The Linguistic Interval-Valued Intuitionistic Fuzzy Aggregation Operators Based on Extended Hamacher T-Norm and S-Norm and Their Application
Abstract
:1. Introduction
2. Preliminaries
- (1)
- if, then ;
- (2)
- if, then ;
- (3)
- the negation ofis defined as .
- (1)
- if, then ;
- (2)
- if , , then .
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- .
- (1)
- the linguistic interval-valued intuitionistic fuzzy weighted average (LIVIFWA) operator is a mapping given by
- (2)
- the linguistic interval-valued intuitionistic fuzzy weighted geometric (LIVIFWG) operator is a mappinggiven by
Extended Hamacher T-Norm and S-Norm
- (1)
- ifsatisfies: (1),; (2),,, thenis called an extended aggregation function.
- (2)
- ifsatisfies, thenandare dual aggregation function with.
- (1)
- Commutativity:;
- (2)
- Associativity:;
- (3)
- Monotonicity:;
- (4)
- Neutral element:.
- (1)
- Commutativity:;
- (2)
- Associativity:;
- (3)
- Monotonicity:;
- (4)
- Neutral element:.
- (1)
- ;
- (2)
3. Linguistic Interval-Valued Intuitionistic Fuzzy Hamacher Aggregation Operators
3.1. Linguistic Interval-Valued Intuitionistic Fuzzy Hamacher Operational Laws
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- .
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- ;
- (5)
- ;
- (6)
- .
3.2. Linguistic Interval-Valued Intuitionistic Fuzzy Hamacher Aggregation Operators
3.2.1. LIVIFHWA Operator and LIVIFHWG Operator
3.2.2. Some Properties of Two LIVIFH Operators
3.3. Relationship between Operator and Parameter
3.3.1. Limiting Cases of LIVIFH Operators
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- ;
- (5)
- ;
- (6)
- ;
- (7)
- ;
- (8)
- .
- (1)
- According to Definition 17 and L’Hospital’s rule,
- (2)
- According to Theorem 6,
- (3)
- According to Definition 17,
- (4)
- According to Definition 17,
- (1)
- When, the LIVIFHWA operator degenerates into the Harmonic weighted average (LIVIFHarWA) operator:
- (2)
- When, the LIVIFHWG operator degenerates into the Harmonic weighted geometric (LIVIFHarWG) operator:
- (3)
- When, the LIVIFHWA operator degenerates into the Algebraic weighted average (LIVIFAWA) operator:
- (4)
- When, the LIVIFHWG operator degenerates into the Algebraic weighted geometric (LIVIFAWG) operator:
- (5)
- When, the LIVIFHWA operator degenerates into the Einstein weighted average (LIVIFEWA) operator:
- (6)
- When, the LIVIFHWG operator degenerates into the Einstein weighted geometric (LIVIFEWG) operator:
- (7)
- When, the LIVIFHWA operator degenerates into the Symmetric weighted average (LIVIFSWA) operator:
- (8)
- When, the LIVIFHWG operator degenerates into the Symmetric weighted geometric (LIVIFSWG) operator:
3.3.2. Monotonicity of Operators with Respect to Their Parameters
- (1)
- The LIVIFHWA operator decreases with increasing parameter;
- (2)
- The LIVIFHWG operator increases with increasing parameter.
4. Multiple Attributes Decision-Making Approach Based on the LIVIFH Operators and Its Application
4.1. Supplier Selection Problem
- (1)
- Supplier alternatives: Let be the set of supplier alternatives, where denotes the th supplier alternative, . The best supplier will be selected from the set of alternatives.
- (2)
- Evaluation attributes: is the set of attributes for the supplier alternatives, where is the th attribute, .
- (3)
- Group of experts: Let be the experts from different research areas, where is the th expert,. is the weight vector of experts, where is the weight of the expert , and satisfies and for . The decision maker aims to coordinate the insights of different experts and select the best one out of supplier alternatives measured on attributes.
- (4)
- Decision-making information matrixes: The experts are requested to express their preferences by using LIVIFNs generated by a discrete linguistic set , which described in Definition 1. For an alternative with respect to an attribute , an expert provides his/her assessments by using LIVIFNs . By collecting each attribute’s evaluation information from expert , the decision-making information matrix is obtained. is continuous virtual linguistic term set with respect to , and ,, , , .
4.2. Decision-Making Method for Solving Supplier Selection Problem
4.3. Case Study
4.3.1. Illustrative Example
4.3.2. Parameter Analysis
- (1)
- decreased with increasing parameter. This is consistent with the Theorem 10 (1);
- (2)
- When , the rank of alternatives was ; when , the rank of alternatives was , which means the rank of alternatives and switched. This is because the decreasing in score value of was relatively smaller compared with that of . Alternative was always the optimal alternative.
- (1)
- increased with increasing parameter . This is consistent with the Theorem 10 (2);
- (2)
- The rank of alternative was always .
- (1)
- The difference between score values of LIVIFHWA operator and LIVIFHarWA operator increased with increasing parameter, while the difference between score values of LIVIFHWA operator and the LIVIFSWA operator decreased.
- (2)
- The difference between score values of LIVIFHWG operator and LIVIFHarWG operator increased with increasing parameter, while the difference between score values of LIVIFHWG operator and LIVIFSWA operator decreased.To summarize, the relationships between the score values of these five operators were:
4.3.3. Comparative Analysis
- (1)
- The adjustable parameter could reflect the DM’s attitude.The parameter analysis has manifested that the LIVIFH aggregation operators were capable of reflecting the DM’s preferences by determining the appropriate values of the adjustable parameter .
- (2)
- The expansion of the domain for evaluation.The WIVLIFMSM operator could deal with MADM problems with LIVIF inputs, but not MAGDM problems. While the LIVAIFWA/LIVAIFWG operators are capable of dealing with MAGDM problems with LIVIF inputs, but they were merely degenerate cases of the proposed LIVIFHWA and LIVIFHWG operators when the adjustable parameter .
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Expert | Alternative | Attribute | |||
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