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Multiple-Attribute Group Decision-Making Based on q-Rung Orthopair Fuzzy Power Maclaurin Symmetric Mean Operators Peide Liu , Shyi-Ming Chen, Fellow, IEEE, and Peng Wang
Abstract—To be able to describe more complex fuzzy uncertainty information effectively, the concept of q-rung orthopair fuzzy sets (q-ROFSs) was first proposed by Yager. The q-ROFSs can dynamically adjust the range of indication of decision information by changing a parameter q based on the different hesitation degree from the decision-makers, where q ≥ 1, so they outperform the traditional intuitionistic fuzzy sets and Pythagorean fuzzy sets. In real decision-making problems, there is often an interaction phenomenon between attributes. For aggregating these complex fuzzy information, the Maclaurin symmetric mean (MSM) operator is more superior by considering interrelationships among attributes. In addition, the power average (PA) operator can reduce the effects of extreme evaluating data from some experts with prejudice. In this paper, we introduce the PA operator and the MSM operator based on q-rung orthopair fuzzy numbers (q-ROFNs). Then, we put forward the q-rung orthopair fuzzy power MSM (q-ROFPMSM) operator and the q-rung orthopair fuzzy power weighed MSM (q-ROFPWMSM) operator of q-ROFNs and present some of their properties. Finally, we present a novel multiple-attribute group decision-making (MAGDM) method based on the q-ROFPWA and the q-ROFPWMSM operators. The experimental results show that the novel MAGDM method outperforms the existing MAGDM methods for dealing with MAGDM problems. Index Terms—Maclaurin symmetric mean (MSM) operator, multiple-attribute group decision-making (MAGDM), power average (PA) operator, q-rung orthopair fuzzy sets (q-ROFSs).
I. I NTRODUCTION ULTIPLE-ATTRIBUTE group decisionmaking (MAGDM) problems can be regarded as a process which ranks alternatives according to the evaluating attribute values of the alternatives from multiple experts
M
Manuscript received May 21, 2018; accepted June 30, 2018. This work was supported in part by the National Natural Science Foundation of China under Grant 71771140 and Grant 71471172, in part by the Special Funds of Taishan Scholars Project of Shandong Province under Grant ts201511045, and in part by the Ministry of Science and Technology, Republic of China, under Grant MOST 104-2221-E-011-084-MY3. This paper was recommended by Associate Editor C. Zhang. (Corresponding author: Peide Liu.) P. Liu and P. Wang are with the School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, China (e-mail:
[email protected];
[email protected]). S.-M. Chen is with the Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan (e-mail:
[email protected]). This paper has supplementary downloadable material available at http://ieeexplore.ieee.org, provided by the author. This includes proofs of theorems given in the paper. This material is 0.128 MB in size. Digital Object Identifier 10.1109/TSMC.2018.2852948
and they have been extensively studied by more and more scholars [5], [22], [44]. MAGDM is a typical and important decision problem. In the broad sense, the decision problems, especially group decision-making (GDM) problems, contain many types, which can be divided from the way of information expression and the way of dealing with problems. The most commonly used definition of GDM problems is based on the four-field notation [3]. The first field X is the evaluation value, which is represented by xi , its domain is (“crisp” and “vague”); the second field CX is the degree of confidence in xi , which is represented by α, its domain is a probability number within (0, 1]; the third field W is the degree of importance of the expert, i.e., the weight of the expert, which is represented by wi , its domain is (crisp and vague); and the fourth field CW is the degree of confidence in wi , which is represented by β, its domain is a probability number within (0, 1]. Based on this notation, most decision situations (S) can be expressed as: S = [crisp/vague, α, crisp/vague, β]. In the existing literatures, many studies have focused on the MAGDM problems based on the situation of [vague, 1, crisp, 1], and indeed, the research contributions and concerns of this paper are based on this problem. For MAGDM problems in fuzzy environments, the description of evaluation information, i.e., the first field X, is crucial, so we need to consider how to effectively express evaluating attribute values of alternatives. Due to the fuzziness of human cognition and the complexity of decision environments, exact values or linguistic variables are not enough to express uncertainty information. In order to deal with this situation, Atanassov [1], [2] first introduced the theory intuitionistic fuzzy sets (IFSs), which is extended from the theory of fuzzy sets (FSs) [42]. The advantage of IFSs is that they can express fuzzy information by a membership degree (MD), a nonmembership degree (NMD), and a hesitancy degree, respectively. The theory of IFSs began to be studied by many researchers and has widely been applied to the fields of multiple-attribute decision-making (MADM) and MAGDM. The main results can be divided into three aspects. 1) The Basic Theoretical Results of Intuitionistic Fuzzy Numbers (IFNs): Such as some operation rules [8], some distance measures [6], [31], some similarity measures [4], [24], the information entropy [11], etc. 2) The Extended Decision Evaluation Methods Based on IFNs: Such as the technique for order preference by similarity to an ideal solution (TOPSIS) method
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based on intuitionistic fuzzy information for credit risk evaluation [29], the intuitionistic fuzzy Elimination Et Choice Translation Reality method for engineering manager choice [27], etc. 3) The Aggregation Operators (AOs) for Intuitionistic Fuzzy Information: For example, Xu [32] presented some basic arithmetic AOs for IFNs, Xu and Yager [34] presented some geometric AOs based on IFSs, Liu [13] presented some Hamacher AOs based on interval-valued IFNs (IVIFNs) for MAGDM problems, etc. However, the information expression range of IFSs is limited. They must meet the restriction that the summation of the MD u and the NMD v is within 1, i.e., u + v ≤ 1. Under this constraint, much complex evaluation information cannot be described due to the fact that some experts may give some evaluating attribute values that exceed the constraint. For example, if the MD and the NMD of an evaluating attribute value given by an expert are 0.8 and 0.6, respectively, then the IFNs are not suitable to be used for this kind of problem. With the further development of fuzzy theory, Yager [37] presented the theory of Pythagorean FSs (PFSs), where an eminent characteristic of PFSs is that the square sum of the MD and the NMD is within 1, i.e., u2 + v2 ≤ 1. As a result, PFSs can be used to deal with a lot of MADM problems and MAGDM problems in which IFSs cannot be used to deal with these problems. Therefore, PFSs are more superior in expressing fuzzy information than IFNs. Based on the theory of PFSs, some corresponding results have emerged for decision-making problems. For example, Yager [37] presented some basic fuzzy weighted operators based on PFNs; Peng and Yang [25] presented a new ranking method for PFNs; Zhang and Xu [43] presented a new distance measure for PFNs and applied it to extended TOPSIS method for dealing with more complicated decision situations; Ren et al. [28] introduced a Pythagorean fuzzy TODIM approach for multicriteria decision-making; and Garg [9], [10] introduced some generalized AOs based on some new operations of PFNs. Although PFSs are widely studied and applied, their scope for expressing fuzzy information is still limited. When the evaluation psychology of decision-makers is too complicated and contradictory, the corresponding decision-making information is still difficult to express with PFNs. Recently, Yager et al. [38], [39] proposed the concept of q-rung orthopair FSs (q-ROFSs), where q ≥ 1 and the summation of the qth power of the MD and the qth power of the NMD cannot be greater than 1, i.e., uq + vq ≤ 1. It is obvious that q-ROFSs are more general than IFSs and PFSs due to the fact that IFSs and PFSs are the special cases of qROFSs when q = 1 and q = 2, respectively. It should be noted that the space of acceptable information can increase with the rung q increases, and more uncertainty information satisfy the bounding constraint. Therefore, q-ROFSs are more suitable to describe fuzzy information than IFSs and PFSs. Especially, q-ROFSs can be used to describe complex and contradictory decision information by adjusting the value of parameter q, where q ≥ 1. The larger the parameter q, the larger the fuzzy information space can be expressed by q-ROFSs. To better understand q-ROFSs, we provide the following example. In
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actual decision, the expert has no confidence in his opinion because of his ambivalence or unsureness about the probability of occurring with a particular event. Assuming that we evaluate the quality of a new mobile phone for a famous brand, according to its production technology, public praise and the customer’s evaluation of previous generations of products, we think its quality is very good and give an evaluation value of 0.9 as the degree of satisfaction. However, with the using of new mobile phones, battery cracking is frequently appearing, so the quality of the new product in this batch is likely to be worrying, in this case, the degree of dissatisfaction we can give is equally high and the value is 0.8. Therefore, the final evaluation result is (0.9, 0.8), then because 0.9 + 0.8 > 1 and 0.92 +0.82 > 1, the evaluation attribute value (0.9, 0.8) cannot be expressed by IFSs and PFSs. In this case, we can see that when q = 5, we can get 0.95 + 0.85 < 1 and the evaluation attribute value can be expressed by q-ROFSs, where q = 5. It is obvious that q-ROFSs have a more powerful capability to describe uncertain information of MAGDM problems than IFSs and PFSs. Information fusion is a process of processing decision information, which contains a variety of tools and models. How to effectively handle and aggregate information has been studied by many scholars [23], [40]. AOs can effectively be used to deal with information fusion problems. In recent years, the topic of information AOs has attracted a lot of attention and has become hot in the problems of MADM and MAGDM. Overall, AOs can be studied from two aspects, i.e., the operations aspect and the functions aspect, described as follows. 1) The Operations Aspect: Many AOs are based on the different operational laws from the t-norm (TN) and tconorm (TCN) family. Especially the Archimedean TN and TCN [15] can deduce most of the TNs and TCNs which each of them has its own characteristics, such as the Hamacher TN and TCN [13] which can generate the arithmetic operation rules and Einstein operation rules; the Dombi TN and TCN [18] which can generate more complex and more flexible operation rules by adjusting a parameter. 2) The functions aspect: The traditional AOs [32], [34] only have an integration function. Recently, some functional operators have been proposed in succession. For example, Xu [33] proposed some power average (PA) operators based on IFNs, which can give an adjusted weight by calculating the support degree of attribute values and eliminate the influence of biased experts; Xu and Yager [35] proposed the Bonferroni mean (BM) operators and Yu and Wu [41] proposed the Heronian mean (HM) operators based on IFNs and IVIFNs, respectively, where both of these two AOs can take into account interrelationships between two aggregated parameters; Liu and Li [17] combined the BM operator with the power operator to propose some new operators for IVIFNs; Liu [14] combined the HM operator with the power operator to propose some new operators for IVIFNs; Qin and Liu [26] proposed some Maclaurin symmetric mean (MSM) operators for IFNs, which consider interrelationships among multiple attributes.
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In actual decision-making environments, the interrelationship between different attributes is ubiquitous. Obviously, the BM operator [35] and the HM operator [41] can successfully deal with this situation. However, the BM and the HM operators just can consider interrelationships between two attributes. Because we need to fully consider interrelationships among multiple attributes or between two attributes, the MSM operator [26] can meet this requirement. The MSM operator can flexibly capture interrelationships among any multiple attributes by adjusting the values of a parameter k, which can deal with MADM or MAGDM problems more flexibly. Moreover, the PA operator [33] can integrate the input data based on the adjusted weighting vector from the different support degrees between evaluation parameters and can eliminate the influence of extreme evaluating attribute values from some biased experts with different preference attitudes. The above operators are very practical in the actual decision-making. With the increase of decision complexity, the actual decision-making problems are often accompanied by the interaction between the attributes. At the same time, some experts may have strong personal bias in the process of evaluating the problem because of their different professional backgrounds and levels of knowledge. We find that in the existing decision-making methods based on AOs, there are few ways to consider both of these factors. They cannot take into account both the correlation among multiple different attributes and unreasonable evaluating attribute values. Moreover, most of AOs are mainly applied to deal with MAGDM problems based on IFNs and PFNs, where they cannot deal with fuzzy information completely and efficiently. Because q-ROFSs can express fuzzy information more complete and more flexible than IFSs and PFSs, we will propose some q-rung orthopair fuzzy power MSM (q-ROFPMSM) operators by combining the MSM operator and the PA operator with q-ROFSs to solve MAGDM problems. So the goals and motivations of this paper are as follows. 1) To present some more functional AOs with the help of compositional operator for q-rung orthopair fuzzy numbers (q-ROFNs). 2) To present a novel MAGDM method based on the proposed operators to solve the decision-making problems of attribute related and unreasonable evaluating data. In this paper, we present the q-rung orthopair fuzzy PA (q-ROFPA) operator, the q-rung orthopair fuzzy power weighted average (q-ROFPWA) operator, the q-ROFPMSM operator, and the q-rung orthopair fuzzy power weighted MSM (q-ROFPWMSM) operator for q-ROFNs and give their corresponding properties. Moreover, we also give a novel MAGDM method on basis of the q-ROFPWA operator and q-ROFPWMSM operator. Finally, we use some examples to elaborate the effectiveness and the superiority of the novel MAGDM method. The experiments confirm that the novel MAGDM method outperforms the existing MAGDM methods [15], [18], [33] for solving MAGDM problems. The remainder of this paper has the following arrangement. In Section II, the basic concepts of q-ROFSs [38], [39],
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Fig. 1.
Geometric space range contrast of IFNs, PFNs, and q-ROFNs.
PA operator [36], and MSM operator [21], [26] are introduced. In Section III, the q-ROFPA operator, the q-ROFPWA operator, the q-ROFMSM operator, the q-ROFPMSM operator, and the q-ROFPWMSM operator of q-ROFSs are proposed, where q ≥ 1. The properties of above operators are also given. In Section IV, a novel MAGDM method based on the q-ROFPWA operator and the q-ROFPWMSM operator is proposed. In Section V, the validity and superiority of the novel method are verified through the application of some practical examples and some comparative analysis with existing MAGDM methods. In Section VI, the conclusions are given. II. P RELIMINARIES To get a better understanding of this paper, the basic concepts of q-ROFSs [38], [39], PA operator [36], and the MSM operator [21], [26] will be introduced in this section. A. q-ROFSs ˆ in the finite universe Definition 1 [38], [39]: A q-ROFS Q of discourse Z is represented by ˆ = z, u ˆ (z), v ˆ (z) |z ∈ Z Q Q Q where uQˆ : Z → [0, 1], vQˆ : Z → [0, 1], uQˆ (z) and vQˆ (z) denote the degree of membership and the degree of nonˆ membership of element z ∈ Z belonging to the q-ROFS Q, q q respectively, with the constraint that 0 ≤ uQˆ (z) + vQˆ (z) ≤ 1, (q ≥ 1). The degree of indeterminacy of element z ∈ Z belongˆ is given as π ˆ (z) = (1 − u ˆ (z)q − ing to the q-ROFS Q Q Q q 1/q vQˆ (z) ) , where q ≥ 1. For convenience, we can call ˜ = u, v, uQˆ (z), vQˆ (z) a q-ROFN [19], represented as Q where q ≥ 1. To understand IFNs, PFNs, and q-ROFNs intuitively, a geometric space range contrast is shown in Fig. 1 [38]. ˜ 2 = u2 , v2 ˜ 1 = u1 , v1 and Q Definition 2 [38], [39]: Let Q be two q-ROFNs, where q ≥ 1. The operational laws between ˜ 2 = u2 , v2 have the following definition: ˜ 1 = u1 , v1 and Q Q ˜1 ∨Q ˜ 2 = max{u1 , u2 }, min{v1 , v2 } (i) Q ˜1 ∧Q ˜ 2 = min{u1 , u2 }, max{v1 , v2 } (ii) Q ˜1 ⊕Q ˜ 2 = uq + uq − uq uq 1/q , v1 v2 (iii) Q 1 2 1 2 ˜1 ⊗Q ˜ 2 = u1 u2 , vq + vq − vq vq 1/q (iv) Q 1 2 1 2
(1) (2) (3) (4)
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(v) (vi)
1/q λ ˜ 1 = 1 − 1 − uq λ λQ , v 1 ,λ > 0 1 1/q ˜ λ = uλ , 1 − 1 − vq λ , λ > 0. Q 1 1 1
(5) (6)
In order to compare q-ROFNs, we introduced the following definitions. ˜ = u, v be a q-ROFN, where Definition 3 [19]: Let Q ˜ of the q-ROFN Q ˜ is defined by q ≥ 1. The score value S(Q)
˜ = uq − vq S Q (7) ˜ ∈ [−1, 1]. where S(Q) ˜ = u, v be a q-ROFN, where Definition 4 [19]: Let Q ˜ of the q-ROFNQ ˜ is q ≥ 1. The accuracy value H(Q) defined by
˜ = uq + vq (8) H Q ˜ ∈ [0, 1]. whereH(Q) Based on Definitions 3 and 4, a comparison method is given, shown as below. ˜ 2 = u2 , v2 be ˜ 1 = u1 , v1 and Q Definition 5 [19]: Let Q ˜ ˜ any two q-ROFNs, let S(Q1 ) and S(Q2 ) be their score val˜ 2 ) be their accuracy ˜ 1 ) and H(Q ues, respectively, and let H(Q values, respectively, where q ≥ 1. Then: ˜ 2 ), then, Q ˜1 < Q ˜ 2; ˜ 1 ) < S(Q 1) if S(Q ˜ ˜ 2) if S(Q1 ) = S(Q2 ), then
˜ 2 , then Q ˜1 < Q ˜2 ˜1 < H Q if H Q
˜ 2 , then Q ˜1 = Q ˜1 = H Q ˜ 2. if H Q ˜ 2 = u2 , v2 be ˜ 1 = u1 , v1 and Q Definition 6 [16]: Let Q any two q-ROFNs, where q ≥ 1. The normalized Hamming ˜ 2 ) between the q-ROFNs Q ˜ 1 and Q ˜ 2 is defined ˜ 1, Q distance d(Q by
uq − uq + vq − vq + π q − π q 1 2 1 2 1 2 ˜2 = ˜ 1, Q (9) d Q 2 ˜ 1, Q ˜ 2 ) ∈ [0, 1], π1 = (1 − uq − vq )1/q , π2 = (1 − where d(Q 1 1 q q 1/q u2 − v2 ) and q ≥ 1.
C. MSM Operators The MSM operator [21], [26] can capture interrelationships among multiple attributes, defined as follows. Definition 8 [21], [26]: Let x1 , x2 , . . ., and xm be nonnegative real numbers. The MSM operator of the nonnegative real numbers x1 , x2 , . . ., and xm is defined by 1/k k x ε j 1≤ε
