. S. H. Kornbluth and R. E. Steuer, Multiple objective linear fractional programming, Management Science 27(9) (1981), 1024 – 1039, DOI: 10.1287/mnsc.27.9.1024.
X. Lin, S. L. Janak and C. A. Floudas, A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty, Computers & Chemical Engineering 28(6-7) (2004), 1069 – 1085, DOI: 10.1016/j.compchemeng.2003.09.020.
I. Nykowski and Z. Zolkiewski, A compromise procedure for the multiple objective fractional programming problem, European Journal of Operational Research 19(1) (1985), 91 – 97, DOI: 10.1016/0377-2217(85)90312-1.
A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research 21(5) (1973), 1154 – 1157, DOI: 10.1287/opre.21.5.1154.
N. A. Sulaiman and B. K. Abdulrahim, Using transformation technique to solve multi-objective linear fractional programming problem, International Journal of Research and Reviews in Applied Sciences 14(3) (2013), 559 – 567, https://www.arpapress.com/Volumes/Vol14Issue3/IJRRAS_14_3_09.pdf.
N. A. Sulaiman, G. W. Sadiq and B. K. Abdulrahim, New arithmetic average technique to solve multiobjective linear fractional programming problem and it is comparison with other techniques, International Journal of Research and Reviews in Applied Sciences18(2) (2014), 122 – 131, https://www.arpapress.com/Volumes/Vol18Issue2/IJRRAS_18_2_03.pdf.
H. J. Zimmermann, Fuzzy mathematical programming, Computer and Operations Research 10(4) (1983), 291 – 298, DOI: 10.1016/0305-0548(83)90004-7.
L. Zukui, D. Ran and A. F. Christodoulos, A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization, Industrial Engineering Chemistry Research 50 (2011), 10567 – 10603, DOI: 10.1021/ie200150p.
eISSN 0975-5748; pISSN 0974-875X