G. Wei has an MSc and a PhD degree in applied mathematics from SouthWest Petroleum University, business administration from school of Economics and Management at SouthWest Jiaotong University, China, respectively. From May 2010 to April 2012, he was a postdoctoral researcher with the School of Economics and Management, Tsinghua University, Beijing, China. He is a professor in the School of Business at Sichuan Normal University. He has published more than 100 papers in journals, books and conference proceedings including journals such as Omega, Decision Support Systems, Expert Systems with Applications, Applied Soft Computing, Knowledge and Information Systems, Computers & Industrial Engineering, Knowledge-Based Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, International Journal of Computational Intelligence Systems and Information: An International Interdisciplinary Journal. He has published one book. He has participated in several scientific committees and serves as a reviewer in a wide range of journals including Computers & Industrial Engineering, International Journal of Information Technology and Decision Making, Knowledge-Based Systems, Information Sciences, International Journal of Computational Intelligence Systems and European Journal of Operational Research. He is currently interested in aggregation operators, decision making and computing with words.
For this article, we shall expand the TODIM model to the MADM with the picture fuzzy numbers (PFNs). Firstly, the concept, comparative method and distance of PFNs are introduced and the traditional TODIM model is presented. Then, the expanded TODIM model is developed to solve MADM problems with PFNs. Finally, a numerical example is given to verify the proposed approach.
Atanassov (1986, 1989) proposed the concept of intuitionistic fuzzy sets (IFSs) based on fuzzy set by Zadeh (1965). Atanassov and Gargov (1989) and Atanassov (1994) defined interval-valued intuitionistic fuzzy sets (IVIFSs). The IFSs and IVIFSs have been investigated by many researchers (Bustince and Burillo, 1995; Atanassov et al., 2005; Ronald, 2015; Xu, 2007; Xu and Yager, 2006; Konwar and Debnath, 2017; Wu and Chiclana, 2014; Wang, 2017; Chen, 2011; Chen and Li, 2011; Chen, 2014, 2016; Chen and Chiou, 2015; Garg, 2016; Li, 2011; Zhao and Wei, 2013; Liu, 2017b; Zhang, 2017; Song and Wang, 2017; Ye, 2009, 2010; Wei and Zhao, 2012; Liu, 2017a). Recently, Cuong (2013) developed picture fuzzy set (PFS) and studied the properties and basic operations laws of PFS. Singh (2014) studied the correlation coefficients for PFSs. Son (2015) and Thong (2015) proposed several clustering algorithms with PFSs. Thong (2015) proposed a hybrid method between PF clustering and IF recommender systems. Wei (2016) proposed the cross-entropy for MADM problems with PFNs. Wei (2017a) investigated the picture fuzzy aggregation operators for MADM problems. Wei et al. (2016b) gave the projection models for MADM with picture fuzzy information. Thong and Son (2016b) considered the improvement of FCM on the PFSs. Thong and Son (2016a) proposed the Automatic Picture Fuzzy Clustering (AFC-PFS). Son (2016) proposed a generalized picture distance measure. Son (2017) proposed the generalized picture distance measures and association measures. Son et al. (2017) proposed the picture inference system (PIS). Son and Thong (2017) developed two hybrid forecast models with picture fuzzy clustering.
Many previous studies have captured the DMs’ attitudinal characters in the MADM problems (Gomes and Lima, 1992; Chen et al., 2012; Liu et al., 2014; Wu and Chiclana, 2014). In order to show the risk and uncertainty of the MADM problems simultaneously, more and more scholars have proposed some fuzzy TODIM approach (Konwar and Debnath, 2017; Fan et al., 2013), the intuitionistic fuzzy TODIM approach (Lourenzutti and Krohling, 2013; Krohling et al., 2013), Pythagorean fuzzy TODIM approach (Ren et al., 2016), multi-hesitant fuzzy linguistic TODIM approach (Wang et al., 2016; Wei et al., 2015), interval type-2 fuzzy sets-based TODIM method (Sang and Liu, 2016), intuitionistic linguistic TODIM method (Wang and Liu, 2017) and 2-dimension uncertain linguistic TODIM method (Liu and Teng, 2016). But until now, no research extend TODIM model for PFNs. Therefore, it is necessary to investigate this issue. The purpose of this paper is to expand TODIM model to MADM with PFNs to overcome this limitation. The rest of this paper is organized as follows. In Section 2, we introduce the concepts of PFNs and classical TODIM model. In Section 3 we develop the TODIM model for MADM with PFNs. In Section 4, an illustrative example for potential evaluation of emerging technology commercialization is pointed out and some comparative analysis is conducted. In Section 5 we conclude this paper.
Preliminaries
Some definitions of PFSs are introduced. The operations of PFNs are also provided as they will be utilized in the rest of the paper. At the same time, the process of traditional TODIM approach in decision making is also presented.
An IFS A in X is given by
A={⟨x,μA(x),νA(x)⟩|x∈X}
where μA:X→[0,1] and νA:X→[0,1], where, 0⩽μA(x)+νA(x)⩽1, ∀x∈X. The number μA(x) and νA(x) represents, respectively, the membership degree and non- membership degree of the element x to the set A.
A picture fuzzy set (PFS) A on the universe X is an object of the form
A={⟨x,μA(x),ηA(x),νA(x)⟩|x∈X}
where μA(x)∈[0,1] is called the “degree of positive membership of A”, ηA(x)∈[0,1] is called the “degree of neutral membership of A” and νA(x)∈[0,1] is called the “degree of negative membership of A”, and μA(x),ηA(x),νA(x) satisfy the following condition: 0⩽μA(x)+ηA(x)+νA(x)⩽1, ∀x∈X. Then for x∈X, πA(x)=1−(μA(x)+ηA(x)+νA(x)) could be called the degree of refusal membership of x in X.
If πA(x)=0, then the picture fuzzy set reduces to the Atanassov’s IFSs theory (Atanassov, 1986, 1989). Thus, the Atanassov’s IFSs theory is a special form of the PFSs (Cuong, 2013).
Let α=(μα,ηα,να) be an accuracy function H of a PFN is:
H(α)=μα+ηα+να,H(α)∈[0,1].
Wei (2018a) gave an order relation between two PFNs. <italic>(See</italic> Wei, <xref ref-type="bibr" rid="j_info1190_ref_048">2017a</xref><italic>).</italic>
Let α=(μα,ηα,να) and β=(μβ,ηβ,νβ) be two PFNs, S(α)=1+μα−να2 and S(β)=1+μβ−νβ2 be the scores of α and β, respectively, and let H(α)=μα+ηα+να and H(β)=μβ+ηβ+νβ be the accuracy degrees of α and β, respectively, then if S(α)<S(β), then α<β; if S(α)=S(β), then (1) if H(α)=H(β), then α=β; (2) if H(α)<H(β), then α<β.
Let α=(μα,ηα,να) and β=(μβ,ηβ,νβ) be two PFNs, then the normalized Hamming distance between α=(μα,ηα,να) and β=(μβ,ηβ,νβ) is:
d(α,β)=12(|μα−μβ|+|ηα−ηβ|+|να−νβ|).
The TODIM Approach
Let G={G1,G2,…,Gn} be the set of attributes, w=(w1,w2,…,wn) be the weight vector of attributes Gj, where wj∈[0,1], j=1,2,…,n„ ∑j=1nwj=1. Let A={A1,A2,…,Am} be a discrete set of alternatives. Suppose that A=(aij)m×n be a decision matrix, where aij is the attribute value, given by an expert, for the alternative Ai∈A with respect to the attribute Gj∈G, i=1,2,…,m, j=1,2,…,n. We define wjr=wj/wr(r,j=1,2,…,n) are the relative weight of the attribute Gj to Gr, and wr=max{wj∣j=12,…,n}, and 0⩽wjr⩽1. Then the traditional TODIM model includes the following steps:
Normalize the A=(aij)m×n into B=(bij)m×n.
Compute the dominance degree of Ai over each alternative At for Gj:
δ(Ai,At)=∑j=1nϕj(Ai,At)(i,t=1,2,…,m)
where
ϕj(Aj,At)=wjr(bij−btj)∑j=1nwjr,ifbij>btj;0,ifbij=btj;−1θ(bij−btj)∑j=1nwjrwjr,ifbij<btj,{b_{tj}};\\ {} 0,\hspace{1em}& if{b_{ij}}={b_{tj}};\\ {} -\frac{1}{\theta }\sqrt{\frac{({b_{ij}}-{b_{tj}}){\textstyle\textstyle\sum _{j=1}^{n}}{w_{jr}}}{{w_{jr}}}},\hspace{1em}& if{b_{ij}}<{b_{tj}},\end{array}\right.\]]]>
and the parameter values θ depict the attenuation factor of the losses. If bij−btj>00$]]>, then ϕj(Ai,At) represents a gain; if bij−btj<0, then ϕj(Ai,At) signifies a loss.
Compute the overall dominance of the alternative Ai with the following formula:
ϕ(Ai)=∑t=1mδ(Ai,At)−mini{∑t=1mδ(Ai,At)}maxi{∑t=1mδ(Ai,At)}−mini{∑t=1mδ(Ai,At)},i=1,2,…,m.
Rank and select the best alternative by the overall values ϕ(Ai)(i=1,2,…,m). The alternative with the minimum value is the worst. Inversely, the maximum value is the most desirable one.
TODIM Method for Picture Fuzzy MADM Problems
The following notations are utilized to show MADM problems with PFNs. Let A={A1,A2,…,Am} be a set of alternatives, and G={G1,G2,…,Gm} be a set of attributes. Let w=(w1,w1,…,w1) be the weight vector of attributes, where wj∈[0,1], j=1,2,…,n, ∑j=1nwj=1. Suppose that R=(rij)m×n=(μij,ηij,νij)m×n be a picture fuzzy decision matrix, where μij indicates the degree of positive membership, ηij indicates the degree of neutral membership, νij indicates the degree of negative membership, μij∈[0,1], ηij∈[0,1], νij∈[0,1], μij+ηij+νij⩽1, i=1,2,…,m, j=1,2,…,n.
Then, we extend the TODIM model to solve the MADM problem with PFNs.
Firstly, we can obtain the relative weight of Gj as:
wjr=wjwr,j,r=1,2,…,n,
where wr=max{wj∣j=1,2,…,n}, and 0⩽wjr⩽1.
We calculate the dominance of Ai over alternative At under attribute Gj:
ϕj(Aj,At)=wjrd(rij,rtj)∑j=1nwjr,ifrij>rtj;0,ifrij=rtj;−1θd(rij,dtj)∑j=1nwjrwjr,ifrij<rtj.{r_{tj}};\\ {} 0,\hspace{1em}& if{r_{ij}}={r_{tj}};\\ {} -\frac{1}{\theta }\sqrt{\frac{d({r_{ij}},{d_{tj}}){\textstyle\textstyle\sum _{j=1}^{n}}{w_{jr}}}{{w_{jr}}}},\hspace{1em}& if{r_{ij}}<{r_{tj}}.\end{array}\right.\]]]>d(rij,rtj)=12(|μij−μtj|+|ηij−ηtj|+|νij−νtj|)
where the parameter θ is the attenuation factor of the losses.
In order to indicate the functions ϕj(Aj,At) clearly, we depict it in a matrix under attribute of Gj as:
where j=1,2,…,n, then we can derive the overall dominance degree of the alternative Ai over alternative Aj by
δ(Ai,Aj)=∑j=1nϕj(Ai,At),i,t=1,2,…,m.
Thus, by Eq. (13), the overall dominance matrix is:
Finally, the overall value of Ai is:
δ(Ai)=∑t=1mδ(Ai,At)−mini{∑t=1mδ(Ai,At)}maxi{∑t=1mδ(Ai,At)}−mini{∑t=1mδ(Ai,At)},i=1,2,…,m,
and rank all alternatives, the greater the overall value δ(Ai)(i=1,2,…,m), the better the alternative Ai.
Numerical Example and Comparative AnalysisNumerical Example
With the rapid development of science and technology, the social life, national politics, the economy and the culture has also taken significant changes. Some theory in the traditional single field has been unable to guide the new practice. The new complex issues that appear in people’s social practice can’t be resolved by relying on the knowledge, theories and tools in a single field. Transdisciplinary research of emerging technologies appears on the scene. Evaluating transdisciplinary research of emerging technologies has important theoretical and practical significance. Thus, we shall give a numerical example for potential evaluation of emerging technology commercialization with PFNs. There are five possible emerging technology enterprises (ETES) Ai(i=1,2,3,4,5) to select. The expert selects four attributes to assess the five possible ETES: (1) G1 is the human resources and financial conditions; (2) G2 is the industrialization infrastructure; (3) G3 is the technical advancement; (4) G4 is the development of science and technology. The five possible ETES Ai(i=1,2,3,4,5) are to be assessed with PFNs according to four attributes (whose weighting vector w=(0.2,0.1,0.3,0.4)T), as listed as follows.
R=(0.89,0.08,0.03)(0.42,0.35,0.18)(0.08,0.89,0.02)(0.80,0.11,0.05)(0.23,0.64,0.11)(0.03,0.82,0.13)(0.73,0.15,0.08)(0.73,0.10,0.14)(0.52,0.26,0.05)(0.04,0.85,0.10)(0.68,0.26,0.06)(0.43,0.13,0.25)(0.74,0.16,0.10)(0.02,0.89,0.05)(0.08,0.84,0.06)(0.85,0.09,0.05)(0.68,0.08,0.21)(0.05,0.87,0.06)(0.13,0.75,0.09)(0.65,0.05,0.02).
In the following, we utilize the approach developed for potential evaluation of emerging technology commercialization of five possible ETEs.
Firstly, since w4=(w1,w2,w3,w4), then G4 is the reference attribute and wr=0.4. Thus, w1r=0.50, w2r=0.25, w3r=0.75 and w4r=1.00. Then, we can calculate the dominance degree of the candidate Ai over each candidate At under Gj(j=1,2,3,4). Let θ=2.5, we get:
Secondly, by Eq. (13), and the overall dominance matrix is:
Then, we can obtain δ(Ai)(i=1,2,3,4,5) by Eq. (14):
δ(A1)=1.0000,δ(A2)=0.0000,δ(A3)=0.1475,δ(A4)=0.4586,δ(A5)=0.3170.
Finally, the order is: A1≻A4≻A5≻A3≻A2, and thus the best ETE is A1.
Comparative Analysis
Then, we compare our method with picture fuzzy weighted averaging (PFWA) operator and picture fuzzy weighted geometric (PFWG) operator proposed by Wei (2017a) as follows:
(See Wei, 2017a.) Let aij=(μij,ηij,νij) be a collection of PFNs, w=(w1,w2,…,wn)T be the weight vector of aij(j=1,2,…,n), and wj>00$]]>, ∑j=1nwj=1, then
ri=(μi,ηi,νi)=PFWAw(ri1,ri2,…,rim)=⨁j=1n(wjrij)=(1−∏j=1n(1−μij)wi,∏j=1n(ηij)wi,∏j=1n(νij)wi),ri=(μi,ηi,νi)=PFWGw(ri1,ri2,…,rim)=⨂j=1n(rij)wj=(∏j=1n(μij)wi,1−∏j=1n(1−ηij)wi,1−∏j=1n(1−νij)wi).
The calculating results are shown in Table 1.
The aggregating results of the ETEs by the PFWA (PFWG).
PFWA
PFWG
A1
(0.6880,0.2170,0.0390)
(0.3840,0.5363,0.0521)
A2
(0.6216,0.2020,0.1120)
(0.4211,0.3729,0.1154)
A3
(0.5121,0.2218,0.1077)
(0.4042,0.3270,0.1431)
A4
(0.6547,0.2482,0.0607)
(0.2801,0.5696,0.0632)
A5
(0.5008,0.1647,0.0561)
(0.3131,0.4816,0.0858)
According to Table 2, the score of the ETEs are listed in Table 3.
The score values of the ETEs.
PFWA
PFWG
A1
0.8245
0.6664)
A2
0.7548
0.6529
A3
0.7022
0.6305
A4
0.7970
0.6084
A5
0.7223
0.6137
According to Table 3, the ordering is in Table 4, and the best ETE is A1.
Ordering of the ETEs.
Ordering
PFWA
A1>A4>A2>A5>A3{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$]]>
PFWG
A1>A2>A3>A4>A1{A_{2}}>{A_{3}}>{A_{4}}>{A_{1}}$]]>
From Table 4, it can be seen that two methods have the same best ETE A1 and two methods’ ranking results are slightly different.
Essentially, these two approaches are discrepant for consideration of the DMs’ psychological behaviours. The PFWA and PFWG operators based on the approaches can’t depict the DMs’ psychological behaviours under risk. The picture fuzzy TODIM model can reasonably show the DMs’ psychological behaviours under risk.
Furthermore, we compare our proposed method with picture fuzzy cross-entropy (Wei, 2016), picture fuzzy projection models (Wei et al., 2016b), generalized picture fuzzy distance measure (Son, 2016), similarity measures for picture fuzzy sets (Wei, 2018b) and cosine similarity measures for picture fuzzy sets (Wei, 2017c) as shown in Table 4.
Ordering of the ETEs by using different methods.
Ordering
Picture fuzzy cross-entropy (Wei, 2016)
A1>A4>A2>A5>A3{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$]]>
Picture fuzzy projection models (Wei et al., 2016b)
Similarity measures for picture fuzzy sets (Wei, 2018b)
A1>A4>A2>A5>A3{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$]]>
Cosine similarity measures for picture fuzzy sets (Wei, 2017c)
A1>A4>A2>A5>A3{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$]]>
Conclusion
In this paper, we expand the TODIM model for MADM with the PFNs. Firstly, the definition, comparative method and distance of PFNs and the calculating steps of the traditional TODIM model are introduced. Then, the extended TODIM model is developed to solve MADM problems in which the attribute values are in the PFNs, and its important characteristic is that it can fully depict the decision makers’ bounded rationality. Finally, an example for potential evaluation of emerging technology commercialization is considered to verify the developed model and a comparative analysis is also given. In subsequent works, more and more models with PFNs need to be investigated in uncertain decision making and risk analysis (Zeng et al., 2016; Wei et al., 2018a; Wei, 2017b; Merigo and Casanovas, 2009; Wei and Wei, 2018; Wei and Lu, 2018b; Zeng, 2017; Wei and Lu, 2017; Wei et al., 2018b; Gao et al., 2018a, 2018b; Tang and Wei, 2018; Wei and Lu, 2018a; Wei et al., 2016a; Wei and Zhang, 2018; Wei et al., 2018c; Wang et al., 2018).
Acknowledgements
This paper is supported by the National Natural Science Foundation of China under Grant No. 71571128.
ReferencesAbdellaoui, M., Bleichrodt, H., Paraschiv, C. (2017). Loss aversion under prospect theory: a parameter-free measurement. Atanassov, K. (1986). Intuitionistic fuzzy sets. Atanassov, K. (1989). More on intuitionistic fuzzy sets. Atanassov, K. (1994). Operators over interval-valued intuitionistic fuzzy sets. Atanassov, K., Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Atanassov, K., Pasi, G., Yager, R.R. (2005). Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision making. Bustince, H., Burillo, P. (1995). Correlation of interval-valued intuitionistic fuzzy sets. Chen, T.Y. (2011). Bivariate models of optimism and pessimism in multi-criteria decision-making based on intuitionistic fuzzy sets. Chen, T.Y. (2014). Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis. Chen, T.Y. (2016). An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis. Chen, S.M., Chiou, C.H. (2015). Multiattribute decision making based on interval-valued intuitionistic fuzzy sets, PSO techniques, and evidential reasoning methodology. Chen, T.Y., Li, C.H. (2011). Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis. Chen, L.H., Hung, C.C., Tu, C.C. (2012). Considering the decision maker’s attitudinal character to solve multi-criteria decision-making problems in an intuitionistic fuzzy environ-ment. Cuong, B. (2013). Picture fuzzy sets-first results. Part 1. In: Fan, Z.P., Zhang, X., Chen, F.D., Liu, Y. (2013). Extended TODIM method for hybrid multiple attribute decision making problems. Gao, H., Lu, M., Wei, G.W., Wei, Y. (2018a). Some novel Pythagorean fuzzy interaction aggregation operators in multiple attribute decision making. Gao, H., Wei, G.W., Huang, Y.H. (2018b). Dual hesitant bipolar fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making. Garg, H. (2016). A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Gomes, L., Lima, M. (1992). TODIM: basics and application to multicriteria ranking of projects with environmental impacts. Konwar, N., Debnath, P. (2017). Continuity and Banach contraction principle in intuitionistic fuzzy n-normed linear spaces. Krohling, R.A., Pacheco, A.G.C., Siviero, A.L.T. (2013). IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making. Li, D.F. (2011). Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information. Liu, P.D. (2017a). Multiple attribute decision-making methods based on normal intuitionistic fuzzy interaction aggregation operators. Liu, P.D. (2017b). Some frank aggregation operators for interval-valued intuitionistic fuzzy numbers and their application to group decision making. Liu, P.D., Teng, F. (2016). An extended TODIM method for multiple attribute group decision-making based on 2-dimension uncertain linguistic variable. Liu, H.C., You, J.X., Fan, X.J., Chen, Y.Z. (2014). Site selection in waste management by the VIKOR method using linguistic assessment. Lourenzutti, R., Krohling, R.A. (2013). A study of TODIM in a intuitionistic fuzzy and random environment. Merigo, J.M., Casanovas, M. (2009). Induced aggregation operators in decision making with the Dempster–Shafer belief structure. Ren, P.J., Xu, Z.S., Gou, X.J. (2016). Pythagorean fuzzy TODIM approach to multi-criteria decision making. Ronald, R.R. (2015). A note on measuring fuzziness for intuitionistic and interval-valued fuzzy sets. Sang, X.Z., Liu, X.W. (2016). An interval type-2 fuzzy sets-based TODIM method and its application to green supplier selection. Singh, P. (2014). Correlation coefficients for picture fuzzy sets. Son, L. (2015). DPFCM: a novel distributed picture fuzzy clustering method on picture fuzzy sets. Son, L.H. (2016). Generalized picture distance measure and applications to picture fuzzy clustering. Son, L.H. (2017). Measuring analogousness in picture fuzzy sets: from picture distance measures to picture association measures. Son, L.H., Thong, P.H. (2017). Some novel hybrid forecast methods based on picture fuzzy clustering for weather nowcasting from satellite image sequences. Song, Y.F., Wang, X.D. (2017). A new similarity measure between intuitionistic fuzzy sets and the positive definiteness of the similarity matrix. Son, L.H., Viet, P.V., Hai, P.V. (2017). Picture inference system: a new fuzzy inference system on picture fuzzy set. Tang, X.Y., Wei, G.W. (2018). Models for green supplier selection in green supply chain management with Pythagorean 2-tuple linguistic information. Thong, N.T. (2015). HIFCF: An effective hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis. Thong, P.H., Son, L.H. (2016a). A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality. Thong, P.H., Son, L.H. (2016b). Picture fuzzy clustering: a new computational intelligence method. Wang, F.X. (2017). Interval-valued intuitionistic fuzzy ideals of B-algebras. Wang, S.W., Liu, J. (2017). Extension of the TODIM method to intuitionistic linguistic multiple attribute decision making. Wang, J., Wang, J.Q., Zhang, H.Y. (2016). A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing. Wang, J., Wei, G.W., Wei, Y. (2018). Models for Green supplier selection with some 2-tuple linguistic neutrosophic number Bonferroni mean operators. Wei, G.W. (2016). Picture fuzzy cross-entropy for multiple attribute decision making problems. Wei, G.W. (2017a). Picture fuzzy aggregation operators and their application to multiple attribute decision making. Wei, G.W. (2017b). Picture uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Wei, G.W. (2017c). Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making. Wei, G.W. (2018a). Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Wei, G.W. (2018b). Some similarity measures for picture fuzzy sets and their applications. Wei, G.W., Lu, M. (2017). Pythagorean hesitant fuzzy Hamacher aggregation operators in multiple attribute decision making. Wei, G.W., Lu, M. (2018a). Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making. Wei, G.W., Lu, M. (2018b). Pythagorean fuzzy power aggregation operators in multiple attribute decision making. Wei, G.W., Wei, Y. (2018). Similarity measures of Pythagorean fuzzy sets based on cosine function and their applications. Wei, G.W., Zhang, Z.P. (2018). Some single-valued neutrosophic bonferroni power aggregation operators in multiple attribute decision making. Wei, G.W., Zhao, X.F. (2012). Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making. Wei, C.P., Ren, Z.L., Rodríguez, R.M. (2015). A hesitant fuzzy linguistic TODIM method based on a score function. Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2016a). Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making. Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2016b). Projection models for multiple attribute decision making with picture fuzzy information. Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A. (2018a). Bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. Wei, G.W., Gao, H., Wei, Y. (2018b). Some q-rung orthopair fuzzy heronian mean operators in multiple attribute decision making. Wei, G.W., Lu, M., Tang, X.Y., Wei, Y. (2018c). Pythagorean hesitant fuzzy hamacher aggregation operators and their application to multiple attribute decision making. Wu, J., Chiclana, F. (2014). A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions. Xu, Z.S. (2007). Intuitionistic fuzzy aggregation operators. Xu, Z.S., Yager, R.R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. Ye, J. (2009). Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Ye, J. (2010). Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Zadeh, L.A. (1965). Fuzzy sets. Zhang, Z.M. (2017). Interval-valued intuitionistic fuzzy Frank aggregation operators and their applications to multiple attribute group decision making. Zhao, X.F., Wei, G.W. (2013). Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making. Zeng, S.Z. (2017). Pythagorean fuzzy multiattribute group decision making with probabilistic information and OWA approach. Zeng, Marqués D. P, S.Z., Zhu, F.C. (2016). A new model for interactive group decision making with intuitionistic fuzzy preference relations.