Cost-Sharing of Ecological Construction Based on Trapezoidal Intuitionistic Fuzzy Cooperative Games
Abstract
:1. Introduction
2. Definition and Arithmetical Operations for Trapezoidal Intuitionistic Fuzzy Numbers
2.1. Definition of Trapezoidal Intuitionistic Fuzzy Numbers
2.2. Arithmetical Operationsfor Positive Trapezoidal Intuitionistic Fuzzy Numbers
3. Quadratic Programming Model for Solving Cooperative Games with Trapezoidal Intuitionistic Fuzzy Numbers
3.1. Cooperative Games with Coalition Values Expressed by Trapezoidal Intuitionistic Fuzzy Numbers
3.2. The Solution Concept of Trapezoidal Intuitionistic Fuzzy Cooperative Games
3.3. A Quadratic Programming Model for Solving the Numerical Value Parts Based on the Least Square Method
3.4. An Improved Model Considering Efficiency and Its Optimal Solution
3.5. A Quadratic Programming Model for Solving the Membership Degrees and Nonmembership Degrees of the Optimal Solution
4. An Example Demonstrating the Cost-Sharing of Ecological Construction in Fujian Province, China
5. Discussion
- (1)
- Modeling. In this paper, we constructed a quadratic programming model to solve the cooperative game with coalition values expressed by trapezoidal intuitionistic fuzzy numbers. It is an expansion of the least square prenucleolus solution concept [31]. The quadratic programming models and methods proposed in this paper always assure that the solutions are positive if all of the coalitions’ values are positive trapezoidal intuitionistic fuzzy numbers.
- (2)
- Calculation complexity. According to the method proposed in this paper, we can easily and quickly obtain all players’ optimal trapezoidal intuitionistic fuzzy payoffs using Equations (30)–(33).
- (3)
- Efficiency. The quadratic programming model proposed in this paper takes into account efficiency, so the allocation scheme is fairly satisfactory for all players. That is to say, the cooperative surplus is distributed thoroughly among the players. In Section 4, it can easily be seen that , (i.e., 30 + 80 + 290 = 400, 46.7 + 126.7 + 476.7 = 650, 58.3 + 183.3 + 658.3 = 900, 96.7 + 246.7 + 756.7 = 1100) which implies that the cost allocation scheme satisfies the efficiency as expected. The optimal trapezoidal intuitionistic fuzzy payoff vector , which is obtained through Equations (30)–(33), is said to be efficient or a preimputation.
- (4)
- Advantages. There exists some information distortion when doing subtraction of trapezoidal intuitionistic fuzzy numbers. In this paper, we construct the optimal mathematical model based on the square of the distance in the numerical value between two trapezoidal intuitionistic fuzzy numbers, which can effectively avoid the distortion of information and enlargement of fuzziness and uncertainty brought about by subtraction of trapezoidal intuitionistic fuzzy numbers.
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Liu, J.; Zhao, W. Cost-Sharing of Ecological Construction Based on Trapezoidal Intuitionistic Fuzzy Cooperative Games. Int. J. Environ. Res. Public Health 2016, 13, 1102. https://doi.org/10.3390/ijerph13111102
Liu J, Zhao W. Cost-Sharing of Ecological Construction Based on Trapezoidal Intuitionistic Fuzzy Cooperative Games. International Journal of Environmental Research and Public Health. 2016; 13(11):1102. https://doi.org/10.3390/ijerph13111102
Chicago/Turabian StyleLiu, Jiacai, and Wenjian Zhao. 2016. "Cost-Sharing of Ecological Construction Based on Trapezoidal Intuitionistic Fuzzy Cooperative Games" International Journal of Environmental Research and Public Health 13, no. 11: 1102. https://doi.org/10.3390/ijerph13111102