With extensive applications of fuzzy numbers, many methods for fuzzy arithmetic especially the basic operations have been developed based on Zadeh’s extension principle. Among these methods, the interval arithmetic approach and the standard approximation method are the most important and commonly used exact and approximate methods, respectively. In this paper, regarding the continuous and strictly monotone functions of triangular fuzzy numbers, we propose an inverse distribution approach to deriving the exact or well approximate membership functions for arithmetic results by embedding the credibility measure of fuzzy sets into fuzzy arithmetic. Besides, some non-complicated and complicated examples are given to illustrate the performance of the new approach, together with a detailed comparison with the interval arithmetic approach and the standard approximate method. Furthermore, the inverse distribution approach is also applied to the fuzzy system reliability calculation based on fault tree compared with several current related researches.
AbbasiF., AllahviranlooT. and AbbasbandyS., A new attitude coupled with fuzzy thinking to fuzzy rings and fields, Journal of Intelligent & Fuzzy Systems29(2) (2015), 851–861.
2.
BanA.I., CoroianuL. and KhastanA., Conditioned weighted L-R approximations of fuzzy numbers, Fuzzy Sets and Systems283 (2016), 56–82.
3.
BaykasogluA. and SubulanK., Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables, Expert Systems with Applications81 (2017), 193–222.
4.
ChenS., Fuzzy system reliability analysis using fuzzy number arithmetic operations, Fuzzy Sets and Systems64(1) (1994), 31–38.
5.
ChenY., FungR.Y.K. and TangJ., Rating technical attributes in fuzzy QFD by integrating fuzzy weighted average method and fuzzy expected value operator, European Journal of Operational Research174(3) (2006), 1553–1566.
6.
ChouC., The canonical representation of multiplication operation on triangular fuzzy numbers, Computers & Mathematics with Applications45(10-11) (2003), 1601–1610.
7.
ChutiaR., MahantaS. and DattaD., Arithmetic of triangular fuzzy variable from credibility theory, International Journal of Energy, Information and Communications2(3) (2011), 9–20.
8.
CoroianuL., GagolewskiM. and GrzegorzewskiP., Nearest piece-wise linear approximation of fuzzy numbers, Fuzzy Sets and Systems233 (2013), 26–51.
9.
DuboisD. and PradeH., Operations on fuzzy numbers, International Journal of Systems Science9(6) (1978), 613–626.
10.
ErdoganY.S. and BakirP.G., Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic, Engineering Applications of Artificial Intelligence26(1) (2013), 357–367.
11.
GiachettiR.E. and YoungR.E., A parametric representation of fuzzy numbers and their arithmetic operators, Fuzzy Sets and Systems91(2) (1997), 185–202.
12.
GuerraM.L. and StefaniniL., Approximate fuzzy arithmetic operations using monotonic interpolations, Fuzzy Sets and Systems150(1) (2005), 5–33.
13.
HaagT., HerrmannJ. and HanssM., Identification procedure for epistemic uncertainties using inverse fuzzy arithmetic, Mechanical Systems and Signal Processing24(7) (2010), 2021–2034.
14.
KaufmannA., GuptaM.M., Introduction to Fuzzy Arithmetic-theory and Applications, Van Nostrand Reinhold Company, New York, 1991.
15.
KumarM., Applying weakest t-norm based approximate in-tuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers to evaluate reliability of PCBA fault, Applied Soft Computing23 (2014), 387–406.
16.
KwongC.K., YeY., ChenY. and ChoyK.L., A novel fuzzy group decision-making approach to prioritising engineering characteristics in QFD under uncertainties, International Journal of Production Research49(19) (2011), 5801–5820.
17.
LiuB., Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg, 2002.
LiuB. and LiuY., Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems10(4) (2002), 445–450.
20.
PiegatA., Cardinality approach to fuzzy number arithmetic, IEEE Transactions on Fuzzy Systems13(2) (2005), 204–215.
21.
PurbaJ.H., LuJ., ZhangG. and PedryczW., A fuzzy reliability assessment of basic events of fault trees through qualitative data processing, Fuzzy Sets and Systems243 (2014), 50–69.
22.
PuriJ. and YadavS.P., A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking in India, Expert Systems with Applications41(14) (2014), 6419–6432.
23.
SingerD., A fuzzy set approach to fault tree and reliability analysis, Fuzzy Sets and Systems34(2) (1990), 145–155.
24.
SriramdasV., ChaturvediS.K. and GargamaH., Fuzzy arithmetic based reliability allocation approach during early design and development, Expert Systems with Applications41(7) (2014), 3444–3449.
25.
TsaurS.H., ChangT. and YenC., The evaluation of airline service quality by fuzzy MCDM, Tourism Management23(2) (2002), 107–115.
26.
WangK., ZhouJ. and RalescuD.A., Arithmetic operations for LR mixed fuzzy random variables via mean chance measure with applications, Journal of Intelligent & Fuzzy Systems32(1) (2017), 451–466.
27.
WangZ., HuangH., LiY., PangY. and XiaoN., An approach to system reliability analysis with fuzzy random variables, Mechanism and Machine Theory52 (2012), 35–46.
28.
ZadehL.A., Fuzzy sets, Information and Control8(3) (1965), 338–353.
29.
ZadehL.A., The concept of a linguistic variable and its application to approximate reasoning, Information Sciences8(3) (1975), 199–249.
30.
ZadehL.A., A theory of approximate reasoning, In: HayesJ., MichieD. and MikulichL., eds., Mathematical Frontiers of the Social andolicy Sciences, Westview Press, Boulder, Cororado, 1979, pp. 69–129.
31.
ZhouJ., YangF. and WangK., Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming, Journal of Intelligent & Fuzzy Systems30(1) (2016), 71–87.
