# Interval-Valued Fuzzy Cooperative Games Based on the Least Square Excess and Its Application to the Profit Allocation of the Road Freight Coalition

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Intervals and the Square Distance between Intervals

#### 2.2. Some Concept and Definition of Fuzzy Cooperative Games

#### 2.3. The Concept of the Square Excess

## 3. The Interval-Valued Least Square Excess Solution of Interval-Valued Cooperative Games

## 4. The Solving Process of the Interval-Valued Least Square Excess Solution

## 5. Profit Allocation Strategy of the Road Freight Coalition Based on the Interval-Valued Least Square Excess Solution

#### 5.1. Cost Accounting of the Road Freight and the Utility Function

#### 5.2. Profit Allocation and Results Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Branzei, R.; Branzei, O.; Alparslan Gök, S.Z.; Tijs, S. Cooperative interval games: A survey. Cent. Eur. J. Oper. Res.
**2010**, 18, 397–411. [Google Scholar] [CrossRef] - Alparslan Gök, S.Z.; Branzei, R.; Tijs, S. The interval Shapley value: An axiomatization. Cent. Eur. J. Oper. Res.
**2010**, 18, 131–140. [Google Scholar] [CrossRef] - Mallozzi, L.; Scalzo, V.; Tijs, S. Fuzzy interval cooperative games. Fuzzy Sets Syst.
**2011**, 165, 98–105. [Google Scholar] [CrossRef] - Alparslan Gök, S.Z.; Palancı, O.; Olgun, M.O. Cooperative interval games: Mountain situations with interval data. J. Comput. Appl. Math.
**2014**, 259, 622–632. [Google Scholar] [CrossRef] - Han, W.B.; Sun, H.; Xu, G.J. A new approach of cooperative interval games: The interval core and Shapley value revisited. Oper. Res. Lett.
**2012**, 40, 462–468. [Google Scholar] [CrossRef] - Li, D.F. Fuzzy Multiobjective Many-Person Decision Makings and Games; National Defense Industry Press: Beijing, China, 2003. (In Chinese) [Google Scholar]
- Branzei, R.; Dimitrov, D.; Tijs, S. Shapley-like values for interval bankruptcy games. Econ. Bull.
**2003**, 3, 1–8. [Google Scholar] - Moore, R. Methods and Applications of Interval Analysis; SIAM Studies in Applied Mathematics; Society for Industrial Mathematics: Philadelphia, PA, USA, 1979. [Google Scholar]
- Kimms, A.; Drechsel, J. Cost sharing under uncertainty: An algorithmic approach to cooperative interval-type games. Bus. Res.
**2009**, 2, 206–213. [Google Scholar] [CrossRef] - Alparslan Gök, S.Z.; Branzei, O.; Branzei, R.; Tijs, S. Set-valued solution concepts using interval-type payoffs for interval games. J. Math. Econ.
**2011**, 47, 621–626. [Google Scholar] [CrossRef] - Branzei, R.; Alparslan Gök, S.Z.; Branzei, O. Cooperation games under interval uncertainty: On the convexity of the interval undominated cores. Cent. Eur. J. Oper. Res.
**2011**, 19, 523–532. [Google Scholar] [CrossRef] - Alparslan Gök, S.Z.; Miquel, S.; Tijs, S. Cooperation under interval uncertainty. Math. Methods Oper. Res.
**2009**, 69, 99–109. [Google Scholar] [CrossRef] - Li, D.F. Linear programming approach to solve interval-type matrix games. Omega Int. J. Manag. Sci.
**2011**, 39, 655–666. [Google Scholar] [CrossRef] - Li, D.F. Models and Methods of Interval-Type Cooperative Games in Economic Management; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Ruiz, L.M.; Valenciano, F.; Zarzuelo, J.M. The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector. Int. J. Game Theory
**1996**, 25, 113–134. [Google Scholar] [CrossRef] - Schmeidler, D. The Nucleolus of a Characteristic Function Game. SIAM J. Appl. Math.
**1969**, 17, 1163–1170. [Google Scholar] [CrossRef] - Li, D.F. A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers. Eur. J. Oper. Res.
**2012**, 223, 421–429. [Google Scholar] [CrossRef] - Chandra, S.; Aggarwal, A. On solving matrix games with pay-offs of triangular fuzzy numbers: Certain observations and generalizations. Eur. J. Oper. Res.
**2015**, 246, 575–581. [Google Scholar] [CrossRef] - Li, D.F. Linear Programming Models and Methods of Matrix Games with Payoffs of Triangular Fuzzy Numbers; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Verma, T.; Kumar, A.; Appadoo, S.S. Modified difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers. J. Intell. Fuzzy Syst.
**2015**, 29, 1607–1618. [Google Scholar] [CrossRef] - Garg, H. New Logarithmic operational laws and their aggregation operators for Pythagorean fuzzy set and their applications. Int. J. Intell. Syst.
**2019**, 34, 82–106. [Google Scholar] [CrossRef] - Garg, H. New exponential operational laws and their aggregation operators for interval-valued Pythagorean fuzzy multicriteria decision-making. Int. J. Intell. Syst.
**2018**, 33, 653–683. [Google Scholar] [CrossRef] - Garg, H. Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process. Int. J. Intell. Syst.
**2018**, 33, 1234–1263. [Google Scholar] [CrossRef] - Grzegorzewski, P. On Separability of Fuzzy Relations. Int. J. Fuzzy Log. Intell. Syst.
**2017**, 17, 137–144. [Google Scholar] [CrossRef] [Green Version] - Jang, L.C.; Lee, J.G.; Kim, H.M. On Jensen-Type and Hölder-Type Inequality for Interval-Valued Choquet Integrals. Int. J. Fuzzy Log. Intell. Syst.
**2018**, 18, 97–102. [Google Scholar] [CrossRef]

Variable | Symbol | Unit | Necessary Illustrations |
---|---|---|---|

Haul distance | $K$ | km | No |

Maintenance and repair fee of vehicles | ${F}_{m}$ | yuan | 8000 for 4.2 m; 14,000 for 9.6 m |

Annual mileage | ${K}_{a}$ | km | ${K}_{a}=26\times 12\times 10\times 80=249,600$ |

Insurance fee of vehicles | ${F}_{i}$ | yuan | 10,000 for 4.2 m; 17,000 for 9.6 m |

Purchase fee of vehicles | ${F}_{p}$ | yuan | 120,000 for 4.2 m; 260,000 for 9.6 m |

Depreciation fee of vehicles | ${F}_{d}$ | yuan | The depreciation period is 8 years |

Driver’s salary | ${F}_{s}$ | yuan | 8000 per month |

Diesel oil price | ${P}_{o}$ | yuan/L | 5.14 for 0# diesel oil |

Oil consumption fee | ${C}_{o}$ | L/100 km | 20 for 4.2 m; 30 for 9.6m |

Highway toll fee | ${T}_{h}$ | yuan/km | 1.54 for 4.2 m; 1.65 for 9.6 m |

Tire wear fee | ${F}_{t}$ | yuan | 2100 yuan per tyre. 6 tyres for 4.2 m; 8 tyres for 9.6 m |

Vehicle interior volume | ${V}_{v}$ | ${\mathrm{m}}^{3}$ | 15 for 4.2 m; 60 for 9.6 m |

Commodity volume | ${V}_{c}$ | ${\mathrm{m}}^{3}$ | No |

Anticipated load factor | ${\tilde{F}}_{e}$ | No | No |

Industrial average load factor | ${\tilde{F}}_{a}$ | No | No |

Transport cost of the single trip | $C$ | yuan | $C={F}_{m}+{F}_{i}+{F}_{d}+{F}_{s}+{C}_{o}+{T}_{h}+{F}_{t}$ |

Actual transport cost of the carried commodity | ${C}_{a}$ | yuan | ${C}_{a}=\frac{{V}_{c}}{{V}_{v}{F}_{e}}C$ |

Average profit rate of the industry of the road freight | ${R}_{p}$ | No | Approximately 6% |

Industrial average non-tax quotation of the carried commodity | ${P}_{q}$ | yuan | ${P}_{q}=\frac{{V}_{c}}{{V}_{v}{F}_{a}}\frac{C}{1-{R}_{p}}$ |

Profit of the carried commodity | $\upsilon $ | yuan | $\upsilon ={P}_{q}-{C}_{a}$ |

Logistics | ${\mathit{V}}_{\mathit{c}}$ | ${\tilde{\mathit{F}}}_{\mathit{e}}$ | Planned Vehicle Type |
---|---|---|---|

1 | 4 | [68%, 71%] | 4.2 m |

2 | 6 | [63%, 66%] | 4.2 m |

3 | 30 | [65%, 70%] | 9.6 m |

Coalition | ${\mathit{V}}_{\mathit{c}}$ | ${\tilde{\mathit{F}}}_{\mathit{e}}$ | Planned Vehicle |
---|---|---|---|

{12} | 10 | [76%, 80%] | 4.2 m |

{13} | 36 | [78%, 83%] | 9.6 m |

{23} | 38 | [85%, 90%] | 9.6 m |

{123} | 44 | [98%, 100%] | 9.6 m |

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**MDPI and ACS Style**

Zhao, W.-J.; Liu, J.-C.
Interval-Valued Fuzzy Cooperative Games Based on the Least Square Excess and Its Application to the Profit Allocation of the Road Freight Coalition. *Symmetry* **2018**, *10*, 709.
https://doi.org/10.3390/sym10120709

**AMA Style**

Zhao W-J, Liu J-C.
Interval-Valued Fuzzy Cooperative Games Based on the Least Square Excess and Its Application to the Profit Allocation of the Road Freight Coalition. *Symmetry*. 2018; 10(12):709.
https://doi.org/10.3390/sym10120709

**Chicago/Turabian Style**

Zhao, Wen-Jian, and Jia-Cai Liu.
2018. "Interval-Valued Fuzzy Cooperative Games Based on the Least Square Excess and Its Application to the Profit Allocation of the Road Freight Coalition" *Symmetry* 10, no. 12: 709.
https://doi.org/10.3390/sym10120709