Computational methods for solving fully fuzzy linear systems
Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modelling, which can formulate uncertainty in actual environment. In addition, this is an impo...
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| Vydáno v: | Applied mathematics and computation Ročník 179; číslo 1; s. 328 - 343 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.08.2006
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| ISSN: | 0096-3003, 1873-5649 |
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| Abstract | Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modelling, which can formulate uncertainty in actual environment. In addition, this is an important sub-process in determining inverse, eigenvalue and some other useful matrix computations, too. One of the most practicable subjects in recent studies is based on LR fuzzy numbers, which are defined and used by Dubois and Prade with some useful and easy approximation arithmetic operators on them. Recently Dehghan et al. [M. Dehghan, M. Ghatee, B. Hashemi, Some computations on fuzzy matrices, submitted for publication.] extended some matrix computations on fuzzy matrices, where a fuzzy matrix appears as a rectangular array of fuzzy numbers. In continuation to our previous work, we focus on fuzzy systems in this paper. It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard. The same result can similarly be derived for fuzzy systems. So we employ some heuristics based methods on Dubois and Prade’s approach, finding some positive fuzzy vector
x
˜
which satisfies
A
˜
x
˜
=
b
˜
, where
A
˜
and
b
˜
are a fuzzy matrix and a fuzzy vector respectively. We propose some new methods to solve this system that are comparable to the well known methods such as the Cramer’s rule, Gaussian elimination, LU decomposition method (Doolittle algorithm) and its simplification. Finally we extend a new method employing Linear Programming (LP) for solving square and non-square (over-determined) fuzzy systems. Some numerical examples clarify the ability of our heuristics. |
|---|---|
| AbstractList | Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modelling, which can formulate uncertainty in actual environment. In addition, this is an important sub-process in determining inverse, eigenvalue and some other useful matrix computations, too. One of the most practicable subjects in recent studies is based on LR fuzzy numbers, which are defined and used by Dubois and Prade with some useful and easy approximation arithmetic operators on them. Recently Dehghan et al. [M. Dehghan, M. Ghatee, B. Hashemi, Some computations on fuzzy matrices, submitted for publication.] extended some matrix computations on fuzzy matrices, where a fuzzy matrix appears as a rectangular array of fuzzy numbers. In continuation to our previous work, we focus on fuzzy systems in this paper. It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard. The same result can similarly be derived for fuzzy systems. So we employ some heuristics based methods on Dubois and Prade’s approach, finding some positive fuzzy vector
x
˜
which satisfies
A
˜
x
˜
=
b
˜
, where
A
˜
and
b
˜
are a fuzzy matrix and a fuzzy vector respectively. We propose some new methods to solve this system that are comparable to the well known methods such as the Cramer’s rule, Gaussian elimination, LU decomposition method (Doolittle algorithm) and its simplification. Finally we extend a new method employing Linear Programming (LP) for solving square and non-square (over-determined) fuzzy systems. Some numerical examples clarify the ability of our heuristics. |
| Author | Ghatee, Mehdi Dehghan, Mehdi Hashemi, Behnam |
| Author_xml | – sequence: 1 givenname: Mehdi surname: Dehghan fullname: Dehghan, Mehdi email: mdehghan@aut.ac.ir – sequence: 2 givenname: Behnam surname: Hashemi fullname: Hashemi, Behnam email: hashemi_am@aut.ac.ir, hoseyn.hashemi@gawab.com – sequence: 3 givenname: Mehdi surname: Ghatee fullname: Ghatee, Mehdi email: ghatee@aut.ac.ir, ghatee@gmail.com |
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| Cites_doi | 10.1016/S0165-0114(97)00140-1 10.1016/j.amc.2005.07.033 10.1080/00207727808941724 10.1016/0020-0255(91)90031-O 10.1016/0165-0114(80)90004-4 10.1016/S0165-0114(00)00096-8 10.1016/S0165-0114(96)00270-9 10.1002/(SICI)1097-0207(19980715)42:5<829::AID-NME386>3.0.CO;2-G 10.1137/0704001 10.1016/S0019-9958(65)90241-X 10.1016/0165-0114(91)90019-M 10.1016/S0165-0114(96)00118-2 |
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| Keywords | Linear programming (LP) Doolittle algorithm Fuzzy approximate arithmetic Gaussian elimination Fully fuzzy linear system (FFLS) Cramer’s rule Fuzzy LU decomposition Over-determined fuzzy linear system of equations LR fuzzy number Uncertainty Complex system Eigenvalue Overdetermined system Direct method Linear equation Fuzzy system Numerical linear algebra Linear programming Cramer's rule Rectangular matrix Engineering design Numerical analysis Linear system Matrix inversion Applied mathematics Heuristic method Matrix calculus |
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| References_xml | – volume: 3 start-page: 37 year: 1980 end-page: 48 ident: bib7 article-title: Systems of linear fuzzy constraints publication-title: Fuzzy Sets and Systems – volume: vol. 10 year: 1998 ident: bib16 publication-title: Computational Complexity and Feasibility of Data Processing and Interval Computations – volume: 123 start-page: 49 year: 2001 end-page: 62 ident: bib19 article-title: Computational aspects of fuzzy arithmetics based on Archimedean t-norms publication-title: Fuzzy Sets and Systems – reference: M. Dehghan, B. 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| SubjectTerms | Algebra Cramer’s rule Doolittle algorithm Exact sciences and technology Fully fuzzy linear system (FFLS) Fuzzy approximate arithmetic Fuzzy LU decomposition Gaussian elimination Linear and multilinear algebra, matrix theory Linear programming (LP) LR fuzzy number Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Numerical methods in mathematical programming Numerical methods in mathematical programming, optimization and calculus of variations Over-determined fuzzy linear system of equations Sciences and techniques of general use |
| Title | Computational methods for solving fully fuzzy linear systems |
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