Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming

In practice, some special LR fuzzy numbers, like the triangular fuzzy number, the Gaussian fuzzy number and the Cauchy fuzzy number, are widely used in many areas to deal with various vague information. With regard to these special LR fuzzy numbers, called regular LR fuzzy numbers in this paper, an operational law is proposed for fuzzy arithmetic, providing a novel approach to analytically and exactly calculating the inverse credibility distribution of some specific arithmetical operations based on the credibility measure. As an application of the operational law, an equivalent form of the expected value operator as well as a theorem for computing the expected value of strictly monotone functions is suggested. Finally, we utilize the operational law to construct a solution framework of fuzzy programming with parameters of regular LR fuzzy numbers, and such type of fuzzy programming problems can be handled by the operational law as the classic deterministic programming without any particular solving techniques.

[1]  Yian-Kui Liu,et al.  Expected value of fuzzy variable and fuzzy expected value models , 2002, IEEE Trans. Fuzzy Syst..

[2]  Jagdeep Kaur,et al.  Mehar’s method for solving fully fuzzy linear programming problems with L-R fuzzy parameters , 2013 .

[3]  Lotfi A. Zadeh,et al.  A Theory of Approximate Reasoning , 1979 .

[4]  George J. Klir,et al.  Fuzzy arithmetic with requisite constraints , 1997, Fuzzy Sets Syst..

[5]  Hans-Jürgen Zimmermann,et al.  Fuzzy Set Theory - and Its Applications , 1985 .

[6]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[7]  Luciano Stefanini,et al.  Some Parametric Forms for LR Fuzzy Numbers and LR Fuzzy Arithmetic , 2009, 2009 Ninth International Conference on Intelligent Systems Design and Applications.

[8]  A. Charnes,et al.  Management Models and Industrial Applications of Linear Programming , 1961 .

[9]  Jinwu Gao,et al.  The Independence of Fuzzy Variables with Applications to Fuzzy Random Optimization , 2007, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[10]  R. Yager.  On the specificity of a possibility distribution , 1992 .

[11]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

[12]  Yian-Kui Liu,et al.  Expected Value Operator of Random Fuzzy Variable, Random Fuzzy Expected Value Models , 2003, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[13]  Baoding Liu,et al.  Chance constrained programming with fuzzy parameters , 1998, Fuzzy Sets Syst..

[14]  Madan M. Gupta,et al.  Fuzzy mathematical models in engineering and management science , 1988 .

[15]  Ch.-Ch Chou.  The canonical representation of multiplication operation on triangular fuzzy numbers , 2003 .

[16]  Baoding Liu,et al.  Theory and Practice of Uncertain Programming , 2003, Studies in Fuzziness and Soft Computing.

[17]  N. Ravi Shankar,et al.  Fuzzy risk analysis based on the novel fuzzy ranking with new arithmetic operations of linguistic fuzzy numbers , 2014, J. Intell. Fuzzy Syst..

[18]  Luciano Stefanini,et al.  Approximate fuzzy arithmetic operations using monotonic interpolations , 2005, Fuzzy Sets Syst..

[19]  Rituparna Chutia,et al.  Arithmetic of Triangular Fuzzy Variable from Credibility Theory , 2011 .

[20]  Hamed Kazemipoor,et al.  Solving fully fuzzy linear programming with symmetric trapezoidal fuzzy numbers using Mehar’s method , 2014 .

[21]  R. Vasuki,et al.  Estimation of Confidence Level 'h' in Fuzzy Linear Regression Analysis using Shape Preserving Operations , 2013 .

[22]  Michael Hanss,et al.  The transformation method for the simulation and analysis of systems with uncertain parameters , 2002, Fuzzy Sets Syst..

[23]  Mehdi Dehghan,et al.  Computational methods for solving fully fuzzy linear systems , 2006, Appl. Math. Comput..

[24]  Shyi-Ming Chen,et al.  Evaluating weapon systems using fuzzy arithmetic operations , 1996, Fuzzy Sets Syst..

[25]  Mohamed Shenify,et al.  The Expected Value of a Fuzzy Number , 2015 .

[26]  Ronald E. Giachetti,et al.  A parametric representation of fuzzy numbers and their arithmetic operators , 1997, Fuzzy Sets Syst..

[27]  Reza Eslamipoor,et al.  An improved ranking method for generalized fuzzy numbers based on Euclidian distance concept , 2015 .

[28]  H. Khademi Zare,et al.  A developed distance method for ranking generalized fuzzy numbers , 2013, Neural Computing and Applications.

[29]  Hsiao-Fan Wang,et al.  Three-parameter Fuzzy Arithmetic Approximation of L-r Fuzzy Numbers for Fuzzy Neural Networks , 2006, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[30]  Stanisław Heilpern,et al.  The expected value of a fuzzy number , 1992 .

[31]  Ronald E. Giachetti,et al.  Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation , 1997, Fuzzy Sets Syst..

[32]  D. Dubois,et al.  The mean value of a fuzzy number , 1987 .

[33]  Shyi-Ming Chen.  FUZZY SYSTEM RELIABILITY-ANALYSIS USING FUZZY NUMBER ARITHMETIC OPERATIONS (VOL 64, PG 31, 1994) , 1994 .

[34]  Reza Eslamipoor,et al.  Proposing a revised method for ranking fuzzy numbers , 2013, J. Intell. Fuzzy Syst..

[35]  Ronald R. Yager,et al.  A procedure for ordering fuzzy subsets of the unit interval , 1981, Inf. Sci..

[36]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[37]  Ching-Hsue Cheng,et al.  A new approach for ranking fuzzy numbers by distance method , 1998, Fuzzy Sets Syst..

[38]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[39]  Didier Dubois,et al.  Possibility theory , 2018, Scholarpedia.

[40]  Baoding Liu,et al.  Uncertainty Theory - A Branch of Mathematics for Modeling Human Uncertainty , 2011, Studies in Computational Intelligence.

[41]  D. Dubois,et al.  An application of fuzzy arithmetic to the optimization of industrial machining processes , 1987 .

[42]  R. Saneifard.  RANKING L-R FUZZY NUMBERS WITH WEIGHTED AVERAGING BASED ON LEVELS , 2009 .