Abstract
We address the full fuzzy linear fractional programming problem with LR fuzzy numbers. Our goal is to revitalize a strict use of extension principle by employing it in all stages of our solution approach, thus deriving results that fully comply to it. Using the \(\alpha \)-cuts of the coefficients we present the linear optimization models that empirically derive the \(\alpha \)-cuts of the optimal objective fuzzy value, and discuss the optimization models able to derive the exact endpoints of the optimal objective values intervals. For initial maximization (minimization) problems the main issue is related to how to solve two stage min-max (max-min) problems to obtain the left (right) most endpoints. Our goals are as it follows: to obtain exact solutions to small-size problems; to obtain relevant information about solutions to large-scale problems that are in accordance to the extension principle; and to provide a procedure able to measure to which extent the solutions obtained by an approach to full fuzzy linear fractional programming comply to the extension principle. We illustrate the theoretical findings reporting numerical results, and including a relevant comparison to the results from the literature.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agarwal, D., Singh, P., Li, X., et al.: Optimality criteria for fuzzy-valued fractional multi-objective optimization problem. Soft Comput. 23, 9049–9067 (2019)
Anukokila, P., Radhakrishnan, B.: Goal programming approach to fully fuzzy fractional transportation problem. J. Taibah Univ. Sci. 13(1), 864–874 (2019)
Arya, R., Singh, P., Kumari, S., Obaidat, M.: An approach for solving fully fuzzy multi-objective linear fractional optimization problems. Soft Comput. 24, 9105–9119 (2020)
Arya, A., Yadav, S.P.: Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic fuzzy input-output targets. Soft Comput. 23, 8975–8993 (2019)
Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Manag. Sci. 17(4), B-141–B-164 (1970)
Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Naval Res. Logist. Q. 9(3–4), 181–186 (1962)
Das, S.K., Mandal, T., Edalatpanah. S.A.: A new approach for solving fully fuzzy linear fractional programming problems using the multi-objective linear programming. RAIRO Oper. Res. 51(1), 285–297 (2017)
Das, S.K., Edalatpanah, S.A., Mandal, T.: Application of linear fractional programming problem with fuzzy nature in industry sector. Filomat 34(15), 5073–5084 (2020)
Diniz, M.M., Gomes, L.T., Bassanezi, R.C.: Optimization of fuzzy-valued functions using Zadeh’s extension principle. Fuzzy Sets Syst. 404, 23–37 (2021)
Dubois, D.: The role of fuzzy sets in decision sciences: old techniques and new directions. Fuzzy Sets Syst. 184(1), 3–28 (2011)
Ebrahimnejad, A., Ghomi, S.J., Mirhosseini-Alizamini. S.M.: A revisit of numerical approach for solving linear fractional programming problem in a fuzzy environment. Appl. Math. Model. 57, 459–473 (2018)
Ebrahimnejad, A., Naser, A.: Fuzzy data envelopment analysis in the presence of undesirable outputs with ideal points. Complex Intell. Syst. 7, 379–400 (2021)
Ghanbari, G., Ghorbani-Moghadam, K., De Baets, B.: Fuzzy linear programming problems: models and solutions. Soft Comput. 24, 10043–10073 (2020)
Khalifa, H.A.E.W., Kumar, P.: A goal programming approach for multi-objective linear fractional programming problem with LR possibilistic variables. Int. J. Syst. Assur. Eng. Manag. 13, 2053–2061 (2022). https://doi.org/10.1007/s13198-022-01618-0
Kaur, J., Kumar, A.: A novel method for solving fully fuzzy linear fractional programming problems. J. Intell. Fuzzy Syst. 33(4), 1983–1990 (2017)
Liu, S.-T., Kao. C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153(3), 661–674 (2004)
Loganathan, T., Ganesan. K.: Solution of fully fuzzy linear fractional programming problems - a simple approach. IOP Conf. Ser. Mater. Sci. Eng. 1377, 012040 (2021)
Pérez-Cañedo, B., Verdegay, J., Miranda Pérez, R.: An epsilon-constraint method for fully fuzzy multiobjective linear programming. Int. J. Intell. Syst. 35(4), 600–624 (2020)
Pop, B., Stancu-Minasian, I.M.: A method of solving fully fuzzified linear fractional programming problems. J. Appl. Math. Comput. 27, 227–242 (2008)
Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization. Nonconvex Optimization and its Applications, vol. 2, pp. 495–608. Kluwer Academic Publishers, Dordrecht (1995)
Stancu-Minasian, I.M.: Fractional Programming: Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht (1997)
Stancu-Minasian, I.M.: A eighth bibliography of fractional programming. Optimization 66(3), 439–470 (2017)
Stancu-Minasian, I.M.: A ninth bibliography of fractional programming. Optimization 68(11), 2125–2169 (2019)
Stanojević. B.: Extension principle-based solution approach to full fuzzy multi-objective linear fractional programming. Soft Comput. 26, 5275–5282 (2022)
Stanojević, B., Dzitac, I., Dzitac. S.: On the ratio of fuzzy numbers - exact membership function computation and applications to decision making. Technol. Econ. Dev. Econ. 21(5), 815–832 (2015)
Stanojević, B., Dzitac, S., Dzitac. I.: Fuzzy numbers and fractional programming in making decisions. Int. J. Inf. Technol. Decis. Mak. 19(4), 1123–1147 (2020)
Stanojević, B., Dzitac, S., Dzitac. I.: Solution approach to a special class of full fuzzy linear programming problems. Procedia Comput. Sci. 162, 260–266 (2019)
Stanojević, B., Stanojević, M.: Analytic description to the fuzzy efficiencies in fuzzy standard data envelopment analysis. Procedia Comput. Sci. 199, 487–494 (2022)
Stanojević, B., Stanojević, M.: Empirical \((\alpha ,\beta )\)-acceptable optimal values to full fuzzy linear fractional programming problems. Procedia Comput. Sci. 199, 34–39 (2022)
Stanojević, B., Stanojević, M., Nădăban, S.: Reinstatement of the extension principle in approaching mathematical programming with fuzzy numbers. Mathematics 9, 1272 (2021)
Stanojević, B., Stanojević, M.: Approximate membership function shapes of solutions to intuitionistic fuzzy transportation problems. Int. J. Comput. Commun. Control 16(1), 4057 (2021)
Stanojević, B., Stanojević, M.: Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems. Fuzzy Optim. Decis. Mak. 15(4), 435–455 (2016). https://doi.org/10.1007/s10700-016-9232-1
Valipour, E., Yaghoobi, M.A.: On fuzzy linearization approaches for solving multi-objective linear fractional programming problems. Fuzzy Sets Syst. 434, 73–87 (2022)
Wu, H., Xu, Z.: Fuzzy logic in decision support: methods, applications and future trends. Int. J. Comput. Commun. Control 16(1), 4044 (2020)
Wu, H.C.: Generalized extension principle. Fuzzy Optim. Decis. Mak. 9, 31–68 (2010)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)
Zimmermann.H.-J.: Applications of fuzzy set theory to mathematical programming. Inf. Sci. 36(1), 29–58 (1985)
Zhou, J., Yang, F., Wang, K.: Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming. J. Intell. Fuzzy Syst. 30(1), 71–87 (2016)
Acknowledgments
This work was supported by the Serbian Ministry of Education, Science and Technological Development through Mathematical Institute of the Serbian Academy of Sciences and Arts and Faculty of Organizational Sciences of the University of Belgrade.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Stanojević, B., Stanojević, M. (2023). Full Fuzzy Fractional Programming Based on the Extension Principle. In: Mihić, M., Jednak, S., Savić, G. (eds) Sustainable Business Management and Digital Transformation: Challenges and Opportunities in the Post-COVID Era. SymOrg 2022. Lecture Notes in Networks and Systems, vol 562. Springer, Cham. https://doi.org/10.1007/978-3-031-18645-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-18645-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-18644-8
Online ISBN: 978-3-031-18645-5
eBook Packages: EngineeringEngineering (R0)