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.