The main objective of the chapter is to introduce a series of picture fuzzy weighted averaging and geometric aggregation operators by using t-norm and t-conorm. In this chapter, they discussed generalized form of weighted averaging and geometric aggregation operator for picture fuzzy information. Further, the proposed aggregation operators of picture fuzzy number are applied to multi-attribute group decision making problems. To implement the proposed models, they provide some numerical applications of group decision making problems. Also compared with previous model, they conclude that the proposed technique is more effective and reliable.
Top1. Introduction
Multiple-criteria decision making (MCDM) means that the best alternative is selected from the limited alternatives set according to the multiple criteria, which can be regarded as cognitive processing. MCDM is a vital division of the decision-making theory which has been widely used in human activities. Because the real decision-making problems were frequently produced from the complicated environment, the evaluation information is usually fuzzy. In general, the fuzzy information takes two forms: one is quantitative and other one is qualitative. The quantitative fuzzy information can be expressed by fuzzy set (FS) (Zadeh, 1965), intuitionistic fuzzy set (IFS) (Atanassov, 1986), Pythagorean fuzzy set (PyFS) [45] picture fuzzy set (PFS) (Beliakov, Bustince, Gosawmi, Mukherjee, & Pal, 2011), spherical fuzzy set (SFS) (Ashraf & Abdullah, 2018) and so on. FS theory proposed by Zadeh (Zadeh, 1965) has been used to describe fuzzy quantitative information which contains only a membership degree. Based on this, Atanassov (1986) presented IFS, which consists of membership and non-membership degrees that fulfills the constraint form that the sum of their degrees is less or equal to 1. For more study about IFS, we refer (Ejegwa, Tyoakaa, & Ayenge, 2016; Ejegwa & Modom, 2015; Ejegwa, 2015; Ejegwa, Akubo, & Joshua, 2014; Ejegwa, Onoja, & Emmanuel, 2014). However, some-times the two degrees don’t meet the constraint, but the sum of squares of the two degrees is less or equal to 1. Yager and Abbasov (2014) introduced the PyFS in which the square sum of membership and non-membership degrees is equal to or less than 1.
Atanassov’s structure of IFSs and Yager’s structure of PyFSs discourses only satisfaction and dissatisfaction degree of elements in a set which is quite insufficient as human nature has some sort of abstain and refusal issues too. The notion of the picture fuzzy set was proposed Cuong (2014), basically (PFS) is the generalization of the intuitionistic fuzzy set (IFS). Outstanding qualities of (PFS) is that it allows to each element the degree of membership, neutral membership and non-membership. Due to its capacity of considerate hesitancy in human decision processes, PFSs have been generally enforced to the field of decision making. PFSs can be applied to the direction that involving human assessment like as, “yes”, “abstain”, “no” and “refusal”. For example, voting is a good example in this position, the voters may be divided in four groups, “vote for”, “abstain”, vote against”, “refusal of voting”. Picture fuzzy obtain thought of many authors in this area. In 2015, Singh gave an interrelationship coefficient for the (PFSs). Son (2016) gave a generalized picture distance measure and applied it to deal with clustering problem under the picture fuzzy surroundings. Wei (2017) conferred a decision-making art depend on the picture fuzzy weighted cross-entropy and apply it to rank the alternatives. In 2017, wang, Zhou, Tu, and Tao used the picture fuzzy information to check the multi-criteria decision-making problems, also introduced some picture fuzzy geometric operators and discuss their elemental properties. Phong and Cuong (2014) also studied multi-attribute group decision making problem using picture linguistic numbers. Ashraf, Mahmood, Abdullaj, and Khan (2018) propose some aggregation operators for picture fuzzy information.