Abstract
The primary aim of this paper is to develop one kind of easy and effective method to solve fuzzy cooperative games with coalition values expressed by triangular fuzzy numbers (TFNs). This method ensures that each player should receive a TFN-typed fuzzy pay-off from the grand coalition because each coalition value is expressed by a TFN. Using the concept of Alpha-cut sets, an arbitrary TFN’s Alpha-cut set can be shown as an interval. If the 1-cut sets and 0-cut sets of the TFN-typed coalition values are known, we can easily gain some important values, such as the means, the lower limits, and the upper limits of the TFN-typed payoffs via the proposed quadratic programming models and method. Furthermore, it is also easy for us to compute the lower and upper limits of Alpha-cut sets at any confidence levels of the TFN-typed payoffs for any TFN-typed cooperative game through solving the constructed quadratic programming models. Hereby the players’ TFN-typed payoffs for the TFN-typed cooperative game can be explicitly solved via the representation theorem for fuzzy sets. It is easy to prove that the proposed solutions of the fuzzy cooperative games with coalition values expressed by TFNs satisfy some useful and important properties, such as symmetry, additivity, and anonymity. Finally, the validity, applicability and advantages of the proposed method is proved and discussed through a numerical example.
Highlights
IntroductionCooperative game theory and methodology with fuzzy coalition values have been a research hotspot in many fields such as management, economics, and business as well as environment [1,2,3,4,5,6,7,8]
The three branches of fuzzy cooperative games are shown as follows: the cooperative games whose coalitions are fuzzy [9,10,11,12], the cooperative games whose coalition values are expressed with fuzzy numbers [1,2,3] and the cooperative games with both fuzzy coalitions and fuzzy coalition values [4,13]
We develop an easy and effective way to solve triangular fuzzy numbers (TFNs)-typed cooperative games based on the quadratic programming method and the square distance, which can bring down the uncertainty magnification and information distortion to a great extent
Summary
Cooperative game theory and methodology with fuzzy coalition values have been a research hotspot in many fields such as management, economics, and business as well as environment [1,2,3,4,5,6,7,8]. The uncertainty and fuzziness are usually expressed by fuzzy sets and/or fuzzy numbers or intuitionistic fuzzy numbers (IFNs) [14,15,16]. Li [2] developed an easy and effective method to solve TFN-typed matrix games. Li’s method was developed on the monotonicity of values in matrix games and Symmetry 2018, 10, 699; doi:10.3390/sym10120699 www.mdpi.com/journal/symmetry
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
