Harmonic Aggregation Operator with Trapezoidal Picture Fuzzy Numbers and Its Application in a Multiple-Attribute Decision-Making Problem
Abstract
:1. Introduction
1.1. Research Background
1.2. Literature Review
1.3. Motivation
1.4. Framework of This Study
2. Preliminaries
2.1. Basics of Picture Fuzzy Number (PFN) and TrPFN
- .
- .
- .
- .
- .
- .
- .
- , .
- , .
2.2. HM and WHM
3. Different WHM Operators for TrPFN
4. Materials and Methods
Algorithm 1: |
Input: To the selection of best possible alternative. Output: Best alternative.
|
5. Numerical Example
- Population of locality : A telecom company expects more customers all the time, because more customers equals greater profit. So, the experts will wish to choose a locality with a larger population. Further, it does not happen that all the people of a locality will be the customers of that telecom company. They may be the customers of another company. So, more population of a locality is important to install a tower.
- Commercial environment : If the commercial environment of a locality is good then it is convenient to do business. Commercial environment means that there is school, college, hospital, shopping mall, etc., around the locality.
- Eco-friendly : The telecom company wants to install the tower without any harm to the environment. The company always takes care of the beauty of the environment so that the tower can be installed. If there are some large trees next to it, they take care of the trees. So, it is important that the locality will be eco-friendly to the company.
- Cost : Before choosing the place, the company should decide how much it will cost and they should fix their maximum budget, including management cost.
6. Result and Discussion
6.1. Decision Process
- At first we calculate the individual ratings of each alternatives by utilizing the TrPFWHM operator given by Equation (5), as follows:
- In this step, we calculate as follows:Similarly the other values are
- Next the score functions of each is calculated as follows:Similarly the other values are obtained. So , , , , , , , , , , .
- In this step, we arrange with respect to each decision makers using their score functions. The arrangement are as follows: , , , .
- In this step, the overall ratings of the decision makers utilizing TrPFHHM operator given by Equation (19) is calculated as follows:Similarly, the other aggregated values are,,and .
- In the last step, the score functions of the alternatives is calculated using Equation (10) , , , and . The arrangement of the alternative is .
6.2. Comparative Study
6.3. Discussion: Advantages and Disadvantages
- The main advantage of the proposed operators is that the presence of neutral membership grades.
- If there is a situation where the object (element) requires a neutrality degree, then Aydin, Kahraman, and Kabak’s [41] method fails.
- A real-life instance of mobile tower site selection is presented utilizing a TrPFWHM and TrPFHHM operator.
- If the values of membership grade, neutral membership grade, and non-membership grade are high in terms of the importance of alternatives, then these operators may not be applicable.
- All the data must be given. However, collection of membership values may not be easy.
6.4. Limitations
- The result of the proposed work are made by using only TrPFWHM and TrPFHHM operators.
- Data collection in real environment may not be easy always.
- If the membership values of the attributes are taken in different environment then this method failed.
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Shit, C.; Ghorai, G.; Xin, Q.; Gulzar, M. Harmonic Aggregation Operator with Trapezoidal Picture Fuzzy Numbers and Its Application in a Multiple-Attribute Decision-Making Problem. Symmetry 2022, 14, 135. https://doi.org/10.3390/sym14010135
Shit C, Ghorai G, Xin Q, Gulzar M. Harmonic Aggregation Operator with Trapezoidal Picture Fuzzy Numbers and Its Application in a Multiple-Attribute Decision-Making Problem. Symmetry. 2022; 14(1):135. https://doi.org/10.3390/sym14010135
Chicago/Turabian StyleShit, Chittaranjan, Ganesh Ghorai, Qin Xin, and Muhammad Gulzar. 2022. "Harmonic Aggregation Operator with Trapezoidal Picture Fuzzy Numbers and Its Application in a Multiple-Attribute Decision-Making Problem" Symmetry 14, no. 1: 135. https://doi.org/10.3390/sym14010135
APA StyleShit, C., Ghorai, G., Xin, Q., & Gulzar, M. (2022). Harmonic Aggregation Operator with Trapezoidal Picture Fuzzy Numbers and Its Application in a Multiple-Attribute Decision-Making Problem. Symmetry, 14(1), 135. https://doi.org/10.3390/sym14010135