Solution of Fuzzy System of Linear Equation Under Different Fuzzy Difference Ideology

Payel Sing; Mostafijur Rahaman; Sankar Prasad Mondal Sankar

doi:10.31181/sor1120244

Authors

Payel Sing Department of Applied Sciences and Humanities, PIET, Parul University, Vadodara 391760, Gujarat, India Author https://orcid.org/0000-0002-6515-0822
Mostafijur Rahaman Department of Mathematics, School of Liberal Arts & Sciences, Mohan Babu University, Tirupati, 517102, Andhra Pradesh, India Author https://orcid.org/0000-0002-5513-4095
Sankar Prasad Mondal Sankar Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata 741249, West Bengal, India Author https://orcid.org/0000-0003-4690-2598

DOI:

https://doi.org/10.31181/sor1120244

Abstract

Fuzzy arithmetic includes distinctive notions regarding differences between fuzzy numbers, which impact the defining characteristics of fuzzy-valued calculus. In this article, we consider the Hukuhara and generalized Hukuhara differences of fuzzy numbers for solving systems of linear equations under fuzzy-rule uncertainty within an analytical framework. Moreover, the discussion of solutions in parametric form includes existence and uniqueness criteria. Numerical examples and graphical representations are provided to illustrate the proposed theory.

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References

Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X

Peeva, K. (1992). Fuzzy linear systems. Fuzzy Sets and Systems, 49(3), 339-355. https://doi.org/10.1016/0165-0114(92)90286-D

Friedman, M., Ming, M., Kandel, A. (1998) Fuzzy linear systems. Fuzzy Sets and Systems, 96(2), 201-209. https://doi.org/10.1016/S0165-0114(96)00270-9

Inearat, L., & Qatanani, N. (2018). Numerical methods for solving fuzzy linear systems. Mathematics, 6(2), 19. https://doi.org/10.3390/math6020019

Mikaeilvand, N., Noeiaghdam, Z., Noeiaghdam, S., & Nieto, J. J. (2020). A novel technique to solve the fuzzy system of equations. Mathematics, 8(5), 850. https://doi.org/10.3390/math8050850

Banerjee, S., & Roy, T. K. (2001). Linear equations and systems in fuzzy environment. Journal of mathematics and computer Science, 15, 23-31. https://doi.org/10.22436/jmcs.015.01.02

Abbasi, F., & Allahviranloo, T. (2022). Solving fully fuzzy linear system: A new solution concept. Information Sciences, 589, 608-635. https://doi.org/10.1016/j.ins.2022.01.004

Pandit, P. (2013). Fuzzy system of linear equations. In Conference Publications (Vol. 2013, No. special, pp. 619-627). Conference Publications. https://doi.org/10.3934/proc.2013.2013.619

Allahviranloo, T., Salahshour, S., Homayoun-Nejad, M., & Baleanu, D. (2013). General solutions of fully fuzzy linear systems. In Abstract and Applied Analysis (Vol. 2013, No. 1, p. 593274). Hindawi Publishing Corporation. https://doi.org/10.1155/2013/593274

Abbasbandy, S., Otadi, M., & Mosleh, M. (2008). Minimal solution of general dual fuzzy linear systems. Chaos, Solitons & Fractals, 37(4), 1113-1124. https://doi.org/10.1016/j.chaos.2006.10.045

Allahviranloo, T. (2003). A comment on fuzzy linear systems. Fuzzy sets and systems, 3(140), 559. http://dx.doi.org/10.1016%2FS0165-0114(03)00139-8

Allahviranloo, T., Ghanbari, M., Hosseinzadeh, A. A., Haghi, E., & Nuraei, R. (2011). A note on “Fuzzy linear systems”. Fuzzy sets and systems, 177(1), 87-92. https://doi.org/10.1016/j.fss.2011.02.010

Allahviranloo, T., & Ghanbari, M. (2012). On the algebraic solution of fuzzy linear systems based on interval theory. Applied mathematical modelling, 36(11), 5360-5379. https://doi.org/10.1016/j.apm.2012.01.002

Allahviranloo, T., & Kermani, M. A. (2006). Solution of a fuzzy system of linear equation. Applied Mathematics and Computation, 175(1), 519-531. https://doi.org/10.1016/j.amc.2005.07.048

Mihailović, B., Jerković, V. M., & Malešević, B. (2018). Solving fuzzy linear systems using a block representation of generalized inverses: The Moore–Penrose inverse. Fuzzy Sets and Systems, 353, 44-65. https://doi.org/10.1016/j.fss.2017.11.007

Allahviranloo, T. (2004). Numerical methods for fuzzy system of linear equations. Applied mathematics and computation, 155(2), 493-502. https://doi.org/10.1016/S0096-3003(03)00793-8

Allahviranloo, T. (2005). Successive over relaxation iterative method for fuzzy system of linear equations. Applied mathematics and computation, 162(1), 189-196. https://doi.org/10.1016/j.amc.2003.12.085

Dehghan, M., & Hashemi, B. (2006). Iterative solution of fuzzy linear systems. Applied mathematics and computation, 175(1), 645-674. https://doi.org/10.1016/j.amc.2005.07.033

Ezzati, R. (2011). Solving fuzzy linear systems. Soft computing, 15, 193-197. https://doi.org/10.1007/s00500-009-0537-7

Abbasbandy, S., & Jafarian, A. (2006). Steepest descent method for system of fuzzy linear equations. Applied Mathematics and Computation, 175(1), 823-833. https://doi.org/10.1016/j.amc.2005.07.036

Inearat, L., & Qatanani, N. (2018). Numerical methods for solving fuzzy linear systems. Mathematics, 6(2), 19. http://dx.doi.org/10.3390/math6020019

Asady, B., Abbasbandy, S., & Alavi, M. (2005). Fuzzy general linear systems. Applied Mathematics and computation, 169(1), 34-40. https://doi.org/10.1016/j.amc.2004.10.042

Senthilkumar, P., & Rajendran, G. (2011). An algorithmic approach to solve fuzzy linear systems. Journal of Information and Computational Science, 8(3), 503-510.

Rahaman, M., Mondal, S. P., Chatterjee, B., & Alam, S. (2021). The solution techniques for linear and quadratic equations with coefficients as Cauchy neutrosphic numbers. Granular computing, 7, 421-439. https://doi.org/10.1007/s41066-021-00276-0

Singh, P., Gazi, K. H., Rahaman, M., Basuri, T., & Mondal, S. P. (2024). Solution strategy and associated result for fuzzy mellin transformation. Franklin Open, 7, 100112. https://doi.org/10.1016/j.fraope.2024.100112

Singh, P., Gazi, K. H., Rahaman, M., Salahshour, S., & Mondal, S. P. (2024). A fuzzy fractional power series approximation and Taylor expansion for solving fuzzy fractional differential equation. Decision Analytics Journal, 10, 100402. https://doi.org/10.1016/j.dajour.2024.100402

Mahata, A., Mondal, S. P., Ahmadian, A., Ismail, F., Alam, S., & Salahshour, S. (2018). Different solution strategies for solving epidemic model in imprecise environment. Complexity, 2018(1), 4902142. https://doi.org/10.1155/2018/4902142

Alamin, A., Rahaman, M., Mondal, S. P., Chatterjee, B., & Alam, S. (2022). Discrete system insights of logistic quota harvesting model: a fuzzy difference equation approach. Journal of uncertain systems, 15(02), 2250007. https://doi.org/10.1142/S1752890922500076

Mondal, S. P., Khan, N. A., Vishwakarma, D., & Saha, A. K. (2018). Existence and stability of difference equation in imprecise environment. Nonlinear Engineering, 7(4), 263-271. https://doi.org/10.1515/nleng-2016-0085

Rahaman, M., Mondal, S. P., Algehyne, E. A., Biswas, A., & Alam, S. (2021). A method for solving linear difference equation in Gaussian fuzzy environments. Granular Computing, 7, 63-76. https://doi.org/10.1007/s41066-020-00251-1

Mahata, A., Mondal, S. P., Ahmadian, A., Alam, S., & Salahshour, S. (2020). Linear fuzzy delay differential equation and its application in biological model with fuzzy stability analysis. In Recent Advances on Soft Computing and Data Mining: Proceedings of the Fourth International Conference on Soft Computing and Data Mining (SCDM 2020), Melaka, Malaysia, January 22–⁠ 23, 2020 (pp. 231-240). Springer International Publishing. https://doi.org/10.1007/978-3-030-36056-6_23

Tudu, S., Mondal, S. P., & Alam, S. (2021). Different solution strategy for solving type-2 fuzzy system of differential equations with application in arms race model. International Journal of Applied and Computational Mathematics, 7, 177. https://doi.org/10.1007/s40819-021-01116-0

Paul, S., Mondal, S. P., & Bhattacharya, P. (2017). Discussion on proportional harvesting model in fuzzy environment: fuzzy differential equation approach. International Journal of Applied and Computational Mathematics, 3, 3067-3090. https://doi.org/10.1007/s40819-016-0283-3

Rahaman, M., Maity, S., De, S. K., Mondal, S. P., & Alam, S. (2021). Solution of an Economic Production Quantity model using the generalized Hukuhara derivative approach. Scientia Iranica, https://doi.org/10.24200/sci.2021.55951.4487

Hukuhara, M. (1967). Integration des applications mesurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj, 10(3), 205-223.

Stefanini, L. (2008). A generalization of Hukuhara difference. In Soft Methods for Handling Variability and Imprecision (pp. 203-210). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_25