Hukuhara difference of Z-numbers

Abstract Real-world problems in decision analysis, economics, optimization and other fields are characterized by fuzzy and partially reliable information. In this regard, Prof. Zadeh suggested the concept of a Z-number, as an ordered pair Z  = ( A, B ) of fuzzy numbers A and B , the first of which describes a linguistic value, and the second one is a fuzzy value of probability measure of the first one, playing a role of reliability of information. Construction of mathematical models to handle Z-number-based information is a new challenge which requires development of Z-number-valued analysis including Z-number-valued equations, Z-number-valued derivative and integral and other concepts. One of the basic concepts in this realm is an inverse of addition operation. In the existing literature devoted to set-valued arithmetic and fuzzy arithmetic this operation is referred to as Hukuhara difference. For interval numbers and fuzzy numbers, the existence conditions of Hukuhara difference are derived. Unfortunately, the concept of Hukuhara difference for Z-numbers is not developed. In this work for the first time we suggest a fundamental approach for development of the concept of Hukuhara difference of Z-numbers. Examples are provided in the paper to show validity of the suggested approach.

[1]  Ali Azadeh,et al.  Z-AHP: A Z-number extension of fuzzy analytical hierarchy process , 2013, 2013 7th IEEE International Conference on Digital Ecosystems and Technologies (DEST).

[2]  Barnabás Bede,et al.  A note on "two-point boundary value problems associated with non-linear fuzzy differential equations" , 2006, Fuzzy Sets Syst..

[3]  T. Allahviranloo,et al.  The Parametric Form of Z‐Number and Its Application in Z‐Number Initial Value Problem , 2017, Int. J. Intell. Syst..

[4]  Tofigh Allahviranloo,et al.  On fuzzy solutions for heat equation based on generalized Hukuhara differentiability , 2015, Fuzzy Sets Syst..

[5]  Rafik A. Aliev,et al.  The Arithmetic of Z-Numbers - Theory and Applications , 2015, The Arithmetic of Z-Numbers.

[6]  Laécio Carvalho de Barros,et al.  Fuzzy differential equation with completely correlated parameters , 2015, Fuzzy Sets Syst..

[7]  Rafik A. Aliev,et al.  Decision Making Under Z-Information , 2014, Human-Centric Decision-Making Models for Social Sciences.

[8]  Phil Diamond.  Position paper on Analysis submissions to Fuzzy Sets and Systems , 2001, Fuzzy Sets Syst..

[9]  Saied Tadayon,et al.  Approximate Z-number Evaluation Based on Categorical Sets of Probability Distributions , 2014 .

[10]  Rafik A. Aliev,et al.  Z‐Number‐Based Linear Programming , 2015, Int. J. Intell. Syst..

[11]  L. Zadeh.  Probability measures of Fuzzy events , 1968 .

[12]  Vasile Lupulescu,et al.  Hukuhara differentiability of interval-valued functions and interval differential equations on time scales , 2013, Inf. Sci..

[13]  Luciano Stefanini,et al.  Some notes on generalized Hukuhara differentiability of interval-valued functions and interval differential equations , 2012 .

[14]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[15]  Laécio C. Barros,et al.  A note on the generalized difference and the generalized differentiability , 2015, Fuzzy Sets Syst..

[16]  Tofigh Allahviranloo,et al.  A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform , 2017, Fuzzy Sets Syst..

[17]  Jaume Casasnovas,et al.  On the addition of discrete fuzzy numbers , 2006 .

[18]  Rafik A Aliev,et al.  Operations on Z-numbers with acceptable degree of specificity , 2017 .

[19]  Lotfi A. Zadeh,et al.  A Note on Z-numbers , 2011, Inf. Sci..

[20]  Michael Hanss,et al.  Applied Fuzzy Arithmetic: An Introduction with Engineering Applications , 2004 .

[21]  Laécio C. Barros,et al.  Fuzzy differential equations with interactive derivative , 2017, Fuzzy Sets Syst..

[22]  D. Dubois,et al.  A NOTE ON MEASURES OF SPECIFICITY FOR FUZZY SETS , 1985 .

[23]  Tofigh Allahviranloo,et al.  Semi-analytical methods for solving fuzzy impulsive fractional differential equations , 2017, J. Intell. Fuzzy Syst..

[24]  Rafik A Aliev,et al.  Decision Theory With Imperfect Information , 2014 .

[25]  Elham Sahebkar Khorasani,et al.  An inference engine toolkit for computing with words , 2013, J. Ambient Intell. Humaniz. Comput..

[26]  Yurilev Chalco-Cano,et al.  Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles , 2015, Fuzzy Sets Syst..

[27]  Ronald R. Yager,et al.  Measures of specificity of fuzzy sets under T-indistinguishabilities , 2006, IEEE Transactions on Fuzzy Systems.

[28]  Sankar K. Pal,et al.  The Z-Number Enigma: A Study through an Experiment , 2013, Soft Computing: State of the Art Theory and Novel Applications.

[29]  Robert C. Williamson,et al.  Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds , 1990, Int. J. Approx. Reason..

[30]  Luciano Stefanini,et al.  On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations , 2007 .

[31]  Barnabás Bede,et al.  Generalized differentiability of fuzzy-valued functions , 2013, Fuzzy Sets Syst..

[32]  P. Diamond,et al.  Extended fuzzy linear models and least squares estimates , 1997 .

[33]  Barnabás Bede,et al.  Generalized fuzzy differentiability with LU-parametric representation , 2014, Fuzzy Sets Syst..

[34]  Phil Diamond,et al.  Brief note on the variation of constants formula for fuzzy differential equations , 2002, Fuzzy Sets Syst..

[35]  John N. Tsitsiklis,et al.  Introduction to Probability , 2002 .

[36]  Yurilev Chalco-Cano,et al.  Calculus for interval-valued functions using generalized Hukuhara derivative and applications , 2013, Fuzzy Sets Syst..

[37]  M. Hukuhara.  INTEGRATION DES APPLICAITONS MESURABLES DONT LA VALEUR EST UN COMPACT CONVEXE , 1967 .

[38]  Jaume Casasnovas,et al.  Weighted Means of Subjective Evaluations , 2012, Soft Computing in Humanities and Social Sciences.

[39]  B. Bede,et al.  Fuzzy Differential Equations in Various Approaches , 2015 .

[40]  Luciano Stefanini,et al.  A generalization of Hukuhara difference and division for interval and fuzzy arithmetic , 2010, Fuzzy Sets Syst..

[41]  Soumitra Dutta,et al.  An Insight Into The Z-number Approach To CWW , 2013, Fundam. Informaticae.

[42]  William Voxman,et al.  Canonical representations of discrete fuzzy numbers , 2001, Fuzzy Sets Syst..

[43]  Luciano Stefanini,et al.  Fuzzy differential equations by fuzzy-transform , 2015, 2015 Annual Conference of the North American Fuzzy Information Processing Society (NAFIPS) held jointly with 2015 5th World Conference on Soft Computing (WConSC).

[44]  Irina Perfilieva,et al.  Fuzzy transforms: Theory and applications , 2006, Fuzzy Sets Syst..

[45]  Yong Deng,et al.  A Method of Converting Z-number to Classical Fuzzy Number , 2012 .

[46]  Bingyi Kang,et al.  Decision Making Using Z-numbers under Uncertain Environment , 2012 .

[47]  Maria Letizia Guerra,et al.  Crisp profile symmetric decomposition of fuzzy numbers , 2016 .

[48]  Phil Diamond.  Theory and applications of fuzzy Volterra integral equations , 2002, IEEE Trans. Fuzzy Syst..

[49]  Ronald R. Yager,et al.  On Z‐valuations using Zadeh's Z‐numbers , 2012, Int. J. Intell. Syst..

[50]  Phil Diamond,et al.  Stability and periodicity in fuzzy differential equations , 2000, IEEE Trans. Fuzzy Syst..

[51]  P. Kloeden,et al.  Metric Spaces Of Fuzzy Sets Theory And Applications , 1975 .

[52]  Rafik A. Aliev,et al.  The arithmetic of discrete Z-numbers , 2015, Inf. Sci..

[53]  Laécio C. Barros,et al.  Fuzzy differential equations: An approach via fuzzification of the derivative operator , 2013, Fuzzy Sets Syst..

[54]  P. R. Nelson.  The algebra of random variables , 1979 .