Advances in Fuzzy Systems

Volume 2018, Article ID 5789192, 12 pages

https://doi.org/10.1155/2018/5789192

## A Multicriteria Decision-Making Approach Based on Fuzzy AHP with Intuitionistic 2-Tuple Linguistic Sets

Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan

Correspondence should be addressed to Tabasam Rashid; moc.liamg@dihsar.masabat

Received 9 May 2018; Revised 26 June 2018; Accepted 9 July 2018; Published 1 August 2018

Academic Editor: Ferdinando Di Martino

Copyright © 2018 Shahzad Faizi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the modern literature related to linguistic decision-making, the 2-tuple linguistic representation model and its useful applications in various fields have been extensively studied and used during the last decade. Recently, some useful multicriteria decision-making (MCDM) methods have been introduced based on fuzzy analytic hierarchy process (AHP) for 2-tuple linguistic representation model. By keeping in mind the importance of this linguistic model, in this paper, we introduce a fuzzy AHP methodology for intuitionistic 2-tuple linguistic sets (I2TLSs) which is a useful extension of the 2-tuple linguistic representation model. This study is comprised of four stages. In the first stage, we define some operational laws for I2TL elements (I2TLEs) and prove some related important properties. In the second stage, intuitionistic 2-tuple linguistic preference relation (I2TLPR) and multiplicative I2TLPR are defined using I2TLSs. In the 3rd stage, a transformation mechanism is introduced which can transform an I2TLPR to a corresponding intuitionistic preference relation (IPR) and vice versa. In the fourth stage, an approach is proposed for checking the consistency of an I2TLPR and presented a method to repair the inconsistent one by using the proposed transformation mechanism. Finally, a numerical example is given and comparative analysis is carried out with the TOPSIS method to verify the validity of the proposed method.

#### 1. Introduction

Herrera and Martínez [1, 2] proposed the 2-tuple fuzzy linguistic representation model which can handle linguistic and numerical information in decision-making effectively without loss and distortion of information which formerly occur during the processing of linguistic information. This useful model is basically developed on the basis of symbolic translation of the linguistic variables and has been extensively used in various MCDM problems [3–6] in recent years. The basic shortcoming of this model is that it can only ensure the accuracy in dealing with uniformly distributed linguistic term sets (LTSs). To make up for the above-mentioned shortcoming, Wang and Hao [5] introduced the proportional 2-tuple fuzzy linguistic representation model, which can ensure the accuracy in dealing with the LTSs that are not uniformly distributed. The studies on MCDM problems in the context of 2-tuple fuzzy linguistic models are growing. For example, Beg and Rashid introduced two important extensions of 2-tuple linguistic representation model, namely, the hesitant 2-tuple linguistic information model [7] and the I2TL information model [8], which are very effective in dealing with fuzziness and uncertainty as compared to the ordinary 2-tuple linguistic arguments. Furthermore, Liu and Chen [9] introduced the extended T-norm and T-conorm with the I2TL information and developed a MAGDM method based on the proposed I2TL generalized aggregation operator.

AHP was originally developed by Satty in [10] which is the most powerful technique to solve complex MCDM problems and help the decision-makers (DMs) to set preferences and make the best decision. In addition, to reduce the biasness of the DMs in the decision-making process, the AHP incorporates a useful technique for checking the consistency of the DM’s evaluations. Recently, extensive studies have been conducted on AHP in fuzzy context, such as, AHP based on 2-tuple linguistic representation model for supplier segmentation by aggregating quantitative and qualitative criteria [11], a hybrid approach based on 2-tuple fuzzy linguistic method and fuzzy AHP for evaluation in-flight service quality [12], and AHP method based on hesitant fuzzy sets for analyzing the factors affecting the performance of different branches of a cargo company [13]. To collect priorities of the DMs in AHP, different kinds of preference relations are used in the literature, but numerical preference relations [14–16] and linguistic preference relations (LPRs) [17, 18] are the two basic preference relations that are often used in MCDM problems. If DMs cannot guess their preferences of one alternative over the other with actual numerical values [19] and are interested in providing their preferences in linguistic values, then they prefer LPRs which are actually a kind of numerical preference relations. The LPRs have been studied as another important tool to collect preferences and have vast applications in MCDM [20–22].

To identify the inconsistency of preference relations, there is a need of a consistency check to avoid the inconsistent solutions during a decision-making process. Saaty [23] developed an idea of consistency ratio () to measure the inconsistency level of numerical preference relations. He observed that the preference relation is of acceptable consistency if ; otherwise, it is inconsistent and it is necessary to return it to the DMs again for the revision of their preferences until acceptable. Extensive studies have been done to measure the degree of inconsistency of numerical preference relations [24–26]. Similar to numerical preference relations, the consistency measure is also a difficult task while using LPRs in various MCDM problems [27]. In order to measure the consistency degree of preference relations, traditional definitions, such as the additive transitivity, the max–min transitivity, and the three-way transitivity, are used. But these definitions are incapable of measuring the consistency degree of LPRs. To make up for the above-mentioned shortcoming, Dong et al. introduced a more flexible method to measure the consistency degree of LPRs in [27]. Xu and Liao [28] proposed a method to check the consistency of an IPR and introduced an interesting procedure to repair the inconsistent IPR without the participation of the DM. Zhu and Xu [29] developed some consistency measures for hesitant fuzzy LPRs and further constructed two optimization methods to improve the consistency of an inconsistent hesitant fuzzy LPR. Zhang and Wu [30] discussed the multiplicative consistency of hesitant fuzzy LPRs and developed a consistency-improving process to adjust hesitant fuzzy LPR with unacceptably multiplicative consistency into an acceptably multiplicative one. Furthermore, Gong et al. [31] introduced the additive consistent conditions of the IPR according to that of intuitionistic fuzzy number preference relation. Wang [32] proved the additive consistency defined in an indirect manner in [31] and proved that the consistency transformation equations' matrix may not always be an IPR.

AHP is a widely used method for solving multicriteria problems in practical situations. The combination of AHP with fuzzy set and 2-tuple representation model can deal with human judgments under fuzzy environment and has no information loss. One of the main strengths of AHP is its ability to deal with subjective opinions of experts and derive a quantitative priority vector that describes the relative importance of each alternative, which makes AHP appealing to a wide variety of MCDM problems [33]. Some authors contend that the applicability of AHP can be attributed to its simplicity, ease of use, and flexibility as well as the possibility of integrating AHP with other techniques such as fuzzy logic and linear programming [34]. Furthermore, the role of AHP is to determine the weights of the criteria in both dimensions. This led to a consistent priority ranking with experts having to make only pairwise comparisons in a decision problem containing number of alternatives. The I2TL information model is a more powerful tool in dealing with vagueness and uncertainty that can assign to each element a membership degree as well as a nonmembership degree in the form of 2-tuple linguistic information. Therefore, the aim of this study is to apply AHP method to solve MCDM problems, where the I2TL information should be collected by a tool. First, this paper has developed some operational laws for I2TLEs and proved some of the important properties related to these operational laws. The concepts of I2TLPRs and multiplicative I2TLPR are then developed to collect the preferences of the DMs as an extension of LPRs along with a transformation function that can transform an I2TLPR to a corresponding intuitionistic preference relation (IPR). Finally, an approach is proposed for checking the consistency of an I2TLPR and presented a method to repair the inconsistent one by using the proposed transformation mechanism.

The rest of the paper is organized in the following way. The preliminary concepts related to the study are briefly reviewed in Section 2. Some operational laws for I2TLEs are defined in Section 3 and their important properties are discussed with proofs. In the same section, distance measure between two I2TLEs and comparison method of I2TLEs, I2TLPR, and multiplicative I2TLPR are proposed along with a procedure to get consistent I2TLPR from the inconsistent one. In Section 4, a numerical example is given and comparative analysis is conducted with the TOPSIS method to verify the effectiveness of the proposed method. Finally, the conclusion is presented in the last section.

#### 2. Preliminaries

In this section, we mainly recall some elementary concepts of LTSs and 2-tuple linguistic representation model as well as the I2TL representation model.

##### 2.1. Intuitionistic Fuzzy Set and Intuitionistic Preference Relation

*Definition 1 (see [35, 36]). *An IFS in is given by , where and with the condition that for every . The numbers , denote, respectively, the degree of membership and nonmembership of the element to the set . For convenience, is called intuitionistic fuzzy element (IFE) and the set of all IFEs. For each IFS in , we will call the degree of indeterminacy of in .

*Definition 2 (see [37]). *An IPR on is defined by a matrix , where for all . For convenience, let be shortly written as , where indicates the degree to which is preferred to , and indicates the degree to which is not preferred to . Furthermore, denotes the indeterminacy degree or a hesitancy degree of the IPR satisfying the conditions ; ;; ; ; for all

##### 2.2. Consistency Checking for Multiplicative IPR

A significant property of preference relations is multiplicative consistency. Xu et al. [26] proposed the definition of multiplicative consistent IPR as follows.

*Definition 3 (see [26]). *An IPR is multiplicative consistent with , if

For IPRs with unacceptable consistency, Xu and Liao [28] proposed a method to measure the consistency of an IPR and then introduced a method to repair the inconsistent IPR. First, they developed an algorithm to build a perfect multiplicative consistent IPR , where and

*Definition 4 (see [28]). *An IPR is called an acceptable multiplicative consistent, if the distance measure between and denoted as is less than , where is the consistency threshold. The distance measure can be determined as follows:

Xu and Liao [28] thought that the transformed IPR cannot represent the initial preferences of the DM for a large value of . Therefore, they fused the IPRs and into a new IPR , where

where is called the controlling parameter of the IPR that is set by the DM only. If is small, then is closer to . For , , and for , .

##### 2.3. Basic Concepts of Linguistic Term Set and 2-Tuple Linguistic Information

*Definition 5 (see [38, 39]). *Let be a finite LTS with odd cardinality, where each represents a possible value for a linguistic variable. The following characteristics for can be defined as follows:(1)Negation operator: , such that ;(2)Ordered set: . Therefore, there exist two operators given as follows:(a)maximization operator: , if ;(b)minimization operator: , if .

Xu [40, 41] introduced the concept of continuous LTS as an extension of discrete term set where . The linguistic term is called the original linguistic term if , and is only used by the DMs to evaluate the alternatives during a decision process. If the linguistic term , then is said to be the virtual linguistic term of and it appears only during the computations.

Herrera and Martínez [2] proposed the 2-tuple linguistic representation model which expresses the linguistic information by a 2-tuple , where and . The basic purpose of this model is to define a transformation mechanism between linguistic 2-tuples and the numerical values.

*Definition 6 (see [2]). *Let be a LTS and a value representing the result of a symbolic aggregation operation. Then, a function which provides a linguistic 2-tuple representing the equivalent information to is defined as follows:

Clearly, is one to one function. The has an inverse function with .

##### 2.4. Intuitionistic 2-Tuple Linguistic Information Model

Beg and Rashid [8] proposed the idea of I2TL information model and some operators based on choquet integral to aggregate the I2TL information. They defined I2TL representation model as follows.

*Definition 7 (see [8]). *For a crisp set and LTS , the set in where is called an intuitionistic LTS if and with the condition that , for all . The linguistic values and represent, respectively, the membership and nonmembership degrees of the element in the set .

*Definition 8 (see [8]). *Let be intuitionistic LTS in and ; an I2TL model can be defined as , where , and are numeric values in denoting the symbolic translation of and , respectively. For our convenience, is called an I2TLE and is the set of all I2TLEs.

In order to avoid any loss of information, Beg and Rashid [8] further presented a computational technique to deal with this model as follows.

*Definition 9 (see [8]). *Let be an I2TLE for a LTS . The function from to an order pair of numerical values is defined as .

It is clear that , with the condition provided and are not simultaneously zero.

The function is used to obtain the I2TL information equivalent to the pair . This function can be defined as , where round, round, , and . The linguistic terms and have the closest index label to and , respectively. Similarly, the values , represent the symbolic translations of and , respectively.

#### 3. Operational Laws of I2TLEs and Consistency Measure

In this section, we define some logical operational laws of I2TLEs and present some properties with proofs. The proposed operational laws for I2TLEs encompass previous operational laws for LTSs and exhibit flexibility. We also define I2TLPR and multiplicative I2TLPR and study a useful method to get a consistent I2TLPR from an inconsistent one. Furthermore, distance measure between two I2TLEs, comparison method of I2TLEs, and a methodology of I2TL AHP method are proposed in the same section to find an optimal alternative in a MCDM problem.

##### 3.1. Some Operational Laws of I2TLEs

Gou and Xu [42] defined some logical operational laws for linguistic variables of a LTS on the basis of two equivalent transformation functions which can avoid the aggregated linguistic values exceeding the bounds of LTSs. They further discussed various related important properties for these operational laws. These operational laws are actually based on a transformation function and inverse transformation function which are defined as follows:

Based on these transformation functions, Gou and Xu [42] introduced the following novel operational laws for linguistic values of a LTS as follows:(1)(2)(3)(4)

Gou and Xu [42] also investigated the following important properties for these novel operational laws:(1)(2)(3)(4)(5)(6)

Motivated by the above operational laws of LTSs, we can also extend these operation laws for I2TLEs as follows.

*Definition 10. *Let be a LTS and , and be three I2TLEs in and . We define(1)(2)(3)(4)

Theorem 11. *Let be a LTS and , , and be three I2TLEs in and . Then*(1)*(2)**(3)**(4)**(5)**(6)*

*Proof. * and are simple, so the proofs of them are omitted here.

*In the following, we put forward the axiom of distance measure for I2TLEs, shown as follows:*

*Definition 12. *Let be a LTS and and be two I2TLEs in . The Euclidean distance between and can be defined as follows:

*Definition 13. *Let ,**; **then the Euclidean distance between and satisfies the following:(1);(2) if and only if ;(3).

*Liu and Chen [9] proposed score and accuracy functions for the comparison of two I2TLEs. We now introduce a new comparison method for I2TLEs, which can be seen as follows.*

*Definition 14. *Let be an I2TLE in where , for . The score function , the accuracy function , and hesitancy degree value of can be defined as follows:

*It can be observed that , . The comparison method of I2TLEs based on score and accuracy functions can be established as follows.*

*Definition 15. *For any two I2FLEs ,(1)if , then ;(2)if , and(a), then ;(b), then

*Example 16. *Let be a LTS and , and be two I2TLEs in . Then using (11), , . This shows that . Similarly, for , in but and This implies

*3.2. Consistency Measure of I2TLPR*

*In preference relations, consistency is an important topic in decision-making and the lack of consistency can lead to inconsistent solutions. Some inconsistencies may typically arise while finding consistent solution to MCDM problems when many pairwise comparisons are performed by the DMs during assessment processes. Saaty [23] proposed a consistency index and a consistency ratio denoted as “” and “”, respectively, in the conventional AHP method to compute the degree or level of consistency for a multiplicative preference relation by using the following formulae:*

*where and are, respectively, the largest eigenvalue and the dimension of the multiplicative preference relation. The term is denoted as random index that completely depends on the value of . The values of for are shown in Table 1. The value of is always equal to zero for a perfectly consistent DM, but small values of inconsistency may be tolerated during a decision process. However, perfect consistency rarely occurs in practice.*