An optimization model for fuzzy nonlinear programming with Beale's conditions using trapezoidal membership functions
DOI:
https://doi.org/10.22199/issn.071762795468Keywords:
Nonlinear optimization – Fuzzy nonlinear programming problem – Beale’s conditions with fuzziness  Trapezoidal fuzzy membership functions.Abstract
Nonlinear Programming (NLP) is an optimization technique for determining the optimum solution to a broad range of research issues. Many times, the objective function is nonlinear, owing to various economic behaviors such as demand, cost, and many others. Since the appearance of Kuhn and Tucker's fundamental theoretical work, a general NLP problem can be resolved using many methods to find the optimum solution. In this chapter, a fuzzy mathematical model based on Beale's condition is proposed to address NLP with inequality constraints in terms of fuzziness. Furthermore, the model demonstrates how quadratic programming problems can be solved using membership functions. The model also describes three stages: that is, mathematical formulation, computational procedures, and numerical illustration with comparative analysis. Likewise, the model illustrates the considered problem using two distinct approaches, namely membership functions (MF) and robust ranking index. Finally, the comparison analysis provides detailed results and discussion that justify the optimal outcome in order to address the vagueness of certain NLPPs.
References
Zadeh, L. Fuzzy Sets. Information and Control.–NY 1965, 8, 338353.
Bellman, R.; Zadeh, L. Management Science, 17, No. 4, December, p. 1970.
Zimmermann, H.J. Fuzzy sets, decision making, and expert systems; Springer Science & Business Media: 2012; Volume 10.
Vasant, P.; Nagarajan, R.; Yaacob, S. Fuzzy linear programming: a modern tool for decision making. In Computational Intelligence for Modelling and Prediction; Springer: 2005; pp. 383401.
Kheirfam, B.; Hasani, F. Sensitivity analysis for fuzzy linear programming problems with fuzzy variables. Advanced Modeling and Optimization 2010, 12, 257272.
Palanivel, K. Contributions to the study on some optimization techniques in fuzzy membership functions. Bharathidasan University, Trichy, Tamil Nadu, India, 2013.
Palanivel, K. Fuzzy commercial traveler problem of trapezoidal membership functions within the sort of $$alpha $$ optimum solution using ranking technique. Afrika Matematika 2016, 27, 263277.
Saranya, R.; Palanivel, K. Fuzzy nonlinear programming problem for inequality constraints with alpha optimal solution in terms of trapezoidal membership functions. International Journal of Pure and Applied Mathematics 2018, 119, 5363.
Tang, J.; Wang, D. A nonsymmetric model for fuzzy nonlinear programming problems with penalty coefficients. Computers & operations research 1997, 24, 717725.
Tang, J.; Wang, D.; Ip, A.; Fung, R. A hybrid genetic algorithm for a type of nonlinear programming problem. Computers and Mathematics with Applications 1998, 36, 1122.
Fung, R.Y.; Tang, J.; Wang, D. Extension of a hybrid genetic algorithm for nonlinear programming problems with equality and inequality constraints. Computers & Operations Research 2002, 29, 261274.
Sarimveis, H.; Nikolakopoulos, A. A line up evolutionary algorithm for solving nonlinear constrained optimization problems. Computers & Operations Research 2005, 32, 14991514, doi:10.1016/j.cor.2003.11.015.
Syau, Y.R.; Stanley Lee, E. Fuzzy convexity and multiobjective convex optimization problems. Computers & Mathematics with Applications 2006, 52, 351362, doi:10.1016/j.camwa.2006.03.017.
Chen, S.P. Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method. European Journal of Operational Research 2007, 177, 445457, doi:10.1016/j.ejor.2005.09.040.
Qin, X.S.; Huang, G.H.; Zeng, G.M.; Chakma, A.; Huang, Y.F. An intervalparameter fuzzy nonlinear optimization model for stream water quality management under uncertainty. European Journal of Operational Research 2007, 180, 13311357, doi:10.1016/j.ejor.2006.03.053.
Kassem, M.A.E.H. Stability achievement scalarization function for multiobjective nonlinear programming problems. Applied Mathematical Modelling 2008, 32, 10441055, doi:10.1016/j.apm.2007.02.028.
Shankar, N.R.; Rao, G.A.; Latha, J.M.; Sireesha, V. Solving a fuzzy nonlinear optimization problem by genetic algorithm. Int. J. Contemp. Math. Sciences 2010, 5, 791803.
AbdElWahed, W.F.; Mousa, A.A.; ElShorbagy, M.A. Integrating particle swarm optimization with genetic algorithms for solving nonlinear optimization problems. Journal of Computational and Applied Mathematics 2011, 235, 14461453, doi:10.1016/j.cam.2010.08.030.
Jameel, A.F.; Sadeghi, A. Solving nonlinear programming problem in fuzzy environment. Int. J. Contemp. Math. Sci 2012, 7, 159170.
Ali H. Ahmadini, A.; Varshney, R.; Mradula; Ali, I. On multivariatemultiobjective stratified sampling design under probabilistic environment: A fuzzy programming technique. Journal of King Saud University  Science 2021, 33, 101448, doi:10.1016/j.jksus.2021.101448.
Khan, M.F.; Modibbo, U.M.; Ahmad, N.; Ali, I. Nonlinear optimization in bilevel selective maintenance allocation problem. Journal of King Saud University  Science 2022, 34, 101933, doi:10.1016/j.jksus.2022.101933.
Gupta, D.; Borah, P.; Sharma, U.M.; Prasad, M. Datadriven mechanism based on fuzzy Lagrangian twin parametricmargin support vector machine for biomedical data analysis. Neural Computing and Applications 2021, doi:10.1007/s00521021058662.
Lin, L.; Lee, H.M. Fuzzy nonlinear programming for production inventory based on statistical data. Journal of Advanced Computational Intelligence and Intelligent Informatics 2016, 20, 512.
Lu, T.; Liu, S.T. Fuzzy nonlinear programming approach to the evaluation of manufacturing processes. Engineering Applications of Artificial Intelligence 2018, 72, 183189, doi:10.1016/j.engappai.2018.04.003.
Saberi Najafi, H.; Edalatpanah, S.A.; Dutta, H. A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Alexandria Engineering Journal 2016, 55, 25892595, doi:10.1016/j.aej.2016.04.039.
Silva, R.C.; Cruz, C.; Verdegay, J.L. Fuzzy costs in quadratic programming problems. Fuzzy Optimization and Decision Making 2013, 12, 231248, doi:10.1007/s1070001391531.
Gani, A.N.; Saleem, R.A. Solving Fuzzy Sequential Quadratic Programming Algorithm for Fuzzy NonLinear Programming. 2018.
Published
How to Cite
Issue
Section
Copyright (c) 2024 Palanivel Kaliyaperumal, Muralikrishna P
This work is licensed under a Creative Commons Attribution 4.0 International License.

Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
 No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.