A note on "using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly"

The aim of this note is to point out and correct some errors in the definition of the four arithmetic operations for triangle intuitionistic fuzzy numbers introduced by Shu et al. [Shu MH, Cheng CH, Chang JR. Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly. Microelectron Reliab 2006;46:2139–48]. 2008 Elsevier Ltd. All rights reserved. Based on the concept of the intuitionistic fuzzy sets introduced by Atanassov [1,2] and Shu et al. [3] defined four arithmetic operations of triangle intuitionistic fuzzy numbers for developing an algorithm of the intuitionistic fuzzy fault-tree analysis to calculate fault interval of system components and finding the most critical system component for the managerial decision-making. It is found that there are the following errors and misprints in the definition of the four arithmetic operations for triangle intuitionistic fuzzy numbers given by Shu et al. [3]. (1) The degrees of non-membership of the results of the four arithmetic operations of A and B are taken as min{tA, tB} instead of max{tA, tB}, where A and B are two triangle intuitionistic fuzzy numbers given by Shu et al. [3], denoted by A 1⁄4 h1⁄2ða1; b1; c1Þ;lA ; 1⁄2ða1; b1; c1Þ; tA i ð1Þ and B 1⁄4 h1⁄2ða2; b2; c2Þ;lB ; 1⁄2ða2; b2; c2Þ; tB i ð2Þ respectively. (2) The result of A B is not a difference. (3) The result of A B does not take into consideration the situations in which A and B are negative or positive. (4) The result of A/B does not take into consideration the situations in which A and B are negative or positive, especially B = 0 which means lB(x) = tB(x) = 0 for any x 61⁄4 0. In this note the correct definition of the four arithmetic operations for triangle intuitionistic fuzzy numbers are given as follows. For two triangle intuitionistic fuzzy numbers A 1⁄4 h1⁄2ða1; b1; c1Þ; lA ; 1⁄2ða1; b1; c1Þ; tA i and B 1⁄4 h1⁄2ða2; b2; c2Þ;lB ; 1⁄2ða2; b2; c2Þ; tB i with lA 61⁄4 lB and tA 61⁄4 tB, define Aþ B 1⁄4 h1⁄2ða1 þ a2; b1 þ b2; c1 þ c2Þ;minflA;lBg ; 1⁄2ða1 þ a2; b1 þ b2; c1 þ c2Þ;maxftA; tBg i ð3Þ and A B 1⁄4 h1⁄2ða1 c2; b1 b2; c1 a2Þ;minflA;lBg ; 1⁄2ða1 c2; b1 b2; c1 a2Þ;maxftA; tBg i ð4Þ Moreover, for A > 0 and B > 0, define A B 1⁄4 h1⁄2ða1a2; b1b2; c1c2Þ;minflA;lBg ; 1⁄2ða1a2; b1b2; c1c2Þ; maxftA; tBg i ð5Þ and A=B 1⁄4 h1⁄2ða1=c2; b1=b2; c1=a2Þ;minflA;lBg ; 1⁄2ða1=c2; b1=b2; c1=a2Þ; maxftA; tBg i ð6Þ When a1 1⁄4 a1, c1 1⁄4 c1 and a2 1⁄4 a2, c2 1⁄4 c2, the above four arithmetic operations (i.e., Eqs. (3)–(6)) will become more easy.