Multi-objective probabilistic fractional programming problem involving two parameters Cauchy distribution
Abstract
The paper presents the solution methodology of a multi-objective probabilistic fractional programming problem, where the parameters of the right hand side constraints follow Cauchy distribution. The proposed mathematical model can not be solved directly. The solution procedure is completed in three steps. In first step, multi-objective probabilistic fractional programming problem is converted to deterministic multi-objective fractional mathematical programming problem. In the second step, it is converted to its equivalent multi-objective mathematical programming problem. Finally, ε -constraint method is applied to find the best compromise solution. A numerical example and application are presented to demonstrate the procedure of proposed mathematical model.
Keyword : multi-objective programming problem, probabilistic programming problem, fractional programming problem, ε -constraint method
How to Cite
Acharya, S., Belay, B., & Mishra, R. (2019). Multi-objective probabilistic fractional programming problem involving two parameters Cauchy distribution. Mathematical Modelling and Analysis, 24(3), 385-403. https://doi.org/10.3846/mma.2019.024
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
H.M. Babul and A. Sumi. Solving LFP by converting it into a single LP. International Journal of Operations Research, 8(3):1–14, 2011.
M. Borza, A.S. Rambely and M. Saraj. Solving linear fractional programming problems with interval coefficients in the objective function. a new approach. Applied Mathematical Sciences, 6(69-72):3443–3459, 2012.
M. Chakraborty and S. Gupta. Fuzzy mathematical programming for multi objective linear fractional programming problem. Fuzzy sets and systems, 125(3):335–342, 2002. https://doi.org/10.1016/S0165-0114(01)00060-4.
V. Charles and D. Dutta. Linear stochastic fractional programming with branchand-bound technique. In Proceedings of the National Conference on Mathematical and Computational Methods, pp. 131–139, 2001.
V. Charles and D. Dutta. Bi-weighted multi-objective stochastic fractional programming problem with mixed constraints. In Proceedings of the 2nd National Conference on Mathematical and Computational Methods, pp. 29–36. Allied New Delhi, India, 2003.
V. Charles and D. Dutta. Linear stochastic fractional programming with sumof-probabilistic-fractional objective. Optimization Online, 2005.
V. Charles and D. Dutta. Extremization of multi-objective stochastic fractional programming problem. Annals of Operations Research, 143(1):297–304, 2006. https://doi.org/10.1016/S0165-0114(01)00060-4.
V. Charles and D. Dutta. Identification of redundant objective functions in multi-objective stochastic fractional programming problems. Asia-Pacific Journal of Operational Research, 23(02):155–170, 2006. https://doi.org/10.1142/S0217595906000863.
A. Charnes and W.W. Cooper. Programming with linear fractional functionals. Naval research logistics (NRL), 9(3-4):181–186, 1962.
R. Enkhbat, Ya. Bazarsad and J. Enkhbayar. Convex–concave fractional minimization problem. J. Mong. Math. Soc, 15:3–10, 2011.
N. Gu¨zel and M. Sivri. Taylor series solution of multi-objective linear fractional programming problem. Trakya Univ J Sci, 6(2):80–87, 2005.
J.S. Kornbluth and R.E. Steuer. Multiple objective linear fractional programming. Management Science, 27(9):1024–1039, 1981. https://doi.org/10.1287/mnsc.27.9.1024.
E. Marchi. Some remarks about the transformation of charnes and cooper. Tersedia di http://www.optimizationonline.org/DBFILE/2006/04/1381.pdf [9 Juli 2008], 2006.
B. Martos and V. Whinston. Hyperbolic programming. Naval Research Logistics (NRL), 11(2):135–155, 1964. https://doi.org/10.1002/nav.3800110204.
A.O. Odior. An approach for solving linear fractional programming problems. International Journal of Engineering & Technology, 1(4):298–304, 2012.
P. Ponnaiah and J. Mohan. On solving linear fractional programming problems. Modern Applied Science, 7(6):90, 2013.
C.F. Ren, P. Guo, M. Li and J.J. Gu. Optimization of industrial structure considering the uncertainty of water resources. Water resources management, 27(11):3885–3898, 2013. https://doi.org/10.1007/s11269-013-0385-1
C.F. Ren, R.H. Li, L.D. Zhang and P. Guo. Multiobjective stochastic fractional goal programming model for water resources optimal allocation among industries. Journal of Water Resources Planning and Management, 142(10):04016036, 2016. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000681
D. Roy. Fractional programming through genetic algorithm. In Computer, Communication, Control and Information Technology (C3IT), 2015 Third International Conference on, pp. 1–6. IEEE, 2015. https://doi.org/10.1109/C3IT.2015.7060175
S. Saha, M. Hossain, M. Uddin and R. Mondal. A new approach of solving linear fractional programming problem (LFP) by using computer algorithm. Open Journal of Optimization, 4(03):74, 2015. https://doi.org/10.4236/ojop.2015.43010
S. Schaible. Fractional programming. i, duality. Management science, 22(8):858– 867, 1976. https://doi.org/10.1287/mnsc.22.8.858
I.M. Stancu-Minasian. A fifth bibliography of fractional programming. Optimization, 45(1-4):343–367, 1999. https://doi.org/10.1080/02331939908844438
I.M. Stancu-Minasian. A sixth bibliography of fractional programming. Optimization, 55(4):405–428, 2006. https://doi.org/10.1080/02331930600819613
S.F. Tantawy. A new procedure for solving linear fractional programming problems. Mathematical and Computer Modelling, 48(5-6):969–973, 2008. https://doi.org/10.1016/j.mcm.2007.12.007
A. Udhayakumar, V. Charles and V.R. Uthariaraj. Stochastic simulationbased genetic algorithm for chance constrained fractional programming problem. International Journal of Operational Research, 9(1):23–38, 2010. https://doi.org/10.1504/IJOR.2010.034359
E. Valipour, MA. Yaghoobi and M. Mashinchi. An iterative approach to solve multiobjective linear fractional programming problems. Applied Mathematical Modelling, 38(1):38–49, 2014.
C.-F. Wen and H.-C. Wu. Using the parametric approach to solve the continuoustime linear fractional max–min problems. Journal of Global Optimization, 54(1):129–153, 2012. https://doi.org/10.1007/s10898-011-9751-9
H. Zhu and G.H. Huang. SLFP: A stochastic linear fractional programming approach for sustainable waste management. Waste Management, 31(12):2612– 2619, 2011. https://doi.org/10.1016/j.wasman.2011.08.009
H. Zhu and G.H. Huang. Dynamic stochastic fractional programming for sustainable management of electric power systems. International Journal of Electrical Power & Energy Systems, 53:553–563, 2013. https://doi.org/10.1016/10.1016/j.ijepes.2013.05.022
M. Borza, A.S. Rambely and M. Saraj. Solving linear fractional programming problems with interval coefficients in the objective function. a new approach. Applied Mathematical Sciences, 6(69-72):3443–3459, 2012.
M. Chakraborty and S. Gupta. Fuzzy mathematical programming for multi objective linear fractional programming problem. Fuzzy sets and systems, 125(3):335–342, 2002. https://doi.org/10.1016/S0165-0114(01)00060-4.
V. Charles and D. Dutta. Linear stochastic fractional programming with branchand-bound technique. In Proceedings of the National Conference on Mathematical and Computational Methods, pp. 131–139, 2001.
V. Charles and D. Dutta. Bi-weighted multi-objective stochastic fractional programming problem with mixed constraints. In Proceedings of the 2nd National Conference on Mathematical and Computational Methods, pp. 29–36. Allied New Delhi, India, 2003.
V. Charles and D. Dutta. Linear stochastic fractional programming with sumof-probabilistic-fractional objective. Optimization Online, 2005.
V. Charles and D. Dutta. Extremization of multi-objective stochastic fractional programming problem. Annals of Operations Research, 143(1):297–304, 2006. https://doi.org/10.1016/S0165-0114(01)00060-4.
V. Charles and D. Dutta. Identification of redundant objective functions in multi-objective stochastic fractional programming problems. Asia-Pacific Journal of Operational Research, 23(02):155–170, 2006. https://doi.org/10.1142/S0217595906000863.
A. Charnes and W.W. Cooper. Programming with linear fractional functionals. Naval research logistics (NRL), 9(3-4):181–186, 1962.
R. Enkhbat, Ya. Bazarsad and J. Enkhbayar. Convex–concave fractional minimization problem. J. Mong. Math. Soc, 15:3–10, 2011.
N. Gu¨zel and M. Sivri. Taylor series solution of multi-objective linear fractional programming problem. Trakya Univ J Sci, 6(2):80–87, 2005.
J.S. Kornbluth and R.E. Steuer. Multiple objective linear fractional programming. Management Science, 27(9):1024–1039, 1981. https://doi.org/10.1287/mnsc.27.9.1024.
E. Marchi. Some remarks about the transformation of charnes and cooper. Tersedia di http://www.optimizationonline.org/DBFILE/2006/04/1381.pdf [9 Juli 2008], 2006.
B. Martos and V. Whinston. Hyperbolic programming. Naval Research Logistics (NRL), 11(2):135–155, 1964. https://doi.org/10.1002/nav.3800110204.
A.O. Odior. An approach for solving linear fractional programming problems. International Journal of Engineering & Technology, 1(4):298–304, 2012.
P. Ponnaiah and J. Mohan. On solving linear fractional programming problems. Modern Applied Science, 7(6):90, 2013.
C.F. Ren, P. Guo, M. Li and J.J. Gu. Optimization of industrial structure considering the uncertainty of water resources. Water resources management, 27(11):3885–3898, 2013. https://doi.org/10.1007/s11269-013-0385-1
C.F. Ren, R.H. Li, L.D. Zhang and P. Guo. Multiobjective stochastic fractional goal programming model for water resources optimal allocation among industries. Journal of Water Resources Planning and Management, 142(10):04016036, 2016. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000681
D. Roy. Fractional programming through genetic algorithm. In Computer, Communication, Control and Information Technology (C3IT), 2015 Third International Conference on, pp. 1–6. IEEE, 2015. https://doi.org/10.1109/C3IT.2015.7060175
S. Saha, M. Hossain, M. Uddin and R. Mondal. A new approach of solving linear fractional programming problem (LFP) by using computer algorithm. Open Journal of Optimization, 4(03):74, 2015. https://doi.org/10.4236/ojop.2015.43010
S. Schaible. Fractional programming. i, duality. Management science, 22(8):858– 867, 1976. https://doi.org/10.1287/mnsc.22.8.858
I.M. Stancu-Minasian. A fifth bibliography of fractional programming. Optimization, 45(1-4):343–367, 1999. https://doi.org/10.1080/02331939908844438
I.M. Stancu-Minasian. A sixth bibliography of fractional programming. Optimization, 55(4):405–428, 2006. https://doi.org/10.1080/02331930600819613
S.F. Tantawy. A new procedure for solving linear fractional programming problems. Mathematical and Computer Modelling, 48(5-6):969–973, 2008. https://doi.org/10.1016/j.mcm.2007.12.007
A. Udhayakumar, V. Charles and V.R. Uthariaraj. Stochastic simulationbased genetic algorithm for chance constrained fractional programming problem. International Journal of Operational Research, 9(1):23–38, 2010. https://doi.org/10.1504/IJOR.2010.034359
E. Valipour, MA. Yaghoobi and M. Mashinchi. An iterative approach to solve multiobjective linear fractional programming problems. Applied Mathematical Modelling, 38(1):38–49, 2014.
C.-F. Wen and H.-C. Wu. Using the parametric approach to solve the continuoustime linear fractional max–min problems. Journal of Global Optimization, 54(1):129–153, 2012. https://doi.org/10.1007/s10898-011-9751-9
H. Zhu and G.H. Huang. SLFP: A stochastic linear fractional programming approach for sustainable waste management. Waste Management, 31(12):2612– 2619, 2011. https://doi.org/10.1016/j.wasman.2011.08.009
H. Zhu and G.H. Huang. Dynamic stochastic fractional programming for sustainable management of electric power systems. International Journal of Electrical Power & Energy Systems, 53:553–563, 2013. https://doi.org/10.1016/10.1016/j.ijepes.2013.05.022