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The larger the accuracy degree is, the greater is. Based on the above score function and the accuracy function, a comparison method of IFNs was proposed by Xu [ 28 ], which is shown as follows. Definition 4 [ 28 ]: Let and be any two IFNs, and be the score functions of and , and and be the accuracy functions of and , respectively, then.

The MM, which was firstly proposed by Muirhead [ 42 ] in , provides a general aggregation function, and it is defined as follows:. Definition 5 [ 42 ]. From the definition 5 and the special cases of MM operator mentioned-above, we can know that the advantage of the MM operator is that it can capture the overall interrelationships among the multiple aggregated arguments and it is a generalization of most existing aggregation operators.

In this section, we will propose some intuitionistic fuzzy Muirhead mean operators for the intuitionistic fuzzy information, and discuss some properties of the new operators. Definition 6.


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Theorem 1. According to i and ii , we can know the aggregation result from 10 is still an IFN. Example 1. Let 0. Property 1 Idempotency. Property 2 Monotonicity. If for all i , then. Let , where and Since , we can get then and Further, and i. Property 3 Boundedness.

According to Properties 1 and 2, we have and So, we have. In the following, we will explore some special cases of IFMM operator with respect to the parameter vector. Lemma 1 [ 49 ]. Theorem 2. The proof this theorem is omitted, please refer to [ 44 ]. In actual decision making, the weights of attributes will directly influence the decision-making results.

However, IFMM operator cannot consider the attribute weights, so it is very important to take into account the weights of attributes for information aggregation. In this subsection, we will propose a weighted IFMM operator as follows. Definition 7. Theorem 3.

Property 4 Monotonicity. Property 5 Boundedness. According to Property 4, we have. According to Eq 20 , we have and So,. In the theory of aggregation operator, there exist two types, i. In this section, we will propose the dual MM operator for intuitionistic fuzzy numbers based on the IFMM operator as follows.

Definition 8. Theorem 5.

1. Introduction

Example 2. Property 6 Idempotency. Property 7 Monotonicity. Property 8 Boundedness. In the following, we will explore some special cases of IFDMM operator with respect to the parameter vector. Theorem 6. Definition 9. Theorem 7. Property 9 Monotonicity. Property 10 Boundedness. Then the goal is to rank the alternatives. In order to show the application of the proposed method in this paper, an illustrative example was cited from reference [ 50 ] which is about the investment selection decision. In the following, we use the proposed methods in Section 4 to select the best company for the investment corporation.

In order to illustrate the influence of the parameter vector P on decision making of this example, we set different parameter vector P to discuss the ranking results, and the results are shown in Tables 9 and As we can see from Tables 9 and 10 , the score functions using the different parameter vector P are different, but the ranking results are the same. The parameter vector P have greater control ability, the values of score function will become greater.

Peter Dybjer: Intuitionistic Type Theory (Lecture I)

However, for the IFDWMM operator, the result is just the opposite, the more interrelationships of attributes we consider, the greater value of score functions will become. The parameter vector P have greater control ability, the values of score function will become small.

To further prove the effectiveness and the prominent advantage of the developed methods in this paper, we solve the same illustrative example by using the three existing MAGDM methods including the intuitionistic fuzzy weighted average IFWA operator in [ 27 ], the weighted intuitionistic fuzzy Bonferroni mean WIFBM operator in [ 48 ], and the weighted intuitionistic fuzzy Maclaurin symmetric mean WIFMSM operator in [ 43 ].

The ranking results are shown in Table From Table 11 , we can find that the rank results are same by using five methods. This shows that the new methods in this paper are effective and feasible. In the following, we will give some comparisons of the three methods and our proposed methods with respect to some characteristic, which are listed in Table According to Table 12 and our further analysis, we can draw the following conclusions. In a word, according to the comparisons and analysis above, the IFWMM operator and the IFDWMM operator proposed in this paper have the advantages 1 they can consider interrelationships among any number of the attributes; 2 they are better and more convenient to deal with the intuitionistic fuzzy information than the existing other methods by a parameter vector P.

Aggregation operators have become a hot issue and an important tool in the decision making fields in recent years. However, they still have some limitations in practical applications. For example, some aggregation operators suppose the attributes are independent of each other.

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Some series of intuitionistic fuzzy interactive averaging aggregation operators

However, the MM operator has a prominent characteristic that it can consider the interaction relationships among any number of attributes by a parameter vector P. Then, the desirable properties were proved and some special cases were discussed. Finally, we used an illustrative example to show the feasibility and validity of the new methods by comparing with the other existing methods. In further research, it is necessary and significant to take the applications of these operators to solve the real decision making problems such as supply chain management, risk assessment, estimate of employment and environmental evaluation, etc.

A NOVEL TRIANGULAR INTERVAL TYPE-2 INTUITIONISTIC FUZZY SETS AND THEIR AGGREGATION OPERATORS

In addition, considering that MM operator has the superiority of compatibility, we can also study some new aggregation operators on the basis of Muirhead mean operator, for example, extend them to linguistic intuitionistic fuzzy sets LIFSs , Pythagorean fuzzy sets PFSs , generalized orthopair fuzzy sets [ 51 ] and trapezoidal intuitionistic fuzzy numbers [ 52 ] and so on.

The authors also would like to express appreciation to the anonymous reviewers and Editors for their very helpful comments that improved the paper. Conceptualization: PL DL. Formal analysis: PL DL. Funding acquisition: PL. Investigation: PL DL. Methodology: PL DL. Project administration: PL DL. Resources: PL DL. Supervision: PL. Validation: DL. Writing — original draft: PL. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field.

Abstract Muirhead mean MM is a well-known aggregation operator which can consider interrelationships among any number of arguments assigned by a variable vector. Introduction Multi-attribute decision making MADM and MAGDM are the important aspects of decision sciences, and they can give the ranking results for the finite alternatives or select best choice from them according to the attribute values of different alternatives [ 1 ]. Preliminaries In this part, we briefly introduce some basic concepts about IFNs and MM operator so that the readers can easily understand this study.

IFSs Definition 1 [ 2 , 3 ]. Definition 4 [ 28 ]: Let and be any two IFNs, and be the score functions of and , and and be the accuracy functions of and , respectively, then If , then ; If , then If , then ; If , then. MM operator The MM, which was firstly proposed by Muirhead [ 42 ] in , provides a general aggregation function, and it is defined as follows: Definition 5 [ 42 ].

If , the MM reduces to which is the geometric averaging operator. Firstly, we prove Eq 10 is kept. Then we will prove that 10 is an IFN. Let Then we need to prove the following two conditions. So, condition i is met. If for all i , then Proof. In the following, we will discuss three situations as follows. According to Properties 1 and 2, we have and So, we have In the following, we will explore some special cases of IFMM operator with respect to the parameter vector.

The intuitionistic fuzzy weighted MM operator In actual decision making, the weights of attributes will directly influence the decision-making results. Where, Proof. Theorem 4. The intuitionistic fuzzy dual MM operator In the theory of aggregation operator, there exist two types, i. Firstly, we prove the Eq 22 is kept. Then we will prove that 22 is an IFN. Let Then we need prove the following two conditions. According to i and ii , we can know the aggregation result from 22 is still an IFN. If for all i , then Property 8 Boundedness.

If for all i , then Property 10 Boundedness. Where, Theorem 8. In real decision, there exist two types of the attributes which are cost type and benefit types. It is necessary to convert them to the same type so as to give the right decision making. Step 5: Ranking all the alternatives. The bigger the IFN z i is, the better the alternative S i is. Download: PPT. An Illustrative Example In order to show the application of the proposed method in this paper, an illustrative example was cited from reference [ 50 ] which is about the investment selection decision.

Table 1. Decision matrix R 1 given by decision maker X 1. Table 2. Decision matrix R 2 given by decision maker X 2. Table 3.

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Decision matrix R 3 given by decision maker X 3. The decision making steps To get the best alternative s , the steps are shown in the following: Step 1: Normalizing the attribute values. All the attribute values are the same type, so they do not need to do the standardization.

Table 6. Table 7. The score function S z i of the comprehensive value Z i by two operators. Table 8.


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  • The ranking results of five alternatives by two operators. The influence of the parameter vector P on decision making result of this example In order to illustrate the influence of the parameter vector P on decision making of this example, we set different parameter vector P to discuss the ranking results, and the results are shown in Tables 9 and Table 9.

    Table Comparing with the other methods To further prove the effectiveness and the prominent advantage of the developed methods in this paper, we solve the same illustrative example by using the three existing MAGDM methods including the intuitionistic fuzzy weighted average IFWA operator in [ 27 ], the weighted intuitionistic fuzzy Bonferroni mean WIFBM operator in [ 48 ], and the weighted intuitionistic fuzzy Maclaurin symmetric mean WIFMSM operator in [ 43 ]. Home Articles List Article Information.

    Articles in Press. Current Issue. Journal Archive. Volume 16 Volume 15 Issue 6. Issue 7. Issue 5. Issue 4. Issue 3. Issue 2. Issue 1. Volume 14 Volume 13 Volume 12 Volume 11 Volume 10 Volume 9 Volume 8 Volume 7 Volume 6 Volume 5 Volume 4 Volume 3 Volume 2 Volume 1 Garg, H. Iranian Journal of Fuzzy Systems , 15 5 , Harish Garg; Sukhveer Singh. Iranian Journal of Fuzzy Systems , 15, 5, , Iranian Journal of Fuzzy Systems , ; 15 5 : Furthermore, based on these operators, an approach to multi-criteria decision-making, in which assessments are in the form of TIT2 intuitionistic fuzzy numbers has been developed.

    A practical example to illustrate the decision-making process has been presented and compared their results with the existing operator results. Keywords Type-2 fuzzy set ; Type-2 intuitionistic fuzzy sets ; Triangular interval type-2 intuitionistic fuzzy sets ; Multi criteria decision-making ; Aggregation operators References [1] K.

    Atanassov and G.