nach oben

Granular Computing

Erschienen in:

18.10.2016 | Original Paper

Concepts of solutions of uncertain equations with intervals, probabilities and fuzzy sets for applied tasks

verfasst von: Boris Kovalerchuk, Vladik Kreinovich

Erschienen in: Granular Computing | Ausgabe 3/2017

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The focus of this paper is to clarify the concepts of solutions of linear equations in interval, probabilistic, and fuzzy sets setting for real-world tasks. There is a fundamental difference between formal definitions of the solutions and physically meaningful concepts of solution in applied tasks, when equations have uncertain components. For instance, a formal definition of the solution in terms of Moore interval analysis can be completely irrelevant for solving a real-world task. We show that formal definitions must follow a meaningful concept of the solution in the real world. The contribution of this paper is the seven formalized definitions of the concept of solution for the linear equations with uncertain components in the interval settings that are interpretable in the real-world tasks. It is shown that these definitions have analogies in probability and fuzzy set terms too. These new formalized concepts of solutions are generalized for difference and differential equations under uncertainty.

Vorheriger Artikel Granular computing: from granularity optimization to multi-granularity joint problem solving

Nächster Artikel Granular computing-based approach for classification towards reduction of bias in ensemble learning

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Allahviranloo T, Salahshour S, Khezerloo M (2011) Maximal-and minimal symmetric solutions of fully fuzzy linear systems. J Comput Appl Math 235(16):4652–4662MathSciNetCrossRefMATH

Allahviranloo T, Keshavarz M, Islam Sh (2016) The prediction of cardiovascular disorders by fuzzy difference equations, In: IEEE 2016 Intern. Conference on Fuzzy Systems, pp 1465–1472

Bridgman PW (1927) The logic of modern physics. Macmillan, NYMATH

Buckley J, Feuring T (2000) Fuzzy differential equations. Fuzzy Sets Syst 11:43–54MathSciNetCrossRefMATH

Di Nola A, Pedrycz W, Sessa S (1985) On measures of fuzziness of solutions of fuzzy relation equations with generalized connectives. J Math Anal Appl 106:443–453MathSciNetCrossRefMATH

Dubois D, Prade H (1984) Fuzzy-set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control Cybern 13(3):129–145MathSciNetMATH

Dubois D, Prade H (2016) Bridging gaps between several forms of granular computing. Granul Comput 1(2):115–126CrossRef

Ghomashi A, Salahshour S, Hakimzadeh A (2014) Approximating solutions of fully fuzzy linear systems: a financial case study. J Intell Fuzzy Syst 26(1):367–378MathSciNetMATH

Girko VL (1992) Systems of linear algebraic equations with random coefficients. Theory Probab Appl 36(2):261–271. doi:10.1137/1136030 MathSciNetCrossRef

Girko VI (1996) Theory of linear algebraic equations with random coefficients, NY

Girko VI (1998) An introduction to statistical analysis of random arrays, VSP

Hisdal E (1988) Are grades of membership probabilities? Fuzzy Sets Syst 25:325–348MathSciNetCrossRefMATH

Horcik R (2008) Solution of a system of linear equations with fuzzy numbers. Fuzzy Sets Syst 159:1788–1810MathSciNetCrossRefMATH

Kac M (1943) On the average number of real roots of a random algebraic equation. Bull Amer Math Soc 49(4):314–320MathSciNetCrossRefMATH

King A, Wallace S (2012) Modeling with stochastic programming. Springer, NYCrossRefMATH

Klement EP, Mesiar R, Pap E (2000) Triangular Norms. Kluwer, DordrechtCrossRefMATH

Kovalerchuk B (1994) Current situation in foundations of fuzzy optimization. In: Delgado M, Kacprzyk J, Verdegay JL, Vila MA (eds) Fuzzy optimization: recent advances. Springer, New York, pp 45–60

Kovalerchuk B (2014) Probabilistic solution of Zadeh’s test problems. In: Laurent A, et al (eds.) IPMU 2014, Part II, CCIS 443, Springer, Switzerland, pp 536–545

Kovalerchuk B (2015) Summation of linguistic numbers. In: Proc. of NAFIPS and World Congress on Soft Computing pp 1–6. doi:10.1109/NAFIPS-WConSC.2015.7284161

Kovalerchuk B, Perlovsky L, Wheeler G (2012) Modeling of phenomena and dynamic logic of phenomena. J App Non-class Log 22(1):51–82

Kreinovich V (2016) Solving equations (and systems of equations) under uncertainty: how different practical problems lead to different mathematical and computational formulations. Granul Comput 1(3):171–179MathSciNetCrossRef

Li CF, Feng YT, Owen DRJ (2006) Explicit solution to the stochastic system of linear algebraic equations (a₁A₁ + a₂A₂ +… + a_mA_m)x = b. Comput Methods Appl Mech Eng 195:6560–6576CrossRefMATH

Pedrycz W, Chen SM (2011) Granular computing and intelligent systems: design with information granules of higher order and higher type. Springer, HeidelbergCrossRef

Pedrycz W, Chen SM (2015a) Information granularity, big data, and computational intelligence. Springer, HeidelbergCrossRef

Pedrycz W, Chen SM (2015b) Granular computing and decision-making: interactive and iterative approaches. Springer, HeidelbergCrossRef

Peeva K (1991) Fuzzy linear systems. Ann Univ Sci Budapest Sect Comput 12:201–207MathSciNetMATH

Piegat A, Pluciński M (2015) Fuzzy number addition with the application of horizontal membership functions. The Scientific World Journal, Article ID 367214, p 16

Popova ED (2013) Improved enclosure for some parametric solution sets with linear shape. Comput Math Appl 66:1655–1665. doi:10.1016/j.camwa.2013.04.007 MathSciNetCrossRefMATH

Provencher S (1982) CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations. Comput Phys Commun 27:229–242CrossRef

Sanchez E (1984) Solution of fuzzy equations with extended operations. Fuzzy Sets Syst 12:237–248MathSciNetCrossRefMATH

Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: Modeling and theory. MPS/SIAM Series on Optimization 9. pp. xvi + 436, Philadelphia

Shary SP (1996) Algebraic approach to the interval linear system identification, tolerance, and control problems, or one more application of Kaucher arithmetic. Reliable Comput 2(1):3–33MathSciNetCrossRefMATH

Shary SP (2002) A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Comput 8:321–418MathSciNetCrossRefMATH

Skalna MV, Rao R, Pownuk A (2008) Systems of fuzzy equations in structural mechanics. J Comput Appl Math 218:149–156MathSciNetCrossRefMATH

Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584MathSciNetCrossRefMATH

Xu Z, Wang H (2016) Managing multi-granularity linguistic information in qualitative group decision making: an overview. Granul Comput 1(1):21–35. doi:10.1007/s41066-015-0006-x CrossRef

Yager R (1979) On solving fuzzy mathematical relationships. Inf Control 41:29–55MathSciNetCrossRefMATH

Yager R (1986) A characterization of the extension principle. Fuzzy Sets Systems 18:205–217MathSciNetCrossRefMATH

Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning. Part I. Inform Sci 8:199–249CrossRefMATH

Zhao R, Govind R (1991) Solutions of algebraic equations involving generalized fuzzy numbers. Info Sci 56(1–3):199–243MathSciNetCrossRefMATH

Titel: Concepts of solutions of uncertain equations with intervals, probabilities and fuzzy sets for applied tasks
verfasst von: Boris Kovalerchuk
Vladik Kreinovich
Publikationsdatum: 18.10.2016
Verlag: Springer International Publishing
Erschienen in: Granular Computing / Ausgabe 3/2017
Print ISSN: 2364-4966
Elektronische ISSN: 2364-4974
DOI: https://doi.org/10.1007/s41066-016-0031-4

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 3/2017

SPIFGIA operators and their applications to decision making

Unified Granular-number-based AHP-VIKOR multi-criteria decision framework

GIFIHIA operator and its application to the selection of cold chain logistics enterprises

Type-2 fuzzy linear systems

Granular computing: from granularity optimization to multi-granularity joint problem solving

A granular computing framework for approximate reasoning in situation awareness