Modern Graph Theory

Recently, bipolar fuzzy sets have been studied and applied a bit enthusiastically and a bit increasingly. In this paper we prove that bipolar fuzzy sets and [0,1](2)-sets (which have been deeply studied) are actually cryptomorphic mathematical notions. Since researches or modelings on real world problems often involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, or/and limit process, we put forward (or highlight) the notion of m-polar fuzzy set (actually, [0,1] (m)-set which can be seen as a generalization of bipolar fuzzy set, where m is an arbitrary ordinal number) and illustrate how many concepts have been defined based on bipolar fuzzy sets and many results which are related to these concepts can be generalized to the case of m-polar fuzzy sets. We also give examples to show how to apply m-polar fuzzy sets in real world problems.

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m-Polar fuzzy sets: an extension of bipolar fuzzy sets.

In this paper, we have investigated the multiple attribute decision-making problems with bipolar fuzzy information. Motivated by the Hamacher operations, we have proposed bipolar fuzzy Hamacher weighted average operator, bipolar fuzzy Hamacher ordered weighted average operator, bipolar fuzzy Hamacher hybrid average operator, bipolar fuzzy Hamacher weighted geometric operator, bipolar fuzzy Hamacher ordered weighted geometric operator, bipolar fuzzy Hamacher hybrid geometric operator. We investigate the characteristics and special cases of these operators. Then, we have utilized these operators to develop some approaches to solve the bipolar fuzzy multiple attribute decision-making problems. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Bipolar Fuzzy Hamacher Aggregation Operators in Multiple Attribute Decision Making

Guiwu Weia,b, Mao Lua, Fuad E. Alsaadib, Tasawar Hayatc,d and Ahmed Alsaedid aSchool of Business, Sichuan Normal University, Chengdu, P.R. China bCommunications Systems and Networks (CSN) Research Group, Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia cDepartment of Mathematics, Quaid-I-Azam University 45320, Islamabad, Pakistan dNonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Pythagorean 2-tuple linguistic aggregation operators in multiple attribute decision making

Recently, bipolar fuzzy graph is a growing research topic as it is the generalization of fuzzy graphs. In this paper, at first we define three new operations on bipolar fuzzy graphs, viz. direct product, semi strong product and strong product. Likewise, sufficient conditions for each one of them to be complete are given. Also, it is proved that any of the products of strong bipolar fuzzy graphs are strong bipolar fuzzy graphs.

Hai-Long Yang

Papers

Notes on “Bipolar fuzzy graphs”