Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets
Abstract
:1. Introduction
2. Preliminaries
- (i)
- is said to be a t-spherical fuzzy set (abbr. ) in .
- (ii)
- are respectively called the membership function, the abstinence function, and the non-membership function of .
- (iii)
- is called the degree of refusal of in .
- i)
- is said to be a single-valued neutrosophic set (abbr. ) in
- ii)
- are called the truth-membership function, the indeterminacy-membership function, and the falsity-membership function of respectively.
- (i)
- (ii)
- (iii)
- (iv)
- (i)
- (ii)
- (iii)
- (iv)
3. Monotonicity, Boundedness, Idempotency, and Commutativity of Operations
- (i)
- (ii)
- (iii)
- , for all being an
- (i)
- implies
- (ii)
- implies
- (iii)
- , for all
- (i)
- implies
- (ii)
- implies
- (iii)
- (i)
- implies
- (ii)
- implies
- (iii)
- , for all
- (i)
- implies
- (ii)
- implies
- (iii)
4. Generalized T-Spherical Fuzzy Subjectively Weighted Interaction Operators
- (i)
- (ii)
- (1)
- With being a positive real number, , and satisfying for all , it follows that:As a result,This further implies thatWe now obtainOn the other hand, it is clear that implies that .Thus
- (2)
- For each , as , so holds by taking in Lemma 2. By following the same procedure as part (1) of this lemma, with and taking place of and respectively:This further implies that
- (i)
- (ii)
- (i)
- The Generalized t-Spherical Fuzzy Weighted Geometric Interaction Function, is defined as for all , whereIn such a case,
- (a)
- is said to be the Generalized t-Spherical Fuzzy -Weighted Geometric Interaction on .
- (b)
- is said to be the weight vector of in .
- (ii)
- The Generalized t-Spherical Fuzzy Weighted Arithmetic Interaction Function, is defined as for all , whereIn such a case,
- (a)
- is said to be the Generalized t-Spherical Fuzzy -Weighted Arithmetic Interaction on .
- (b)
- is said to be the weight vector of in .
- (i)
- If for all , then both and
- (ii)
- If for all , then both and
- (iii)
- If for all , then both and
- (i)
- is said to be the score value of .
- (ii)
- is said to be the accuracy value of .
- (i)
- is said to be the score function of .
- (ii)
- is said to be the accuracy function of .
- (i)
- is said to be the Generalized t-Spherical score value (abbr. - score value) of .
- (ii)
- is said to be the Generalized t-Spherical accuracy value (abbr. - accuracy value) of .
- (i)
- (ii)
- but.
- (i)
- for all.
- (ii)
- Ifis such thatfor allthen.
- (i)
- for all.
- (ii)
- Ifis such thatfor all, then.
5. Algorithms for Multi-Attribute Multi-Perception Decision-Making Based on and
5.1. Prologue: The Derivation of from a Raw Dataset
5.2. Algorithm for Based Multi-Attribute Multi-Perception Decision Making
5.3. Algorithm for Based Multi-Attribute Multi-Perception Decision-Making
6. Application of the Proposed Algorithms to Air Pollution in China
6.1. An Overview of the Scenario—Air Pollution in China
6.1.1. The Two Major Smog Outbreaks in 2013
6.1.2. The Sources of Pollution
6.1.3. Actions Taken to Combat Pollution in China
6.2. The Multiple Perception of Comparing the Severity of Pollution
- (i)
- by pinpointing which city registered the highest daily PM2.5 concentrations recorded
- (ii)
- by pinpointing which city registered the lowest daily PM2.5 concentrations recorded
6.2.1. From the View of Environmental Management
6.2.2. From the View of Tourism Marketing
6.2.3. On Dealing with the Complete Absence of Data
6.3. Application of Our Proposed Method Using a Real Life Dataset
6.3.1. A Brief Description of the Dataset
6.3.2. Notations Used in the Dataset
- (a)
- Denote to be the six years of concern, where represent the years 2010, 2011, 2012, 2013, 2014 and 2015, respectively.
- (b)
- For each year :
- Denote to be the 5 cities in that year, where represents the city of Beijing, Chengdu, Guangzhou, Shanghai, and Shenyang, respectively.
- Denote to be all the hours within that year, starting with the 00:00 of 1st January, till 23:00 of 31st December. It is to be noted that , but as the year 2012 is a leap year.
- Denote all the PM2.5 readings within that year by a matrix whose elements are ordered sets of the following form:
- The rows of from the 1st to the 5th represents the readings from the city of , respectively.
- The columns of from the 1st to the -th represents the readings from the hour , respectively.
- For each : if the reading exists in the dataset for the th station in the city of during the hour of , then is taken to be that reading, otherwise, is assigned to be .
- (c)
- For each in :
- Denote the set .
- Denote and to be the maximum and minimum value of .
- Denote to be the population variance of PM2.5 concentration for city during the hour of the year , which we can never know.
- Denote to be the unbiased estimate of using elements of .
6.4. The Objectives
6.5. The Chosen Method of Obtaining the SVNn
6.5.1. The Formulas
- (1)
- For the based approach used by the environment management sector, take For the based approach used by the tourism marketing sector, take . (See Section 6.2.3.)
- (2)
- (3)
- (4)
- where .
6.5.2. Motive behind the Choices of Formulas
6.6. Results for Some Values of t
7. Compliance Tests to Investigate the Accuracy of our Algorithm
7.1. The Range of the Values of t to Be Investigated
7.2. Test 1: Test for Small Values [46]
7.2.1. The Test Inputs
7.2.2. The Criteria of Compliance
7.2.3. The Results of Our Algorithm
7.3. Test 2: Priority Test for Subjective Weights [46]
7.3.1. The Test Inputs
7.3.2. The Criteria of Compliance
7.3.3. The Results of Our Algorithm
7.4. Test 3: Priority Test for Objective Weights [46]
7.4.1. The Test Inputs
7.4.2. The Criteria of Compliance
7.4.3. The Results of Our Algorithm
7.5. Test 4: t–Dependence Test
7.5.1. t-Dependence Test of Type-A: For Environmental Management
The Test Inputs
The Criteria of Compliance
- and whenever
- and whenever
- whenever
- whenever
- whenever
- whenever
- All the 10 days of City S are “slightly polluted”: . Thus, the low values of should produce an “overview” or “generalizing” perception, where a general value or trend of the degree of pollution across all the days is of primary concern. This perception consequently results in the deduction of City S as the most polluted city, for low values of .
- City P and City Q contains one day that is “severely polluted”: . Thus, the high values of should produce a “pinpointing” perception, where the highest degree of pollution recorded on a particular day is of primary concern. This perception consequently results in the deduction of City P and City Q as the two most polluted cities, for high values of .Remark 13:The most polluted city, out of City P and City Q depends on the objective weight which was already dealt by Test 3 from Section 7.4.
- City contains three days that is “medially polluted”: . Thus, the medium values of should produce a perception that is between “overview” and “pinpointing” in nature. This perception consequently results in the selection of City R as the most polluted city, for medium values of .
The Results of Our Algorithm
7.5.2. -Dependence Test of Type-B: For Tourism Marketing
The Test Inputs
The Criteria of Compliance
- and whenever .
- and whenever .
- whenever .
- whenever .
- whenever .
- whenever .
- All the 10 days of City are “slightly polluted”: . Thus, the low values of should produce an “urgent” perception, where time is at stake and therefore the photographic team must quickly take photographs of a city. This perception consequently results in the deduction of City as the least polluted city for low values of .
- City and City contain one day that is “good”: . Thus, the high values of should produce a “quality” perception, where the photographic team must wait for the clearest possible sky to produce the best possible photographs to market China tourism. This perception consequently results in the deduction of City and City as the two least polluted cities, for high values of .Remark 14:The least polluted city, out of City and City depends on the objective weight which was already dealt by Test 3 from Section 7.4.
- City contains three days that is “okay”: . Thus, the medium values of should produce a perception that is between “urgent” and “quality” in nature. This perception consequently results in the deduction of City as the least polluted city for medium values of .
The Results of Our Algorithm
8. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Type of Entity in Datasets | Examples | |
---|---|---|
Qualitative | Nominal | Nationality (Malaysia, Singapore, China …) |
Ordinal | Customer Feedback (Poor, Fair, Good …) | |
Quantitative | Discrete | Number of stations (1, 2, 3, …) |
Continuous | Measurement (12.3 kg, 34.1 m, …) |
(Environment Management/Tourism Marketing) | ||
---|---|---|
1/20 (very overviewing/very urgent) | ||
¼ (rather overviewing/rather urgent) | ||
1 (balanced) | ||
4 (rather pin-pointing/rather patient) | ||
20 (very pin-pointing/very patient) |
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Quek, S.G.; Selvachandran, G.; Munir, M.; Mahmood, T.; Ullah, K.; Son, L.H.; Thong, P.H.; Kumar, R.; Priyadarshini, I. Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets. Mathematics 2019, 7, 780. https://doi.org/10.3390/math7090780
Quek SG, Selvachandran G, Munir M, Mahmood T, Ullah K, Son LH, Thong PH, Kumar R, Priyadarshini I. Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets. Mathematics. 2019; 7(9):780. https://doi.org/10.3390/math7090780
Chicago/Turabian StyleQuek, Shio Gai, Ganeshsree Selvachandran, Muhammad Munir, Tahir Mahmood, Kifayat Ullah, Le Hoang Son, Pham Huy Thong, Raghvendra Kumar, and Ishaani Priyadarshini. 2019. "Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets" Mathematics 7, no. 9: 780. https://doi.org/10.3390/math7090780
APA StyleQuek, S. G., Selvachandran, G., Munir, M., Mahmood, T., Ullah, K., Son, L. H., Thong, P. H., Kumar, R., & Priyadarshini, I. (2019). Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets. Mathematics, 7(9), 780. https://doi.org/10.3390/math7090780