A New Multi-Attribute Decision-Making Method Based on m-Polar Fuzzy Soft Rough Sets
Abstract
:1. Introduction
2. Soft Rough -Polar Fuzzy Sets
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
- From Definition 5, we haveIt follows that
- It can be easily proved by Definition 5.
- By Definition 5,Hence,
- From Definition 5,Hence,
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
3. F Soft Rough Sets
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
- From Definition 8,
- It can be proved directly by Definition 8.
- By Definition 8,
- Using Definition 8,
- denotes the Fuel efficiency,
- denotes the Price,
- denotes the Technology.
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
- 1.
- 2.
- For all ,By similar arguments, we can compute
- By Definitions 9 and 10, we directly verified that Now, it is sufficient to show that andFor all , we have . By Definition 8, . Then, there exists , such that , that is, and . Thus, and . It follows that . By Definition 4, we have . Hence, .To prove , let an arbitrary , we have . Since , we obtain . Hence,
- 1.
- 2.
- for all
4. Applications to Decision-Making
4.1. Selection of a Hotel
- ‘’ represents the Location,
- ‘’ represents the Meal Options,
- ‘’ represents the Services.
- The “Location” of the hotel include close to main road, in the green surroundings, in the city center.
- The “Meal options” of the hotel include fast food, fast casual, casual dining.
- The “Services” of the hotel include Wi-Fi connectivity, fitness center, room service.
Algorithm 1: Selection of a suitable hotel |
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4.2. Selection of a Place
- ‘’ represents the Environment,
- ‘’ represents the Tour Cost.
- The “Environment” of the place includes built environment, natural environment, and social environment.
- The “Tour Cost” of the place may be low, medium, or high.
Algorithm 2: Selection of a suitable place |
|
4.3. Selection of a House
- ‘’ represents the Size,
- ‘’ represents the Location,
- ‘’ represents the Price.
- The “Size” of the house include small , large, and very large.
- The “Location” of the house include close to the main road, in the green surroundings, and in the city center.
- The “Price” of the house includes low, medium, and high.
Algorithm 3: Selection of a suitable house |
|
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Akram, M.; Ali, G.; Alshehri, N.O. A New Multi-Attribute Decision-Making Method Based on m-Polar Fuzzy Soft Rough Sets. Symmetry 2017, 9, 271. https://doi.org/10.3390/sym9110271
Akram M, Ali G, Alshehri NO. A New Multi-Attribute Decision-Making Method Based on m-Polar Fuzzy Soft Rough Sets. Symmetry. 2017; 9(11):271. https://doi.org/10.3390/sym9110271
Chicago/Turabian StyleAkram, Muhammad, Ghous Ali, and Noura Omair Alshehri. 2017. "A New Multi-Attribute Decision-Making Method Based on m-Polar Fuzzy Soft Rough Sets" Symmetry 9, no. 11: 271. https://doi.org/10.3390/sym9110271