Generalized addition and scalar multiplication operations on Liu's generalized intuitionistic fuzzy numbers

Liu's generalized intuitionistic fuzzy number (L-GIFN) was proposed by author's previous work, in which, we know that if the value of extension index L equals zero, then L-GIFN is just Atanassov's intuitionistic fuzzy number (A-IFN). However, both of Atanassov's addition operation and scalar multiplication operation can only be used for the set of all L-GIFNs with the value of extension index zero or one, but not for other sets of all L-GIFNs with different values of extension index L. In this paper, a generalized addition operation and a scalar multiplication operation are proposed. Both of them can be used for all kinds of set of L-GIFNs with different values of extension index L. On the other hand, it is proved that Liu's partial order is order-preserving not only for Atanassov's addition operation and scalar multiplication operation but also for these two new operations, which can be used for constructing an ordered poset of Liu's generalized intuitionistic fuzzy numbers to handle intuitionistic fuzzy multi-criteria fuzzy decision making problems.