American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 619-627. doi: 10.3934/proc.2013.2013.619

Fuzzy system of linear equations

Purnima Pandit 1,

1. 

Department of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara, Gujarat, India

Received  September 2012 Revised  March 2013 Published  November 2013

Real life applications arising in various fields of Engineering and Sciences like Electrical, Civil, Economics, Dietary etc. can be modeled using system of linear equations. In such models it may happen that the values of the parameters are not known or they cannot be stated precisely only their estimation due to experimental data or experts knowledge is available. In such situation it is convenient to represent such parameters by fuzzy numbers (refer [22]). Klir, [15] gave the results for the existence of solution of linear algebraic equation involving fuzzy numbers. The method to obtain solution of system of linear equations with all the involved parameters being fuzzy is proposed here. The $\alpha$-cut technique is well known in obtaining weak solutions, (refer [7]) for fully fuzzy systems of linear equations (FFSL). In this paper, the conditions for the existence and uniqueness of the fuzzy solution are proved.
Keywords: level cut, $\alpha$-cut, Fuzzy numbers, fully fuzzy linear equations, parallel computing., weak solutions.
Mathematics Subject Classification: Primary: 03E7.
Citation: Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619
References:
[1]

S. Abbasbandy S., A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric positive-definite system of linear equations,, Applied Mathematics and Computation, 171 (2005), 1184.   Google Scholar

[2]

S. Abbasbandy S., R. Ezzati and A. Jafarian, LU decomposition method for solving fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{172} (2006), 172 (2006), 633.   Google Scholar

[3]

S. Abbasbandy and A. Jafarian, Steepest descent method for system of fuzzy linear equations,, Applied Mathematics and Computation \textbf{175} (2006), 175 (2006), 823.   Google Scholar

[4]

T. Allahviranloo, Numerical methods for fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{155} (2004), 155 (2004), 493.   Google Scholar

[5]

T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{162} (2004), 162 (2004), 189.   Google Scholar

[6]

T. Allahviranloo, The adomian decomposition method for fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{163} (2005), 163 (2005), 553.   Google Scholar

[7]

T. Allahviranloo, M. Ghanbari, A. A. Hosseinzadeh, E. Haghi and R. Nuraei, A note on fuzzy linear systems,, Fuzzy Sets and Systems \textbf{177}(1) (2011), 177 (2011), 87.   Google Scholar

[8]

T. Allahviranloo, E. Ahmady and N. Ahmady, A method for solving nth order fuzzy linear differential equations,, International Journal of Computer Mathematics, 86 (2009), 730.   Google Scholar

[9]

D. Behera and S. Chakraverty, Solution of Fuzzy System of Linear Equations with Polynomial Parameteric form,, Applications and Applied Mathematics, 7 (2012), 648.   Google Scholar

[10]

J. J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations,, Fuzzy Sets and Systems, 38 (1990), 43.   Google Scholar

[11]

M. Dehghan, B. Hashemi and M. Ghatee, Computational methods for solving fully fuzzy linear systems,, Applied Mathematics and Computation, 179 (2006), 328.   Google Scholar

[12]

M. Dehghan and B. Hashemi, Solution of the fully fuzzy linear system using the decomposition procedure,, Applied Mathematics and Computation, 182 (2006), 1568.   Google Scholar

[13]

D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications,, Academic Press, (1980).   Google Scholar

[14]

M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems,, Fuzzy Sets and Systems, 96 (1998), 201.   Google Scholar

[15]

G. Klir and B.Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications,, Prentice Hall, (1997).   Google Scholar

[16]

M. Matinfar, S. H. Nasseri and M. Sohrabi, Solving Fuzzy Linear System of Equations by Using Householder Decomposition Method,, Applied Mathematical Sciences, 2 (2008), 2569.   Google Scholar

[17]

S. H. Nasseri, M. Abdi and B. Khabiri, An Application of Fuzzy linear System of Equations in Economic Sciences,, Australian Journal of Basic and Applied Sciences, 5 (2011), 7.   Google Scholar

[18]

S. H. Nasseri and M. Sohrabi, Gram-Schmidt approach for linear System of Equations with fuzzy parameters,, The Journal of Mathematics and Computer Science, 1 (2010), 80.   Google Scholar

[19]

M. J. Quinn, Parallel Computing Theory and Practice,, Oregon State University, (2002).   Google Scholar

[20]

T. Rahgooy, H. Sadoghi and R. Monsefi, Fuzzy Complex System of linear equations Applied to Circuit Analysis,, International Journal of Computer and Electrical Engineering, 1 (2009), 535.   Google Scholar

[21]

A. Sadeghi, I. M. Ahmad and A. F. Jameel, Solving Systems of Fuzzy Differential Equation,, International Mathematical Forum, 6 (2011), 2087.   Google Scholar

[22]

L. A. Zadeh, Fuzzy sets,, Information and Control, 8 (1965), 338.   Google Scholar

show all references

References:
[1]

S. Abbasbandy S., A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric positive-definite system of linear equations,, Applied Mathematics and Computation, 171 (2005), 1184.   Google Scholar

[2]

S. Abbasbandy S., R. Ezzati and A. Jafarian, LU decomposition method for solving fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{172} (2006), 172 (2006), 633.   Google Scholar

[3]

S. Abbasbandy and A. Jafarian, Steepest descent method for system of fuzzy linear equations,, Applied Mathematics and Computation \textbf{175} (2006), 175 (2006), 823.   Google Scholar

[4]

T. Allahviranloo, Numerical methods for fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{155} (2004), 155 (2004), 493.   Google Scholar

[5]

T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{162} (2004), 162 (2004), 189.   Google Scholar

[6]

T. Allahviranloo, The adomian decomposition method for fuzzy system of linear equations,, Applied Mathematics and Computation \textbf{163} (2005), 163 (2005), 553.   Google Scholar

[7]

T. Allahviranloo, M. Ghanbari, A. A. Hosseinzadeh, E. Haghi and R. Nuraei, A note on fuzzy linear systems,, Fuzzy Sets and Systems \textbf{177}(1) (2011), 177 (2011), 87.   Google Scholar

[8]

T. Allahviranloo, E. Ahmady and N. Ahmady, A method for solving nth order fuzzy linear differential equations,, International Journal of Computer Mathematics, 86 (2009), 730.   Google Scholar

[9]

D. Behera and S. Chakraverty, Solution of Fuzzy System of Linear Equations with Polynomial Parameteric form,, Applications and Applied Mathematics, 7 (2012), 648.   Google Scholar

[10]

J. J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations,, Fuzzy Sets and Systems, 38 (1990), 43.   Google Scholar

[11]

M. Dehghan, B. Hashemi and M. Ghatee, Computational methods for solving fully fuzzy linear systems,, Applied Mathematics and Computation, 179 (2006), 328.   Google Scholar

[12]

M. Dehghan and B. Hashemi, Solution of the fully fuzzy linear system using the decomposition procedure,, Applied Mathematics and Computation, 182 (2006), 1568.   Google Scholar

[13]

D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications,, Academic Press, (1980).   Google Scholar

[14]

M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems,, Fuzzy Sets and Systems, 96 (1998), 201.   Google Scholar

[15]

G. Klir and B.Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications,, Prentice Hall, (1997).   Google Scholar

[16]

M. Matinfar, S. H. Nasseri and M. Sohrabi, Solving Fuzzy Linear System of Equations by Using Householder Decomposition Method,, Applied Mathematical Sciences, 2 (2008), 2569.   Google Scholar

[17]

S. H. Nasseri, M. Abdi and B. Khabiri, An Application of Fuzzy linear System of Equations in Economic Sciences,, Australian Journal of Basic and Applied Sciences, 5 (2011), 7.   Google Scholar

[18]

S. H. Nasseri and M. Sohrabi, Gram-Schmidt approach for linear System of Equations with fuzzy parameters,, The Journal of Mathematics and Computer Science, 1 (2010), 80.   Google Scholar

[19]

M. J. Quinn, Parallel Computing Theory and Practice,, Oregon State University, (2002).   Google Scholar

[20]

T. Rahgooy, H. Sadoghi and R. Monsefi, Fuzzy Complex System of linear equations Applied to Circuit Analysis,, International Journal of Computer and Electrical Engineering, 1 (2009), 535.   Google Scholar

[21]

A. Sadeghi, I. M. Ahmad and A. F. Jameel, Solving Systems of Fuzzy Differential Equation,, International Mathematical Forum, 6 (2011), 2087.   Google Scholar

[22]

L. A. Zadeh, Fuzzy sets,, Information and Control, 8 (1965), 338.   Google Scholar

[1]

Fanghua Lin, Dan Liu. On the Betti numbers of level sets of solutions to elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4517-4529. doi: 10.3934/dcds.2016.36.4517
[2]

Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263
[3]

G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100.

[4]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118
[5]

Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261
[6]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435
[7]

Wenjuan Jia, Yingjie Deng, Chenyang Xin, Xiaodong Liu, Witold Pedrycz. A classification algorithm with Linear Discriminant Analysis and Axiomatic Fuzzy Sets. Mathematical Foundations of Computing, 2019, 2 (1) : 73-81. doi: 10.3934/mfc.2019006
[8]

Blaine Keetch, Yves Van Gennip. A Max-Cut approximation using a graph based MBO scheme. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6091-6139. doi: 10.3934/dcdsb.2019132
[9]

Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial & Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581
[10]

Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
[11]

Jun Xie, Jinlong Yuan, Dongxia Wang, Weili Liu, Chongyang Liu. Uniqueness of solutions to fuzzy relational equations regarding Max-av composition and strong regularity of the matrices in Max-av algebra. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1007-1022. doi: 10.3934/jimo.2017087
[12]

Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699
[13]

Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015
[14]

Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008
[15]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[16]

Wei Wang, Xiao-Long Xin. On fuzzy filters of Heyting-algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1611-1619. doi: 10.3934/dcdss.2011.4.1611
[17]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[18]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
[19]

Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203
[20]

Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559

 Impact Factor: 

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences