# Triangular Fuzzy Number-Typed Fuzzy Cooperative Games and Their Application to Rural E-Commerce Regional Cooperation and Profit Sharing

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## Abstract

**:**

## 1. Introduction

## 2. TFNs and Alpha-Cut Sets

#### 2.1. The Arithmetical Operations of the TFNs

#### 2.2. Alpha-Cut Sets and the Representation Theorem

## 3. Quadratic Programming Methods Based on Alpha-Cut Sets of TFN-Typed Coalition Values

#### 3.1. A Quadratic Programming Model with TFN-Typed Coalition Values and Its Optimal Solution

#### 3.2. A Quadratic Programming Model Considering Efficiency

- (1)
- ${\mathit{x}}_{}^{L1}={\mathit{x}}^{L}{}^{\ast}(\alpha )$
- (2)
- ${M}_{}^{L1}=\{j\in N/{x}_{i}^{L}{}^{\ast}(\alpha )<0\}$; ${M}_{}^{L0}=\varnothing $
- (3)
- ${x}_{i}^{L(k+1)}=\{\begin{array}{ll}{x}_{i}^{Lk}+\frac{{x}^{Lk}({M}_{}^{Lk})}{n-{m}_{k}^{L}}& (\forall j\notin {M}_{}^{Lk})\\ 0& (\forall j\in {M}_{}^{Lk})\end{array}$, where ${m}_{k}^{L}$ denotes the size of ${M}_{}^{Lk}$ and ${M}_{}^{L(k+1)}={M}_{}^{Lk}\cup \{j\in N/{x}_{i}^{L(k+1)}<0\}$
- (4)
- The algorithm processes stop once ${M}_{}^{Lk}={M}_{}^{L(k-1)}.$

## 4. A Numerical Example and Computational Result Analysis

#### 4.1. Computational Results Obtained by the Proposed Method

#### 4.2. Discussion and the Superiority of the Proposed Method

- (1)
- Rationality. In many real management situations, the prospective returns of cooperation are inevitably imprecise or not totally reliable owing to the limitations of human expertise, experience, and knowledge. The TFN-typed value can appropriately express the uncertainty and fuzziness. The models and methods proposed in this paper can effectively solve the TFN-typed cooperative games.
- (2)
- Superiority. In this paper, we develop an easy and effective way to solve TFN-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.
- (3)
- Computational complexity. The proposed model and method in this paper are simpler and more convenient than other methods in term of the computational complexity. Players’ TFN-typed payoffs can be obtained simultaneously through the proposed method in this paper. However, other methods can only be used to solve the imputations of players one by one.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Zhao, W.-J.; Liu, J.-C.
Triangular Fuzzy Number-Typed Fuzzy Cooperative Games and Their Application to Rural E-Commerce Regional Cooperation and Profit Sharing. *Symmetry* **2018**, *10*, 699.
https://doi.org/10.3390/sym10120699

**AMA Style**

Zhao W-J, Liu J-C.
Triangular Fuzzy Number-Typed Fuzzy Cooperative Games and Their Application to Rural E-Commerce Regional Cooperation and Profit Sharing. *Symmetry*. 2018; 10(12):699.
https://doi.org/10.3390/sym10120699

**Chicago/Turabian Style**

Zhao, Wen-Jian, and Jia-Cai Liu.
2018. "Triangular Fuzzy Number-Typed Fuzzy Cooperative Games and Their Application to Rural E-Commerce Regional Cooperation and Profit Sharing" *Symmetry* 10, no. 12: 699.
https://doi.org/10.3390/sym10120699