Some Improved <i>q</i>-Rung Orthopair Fuzzy Aggregation Operators and Their Applications to Multiattribute Group Decision-Making

Xu, Lei; Liu, Yi; Liu, Haobin

doi:https://doi.org/10.1155/2019/2036728

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 2036728 | https://doi.org/10.1155/2019/2036728

Some Improved q-Rung Orthopair Fuzzy Aggregation Operators and Their Applications to Multiattribute Group Decision-Making

Lei Xu,^1,2Yi Liu ,^1,2and Haobin Liu^1,2

Academic Editor: Francesco Soldovieri

Received13 Aug 2019

Revised11 Nov 2019

Accepted26 Nov 2019

Published27 Dec 2019

Abstract

As a generalization of intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS), q-rung orthopair fuzzy set (q-ROFS) is a new concept in describing complex fuzzy uncertainty information. The present work focuses on the multiattribute group decision-making (MAGDM) approach under the q-rung orthopair fuzzy information. To begin with, some drawbacks of the existing MAGDM methods based on aggregation operators (AOs) are firstly analyzed. In addition, some improved operational laws put forward to overcome the drawbacks along with some properties of the operational law are proved. Thirdly, we put forward the improved q-rung orthopair fuzzy-weighted averaging (q-IROFWA) aggregation operator and improved q-rung orthopair fuzzy-weighted power averaging (q-IROFWPA) aggregation operator and present some of their properties. Then, based on the q-IROFWA operator and q-IROFWPA operator, we proposed a new method to deal with MAGDM problems under the fuzzy environment. Finally, some numerical examples are provided to illustrate the feasibility and validity of the proposed method.

1. Introduction

Decision-making is an inseparable science in human daily life, and its ideas and methods have been applied to many aspects of real life such as systems engineering, regional planning and design, and operational research.

In the process of decision-making, because of the fuzziness of human thinking, the uncertainty of decision-making information and the complexity of objective things, it is difficult for decision makers to give accurate evaluation values for different schemes. Since Zadeh [1] put forward the concept of fuzzy set (FS), the discussions on MAGDM under the condition of fuzzy information have never stopped. However, the fuzzy set cannot reasonably express all the decision-making problems. For example, in the process of site selection, some evaluators agree with a scheme, while others oppose or even abstain from it. Therefore, how to use them to effectively deal with the fuzzy decision-making problem is directly related to people’s production, life, and social progress.

For this, Bulgarian scholar Atanassov [2] proposed the intuitionistic fuzzy set (IFS) which is extended from FS. The IFS describes the fuzziness of things from three aspects, membership degree (MD), nonmembership degree (NMD), and hesitation degree; therefore, it has higher flexibility in fuzziness and uncertainty. After the introduction of IFS, scholars have conducted extensive and in-depth research [3–6]. The main results are as follows: (1) the related properties of IFS [7], operational rules (ORs) [8], similarity measurement [9, 10], distance measurement [11–13], information entropy [14], and so on. (2) Some extended decision-making method based on IFS, such as ELECTRE-III [15], Taguchi and the VIKOR methods [16, 17], and other decision-making approaches [18–22]. (3) The AOs, for example, IFWA operator and IFWG operator proposed by Xu [23, 24], Hamacher aggregation operators by Liu [25], and Frank aggregation operators by Zhang et al. [26].

However, the application scope of the IFS is limited. MD u and NMD satisfy the formula . Therefore, some special decision evaluation information cannot be expressed. For example, if MD and NMD are equal to 0.7 and 0.5, respectively, it is clear that the IFS cannot be used. After that, Yager and Abbasov [27] proposed the Pythagorean fuzzy set (PFS) to solve the above problem. Compared with the IFS, the PFS generalizes the MD and the NMD to the square sum less than or equal to 1. Therefore, it can solve more problems than the IFS [16, 28]. However, with the increase of the complexity of decision-making problems and the contradictory psychology of policy makers, it is still difficult for PFS to express corresponding information.

Recently, Yager [29] again proposed the concept of the q-ROFS, in which MD u and NMD satisfy (). We can see that the IFS and PFS are special cases of q-ROFS. As q-rung increases, the range of processing fuzzy information increases. The acceptable space for different types of fuzzy numbers is shown in the Figure 1. IFN, PFN, and q-ROFN are abbreviations of intuitionistic fuzzy number, Pythagorean fuzzy number, and q-rung orthopair fuzzy number, respectively. Therefore, it can be seen that q-ROFS has stronger capability in dealing with uncertain information than the IFS and PFS.

In recent years, the topic of information aggregation has attracted a lot of attention and is one of the key research issues in the problems of MAGDM. As far as q-ROFS is concerned, different aggregation operators have been introduced and applied, such as q-ROFWA and q-ROFWG operator [31], q-ROFPA and q-ROFPMSM operator [32], q-ROFGBM and q-ROFWGBM operator [33], IFMM operator [34], q-ROFWPHM operator [30], and q-ROFIDHM operator [35].

Although different aggregation operators have been introduced, these operators have some drawbacks. The above aggregation operators are carried out based on traditional arithmetic rules, i.e., Einstein ORs [36], Hamacher ORs [25], and Frank ORs [26]. However, these operational rules have some drawbacks. When NMD of any an IFN is 0, the aggregated result will be 0 no matter what value of another NMD is. In addition, when MD of any IFN is 1, the aggregated result will be 1 no matter what value of another MD is. Furthermore, aggregated results are obtained independently. The change of MD will not affect the aggregated result of NMD. Obviously, this is unreasonable.

Motivated by the new operational laws developed by He et al. [37], we propose some improved operational laws for q-ROFS to surmount the drawbacks firstly, and then develop an improved q-rung orthopair fuzzy-weighted averaging (q-IROFWA) aggregation operator. Considering that the PA operator can reduce the impact of irrational data given by biased decision makers, we extend the power average (PA) operator [38] to deal with q-ROFS and based on the q-IROFWA and q-IROFWPA operator we proposed a new method to deal with MAGDM problems.

The contributions of this paper are as follows: (1) some drawbacks of the existing aggregation operator are reviewed and analyzed; (2) proposed new aggregation operators containing the q-IROFWA operator and q-IROFWPA operator; (3) proposed a new method to deal with MAGDM problems based on the q-IROFWA and q-IROFWPA operator; (4) some numerical examples are provided to verify the feasibility and practicability of the new method.

The remaining of this paper is organized as follows. In Section 1, we briefly introduce the preliminary conceptions of q-ROFS, PA operator, and score function. In Section 2, we analyze some drawbacks of existing q-rung orthopair fuzzy aggregation operators. In Section 3, we propose a new q-IROFWA and q-IROFWPA operator based on new operational rules. In Section 4, we will give some examples to evaluate the feasibility and validity of the proposed method. In Section 5, the conclusions are given.

2. Preliminaries

In this section, some basic theories related to q-ROFS, PA operator, and score function are reviewed.

2.1. q-ROFS

Definition 1. Let be a finite universe of discourse, a q-ROFS A in X characterized by a membership function , and a nonmembership function , which satisfies that , (). A q-ROFS is represented aswhere is the degree of indeterminacy, and we called a , denoted by .

2.2. The Power Average Operator

Definition 2. Let be a set of crisp numbers; the PA operator can be expressed aswhere represents the support degree for from , which meets(1);(2);(3), if .

2.3. Score Function of Decision-Making Problem

Definition 3 (see [31])]. Let be a q-ROFN (). The score function S of α is defined as follows:where .

Definition 4 (see [31]). Let be a q-ROFN (). The accuracy function H of α is defined as follows:where .

Definition 5 (see [31]). Let and be any two q-ROFN (). and be the score functions of and , and and be the accuracy functions of and .(1), then (2) and , then , then

3. Analysis of the Existing q-ROFS Operators

In this section, we will review the formal representation of the typical q-ROFS operators and analyze their drawbacks; then in Section 3, we will introduce methods in order to overcome those drawbacks.

Definition 6 (see [31]). Let is a collection of q-ROFN () and be the weight vector of , such that and . Then, the aggregation result is still a q-ROFN and has

Example 1. Let α₁ = (0.6, 0), α₂ = (0.5, 0.3), α₃ = (0.7, 0.2), and α₄ = (0.3, 0.5) be four q-ROFN and be the corresponding weight vector (WV), then suppose q = 3 and according to equation (6) we haveDrawback A: as can be seen from Example 1, although the degree of NMD of three q-ROFNs , , and are not equal to 0, the aggregated result is still 0. Obviously, this is an unreasonable result. Considering equation (2), if there only one NMD of q-ROFNs is 0, the aggregated result will be 0 even if the other q-ROFNs are not 0. Therefore, the q-ROFWA operator in [31] is not well defined. Accordingly using this ill-defined q-ROFWA operator will get an unreasonable ranking result of the alternatives in some situations.
In addition, from equation (11), we know that the MD and NMD of aggregated results are obtained independently from the MDs and NMDs of n q-ROFNs, respectively. It is shown as Example 2.

Example 2. Let α₁ = (0.6, 0.3), α₂ = (0.5, 0.3), α₃ = (0.7, 0.2), and α₄ = (0.3, 0.5) be four q-ROFNs and be the corresponding WV. Then, based on equation (6) and suppose q = 3, we haveWhen we change the value of and to α₃ = (0.8, 0.2) and α₄ = (0.5, 0.5), then the aggregated result isDrawback B: from Example 2 we can see that, the change of MD will not affect the aggregated result of NMD. In the calculation process, MD and NMD are independent. Therefore, the aggregated result cannot provide the right information for the decision makers.
In Section 3 below, two new methods are proposed to surmount the abovementioned drawbacks of the existing methods.

4. Improved q-ROFS Operators

In this section, two operators q-IROFWA and q-IROFWPA are proposed to overcome the drawbacks presented above.

Because the aggregation operation is derived from the operational rules, the first step to improve the aggregation operator is to improve the operation rules. Some improved rules based on He et al. [37] are proposed as follows.

Let , , and be any two q-ROFN () and , then

Theorem 1. Let , , and be any three q-ROFN () and , then

Proof of Theorem 1. (1)According to rules (11) and (12), it is obvious that equations (15)and (16) are kept.(2)Then, we prove equations (17), (19), and (21), and the proof of others are similar to equations (17) and (19), so omitted here.(3)For the left-hand side of equation (17), we have and for the right-hand side of equation (17), we have Thus, equation (17) is kept.(4)The proof of equation (18) is similar to that of equation (17), so it is omitted.(5)For the right-hand side of equation (19), we have and for the left-hand side of equation (19), we have Thus, equation (19) is kept.(6)For the right-hand side of equation (21), we get and for the left-hand side of equation (21), we have Thus, equation (21) is kept.(7)The proof of equation (20) is similar to that of equation (21), so it is omitted.

Example 3. Let α₁ = (0.6, 0) and α₂ = (0.5, 0.3) be two q-ROFN, , then we can get

5. Improved q-ROFS Aggregation Operators for MADM Problems

5.1. Improved q-Rung Orthopair Fuzzy-Weighted Averaging Operator

In the section, we will introduce and analyze the q-IROFWA operator based on the new ORs in Section 3.

Theorem 2. Let be a collection of q-ROFN () and be the corresponding WV, and meet and . Then, the aggregation result is still a q-ROFN and has

Proof of Theorem 2. Using mathematical induction, it can be proved as follows:(1)When n = 1, for the left-hand side of equation (30), we obtain For the right-hand side of equation (30), we have So, when n = 1, equation (30) is kept.(2)When n = 2, So, when n = 2, equation (30) is kept.(3)Suppose , equation (30) is kept, that is, Then, when , we haveSo, when , equation (30) is kept.
Therefore, according to steps 1–3, equation (30) is kept for any k.
In the following, we will prove that is still a .
Let , then if is a , it meets the following two conditions:(1)(2)(1)Because , we have So, we get . In addition, since , we have Moreover, because , we have , so , and we get . Therefore, condition (1) is proved.(2)Since , we have Therefore, condition (2) is proved. Thus, is still a .

Theorem 3 (idempotency). Suppose is a collection of q-ROFNs, if , then

Proof of Theorem 3.

Theorem 4 (boundedness). Suppose is a collection of q-ROFNs and and , then we have , where

Proof of Theorem 4. (1)For the MD of , we have(2)For the NMD, we haveBecausethen

Example 4. For Example 2, we haveWhen we change the value of and to and , then the aggregated result is (0.5861, 0.4428).
Obviously, as the membership degrees of the aggregated parameters are changed, then the MD and NMD of the aggregated result are also changed simultaneously.

5.2. Improved q-Rung Orthopair Fuzzy Power Averaging Operator

Definition 7. Let be a set of q-ROFN (), then is defined as follows:where expresses the support degree of jth ROFN from all the other ROFNS, the closer the value, the more they support each other.

Definition 8. Let and be any two q-ROFNS (). Then, the generalized hesitance degree-preference distance of and is defined as follows:where .
For equation (47), we can get some different distance measures when setting different parameter values of .

Case 1. When , the generalized hesitance degree-preference distance will reduce to the Hamming degree-preference distance:

Case 2. When , the generalized hesitance degree-preference distance will reduce to the Hamming-indeterminacy degree-preference distance. Its normalization form is defined by

Theorem 5. Let be a set of q-ROFN (), then the aggregation result is still a q-ROFN and has

Proof of Theorem 5. Let , then the proof is similar to the proof of Theorem 2, so it is omitted here.
Obviously, the weighted vector is not considered in Definition 8. In many actual decision cases, the weights of attributes can affect the result. Therefore, we need to propose the q-IROFWPA operator as follows.

Definition 9. Let be a set of q-ROFN (), then is defined as follows:In order to simplify the proof, we let and be the weight vector of , such that and . Then, equation (51) has the following expression:

Theorem 6. Suppose be a set of q-ROFN (), then the aggregation result from Definition 8 is still a q-ROFN:

Proof of Theorem 6. The proof is similar to Theorem 2, and it can be omitted here.

Theorem 7 (idempotency). Let is a collection of q-ROFNs, if , then .

Proof of Theorem 7. The proof is similar to Theorem 3, and it can be omitted here.

Theorem 8 (boundedness). Let and , then we have , where is consistent with the definition in Theorem 4.

Proof of Theorem 8. The proof is similar to Theorem 4, and it can be omitted here.

6. Novel MAGDM Method Based on the Proposed Operators

In this section, based on the q-IROFWA operator and the q-IROFWPA operator, we proposed a novel method to deal with MAGDM problems. Let be the set of experts (decision makers), be all alternatives, and be all attributes. represents the value of attribute from alternative given by the expert . Let be the decision matrix from experts . Let be the WV of experts, , and . Then, the details of the novel method are described as follows: Step 1: normalize the decision matrix. In MAGDM problem, the attribute values may have different types, such as benefit and cost. Since different types of attributes may be neutralized during the aggregating process, it needs to convert different attribute types into the same. Owing to cognitive habits, cost-based attributes are usually transformed into benefit ones by the following formula: Step 2: aggregate all attribute values to a comprehensive value by the q-IROFWA operator as follows: Step 3: calculate the supports: Step 4: calculate the supports and weights : where and Step 5: use the q-IROFWPA operator expressed by equation (53) to get the collective preference values : Step 6: use score function and accuracy function to rank . Step 7: select the best alternative based on the value of .

The flow chart of the novel MAGDM method is shown in Figure 2.

7. Case Study

In this section, an example is used to illustrate the application of the method in detail, and then some examples are further adopted to verify the effectiveness and superiority of the proposed method with some existing MAGDM method.

7.1. The Application of the Proposed Method

Example 5. This example is from [39]. There are five alternative companies which are evaluated by three experts with respect to four attributes: the enterprise management level , the business growth ability , the economic benefit , and the corporate reputation , respectively. be the decision matrix obtained from expert by using , as shown in Table 1.
Let be of the three experts and be of the attributes. The detail of the proposed MAGDM method to select the best company is as below.
To further verify the effect of parameter q on the results, experiments were carried out using different q. The ranking results are shown in Table 3.
As can be seen from Table 3 that although the aggregation result will change with the parameter q, the optimal result remains unchanged. Further, we can see that the value of the score function becomes smaller as q increases, so q is better in the interval 2 to 5.

Step 1: normalize the decision matrices , , and . In this example, the type of the four attributes is the same, so it can be omitted. Step 2: according to the operator in equation (55), all the attributes matrices are aggregated into the integrated matrices , as shown in Table 2.

Step 3: according to equation (56), we can get the supports as follows: Step 4: according to equations (57) and (58), we can get the supports and the weights as follows: Similarly, we have Step 5: according to the operator in equation (59), we can get the collective preference values as follows: Step 6: obtain the comprehensive evaluation value by the score function of equation (4) as follows: Step 7: sort the calculation results of the score function, and we obtain , so is the best choice.

7.2. The Verification of the Effectiveness

Example 6. There are five companies as alternatives and four attributes: the risk analysis (), the growth analysis (), the social political impact analysis (), and the environmental impact analysis () to be evaluated by three decision makers. The decision matrices (see [40]) are listed in Table 3. The experimental results are listed in Table 4. It can be seen from Table 4 that all methods have the same result, thus demonstrating that the method proposed in this paper is effective.

7.3. Further Comparison with Other Methods

In the following, in order to explain the advantages of the proposed method compared with the existing method, such as q-ROFWA and q-ROFWG operator [31] and q-ROFPA and q-ROFPMSM operator [32], we will use a new example to illustrate it.

Example 7. This example is about a investment problem in renewable energy. There are three alternatives : (1) is geothermal; (2) is solar; and (3) is biomass. The following three attributes are to be evaluated by three decision makers, where denotes the risk factor, denotes the growth rate in the sector, and denotes the payback reliability. The three possible alternatives are to be evaluated using the q-ROFN by the DM, as shown in Table 5. Let be WV of decision makers and be WV of the attributes .
The details of the calculation are the same as in Example 1, so skipped. In practical applications, it is often unavoidable that some NMD are 0 in q-ROFS. In this situation, the methods proposed by Liu and Wang [31] are not able to get an effective or even the wrong ranking results. The comparison results are shown in Table 6.
From the comparison results in Table 6, we can see that when there is 0 in the NMD of q-ROFs, the existing method cannot be solved very well. The ranking results will change as the parameter q changes, and when q is equal to 3, there is a situation that cannot be distinguished. Obviously, this is unreasonable. However, for the method in this paper, the ranking result is the same. Therefore, the developed method in this paper can overcome Drawback A. In addition, regardless of how the q value changes, the proposed method can distinguish the priority of the alternative and the result will not change, and furthermore the ranking result is the same as the ones presented in [41].

Example 8. In this example, the decision matrix is shown in Table 7. The other data are identical with Example 3. The comparison results of different methods are shown in Table 8. From Table 8, we can get conclusion that in some special cases the existing method cannot distinguish the preference order of alternatives. The root of this problem is the related aggregation operators. Compared with the existing methods, the MD and NMD in this paper are not independent during the aggregation process. It is also shown that our method can overcome effectively the Drawback B.
In the following, we analyze the relative shortcoming of Xu’s IFPWA [38], Liu’s q-ROFWA [31], and Liu et al.’s ROFPWMSW Operator [32] is shown as follows:(1)The shortcoming of Xu’s IFPWA Operator [38]: first, information is described by IFN, which must meet the restriction that MD u and NMD satisfy the formula . Second, it cannot consider the interrelationship between multiple attributes. Third, it cannot overcome the Drawbacks A and B mentioned in the paper.(2)The shortcoming of Liu and Wang’s q-ROFWA [31]: first, it cannot overcome the Drawbacks A and B mentioned in the paper. Second, it cannot properly deal extreme data given by biased decision makers.(3)The shortcoming of Liu et al.’s ROFPWMSW Operator [32]: although the method integrates the advantages of the MSM operator (can consider the interrelationship between multiple attributes), PA operator (can eliminate the effects of extreme values), and q-ROFS operator (can express a wider range of information), MD and NMD are independent. Therefore, in some special cases the ROFPWMSW operator in [32] cannot distinguish the preference order of alternatives.More visual comparison results can be easily obtained from Table 9.

8. Conclusions

For express fuzzy information, q-ROFS is a good tool. It has a parameter q, so it holds a wider range of fuzzy information than IFS and PFS. In this paper, we reviewed and analyzed some drawbacks of the existing MAGDM methods. We improved the traditional operation rules to overcome these drawbacks, proposed a new method to deal with the MAGDM method based on the proposed q-IROFWA and q-IROFWPA operator. At the same time, some features of the newly proposed operators are proved. Moreover, some numerical examples are provided to show the application of the method in detail and further to verify the feasibility and the superiority of the proposed method. From the example results shown in Tables 6 and 8, we can come to the conclusion that the proposed methods can surmount the drawbacks of some existing MAGDM methods. For future research, we will apply the improved operation rules to more aggregation operators. In addition, we will introduce granular computing techniques [42] to MAGDM and develop the new MAGDM method.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Sichuan Province Youth Science and Technology Innovation Team (No. 2019JDTD0015); Application Basic Research Plan Project of Sichuan Province (No. 2017JY0199); Scientific Research Project of Department of Education of Sichuan Province (Nos. 17ZB0220 and 18ZA0273); Scientific Research Innovation Team of Neijiang Normal University (No. 18TD008); and open fund of Data Recovery Key Lab of Sichuan Province (No. DRN19018).

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Copyright © 2019 Lei Xu 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.

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