Research Article | Open Access

# Intuitionistic Linguistic Weighted Bonferroni Mean Operator and Its Application to Multiple Attribute Decision Making

**Academic Editor:**Ching-Ter Chang

#### Abstract

The intuitionistic linguistic variables are easier to describe the fuzzy information which widely exists in the real world, and Bonferroni mean can capture the interrelationship of the individual arguments. However, the traditional Bonferroni mean can only process the crisp number. In this paper, we will extend Bonferroni mean to the intuitionistic linguistic environment and propose a multiple attribute decision making method with intuitionistic linguistic information based on the extended Bonferroni mean which can consider the interrelationship of the attributes. Firstly, score function and accuracy function of intuitionistic linguistic numbers are introduced. Then, an intuitionistic linguistic Bonferroni mean (ILBM) operator and an intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator are developed, and some desirable characteristics of them are studied. At the same time, some special cases with respect to the parameters and in Bonferroni are analyzed. Based on the ILWBM operator, the approach to multiple attribute decision making with intuitionistic linguistic information is proposed. Finally, an illustrative example is given to verify the developed approach and to demonstrate its effectiveness.

#### 1. Introduction

Since the object things are fuzzy and uncertain, the attributes involved in the multiple attribute decision making (MADM) problems are not always expressed as crisp numbers, and some of them are more suitable to be denoted by fuzzy numbers, such as interval number, linguistic variable, and intuitionistic fuzzy number. Atanassov [1, 2] proposed the intuitionistic fuzzy set (IFS) characterized by a membership function and a nonmembership function, which is a generalization of the concept of fuzzy set proposed by Zadeh [3]. Later, Atanassov and Gargov [4] and Atanassov [5] further introduced the interval-valued intuitionistic fuzzy set (IVIFS), and Xu [6] and Wang [7] proposed the decision-making methods based on IVIFS. Zhang and Liu [8] defined the triangular intuitionistic fuzzy number, and they proposed the relevant decision making methods separately. Wang [9] gave the definition of intuitionistic trapezoidal fuzzy number and interval intuitionistic trapezoidal fuzzy number; then, some decision making methods based on the intuitionistic triangular fuzzy number had been proposed [10]. On the other hand, because linguistic variables are easy to express the qualitative information in evaluating the attributes, the decision making methods based on the linguistic variables have been a rapid development and a wide range of applications [11â€“14]. Furthermore, Wang and Li [15] proposed intuitionistic linguistic sets which combine intuitionistic fuzzy sets and linguistic variables, intuitionistic linguistic numbers, intuitionistic two-semantics, and the Hamming distance between two intuitionistic two-semantics and rank the alternatives by calculating the comprehensive membership degree to the ideal solution for each alternative. Obviously, intuitionistic linguistic sets are better to express the fuzzy information by integrating the advantages of intuitionistic fuzzy sets and linguistic variables, and they are receiving wide concerns. Liu [16] developed an intuitionistic linguistic generalized dependent ordered weighted average (ILGDOWA) operator and an intuitionistic linguistic generalized dependent hybrid weighted aggregation (ILGDHWA) operator. Liu and Wang [17] proposed an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and an intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator. On the basis of intuitionistic linguistic variables, Liu and Jin [18] further proposed the concept of intuitionistic uncertain linguistic variables (IULVs) and defined the operations on them, further developing some geometric average operators based on IULVs. Liu et al. [19] proposed the intuitionistic uncertain linguistic arithmetic Heronian mean (IULAHM) operator, intuitionistic uncertain linguistic weighted arithmetic Heronian mean (IULWAHM) operator, intuitionistic uncertain linguistic geometric Heronian mean (IULGHM) operator, and intuitionistic uncertain linguistic weighted geometric Heronian mean (IULWGHM) operator. Liu [20] proposed the concept of interval valued intuitionistic uncertain linguistic variables (IVIULVs) and defined the operations on them, further developing some geometric average operators based on IVIULVs. Obviously, now there are no researches on intuitionistic linguistic variables being applied to Bonferroni mean.

The information aggregation operators are an interesting and important research topic, which are receiving increasing concerns [16â€“28]. Bonferroni [22] originally proposed a Bonferroni mean (BM) operator, which has a desirable characteristic; that is, it can capture the expressed interrelationship of the individual arguments. Recently, Yager [23] further studied the BM and proposed an OWA variation of Bonferroni means, weighted Bonferroni aggregation, and Bonferroni choquet aggregation operator, and these generalizations enhance their modeling capability. Later, Beliakov et al. [24] proposed the generalized Bonferroni mean and discussed several interesting special cases with quite an intuitive interpretation for application. Xu and Yager [25] investigated the BM under intuitionistic fuzzy environments and developed an intuitionistic fuzzy BM (IFBM) and discussed its variety of special cases. Furthermore, they applied the weighted IFBM to multicriteria decision making and gave some numerical examples to illustrate related results. Beliakov and James [26] proposed the extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts. Wei et al. [27] proposed the uncertain linguistic weighted Bonferroni mean operator (ULWBM) and the uncertain linguistic weighted geometric Bonferroni mean operator (ULWGBM). Liu and Jin [28] proposed some extended Bonferroni mean operators for trapezoid fuzzy linguistic variables, including a trapezoid fuzzy linguistic weighted Bonferroni mean operator (TFLWBM) and a trapezoid fuzzy linguistic weighted Bonferroni OWA operator (TFLWBOWA). Obviously, Bonferroni mean had been extended to intuitionistic fuzzy sets, uncertain linguistic variables, and trapezoid fuzzy linguistic variables. However, now Bonferroni mean has not been extended to intuitionistic linguistic variables.

The intuitionistic linguistic variables are very suitable to be used for depicting uncertain or fuzzy information, and Bonferroni mean can capture the interrelationship of the individual arguments. Motivated by the idea of IFBM operator proposed by Xu and Yager [25], this paper is to propose some Bonferroni operators, such as an intuitionistic linguistic Bonferroni mean (ILBM) operator and an intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator, and some desirable properties of these operators are studied. At the same time, some special cases in these operators are analyzed. Based on the ILWBM operator, the approach to multiple attribute decision making with intuitionistic linguistic information is proposed.

The remainder of this paper is organized as follows. In Section 1, we give an introduction of the research background. Section 2 briefly reviews some basic concepts and operations related to the intuitionistic linguistic variables and BM. In Section 3, an intuitionistic linguistic BM (ILBM) and an intuitionistic linguistic weighted BM (ILWBM) are developed, and some special cases are discussed. Section 4 introduces a procedure for multiple attribute decision making based on ILWBM operator. Section 5 gives an example to illustrate the decision making steps based on the proposed method and to analyze the affect on the decision-making results of the different parameters. Section 6 ends this paper with some concluding remarks.

#### 2. Preliminaries

##### 2.1. The Linguistic Set and Its Extension

Suppose that is a finite and totally ordered discrete term set, where is the odd value. In real situation, is equal to 3, 5, 7, 9, and so forth. For example, when , a set could be given as follows: = {extremely poor, very poor, poor, slightly poor, fair, slightly good, good, very good, extremely good}.

Usually, for any linguistic set , it is required that and must satisfy the following additional characteristics [11, 12].(1)The set is ordered: , if and only if .(2)There is the negation operator: .(3)Maximum operator: , if .(4)Minimum operator: , if .

Furthermore, in order to preserve all the given information, Herrera et al. [11] proposed that the discrete linguistic label is extended to a continuous linguistic label which satisfied the above characteristics.

For any linguistic variables , the operations are defined as follows [13, 14]:

##### 2.2. The Intuitionistic Linguistic Set (ILS)

*Definition 1 (see [15]). *An ILS in is defined as
where , , and , with the condition , for all . The numbers and represent, respectively, the membership degree and nonmembership degree of the element to linguistic index .

For each ILS in , if , for all , then is called a hesitancy degree of to linguistic index . It is obvious that , for all .

*Definition 2 (see [15]). *Let be ILS; the ternary group is called an intuitionistic linguistic number, and can also be viewed as a collection of the intuitionistic linguistic number (ILN). So, it can also be expressed as . In addition, represents the hesitancy degree, and it can also be called the intuitionistic linguistic fuzzy degree. For convenience, an ILN is denoted by , where , .

Let and be two ILNs and ; then, the operations of ILNs are defined as follows [15]:

*Definition 3 (see [16]). *Let be an ILN; a score function of an ILN can be represented as follows:

*Definition 4 (see [16]). *Let be an ILN; an accuracy function of an ILN can be represented as follows:

*Definition 5 (see [16]). *If and are any two ILNs, then(1)if , then ;(2)if , then if , then ; if , then .

##### 2.3. Bonferroni Mean (BM)

The BM was originally proposed by Bonferroni in [22], which was defined as follows.

*Definition 6 (see [22]). *Let , and let be a collection of nonnegative numbers. If
then is called the Bonferroni mean (BM).

Obviously, the BM has the following properties.(1).(2), if , for all .(3); that is, is monotonic, if , for all .(4).If and , then (9) reduces to the following: If , (9) reduces to the following: which is a generalized mean operator; particularly the following cases hold.(1)If and , then (11) reduces to the usual average (2)If and , then (11) reduces to the geometric mean operator

#### 3. The Intuitionistic Linguistic Weighted Bonferroni Mean Operators

The Bonferroni mean (BM) has a significant advantage of capturing the interrelationship of the individual arguments; however, the traditional BM can only process the crisp numbers and cannot deal with intuitionistic linguistic. In this section, we will extend BM to deal with intuitionistic linguistic information and develop an intuitionistic linguistic Bonferroni mean (ILBM) operator and an intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator. Further, we will discuss some desirable characteristics of them and some special cases with respect to the parameters and in Bonferroni.

*Definition 7. *Let be a collection of ILNs, and ; if
where is the set of all intuitionistic linguistic numbers and for any , then is called the intuitionistic linguistic Bonferroni mean (ILB).

According to the operations of ILNs, we can get the following result.

Theorem 8. *Let , and let be a collection of ILNs. Then, the aggregated result by formula (14) is also an ILN, and
*

We use mathematical induction to prove this theorem shown as follows.

*Proof. *(1) Firstly, we need to prove that
By the operations of ILNs defined in (3)â€“(6), we have

(a) When , by formulas (18) and (3), we can get
that is, when , formula (16) is right.

(b) Suppose that when , formula (16) is right; that is,
then, when , we have

Firstly, we prove that

We also use the mathematical induction on as follows.

(i) When , we have

(ii) Suppose that , then formula (22) is right; that is,

Then, when , we have
that is, for , formula (22) is also right.

(iii) So, for all , formula (22) is right.

Similarly, we can prove that
So, by formulas (20), (22), and (26), formula (21) can be transformed as
So, when , formula (16) is also right.

Thus, formula (16) is right, for all .

(2) Then, we can prove that formula (15) is right.

By formula (16), we can get

*Example 9. *Suppose that there are three intuitionistic linguistic numbers , , and , and suppose that and ; then, we can calculate the shown as follows:
where
Replace the data of , , and ; we can get

In the following, we will discuss some special cases of the operator shown as follows.

(1) When , formula (15) reduces to an intuitionistic linguistic generalized mean operator; it follows that

(2) If and , then (15) reduces to an intuitionistic linguistic average operator

(3) If and , then (15) reduces to an intuitionistic linguistic geometric mean operator

The traditional BM has the properties of commutativity, idempotency, monotonicity, and boundedness; in the following, we will prove that ILB also has these properties.

Theorem 10 (commutativity). *Let be any permutation of ; then,
*

*Proof. *Let
Since is any permutation of , we have
thus,

Theorem 11 (idempotency). *Let , ; then .*

*Proof. *Since , for all , we have

Theorem 12 (monotonicity). *Let and be two collections of IFNs. If , for all , then
*

*Proof. *Since for all , we have

So,
that is, .

Theorem 13 (boundedness). *The operator lies between the and operators:
*

*Proof. *Let and .

Since , then
That is,
that is, .

In operator, we only consider the input parameters and their interrelationships and do not consider the importance of each input parameter itself. However, in many practical situations, the weight of input data is also an important parameter. So, we can define an intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator.

*Definition 14. *Let be a collection of ILNs, and , if
where is the set of all intuitionistic linguistic numbers and is the weight vector of , , . ; is a balance parameter. Then, ILWB is called the intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator.

Theorem 15. *ILB operator is a special case of the ILWB operator.*

*Proof. *If , then :

Theorem 16. *Let and be a collection of ILNs, and is the weight vector of , , . Then, the aggregated result by formula (46) is also an ILN, and
**
Similar to Theorem 8, it can be proved by using mathematical induction on .*

*Example 17. *Suppose that there are three intuitionistic linguistic numbers , , and , and the weight vector (suppose and ); then, we can calculate the shown as follows:
where

Replace the data of , , and ; we can get
then,

It is easy to prove that the operator hasthe properties of commutativity and monotonicity, but it has not the property of idempotency.

#### 4. An Approach to Multiple Attribute Decision Making Based on the Intuitionistic Linguistic Numbers

In the previous section, we extended BM to intuitionistic linguistic information and proposed ILBM and ILWBM operators. In this part, we will apply these extended BM operators to solve the multiple attribute decision making problems with intuitionistic linguistic information and give the detail decision making steps. The advantage of the proposed method is that it can consider the interrelationship of the attributes.

Consider a multiple attribute decision making with intuitionistic linguistic information: let be a discrete set of alternatives, and let be the set of attributes; is the weighting vector of the attribute , where , . Suppose that is the decision matrix, where takes the form of the intuitionistic linguistic number, and , , , . Then, the ranking of alternatives is required.

In the following, we apply operator to multiple attribute decision making based on intuitionistic linguistic information.

The methods involve the following steps.

*Step 1 (normalization). *Generally, there are two attribute types in multiple attribute decision making: they are benefit type (the bigger the performance values the better) and cost type (the smaller the performance values the better); we need normalization in order to transform the performance values of the cost type into the performance values of the benefit type. Then, will be transformed into the matrix , where (1), for benefit type of ;(2)