A Method to Multi-Attribute Group Decision-Making Problem with Complex q-Rung Orthopair Linguistic Information Based on Heronian Mean Operators

Peide Liu; Zeeshan Ali; Tahir Mahmood

doi:10.2991/ijcis.d.191030.002

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 12, Issue 2, 2019, Pages 1465 - 1496

A Method to Multi-Attribute Group Decision-Making Problem with Complex q-Rung Orthopair Linguistic Information Based on Heronian Mean Operators

Authors

Peide Liu¹^{, 3}^{, *}, Zeeshan Ali², Tahir Mahmood²

¹School of Economics and Management, Civil Aviation University of China, Tianjin, China

²Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan

³School of Management Science and Engineering, Shandong University of Finance and Economics, Shandong Province, China

^*Corresponding author. Email: peide.liu@gmail.com

Corresponding Author

Peide Liu

Received 29 September 2019, Accepted 25 October 2019, Available Online 25 November 2019.

DOI: 10.2991/ijcis.d.191030.002 How to use a DOI?
Keywords: Complex q-rung orthopair fuzzy sets; Linguistic sets; Complex q-rung orthopair linguistic sets; Heronian mean operators; Geometric Heronian mean operators
Abstract: The notions of complex q-rung orthopair fuzzy sets (Cq-ROFSs) and linguistic sets (LSs) are two different concepts to deal with uncertain information in multi-attribute group decision-making (MAGDM) problems. The Heronain mean (HM) and geometric Heronain mean (GHM) operators are an effective tool used to aggregate some q-rung orthopair linguistic fuzzy numbers (q-ROLFNs) into a single element. The purpose of this manuscript is to propose a new concept called complex q-rung orthopair linguistic sets (Cq-ROLSs) to cope with complex uncertain information in real decision-making problems. Then the fundamental laws and their examples of the Cq-ROLSs are also given. Furthermore, the notions of complex q-rung orthopair linguistic Heronian mean (Cq-ROLHM) operator, complex q-rung orthopair linguistic weighted Heronian mean (Cq-ROLWHM) operator, complex q-rung orthopair linguistic geometric Heronian mean (Cq-ROLGHM) operator, complex q-rung orthopair linguistic weighted geometric Heronian mean (Cq-ROLWGHM) operator are proposed and their basic properties are also discussed. Moreover, we develop a novel approach to MAGDM using proposed operators and a numerical example is used to describe the flexibility and explicitly of the initiated operators. In last, the comparison between proposed method and existing work is also discussed in detail.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The framework of the complex fuzzy set (CFS) was proposed by Ramot et al. [1], which is a generaeronian mean lization of a fuzzy set (FS) [2]. The difference between CFS and FS is that the range of CFS is not restricted to [0, 1], but is extended into a unit disc in a complex plane. The CFS has received more attention in the environment of FS theory. While, Alkouri and Salleh [3] proposed the notions of linguistic variable, hedges and several distances on CFS. Yazdanbakhsh and Dick [4] proposed time-series forecasting via complex fuzzy logic and a systematic review of CFS. Recently, Bi et al. [5] proposed complex fuzzy geometric aggregation operators. Because of its merits and advantages, CFS has been extensively applied to decision-making problems and other fields [6–7]. Because FS and CFS can only describe the membership degree and complex-valued membership degree, and cannot express the non-membership degree and complex-valued non-membership degree. Then the framework of intuitionistic fuzzy set (IFS) is introduced by Atanassov [8] as a generalization of FS by including non-membership degree. The IFS is characterized by two different degrees such as membership and non-membership grades, and their sum is limited to [0, 1]. IFS has been extensively used in different fields [9–10]. Further, Alkouri et al. [11] proposed the framework of complex intuitionistic fuzzy set (CIFS) as a generalization of FS to deal with uncertain and unpredictable information in real-life problems. The CIFS is characterized by complex-valued membership grade and complex-valued non-membership grade in the form of polar coordinates. Because there is a restrict condition in IFS, further, Yager [12] initiated the idea of Pythagorean fuzzy set (PFS) as an effective tool to describe the uncertainty for the multi-attribute group decision making (MAGDM) problems. The notion of PFS is more general than IFS and FS to cope with difficult information in real decision problems. When a decision maker provides (0.6,0.7) for membership and non-membership grades, i.e., 0.6+0.7=1.3≥1, the IFS cannot describe it effectively, but the PFS can describe such kinds of information effectively, i.e., 0.62+0.72=0.36+0.49=0.85<1. Based on PFS, Garg [13,14] proposed a novel correlation coefficient between PFSs, and a new generalized Pythagorean fuzzy Einstein aggregation operators and their application to decision-making were developed. Dick et al. [15] introduced Pythagorean and complex fuzzy operations. Garg [16] further proposed a novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multi-attribute decision-making (MADM) problem. Further, some new MADM methods and operators about PFSs were developed. Ren et al. [17] developed the notion of Pythagorean fuzzy TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision-Making) approach to MADM. Garg [18] proposed a new improved score function of an interval-valued PFSs based TOPSIS method. Wei and Wei [19] proposed the concept of similarity measures for PFSs based on the cosine function and their applications. Garg [20], proposed generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for the multi-criteria decision-making process. Wei [21] proposed the Pythagorean fuzzy interaction aggregation operators and their application to MADM problems.

In some particular cases, the IFS and PFS are failed, if a decision maker provides 0.9,0.7 for membership and non-membership degrees, i.e., 0.9+0.7=1.6≥1 and 0.92+0.72=0.81+0.49=1.30≥1, the IFS and PFS cannot describe effectively such kinds of information. To precisely cope with such kind of problems, Yager [22] proposed the framework of q-rung orthopair fuzzy set (q-ROFS) whose restriction is that the sum of q-power of membership and q-power of non-membership grade is belonging to [0,1]. Obviously, the q-ROFS can describe effectively such kinds of information, i.e., 0.94+0.74=0.7+0.24=0.94≤1. The FS, IFS, and PFS all are the special cases of q-ROFS, this characteristic makes q-ROFS more general than existing FSs. For example, if q=1 and non-membership equals to zero then, the q-ROFS is converted to FS. If q=1, then the q-ROFS is converted to IFS. If q=2, then the q-ROFS is converted to PFS. To better understand the relationship among q-ROFS, PFS, and IFS, please see Figure 1.

Further, Liu and Liu [23] initiated the concept of some q-rung orthopair fuzzy Bonferroni mean operators and their application to MAGDM. Wei et al. [24] proposed some q-rung orthopair fuzzy Heronian mean (HM) operators. Liu and Wang [25,26] proposed some q-rung orthopair fuzzy aggregation operators and their application to MADM based on Archimedean Bonferroni operators of q-rung orthopair fuzzy numbers. Further, the power maclaurin symmetric mean [27], partitioned maclaurin symmetric mean [28], as a powerful operator to aggregate the interrelation among q-ROFNs, were developed. Li et al. [29] proposed q-rung orthopair linguistic HM operators with their application to MAGDM.

Moreover, the linguistic variable (LV), proposed by Zadeh [30], can easily express the qualitative information. Many researchers combined the notion of LV with IFS, PFS, q-ROFS, and proposed the novel concepts of intuitionistic linguistic fuzzy numbers [31], linguistic Pythagorean FS [32] and q-rung orthopair linguistic HM operators [29]. Obviously, these combinations can easily describe the complex fuzzy information.

Consequently, motivated by the idea from IFS to CIFS, it is necessary to extend q-ROFS to complex q-ROFS (Cq-ROFS) because Cq-ROFS is a powerful idea to cope with uncertain and unpredictable information, and it is also a generalization of CFS and FS, whose constraint is like q-ROFS, but the range of membership and non-membership grades are bounded to unit disc in a complex plane instead of [0,1]. The complex-valued membership and complex-valued non-membership grades are represented in the polar form. The q-ROFS copes with one-dimension information at a time in a single elements, which results in data loss sometimes. But, the Cq-ROFS is a powerful tool to deal with uncertain information as compared to q-ROFS, because it contains two-dimension information in a single elements. So by introducing the second dimension to the grade of membership and non-membership, loss of data can be avoided. At the same time, motivated by combining the LV with q-ROFS, it is meaningful to combine the LV with Cq-ROFS, and propose complex q-rung orthopair linguistic number (Cq-ROLN) which is more general than existing fuzzy sets, such as complex Pythagorean linguistic set (CPYLS) and complex intuitionistic linguistic set (CILS). If we take the imaginary part is zero, in the terms of membership grade and non-membership grade, then the Cq-ROLS is convert into q-rung orthopair linguistic set (q-ROLS), i.e., q-ROLS is its special case. If we set the value of parameter q=1 in the environment of q-ROLS, then the q-ROLS is converted for intuitionistic linguistic set (ILS). Similarly, if we take the value of parameter q=2 in the environment of q-ROLS, then the q-ROLS is converted for Pythagorean linguistic set (PYLS). The ILS and PYLS are the particular cases of the Cq-ROLSs. Moreover, Heronain mean (HM) can consider the relationship between any two attributes, compared with the BM which has the same function as HM, however, HM can reduce the operational amount to half of BM. So it is necessary to extend HM to Cq-ROFS and Cq-ROLN, and then to propose a new MADM method based on the proposed operators. Therefore, the motivation and goal of this paper are shown as follows:

Propose the notion of Cq-ROFS and some operational laws, and then explain their characteristics and comparison method.
Propose the notion of Cq-ROLN and some operational laws, and then explain their characteristics and comparison method.
Develop some extended HM operators, such as complex q-rung orthopair linguistic Heronian mean operator (Cq-ROLHM), complex q-rung orthopair linguistic weighted Heronian mean operator Cq-(ROLWHM), complex q-rung orthopair linguistic geometric Heronian mean operator (Cq-ROLGHM), complex q-rung orthopair linguistic weighted geometric Heronian mean operator(Cq-ROLWGHM), and then verify their some properties.
Develop a new MADM method based on the proposed operators.
Give some examples to show the flexibility and superiority of the developed method.

The construct of this manuscript is as follows: in Section 1, we introduce some basic theories, and propose notion of Cq-ROFS; Section 3 proposes the notion of Cq-ROLN and their operational laws. In Section 4, we propose the Cq-ROLHM, Cq-ROLWHM, Cq-ROLGHM, Cq-ROLWGHM operators and their properties. In Section 5, we propose a new method to solve MAGDM problem based on the proposed operators. In Section 6, some examples are given to show the flexibility and superiority of our proposed operators. The conclusion is discussed in Section 7.

2. PRELIMINARIES

In this section, we will review the existing concepts and initiate the idea of Cq-ROFSs. The operational laws of Cq-ROFSs are also discussed in detail.

2.1. The q-ROFS

In this sub-section, we review some basic concepts of q-ROFS, LV, HM, GHM and their operations.

Definition 1.

[22] For ordinary fixed set X, the q-ROFS is given by

(1)P={〈x,t′(x),f′(x)〉/x∈X}

where t′x,f′x:X→0,1 denoted the membership and non-membership degrees respectively, satisfying the condition 0≤t′qx^+f′qx≤1q≥1. The hesitancy degree is defined by μPx=t′qx+f′qx−t′qxf′qx1q. Further, t′x,f′x is called q-ROFN. The geometrical interpretation of Cq-ROFS is shown in Figure 1.

Definition 2.

[22] For two q-ROFNs P1=t′1x,f′1x and P2=t′2x,f′2x, their operational laws are defined by (δ≥1 is a positive number)

(2)P1⊗P2=t′q1+t′q2−t′q1t′q21q,f′1f′2
(3)P1⊗P2=t′1t′2,f′1q+f′2q−f′1qf′2q1q
(4)δP1=1−1−t′q1δ1q,f′1δ
(5)P1δ=t′1δ,1−1−f′1qδ1q

Definition 3.

[22] For q-ROFS P=t′x,f′x, the score and accuracy functions are defined by

(6)SP=t′q−f′q

(7)HP=t′q+f′q

For any two q-ROFNs P1=t′1x,f′1x and P2=t′2x,f′2x, then we have

If SP1>SP2, then P1>P2.
If SP1=SP2, then
1. If HP1>HP2, then If P1>P2.
2. If HP1=HP2, then If P1=P2.

2.2. Linguistic Term Set and HM

Definition 4.

[30] For a linguistic term set S=Si/i=1,2,..,z with odd cardinality, where, z is the cardinality of S, and Si is a linguistic variable. A possible linguistic term set is given by

S=S1,S2,S3,S4,S5,S6,S7=very poor, poor,slightly poor,fair,slightly good,good,very good. The linguistic terms are expressed by PFSs for 5 or 7 terms, which are shown in Figure 2.

HM and geometric Heronian mean (GHM) are a more generalized operators than existing operators like averaging mean operator, geometric mean operator, weighted averaging mean operator, weighted geometric mean operator and more others. The operators which are discussed in [5,20,24,25,29,31,33,34] are all the special cases of the proposed operators. In this article, we will use HM and GHM operators to propose the complex q-rung orthopair linguistic Heronian mean and complex q-rung orthopair linguistic geometric Heronian mean operators.

Definition 5.

[29] For a set of crisp numbers Pii=1,2,..,n withs,t>0 , Heronian mean (HM) is given by

(8)HMs,tP1,P2,..,Pn=2nn+1∑i=1n∑j=inPisPjt1s+t

Definition 6.

[29] For a family of crisp numbers Pii=1,2,..,nwiths,t>0 , the Geometric Heronian mean (GHM) is given by

(9)GHMs,t(P1,P2,..,Pn)=(1s+t∏i=1n∏j=in(sPi+tPj)2n(n+1))

2.3. The Complex q-Rung Orthopair Fuzzy Set

In this section, we will propose the notion of complex q-rung orthopair fuzzy set (Cq-ROFS) and their operations.

Definition 7.

For ordinary fixed set X, the Cq-ROFS is given by

(10)P={〈x,t′(x),f′(x)〉/x∈X}

where t′x=txei2πWtx and f′x=fxei2πWfx denoted complex-valued membership and non-membership degrees respectively, satisfying the condition 0≤tqx+fqx≤1 and 0≤Wtxq+Wfxq≤1,q≥1. The hesitancy degree is defined by μx=1−tqx+fqx1qei2π1−Wtxq+Wfxq1q. Moreover, txei2πWtx,fxei2πWfx is called complex q-rung orthopair fuzzy number (Cq-ROFN). Simply we write tei2πWt,fei2πWf.

Definition 8.

For two Cq-ROFNs P1=t1ei2πWt1,f1ei2πWf1 and P2=t2ei2πWt2,f2ei2πWf2, the operational laws are defined by ( δ≥1 is a positive number)

(11)P1⊗P2=t1q+t2q−t1qt2q1qei2πWt1q+Wt2q−Wt1qWt2q1q,f1f2ei2πWf1Wf2
(12)P1⊗P2=(t1t2ei2πWt1Wt2,(f1q+f2q−f1qf2q)1qei2π(Wf1q+Wf2q−Wf1qWf2q)1q)
(13)δP1=((1−(1−t1q)δ)1qei2π(1−(1−Wt1q)δ)1q,f1δei2πWf1δ)
(14)P1δ=(t1δei2πWt1δ,(1−(1−f1q)δ)1qei2π(1−(1−Wf1q)δ)1q)

Next, we will examine a numerical example for Cq-ROFNs as follows:

Example 1

We consider the two Cq-ROFNs for q=2 and λ=3 such that P1=0.86ei2π0.99,0.10ei2π0.03 and P2=0.99ei2π0.99,0.01ei2π0.09, then we have

P1⊕P2=0.862+0.992−0.8620.99212ei2π0.992+0.992−0.9920.99212,0.10.01ei2π0.090.03=0.997ei2π0.9998,0.001ei2π0.003
P1⊗P2=0.860.99ei2π0.990.99,0.12+0.012−0.120.01212ei2π0.032+0.092−0.0320.09212=0.85ei2π0.001,0.1ei2π0.09
λP1=1−1−0.862312ei2π1−1−0.992312,0.13ei2π.033=0.99ei2π0.99,0.001ei2π0.000027
P1λ=0.863ei2π.993,1−1−0.12312ei2π1−1−.032312=0.64ei2π0.97,0.17ei2π0.05

Definition 9.

For a Cq-ROFN P=tei2πWt,fei2πWf, the score and accuracy functions are defined by

(15)S(P)=12(tq−fq+Wtq−Wfq)

(16)H(P)=12(tq+fq+Wtq+Wfq)

Definition 10.

For any two Cq-ROFNs P1=t1ei2πWt1,f1ei2πWf1 and P2=t2ei2πWt2,f2ei2πWf2, then we have the comparison method as follows:

If SP1>SP2, then P1>P2.
If SP1=SP2, then
1. If HP1>HP2, then If P1>P2
2. If HP1=HP2, then If P1=P2

Example 2

We consider the two Cq-ROFNs for q=2 and such that P1=0.86ei2π0.99,0.10ei2π0.03 and P2=0.99ei2π0.99,0.01ei2π0.09. Then the score values of the P1 and P2 are calculated as follows:

S(P1)=12(0.862−0.12+0.992−0.032)=12(0.74−0.01+0.98−0.0009)=0.86S(P1)=0.86

Similarly

S(P2)=12(0.992−0.012+0.992−0.092)=12(0.98−0.0001+0.98−0.0081)=0.98S(P2)=0.98

So it is clear that SP2>SP1, then we say that P2>P1. If P2=SP1, the we will use the accuracy function of the Cq-ROFNs.

3. COMPLEX q-RUNG ORTHOPAIR LINGUISTIC SET

Motivated by the notion of Cq-ROFS and LV, we will initiate the novelty of Cq-ROLS by combing the two different concepts. Throughout this article, S¯ is represented the continuous linguistic term set of S=Si/i=1,2,..,z.

Definition 11.

A Cq-ROLS on a universal fixed set X is given by

(17)P={〈x,(Sθ(x),(t′(x),f′(x)))〉/x∈X}

where Sθx∈S¯, t′x=txei2πWtx and f′x=fxei2πWfx denoted positive and negative complex-valued degrees respectively, holds 0≤tqx+fqx≤1 and 0≤Wtxq+Wfxq≤1,q≥1. Further, the refusal degree is defined by: μx=1−tqx+fqx1qei2π1−Wtxq+Wfxq1q. Moreover, Sθx,txei2πWtx,fxei2πWfx is called complex q-rung orthopair linguistic number (Cq-ROLN). Simply, we write Sθ,tei2πWt,fei2πWf.

Next, we defined some operations for Cq-ROLNs.

Definition 12.

Let P=Sθ,tei2πWt,fei2πWf,P1=Sθ1,t1ei2πWt1,f1ei2πWf1 and P2=Sθ2,t2ei2πWt2,f2ei2πWf2 be a three Cq-ROLNs with λ≥1, then we have

(18)P1⊕P2=(Sθ1+θ2,((t1q+t2q−t1qt2q)1qei2π(Wt1q+Wt2q−Wt1qWt2q)1q,(f1f2)ei2π(Wf1Wf2)))
(19)P1⊗P2=Sθ1×θ2,t1t2ei2πWt1Wt2,f1q+f2q−f1qf2q1qei2πWf1q+Wf2q−Wf1qWf2q1q
(20)λP=Sλ×θ,1−1−tqλ1qei2π1−1−Wtqλ1q,fλei2πWfλ
(21)Pλ=Sθλ,tλei2πWtλ,1−1−fqλ1qei2π1−1−Wfqλ1q

First, we give the numerical example for Cq-ROLNs and the four points of Definition 8. Then, we will proposed the ideas of score function and accuracy function for compression between two Cq-ROFLNs.

Example 3

We will consider the two Cq-ROLNs P1=S4.6,0.86ei2π0.99,0.10ei2π0.03 and P2=(S2.4,(0.99ei2π(0.99),0.01ei2π(0.09))) with q=2 and λ=3 , then we can get

P1⊕P2=S4.6+2.4,0.862+0.992−0.8620.99212ei2π0.992+0.992−0.9920.99212,0.10.01ei2π0.090.03=S7,0.997ei2π0.9998,0.001ei2π0.003,
P1⊗P2=S4.6×2.4,0.860.99ei2π0.990.99,0.12+0.012−0.120.01212ei2π0.032+0.092−0.0320.09212=S11.04,0.85ei2π0.001,0.1ei2π0.09,
λP1=S3×4.6,1−1−0.862312ei2π1−1−0.992312,.13ei2π.033=S13.8,0.99ei2π0.99,0.001ei2π0.000027,
P1λ=S4.63,0.863ei2π.993,1−1−0.12312ei2π1−1−.032312=S13.8,0.64ei2π0.97,0.17ei2π0.05

Definition 13.

The score function and accuracy function of Cq-ROLN P=Sθ,tei2πWt,fei2πWf are defined as

(22)SP=12tq−fq+Wtq−Wfq×θ

(23)HP=12tq+fq+Wtq+Wfq×θ

Definition 14.

Let P1=Sθ1,t1ei2πWt1,f1ei2πWf1 and P2=Sθ2,t2ei2πWt2,f2ei2πWf2 be a two Cq-ROLNs, then

If SP1>SP2, then P1>P2.
If SP1=SP2, then
1. If HP1>HP2, then P1>P2.
2. If HP1=HP2, then P1=P2.

Example 4

We will consider the two Cq-ROLNs P1=S4.6,0.86ei2π0.99,0.10ei2π0.03 and P2=S2.4,0.99ei2π0.99,0.01ei2π0.09 for q=2 . Then the score values of the P1 and P2 are calculated as

S(P1)=12((0.862−.12)+(0.992−0.032))×4.6=0.86×4.6S(P1)=3.86

Similarly

S(P2)=12((0.992−.012)+(0.992−0.092))×2.4=0.98×4.6S(P2)=2.55

So it is clear that SP1>SP2, then we say that P1>P2. If SP2=SP1, then we will use the accuracy function of the Cq-ROFLNs.

4. COMPLEX q-RUNG ORTHOPAIR LINGUISTIC HM OPERATORS

In this section, we generalize the HM operator to Cq-ROLS and propose the concepts of Cq-ROLHM, Cq-ROLWHM, Cq-ROLGHM, Cq-ROLWGHM operators and discuss their properties in detailed, where s,t.

4.1. Complex q-Rung Orthopair Linguistic Heronian Mean (Cq-Rolhm) Operators

Definition 15.

Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, then the Cq−ROLHMs,t is defined: Cq−ROLHMs,t:ξn→ξ by

(24)Cq−ROLHMs,tP1,P2,..,Pn=2nn+1∑i=1n∑j=InPisPjt1s+t

where ξn denotes the family of all Cq-ROLNs.

According to the operational laws of Cq-ROLNs, we can get the following results.

Theorem 1

Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, we can get

(25)Cq−ROLHMs,t(P1,P2,..,Pn)=S(2n(n+1)∑i=1n∑j=Inθisθjt)1s+t,(1−∏i=1n∏j=In(1−tisqtjtq)2n(n+1))1q(s+t)ei2π1−∏i=1n∏j=In(1−WtisqWtjtq)2n(n+1)1q(s+t),1−(1−∏i=1n∏j=In(1−(1−fiq)s(1−fjq)t)2qn(n+1))1s+t1qei2π1−1−∏i=1n∏j=In(1−(1−Wfiq)s(1−Wfjq)t)2qn(n+1)1s+t1q

Proof:

Using the Definition 12, we get

Pis=(Sθis,(tisei2πWtis,(1−(1−fiq)s)1qei2π(1−(1−Wfiq)s)1q))

Pjt=(Sθjt,(tjtei2πWtjt,(1−(1−fjq)t)1qei2π(1−(1−Wfjq)t)1q))

PisPjt=Sθisθjt,tistjtei2πWtisWtjt,1−1−fiqs1−fjqtei2π1−1−Wfiqs1−Wfjqt

∑j=1nPisPjt=S∑j=1nθisθjt,(1−∏j=1n(1−tisqtjtq))1qei2π(1−∏j=1n(1−WtisqWtjtq))1q,∏j=1n(1−(1−fiq)s(1−fjq)t)ei2π∏j=1n(1−(1−Wfiq)s(1−Wfjq)t)

then

∑i=1n∑j=1nPisPjt=S∑i=1n∑j=1nθisθjt,(1−∏i=1n(∏j=In(1−tisqtjtq)))1qei2π(1−∏i=1n(∏j=In(1−WtisqWtjtq)))1q,∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t)ei2π∏i=1n∏j=in(1−(1−Wfiq)s(1−Wfjq)t)

2n(n+1)∑i=1n∑j=1nPisPjt=S2n(n+1)∑i=1n∑j=1nθisθjt,(1−(∏i=1n∏j=in(1−tisqtjtq))2n(n+1))1qei2π(1−(∏i=1n∏j=in(1−WtisqWtjtq))2n(n+1))1q,(∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t))2n(n+1)ei2π(∏i=1n∏j=in(1−(1−Wfiq)s(1−Wfjq)t))2n(n+1)

So, we obtained

Cq−ROLHMs,t(P1,P2,..,Pn)=(2n(n+1)∑i=1n∑j=1nPisPjt)1s+t=S(2n(n+1)∑i=1n∑j=inθisθjt)1s+t,(1−∏i=1n∏j=in(1−tisqtjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=in(1−WtisqWtjtq)2n(n+1))1q(s+t),(1−(1−∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t)2qn(n+1))1s+t)1qei2π1−1−∏i=1n∏j=in(1−(1−Wfiq)s(1−Wfjq)t)2qn(n+1)1s+t1q

Further, we discuss the properties of Cq-ROLNs as follows.

Theorem 2.

(Monotonicity) Let Pi=Sθi,tiei2πWti,fiei2πWfi and Pi′=Sθi′,ti′ei2πWti′,fi′ei2πWfi′,i=1,2,…,n be two families of Cq-ROLNs, if Pi≤Pi′⇔θi≤θi′,ti≤ti′,Wti≤Wti′ and fi≤fi′, Wfi≤Wfi′ for all i=1,2,…,n. Then

Cq−ROLHMs,tP1,P2,..,Pn≤Cq−ROLHMs,tP1,P2,..,Pn

Proof.

Since Pi≤Pi′⇔θi≤θi′,ti≤ti′,Wti≤Wti′ and fi≥fi′, Wfi≥Wfi′ and Pj≤Pj′⇔θj≤θj′,tj≤tj′,Wtj≤Wtj′ and fj≥fj′, Wfj≥Wfj′ for all i=1,2,…,n and j=i,i+1,..,n. Then it is clear that for linguistic number

(2n(n+1)∑i=1n∑j=inθisθjt)1s+t≤(2n(n+1)∑i=1n∑j=inθi′sθj′t)1s+t

Nest we will check the real-valued membership grade such that ti≤ti′,Wti≤Wti′ and tj≤tj′,Wtj≤Wtj′, then

tisqtjtq≤ti′sqtj′tq⇒1−tisqtjtq≥1−ti′sqtj′tq⇒1−tisqtjtq1nn+1≥1−ti′sqtj′tq1nn+1

⇒∏i=1n∏j=in1−tisqtjtq2nn+1≥∏i=1n∏j=1n1−ti′sqtj′tq1nn+1

⇒1−∏i=1n∏j=in1−tisqtjtq2nn+11qs+t≤1−∏i=1n∏j=1n1−ti′sqtj′tq1nn+11qs+t

Similarly procedure for imaginary-valued membership grades, we get

⇒1−∏i=1n∏j=in1−WtisqWtjtq2nn+11qs+t≤1−∏i=1n∏j=in1−Wti′sqWtj′tq2nn+11qs+t

Combined both values, we have

(1−∏i=1n∏j=in(1−tisqtjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=in(1−WtisqWtjtq)2n(n+1))1q(s+t)≤(1−∏i=1n∏j=in(1−ti′sqtj′tq)1n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=in(1−Wti′sqWtj′tq)2n(n+1))1q(s+t)

Nest we will describe the real-valued non-membership grade such that fi≥fi′, Wfi≥Wfi′ and fj≥fj′, Wfj≥Wfj′. Then

fiqfjq≥fi′qfj′q⇒1−fiqs1−fjqt≥1−fi′qs1−fj′qt

⇒1−1−fiqs1−fjqt2qnn+1≤1−1−fi′qs1−fj′qt2qnn+1

⇒∏i=1n∏j=in1−1−fiqs1−fjqt2qnn+1≤∏i=1n∏j=in1−1−fi′qs1−fj′qt2qnn+1

⇒1−∏i=1n∏j=in1−1−fiqs1−fjqt2qnn+1≥1−∏i=1n∏j=in1−1−fi′qs1−fj′qt2qnn+1

⇒1−∏i=1n∏j=in1−1−fiqs1−fjqt2qnn+11s+t≥1−∏i=1n∏j=in1−1−fi′qs1−fj′qt2qnn+11s+t

⇒1−1−∏i=1n∏j=in1−1−fiqs1−fjqt2qnn+11s+t≤1−1−∏i=1n∏j=in1−1−fi′qs1−fj′qt2qnn+11s+t

⇒1−1−∏i=1n∏j=in1−1−fiqs1−fjqt2qnn+11s+t1q≤1−1−∏i=1n∏j=in1−1−fi′qs1−fj′qt2qnn+11s+t1q

Similarly procedure for imaginary-valued non-membership grades, we get

⇒1−1−∏i=1n∏j=in1−1−Wfiqs1−Wfjqt2qnn+11s+t1q≤1−1−∏i=1n∏j=in1−1−Wfi′Qs1−Wfj′Qt2qnn+11s+t1q

Combined both values, we have

1−(1−∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t)2qn(n+1))1s+t1qei2π1−1−∏i=1n∏j=in(1−(1−Wfiq)s(1−Wfjq)t)2qn(n+1)1s+t1q

So by combining the values of complex-valued membership and complex-valued non-membership grades, then we get

S2n(n+1)∑i=1n∑j=inθisθjt1s+t,1−∏i=1n∏j=in(1−tisqtjtq)2n(n+1)1q(s+t)ei2π1−∏i=1n∏j=in(1−WtisqWtjtq)2n(n+1)1q(s+t),1−1−∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t)2qn(n+1)1s+t1qei2π1−1−∏i=1n∏j=in(1−(1−Wfiq)s(1−Wfjq)t)2qn(n+1)1s+t1q

≤S2nn+1∑i=1n∑j=inθi′sθj′t1s+t,1−∏i=1n∏j=in1−ti′sqtj′tq2nn+11qs+tei2π1−∏i=1n∏j=in1−Wti′sqWtj′tq2nn+11qs+t,1−1−∏i=1n∏j=in1−1−fi′qs1−fj′qt2qnn+11s+t1qei2π1−1−∏i=1n∏j=in1−1−Wfi′qs1−Wfj′qt2qn(n+1)1s+t1q

(2n(n+1)∑i=1n∑j=inPisPjt)1s+t≤(2n(n+1)∑i=1n∑j=inQisQjt)1s+t

i.e., Cq−ROLHMs,tP1,P2,..,Pn≤Cq−ROLHMs,tQ1,Q2,..,Qn

Hence proved the result.

Theorem 3

(Idempotency) Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, if Pi=P for all i=1,2,…,n. Then

Cq−ROLHMs,tP1,P2,..,Pn=P

Proof.

If Pi=P, for all i=1,2,…,n, then

Cq−ROLHMs,t(P1,P2,..,Pn)=(S(2n(n+1)∑i=1n∑j=inθisθjt)1s+t,((1−∏i=1n∏j=in(1−tisqtjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=in(1−WtisqWtjtq)2n(n+1))1q(s+t),(1−(1−∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=in(1−(1−Wfiq)s(1−Wfjq)t)2qn(n+1))1s+t)1q))

=(S(2n(n+1)∑i=1n∑j=inθsθt)1s+t,((1−∏i=1n∏j=in(1−tsqttq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=in(1−WtsqWttq)2n(n+1))1q(s+t),(1−(1−∏i=1n∏j=in(1−(1−fq)s(1−fq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=in(1−(1−Wfq)s(1−Wfq)t)2qn(n+1))1s+t)1q))

=(S(θsθt)1s+t,((tstt)1(s+t)ei2π(WtsWtt)1(s+t),(1−(1−fq)s(1−fq)t)1s+tei2π(1−(1−Wfq)s(1−Wfq)t)1s+t))=PsPt1s+t=Ps+t1s+t=P

Hence proved the result.

Theorem 4

(Boundedness) The Cq-ROLHM operator lies between the max and min operators

minP1,P2,..,Pn≤Cq−ROLHMs,tP1,P2,..,Pn≤maxP1,P2,..,Pn

Proof.

When, we consider a=minP1,P2,..,Pn and b =maxP1,P2,..,Pn, then using the result of monotonicity, we have

mina,a,a,..,a≤Cq−ROLHMs,tP1,P2,..,Pn≤maxb,b,b,..,b

Moreover, mina,a,a,..,a=a and maxb,b,b,..,b=b, then

a≤Cq−ROLHMs,tP1,P2,..,Pn≤b

That is

minP1,P2,..,Pn≤Cq−ROLHMs,tP1,P2,..,Pn≤maxP1,P2,..,Pn

Hence completed the result.

4.2. Special Cases

In this sub-section, the particular cases of Cq-ROLHM operator is discuss about the parameters s and t.

When t→0, we have
Cq−ROLHMs,0(P1,P2,..,Pn)=limt→0(S(2n(n+1)∑i=1n∑j=inθisθjt)1s+t,((1−∏i=1n∏j=in(1−tisqtjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−Wti sqWtj tq)2n(n+1))1q(s+t),(1−(1−∏i=1n∏j=in(1−(1−fiq)s(1−fjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wfi q)s(1−Wfj q)t)2qn(n+1))1s+t)1q))

=(S(2n(n+1)∑i=1n(n+1−i)θis)1s,((1−(∏i=1n(1−tisq)(n+1−i))2n(n+1))1qsei2π(1−(∏i=1n(1−Wtisq)(n+1−i))2n(n+1))1qs,(1−(1−(∏i=1n(1−(1−fiq)s)(n+1−i))2qn(n+1))1s)1qei2π(1−(1−(∏i=1n(1−(1−Wfiq)s)(n+1−i))2qn(n+1))1s)1q))
and it is called q-rung orthopair linguistic generalized linear descending weighted mean (q-ROLGLDWM) operator.
When s→0, then
Cq−ROLHM0,t(P1,P2,..,Pn)=lims→0(S(2n(n+1)∑i=1n∑j=1nθisθjt)1s+t,((1−∏i=1n∏j=1n(1−tisqtjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−WtisqWtjtq)2n(n+1))1q(s+t),(1−(1−∏i=1n∏j=1n(1−(1−fiq)s(1−fjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wfiq)s(1−Wfjq)t)2qn(n+1))1s+t)1q))

=S(2n(n+1)∑i=1niθit)1t,(1−(∏i=1n(1−titq)i)2n(n+1))1qtei2π(1−(∏i=1n(1−Wtitq)i)2n(n+1))1qt,(1−(1−(∏i=1n(1−(1−fiq)t)i)2qn(n+1))1t)1qei2π(1−(1−(∏i=1n(1−(1−Wfiq)t)i)2qn(n+1))1t)1q
and it is called q-rung orthopair linguistic generalized linear ascending weighted mean (q-ROLGLAWM) operator.
When s=t=1, then
Cq−ROLHM1,1(P1,P2,..,Pn)=(S(2n(n+1)∑i=1n∑j=1nθi1θj1)12,((1−∏i=1n∏j=1n(1−tiqtjq)2n(n+1))12qei2π(1−∏i=1n∏j=1n(1−WtiqWtjq)2n(n+1))12q,(1−(1−∏i=1n∏j=1n(1−(1−fiq)1(1−fjq)1)2qn(n+1))12)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wfiq)1(1−Wfjq)1)2qn(n+1))12)1q))
is called q-rung orthopair linguistic line Heronian mean (q-ROLLHM) operator.
When q=2, then
Cq−ROLHMs,t(P1,P2,..,Pn)=(S(2n(n+1)∑i=1n∑j=1nθisθjt)1s+t,((1−∏i=1n∏j=1n(1−ti2stj2t)2n(n+1))12(s+t)ei2π(1−∏i=1n∏j=1n(1−Wti2sWtj2t)2n(n+1))12(s+t),(1−(1−∏i=1n∏j=1n(1−(1−fi2)s(1−fj2)t)4n(n+1))1s+t)12ei2π(1−(1−∏i=1n∏j=1n(1−(1−Wfi2)s(1−Wfj2)t)4n(n+1))1s+t)12))
is called pythagorean linguistic Heromian mean (PLHM) operator.
When q=1, then
Cq−ROLHMs,t(P1,P2,..,Pn)=(S(2n(n+1)∑i=1n∑j=1nθisθjt)1s+t,((1−∏i=1n∏j=1n(1−tistjt)2n(n+1))1(s+t)ei2π(1−∏i=1n∏j=1n(1−WtisWtjt)2n(n+1))1(s+t),(1−(1−∏i=1n∏j=1n(1−(1−fi1)s(1−fj1)t)2n(n+1))1s+t)1ei2π(1−(1−∏i=1n∏j=1n(1−(1−Wfi1)s(1−Wfj1)t)2n(n+1))1s+t)1))
is called intuitionistic linguistic Heromian mean (ILHM) operator.

4.3. Complex q-Rung Orthopair Linguistic Weighted Heronian Mean (Cq-Rolwhm) Operators

Last sub-section, we proposed the Cq-ROFHM operator without weight vectors. Therefore, we propose the weighted form of Cq-ROFHM called Cq-ROFWHM operator. Further, ω=ω1,ω2,..,ωnT represented the weight vectors with ∑i=1nωi=1.

Definition 16

Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, then the Cq−ROLWHMs,t is defined: Cq−ROLWHMs,t:ξn→ξ by

(26)Cq−ROLWHMs,tP1,P2,..,Pn=2nn+1∑i=1n∑j=1nnωiPisnωjPjt1s+t

where ξn is denoted the family of all Cq-ROLNs.

According to the operational laws of Cq-ROLNs, we can get the following result.

Theorem 5

Let Pi=Sθi,tiei2πWti,fiei2πWfi,(i=1,2,…,n) be a family of Cq-ROLNs. According the Definition 16 and the operational laws of Cq-ROLNs, we have

(27)Cq−ROLWHMs,t(P1,P2,..,Pn)=(S(2n(n+1)∑i=1n∑j=1n(nωiθi)s(nωjθj)t)1s+t,((1−∏i=1n∏j=1n(1−(1−(1−tiq)nωi)2sn(n+1)(1−(1−tjq)nωj)2tn(n+1)))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−(1−(1−Wtiq)nωi)2sn(n+1)(1−(1−Wtjq)nωj)2tn(n+1)))1q(s+t),(1−(1−∏i=1n∏j=1n(1−(1−finωiq)s(1−fjnωjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wfinωiq)s(1−Wfjnωjq)t)2qn(n+1))1s+t)1q))

Proof:

Similar to Theorem 1.

Further, we explore some properties of Cq-ROLNs as follows:

Theorem 6

(Monotonicity) Let Pi=Sθi,tiei2πWti,fiei2πWfi and Qi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be two families of Cq-ROLNs, if Pi≤Qi for all i=1,2,…,n. Then

Cq−ROLWHMs,tP1,P2,..,Pn≤Cq−ROLWHMs,tQ1,Q2,..,Qn

Proof:

Similar to Theorem 2.

Theorem 7

(Idempotency) Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, if Pi=P for all i=1,2,…,n. Then

Cq−ROLWHMs,tP1,P2,..,Pn=P

Proof:

Similar to Theorem 3.

Theorem 8

(Boundedness) The Cq-ROLWHM operator lies between the max and min operators

minP1,P2,..,Pn≤Cq−ROLWHMs,tP1,P2,..,Pn≤maxP1,P2,..,Pn

Proof:

Similar to Theorem 3.

4.4. Complex q-Rung Orthopair Linguistic Geometric Heronian Mean (Cq-Rolghm) Operators

Definition 17.

Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, then the Cq−ROLGHMs,t is defined: Cq−ROLGHMs,t:ξn→ξ by

(28)Cq−ROLGHMs,t(P1,P2,..,Pn)=(1s+t∏i=1n∏j=in(sPi+tPj)2n(n+1))

where ξn is denoted the family of all Cq-ROLNs.

According to the operational laws of Cq-ROLNs, we can get the following result.

Theorem 9

Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, we can get

(29)Cq−ROLGHMs,t(P1,P2,..,Pn)=(S(1s+t∏i=1n∏j=1n(sθi+tθj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−tiq)s(1−tjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wtiq)s(1−Wtjq)t)2qn(n+1))1s+t)1q,(1−∏i=1n∏j=1n(1−fisqfjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−WfisqWfjtq)2n(n+1))1q(s+t)))

Proof.

Similar to Theorem 1.

Further, we discuss some properties of Cq-ROLNs as follows:

Theorem 10

(Monotonicity) Let Pi=Sθi,tiei2πWti,fiei2πWfi and Qi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be two families of Cq-ROLNs, if Pi≤Qi for all i=1,2,…,n. Then

Cq−ROLGHMs,tP1,P2,..,Pn≤Cq−ROLGHMs,tQ1,Q2,..,Qn

Proof:

Similar to Theorem 2.

Theorem 11

(Idempotency) Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, if Pi=P for all i=1,2,…,n. Then

Cq−ROLGHMs,tP1,P2,..,Pn=P

Proof:

Similar to Theorem 3.

Theorem 12

(Boundedness) The Cq-ROLWHM operator lies between the max and min operators

minP1,P2,..,Pn≤Cq−ROLGHMs,tP1,P2,..,Pn≤maxP1,P2,..,Pn

Proof.

Similar to Theorem 4.

4.5. Special Cases

In this sub-section, the particular cases of Cq-ROLGHM operator is discussed based on the parameters s and t.

When t→0, then
Cq−ROLGHMs,0(P1,P2,..,Pn)=limt→0(S(1s+t∏i=1n∏j=1n(sθi+tθj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−tiq)s(1−tjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wtiq)s(1−Wtjq)t)2qn(n+1))1s+t)1q,(1−∏i=1n∏j=1n(1−fisqfjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−WfisqWfjtq)2n(n+1))1q(s+t)))

=(S(1s(∏j=in(sθi)n+1−i)2n(n+1)),((1−(1−(∏j=1n(1−(1−tiq)s)n+1−i)2n(n+1))1s+t)1qei2π(1−(1−(∏j=1n(1−(1−Wtiq)s)n+1−i)2n(n+1))1s+t)1q,(1−(∏j=1n(1−fisq)n+1−i)2n(n+1))1q(s+t)ei2π(1−(∏j=1n(1−Wfisq)n+1−i)2n(n+1))1q(s+t)))
is called q-rung orthopair linguistic generalized geometric linear descending weighted mean (q-ROLGGLDWM) operator.
When t→0, then
Cq−ROLGHM0,t(P1,P2,..,Pn)=lims→0(S(1s+t∏i=1n∏j=1n(sθi+tθj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−tiq)s(1−tjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wtiq)s(1−Wtjq)t)2qn(n+1))1s+t)1q,(1−∏i=1n∏j=1n(1−fisqfjtq)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−WfisqWfjtq)2n(n+1))1q(s+t)))

=(S(1t(∏j=1n(tθj)i)2n(n+1)),((1−(1−(∏i=1n(1−(1−tjq)t)i)2n(n+1))1t)1qei2π(1−(1−(∏i=1n(1−(1−Wtjq)t)i)2n(n+1))1t)1q,(1−(∏i=1n(1−fjtq)i)2n(n+1))1q(t)ei2π(1−(∏i=1n(1−Wfjtq)i)2n(n+1))1q(t)))
is called q-rung orthopair linguistic generalized geometric linear ascending weighted mean (q-ROLGGLAWM) operator.
When s=t=1, then
Cq−ROLGHM1,1(P1,P2,..,Pn)=(S(1s+t∏i=1n∏j=1n(sθi+tθj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−tiq)1(1−tjq)1)2qn(n+1))12)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wtiq)1(1−Wtjq)1)2qn(n+1))12)1q,(1−∏i=1n∏j=1n(1−fiqfjq)2n(n+1))12qei2π(1−∏i=1n∏j=1n(1−WfiqWfjq)2n(n+1))12q))
is called q-rung orthopair linguistic geometric line Heronian mean (q-ROLGLHM) operator.
When q=2, then
Cq−ROLGHMs,t(P1,P2,..,Pn)=(S(1s+t∏i=1n∏j=1n(sθi+tθj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−ti2)s(1−tj2)t)2qn(n+1))1s+t)12ei2π(1−(1−∏i=1n∏j=1n(1−(1−Wti2)s(1−Wtj2)t)2qn(n+1))1s+t)12,(1−∏i=1n∏j=1n(1−fi2sfj2t)2n(n+1))12(s+t)ei2π(1−∏i=1n∏j=1n(1−Wfi2sWfj2t)2n(n+1))12(s+t)))
is called pythagorean linguistic geometric Heromian mean (PLGHM) operator.
When q=1, then
Cq−ROLGHMs,t(P1,P2,..,Pn)=(S(1s+t∏i=1n∏j=1n(sθi+tθj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−ti1)s(1−tj1)t)2qn(n+1))1s+t)1ei2π(1−(1−∏i=1n∏j=1n(1−(1−Wti1)s(1−Wtj1)t)2qn(n+1))1s+t)1,(1−∏i=1n∏j=1n(1−fisfjt)2n(n+1))1(s+t)ei2π(1−∏i=1n∏j=1n(1−WfisWfjt)2n(n+1))1(s+t)))
is called intuitionistic linguistic geometric Heronian mean (ILGHM) operator.

4.6. Complex q-Rung Orthopair Linguistic Weighted Geometric Heronian Mean (Cq-Rolwghm) Operators

Last sub-section, we proposed the Cq-ROLGHM operator without weight vectors. Therefore, we will propose the weighted form of the Cq-ROLGHM called Cq-ROLWGHM operator. Further, ω=ω1,ω2,..,ωnT represented the weight vectors with ∑i=1nωi=1.

Definition 18.

Let Pi=Sθi,tiei2πWti,fiei2πWfi,(i=1,2,…,n) be a family of Cq-ROLNs, then the Cq−ROLWGHMs,t is defined: Cq−ROLWGHMs,t:ξn→ξ by

(30)Cq−ROLWGHMs,t(P1,P2,..,Pn)=(1s+t∏i=1n∏j=in(sPinωi+tPjnωj)2n(n+1))

where ξn is denoted the family of all Cq-ROLNs.

According to the operational laws of Cq-ROLNs, we can get the following result.

Theorem 13

Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs. Then we have

(31)Cq−ROLWGHMs,t(P1,P2,..,Pn)=(S(1s+t∏i=1n∏j=1n(sθinωi+tθjnωj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−tinωiq)s(1−tjnωjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wtinωiq)s(1−Wtjnωjq)t)2qn(n+1))1s+t)1q,(1−∏i=1n∏j=1n(1−(1−(1−fiq)nωi)s(1−(1−fjq)nωj)t)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−(1−(1−Wfiq)nωi)s(1−(1−Wfjq)nωj)t)2n(n+1))1q(s+t)))

Proof.

Similar to Theorem 1.

Example 5

We consider the four Cq-ROLNs P1=S4.6,0.97ei2π0.97,0.22ei2π0.22, P2=S2.3,0.02ei2π0.01,0.12ei2π0.25, P3=S2.05,0.95ei2π0.98,0.20ei2π0.13 and P4=S2.00,0.96ei2π0.95,0.21ei2π0.20, and suppose the parameters q=5,s=t=1, n=4, then we have

A=Cq−ROLWGHMs,t(P1,P2,..,Pn)=(S(1s+t∏i=1n∏j=1n(sθinωi+tθjnωj)2n(n+1)),((1−(1−∏i=1n∏j=1n(1−(1−tinωiq)s(1−tjnωjq)t)2qn(n+1))1s+t)1qei2π(1−(1−∏i=1n∏j=1n(1−(1−Wtinωiq)s(1−Wtjnωjq)t)2qn(n+1))1s+t)1q,(1−∏i=1n∏j=1n(1−(1−(1−fiq)nωi)s(1−(1−fjq)nωj)t)2n(n+1))1q(s+t)ei2π(1−∏i=1n∏j=1n(1−(1−(1−Wfiq)nωi)s(1−(1−Wfjq)nωj)t)2n(n+1))1q(s+t)))

=S2.415,0.99ei2π0.99,0.009ei2π0.008

When we use the definition of score function, we get SA=2.39

Further, we discuss some properties of Cq-ROLNs as follows:

Theorem 14

(Monotonicity) Let Pi=Sθi,tiei2πWti,fiei2πWfi and Qi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be two families of Cq-ROLNs, if Pi≤Qi for all i=1,2,…,n. Then

Cq−ROLWGHMs,tP1,P2,..,Pn≤Cq−ROLWGHMs,tQ1,Q2,..,Qn

Proof:

Similar to Theorem 2.

Theorem 15

(Idempotency) Let Pi=Sθi,tiei2πWti,fiei2πWfi,i=1,2,…,n be a family of Cq-ROLNs, if Pi=P for all i=1,2,…,n. Then

Cq−ROLWGHMs,tP1,P2,..,Pn=P

Proof:

Similar to Theorem 3.

Theorem 16

(Boundedness) The Cq-ROLWGHM operator lies between the max and min operators

minP1,P2,..,Pn≤Cq−ROLWGHMs,tP1,P2,..,Pn≤maxP1,P2,..,Pn

Proof:

Similar to Theorem 4.

5. A NEW MULTI-ATTRIBUTE GROUP DECISION-MAKING (MAGDM) METHOD

In this section, we would propose a new decision-making method with complex q-rung orthopair linguistic information. Consider the set of alternatives and the set of attributes with respect to weight vectors, i.e., X=x1,x2,..,xm is set of alternatives, Y=y1,y2,..,yn is set of attributes, and ω=ω1,ω2,..,ωnT is the weight vector of the attributes such that ∑i=1nωi=1. Suppose decision makers are D=D1,D2,..,Dp, and decision maker Dk gives the evaluation value of attribute yjfor alternative x2i which is expressed by Cq-ROLN is Pijk=Sθijk,tijkei2πWtijk,fijkei2πWfijk, and the complex q-rung orthopair linguistic decision matrices is represented by Ak=Pijkm×n. Then we will use two different operators to solve this problem. The procedure of the MAGDM is shown as follows:

Construct the decision matrices, it is necessary to consider two kinds of attribute like cost and benefits. The decision matrices is obtained by
(32)Pijk=Sθijk,tijkei2πWtijk,fijkei2πWfijkyj∈I1Sθijk,fijkei2πWfijk,tijkei2πWtijkyj∈I2

The symbol I1 and I2 represent the benefits and cost attributes.
Use the Cq-ROLWHM operator
Pij=Cq−ROLWHMs,tPij1,Pij2,..,Pijp

Or the Cq-ROLWHM operator
Pij=Cq−ROLWGHMs,tPij1,Pij2,..,Pijp

To aggregate the decision matrices Ak=Pijkm×n into a single matrix A=Pijm×n.
Use the Cq-ROLWHM operator
Pi=Cq−ROLWHMs,tPi1,Pi2,..,Pin

Or the Cq-ROLWHM operator
Pi=Cq−ROLWGHMs,tPi1,Pi2,..,Pin

To aggregate the decision matrices Ak=Pijkm×n into a single value Cq-ROLN.
Calculate the score function and accuracy function of Cq-ROLNs.
Rank to all Cq-ROLNs and choose the best alternative.
End

Example 6

In this sub-section, we adopted a numerical example from [29] to show the application of the proposed method. The saving enterprise wants to invest its share with another enterprise. After search, there are four possible enterprises in the list of applicants which are

A1: Car enterprise.
A2: Computer enterprise.
A3: TV enterprise.
A4: Food enterprise.

The decision experts D1,D2, and D3 are invited to examine the candidates with respect attributes which are

C1: Risk analysis.
C2: Growth analysis.
C3: Social–political impact analysis.
C4: Environmental impact analysis.

Suppose the weight vector for attributes is ω=0.34,0.32,0.11,0.23T and the weight vector for decision experts is ℧=0.45,0.35,0.20T. The decision experts adopt linguistic term set: S = {S₀ = very poor, S₁ = poor, S₂ = slightly poor, S₃ = fair, S₄ = slightly good, S₅ = good, S₆ = very good} to give the evaluation information shown in Tables 1–3.

Data Analysis	C1	C2	C3	C4
A1	(S5,(0.2ei2π0.6,0.5ei2π0.3))	(S5,(0.5ei2π0.7,0.45ei2π0.3))	(S5,(0.7ei2π0.5,0.2ei2π0.4))	(S6,(0.4ei2π0.7,0.45ei2π0.2))
A2	(S3,(0.3ei2π0.5,0.7ei2π0.4))	(S6,(0.55ei2π0.45,0.35ei2π0.55))	(S4,(0.34ei2π0.54,0.6ei2π0.4))	(S5,(0.3ei2π0.45,0.66ei2π0.5))
A3	(S3,(0.1ei2π0.2,0.8ei2π0.6))	(S1,(0.45ei2π0.6,0.5ei2π0.3))	(S6,(0.22ei2π0.53,0.56ei2π0.45))	(S4,(0.1ei2π0.6,0.9ei2π0.3))
A4	(S2,(0.4ei2π0.6,0.5ei2π0.3))	(S4,(0.34ei2π0.6,0.5ei2π0.4))	(S1,(0.1ei2π0.56,0.77ei2π0.34))	(S3,(0.4ei2π0.6,0.5ei2π0.3))

Table 1

Complex q-rung orthopair linguistic decision matrix R1 by D1

Data Analysis	C1	C2	C3	C4
A1	(S4,(0.1ei2π0.7,0.1ei2π0.2))	(S6,(0.55ei2π0.71,0.44ei2π0.29))	(S6,(0.72ei2π0.52,0.18ei2π0.4))	(S2,(0.3ei2π0.4,0.7ei2π0.2))
A2	(S6,(0.55ei2π0.45,0.42ei2π0.5))	(S4,(0.53ei2π0.46,0.34ei2π0.54))	(S5,(0.46ei2π0.57,0.34ei2π0.44))	(S4,(0.5ei2π0.3,0.45ei2π0.5))
A3	(S4,(0.45ei2π0.6,0.34ei2π0.4))	(S1,(0.46ei2π0.62,0.49ei2π0.29))	(S4,(0.44ei2π0.54,0.54ei2π0.45))	(S4,(0.2ei2π0.1,0.6ei2π0.3))
A4	(S5,(0.34ei2π0.6,0.22ei2π0.3))	(S3,(0.35ei2π0.61,0.49ei2π0.39))	(S2,(0.12ei2π0.58,0.74ei2π0.13))	(S6,(0.3ei2π0.4,0.6ei2π0.3))

Table 2

Complex q-rung orthopair linguistic decision matrix R2by D2

Data Analysis	C1	C2	C3	C4
A1	(S5,(0.2ei2π0.7,0.45ei2π0.3))	(S5,(0.3ei2π0.4,0.7ei2π0.2))	(S6,(0.7ei2π0.5,0.2ei2π0.4))	(S5,(0.72ei2π0.52,0.18ei2π0.4))
A2	(S6,(0.55ei2π0.45,0.35ei2π0.55))	(S6,(0.5ei2π0.3,0.45ei2π0.5))	(S4,(0.34ei2π0.54,0.6ei2π0.5))	(S3,(0.46ei2π0.57,0.34ei2π0.44))
A3	(S4,(0.45ei2π0.6,0.5ei2π0.3))	(S1,(0.2ei2π0.1,0.6ei2π0.3))	(S4,(0.22ei2π0.53,0.56ei2π0.45))	(S6,(0.44ei2π0.54,0.54ei2π0.45))
A4	(S6,(0.34ei2π0.6,0.5ei2π0.4))	(S4,(0.3ei2π0.4,0.6ei2π0.3))	(S1,(0.1ei2π0.56,0.77ei2π0.34))	(S1,(0.12ei2π0.58,0.74ei2π0.13))

Table 3

Complex q-rung orthopair linguistic decision matrix R3 by D3

5.1. Decision-Making Process

The steps of this decision-making problem are given as

The four attributes are all benefits types, so we cannot normalize the decision matrix.

We will consider the Cq-ROLWHM operator

Pij=Cq−ROLWHMs,tPij1,Pij2,..,Pijp

To aggregate the decision matrices Ak=Pijkm×n into a single matrix A=Pijm×n which is shown in Table 4 for q=3.

Data Analysis	C1	C2	C3	C4
A1	(S4.6,(0.86ei2π0.99,0.10ei2π0.03))	(S2.3,(0.01ei2π0.03,0.25ei2π0.03))	(S2.4,(0.99ei2π0.99,0.01ei2π0.09))	(S2.04,(0.97ei2π0.97,0.18ei2π0.04))
A2	(S4.6,(0.97ei2π0.97,0.22ei2π0.22))	(S2.3,(0.02ei2π0.01,0.12ei2π0.25))	(S2.05,(0.95ei2π0.98,0.20ei2π0.13))	(S2.00,(0.96ei2π0.95,0.21ei2π0.20))
A3	(S3.50,(0.93ei2π0.97,0.34ei2π0.15))	(S0.99,(0.007ei2π0.006,0.25ei2π0.06))	(S2,15,(0.92ei2π0.98,0.24ei2π0.14))	(S2.08,(0.89ei2π0.94,0.53ei2π0.08))
A4	(S3.8,(0.95ei2π0.99,0.15ei2π0.09))	(S1.9,(0.005ei2π0.024,0.25ei2π0.09))	(S1.13,(0.81ei2π0.99,0.54ei2π0.033))	(S1.8,(0.92ei2π0.97,0.37ei2π0.04))

Table 4

Complex q-rung orthopair linguistic decision matrix after using the Cq-ROLWHM operator

We use the Cq-ROLWGHM operator

Pi=Cq−ROLWGHMs,tPi1,Pi2,..,Pin

To aggregate the decision matrix (in Table 4) and get the comprehensive value of four alternatives which is listed in Table 5.

Data Analysis	Cq−ROLNs
A1	(S3.06,(0.99ei2π0.99,0.021ei2π0.09))
A2	(S2.95,(0.99ei2π0.99,0.06ei2π0.06))
A3	(S2.21,(0.99ei2π0.99,0.18ei2π0.02))
A4	(S2.4,(0.99ei2π0.99,0.16ei2π0.006))

Table 5

The comprehensive value of four alternatives

Calculate the score functions of four alternatives which is listed in Table 6.

Cq−ROLNs Score Function Ranking

A1 SA1=3.02 first

A2 SA2=2.76 second

A3 SA3=1.99 fourth

A4 SA4=2.16 thrid

Table 6
The score function for four alternatives
Rank all Cq-ROLNs and choose the best alternative.
A1≥A2≥A4≥A3

So, A1 is the best alternative.
End.

In order to explain the validity of the proposed method, we use the method for CILS and the method for CPYLS, the ranking results are listed in Table 7.

Methods	Score Function	Ranking
CIFPA operator proposed by Rani and Garg [35]	SA1=0.75,SA2=0.73, SA3=0.64,SA4=0.67	A1>A2>A4>A3
CIFWA operator proposed by Garg and Rani [33]	SA1=1.99,SA2=1.98, SA3=1.94,SA4=1.95	A1>A2>A4>A3
WDM for PYFS proposed by Ullah et al. [36]	SA1=0.56,SA2=0.093, SA3=0.089,SA4=0.09	A1>A2>A4>A3
Method based on Cq-ROLS in this paper for q=1	SA1=4.86,SA2=3.6, SA3=2.19,SA4=2.57	A1>A2>A4>A3
Method based on Cq-ROLS in this paper for q=2	SA1=3.6,SA2=3.04, SA3=2.05,SA4=2.3	A1>A2>A4>A3
Method based on Cq-ROLS in this paper	SA1=2.47,SA2=2.39, SA3=1.88,SA4=1.99	A1>A2>A4>A3

Table 7

Validation test

The geometrical interpretation of the proposed method with existing methods are discussed in Figure 3.

From Table 7, we can get the same ranking result, it can explain the validity of the proposed method.

6. ADVANTAGES AND COMPARATIVE ANALYSIS

6.1. The Influence of Parameters on Ranking Results

The parameters in the proposed operators play a key role on the final ranking results. By example 6, we assign different values to parameters s and t, and discuss the ranking results which are shown in Table 8.

From Table 8, we can see that although the best choice is the same, the ranking order is different, this can explain the parameters s and t can affect the ranking results.

In order to show clearly the ranking results, we consider the values of parameters for s=t, then the score values of alternatives Aii=1,2,3,4 are shown in Figure 4.

Parameters Value	Using Score Function	Ranking	Best Alternative
s→0,t=1	SA1=30.85,SA2=27.53, SA3=11.54,SA4=14.00	A1>A2>A4>A3	A1
s=1,t→0	SA1=21.97,SA2=20.43, SA3=9.49,SA4=11.68	A1>A2>A4>A3	A1
s=1,t=1	SA1=3.02,SA2=2.76, SA3=1.99,SA4=2.16	A1>A2>A4>A3	A1
s=1,t=0.5	SA1=5.12,SA2=4.76, SA3=3.07,SA4=3.45	A1>A2>A4>A3	A1
s=1,t=2	SA1=1.94,SA2=1.69, SA3=1.33,SA4=1.42	A1>A2>A4>A3	A1
s=2,t=3	SA1=1.32,SA2=1.07, SA3=0.93,SA4=1.03	A1>A2>A4>A3	A1
s=3,t=4	SA1=1.12,SA2=0.86, SA3=0.79,SA4=0.90	A1>A4>A2>A3	A1
s=t=5	SA1=0.98,SA2=0.72, SA3=0.70,SA4=0.81	A1>A4>A2>A3	A1
s=t=6	SA1=0.93,SA2=0.664, SA3=0.660,SA4=0.78	A1>A4>A2>A3	A1
s=t=7	SA1=0.89,SA2=0.628, SA3=0.635,SA4=0.75	A1>A4>A2>A3	A1

Table 8

Ranking results for different values of parameters

6.2. Advantages of the Proposed Cq-ROLS with the Existing CFSs

The HM operators for CILS and CPYLS are also the special cases of our proposed method. The following examples can explain the generalization of the proposed Cq-ROLS.

Example 7

In some practical examples, the CILS cannot described effectively, because the restriction of of CILS is that the sum of membership (for real part and imaginary part) and non-membership (for real part and imaginary part) are limited to 1. So we considered the complex Pythagorean linguistic kinds of information, and solved by our proposed methods and then compared with existing methods. The weight vectors are given by ω=0.34,0.32,0.11,0.23T. The complex Pythagorean linguistic decision matrix R shown in Table 9.

Data Analysis	C1	C2	C3	C4
A1	(S1.34,(0.67ei2π0.91,0.2ei2π0.1))	(S4,(0.8ei2π0.66,0.4ei2π0.13))	(S3,(0.66ei2π0.78,0.67ei2π0.34))	(S2,(0.5ei2π0.7,0.6ei2π0.5))
A2	(S1.33,(0.92ei2π0.12,0.13ei2π0.5))	(S5,(0.88ei2π0.89,0.33ei2π0.3))	(S4,(0.56ei2π0.68,0.5ei2π0.6))	(S5,(0.6ei2π0.9,0.5ei2π0.12))
A3	(S1.27,(0.78ei2π0.93,0.15ei2π0.11))	(S3,(0.8ei2π0.78,0.23ei2π0.4))	(S5,(0.45ei2π0.67,0.7ei2π0.4))	(S4,(0.7ei2π0.8,0.4ei2π0.4))
A4	(S1.29,(0.9ei2π0.91,0.2ei2π0.1))	(S2,(0.78ei2π0.67,0.34ei2π0.5))	(S1,(0.7ei2π(0.6),0.4ei2π(0.5)))	(S3,(0.56ei2π0.6,0.5ei2π0.6))

Table 9

Decision matrix for complex Pythagorean linguistic information’s

The aggregation results for different approaches shown in Table 10.

Methods	Score Function	Ranking
CIFPA operator proposed by Rani and Garg [35]	Cannot be calculated	Cannot be calculated
CIFWA operator proposed by Garg and Rani [33]	Cannot be calculated	Cannot be calculated
WDM for CPYFS proposed by Ullah et al. [36]	SA1=0.76,SA2=0.67, SA3=0.45,SA4=0.54	A1>A2>A4>A3
Method based on Cq-ROLS in this paper for q=2	SA1=0.60,SA2=0.75, SA3=0.66,SA4=0.50	A2>A3>A1>A4
Method based on Cq-ROLS in this paper	SA1=0.8,SA2=1.01, SA3=0.91,SA4=0.64	A2>A3>A1>A4

cq-ROLS, complex q-rung orthopair linguistic set.

Table 10

Ranking results for proposed and existing methods to solve Example 7

From Table 10, we can get (1) CILS cannot express the information described by CPYLS; (2) the proposed method in this paper can the same ranking results as method in [33], which can show the effectiveness of the proposed method because the Cq-ROLS is reduced into CPYFS when q=2,s=t=5.

Example 8

In this example, we consider the information is expressed by Cq-ROLNs which is listed in Table 11, and the weight vectors is taken from example 6.

Data Analysis	C1	C2	C3	C4
A1	(S4.6,(0.86ei2π0.99,0.10ei2π0.03))	(S2.3,(0.01ei2π0.03,0.25ei2π0.03))	(S2.4,(0.99ei2π0.99,0.01ei2π0.09))	(S2.04,(0.97ei2π0.97,0.18ei2π0.04))
A2	(S4.6,(0.97ei2π0.97,0.22ei2π0.22))	(S2.3,(0.02ei2π0.01,0.12ei2π0.25))	(S2.05,(0.95ei2π0.98,0.20ei2π0.13))	(S2.00,(0.96ei2π0.95,0.21ei2π0.20))
A3	(S3.50,(0.93ei2π0.97,0.34ei2π0.15))	(S0.99,(0.007ei2π0.006,0.25ei2π0.06))	(S2,15,(0.92ei2π0.98,0.24ei2π0.14))	(S2.08,(0.89ei2π0.94,0.53ei2π0.08))
A4	(S3.8,(0.95ei2π0.99,0.15ei2π0.09))	(S1.9,(0.005ei2π0.024,0.25ei2π0.09))	(S1.13,(0.81ei2π0.99,0.54ei2π0.033))	(S1.8,(0.92ei2π0.97,0.37ei2π0.04))

Table 11

The decision matrix from Example 8

Then the ranking results are listed in Table 12 for q=5,s=t=1.

Methods	Score Function	Ranking
Complex intuitionistic fuzzy power averaging (CIFPA) aggregation operator proposed by Rani and Garg [35]	Cannot be calculated	Cannot be calculated
Complex intuitionistic fuzzy weighted averaging (CIFWA) operator proposed by Garg and Rani [33]	Cannot be calculated	Cannot be calculated
Weighted distance measure (WDM) for complex pythagorean fuzzy set (CPYFS) proposed by Ullah et al. [36]	Cannot be calculated	Cannot be calculated
Method based on Cq-ROLS in this paper	SA1=2.48,SA2=2.39, SA3=1.88,SA4=1.99	A1>A2>A4>A3

cq-ROLS, complex q-rung orthopair linguistic set.

Table 12

Ranking results from different complex fuzzy sets for Example 8

From Table 12, we can know the Cq-ROLS is more generalized than existing CFSs, so we easily find that our proposed method is more superior and more reliable than existing methods.

Example 9

In this example, we consider the information expressed by Pythagorean linguistic sets, which is listed in Table 13, and the weight vectors is taken from example 6. The information discussed in this example is taken from [29].

We will convert the Table 13 into Table 14, and we also clear that about e0=1.

Then the ranking results are listed in Table 15 for q=5,s=t=1.

Data Analysis	C1	C2	C3	C4
A1	(S4.7038,(0.1820,0.6711))	(S2.6020,(0.3377,0.6650))	(S4.1372,(0.4235,0.5997))	(S5.0751,(0.3067,0.5992))
A2	(S4.3333,(0.3796,0.5992))	(S4.4066,(0.3515,0.5672))	(S3.7082,(0.1533,0.7358))	(S3.4366,(0.4235,0.5999))
A3	(S3.6013,(0.2002,0.6686))	(S4.2846,(0.2396,0.6711))	(S2.6550,(0.3237,0.6979))	(S4.1372,(0.2450,0.7011))
A4	(S4.9334,(0.4084,0.5621))	(S2.9382,(0.3018,0.6743))	(S2.9832,(0.2703,0.6020))	(S3.8820,(0.3199,0.5710))

Table 13

The decision matrix from Example 9

Data Analysis	C1	C2	C3	C4
A1	(S4.7038,(0.1820ei2π0.0,0.6711ei2π0.0))	(S2.6020,(0.3377ei2π0.0,0.6650ei2π0.0))	(S4.1372,(0.4235ei2π0.0,0.5997ei2π0.0))	(S5.0751,(0.3067ei2π0.0,0.5992ei2π0.0))
A2	(S4.3333,(0.3796ei2π0.0,0.5992ei2π0.0))	(S4.4066,(0.3515ei2π0.0,0.5672ei2π0.0))	(S3.7082,(0.1533ei2π0.0,0.7358ei2π0.0))	(S3.4366,(0.4235ei2π0.0,0.5999ei2π0.0))
A3	(S3.6013,(0.2002ei2π0.0,0.6686ei2π0.0))	(S4.2846,(0.2396ei2π0.0,0.6711ei2π0.0))	(S2.6550,(0.3237ei2π0.0,0.6979ei2π0.0))	(S4.1372,(0.2450ei2π0.0,0.7011ei2π0.0))
A4	(S4.9334,(0.4084ei2π0.0,0.5621ei2π0.0))	(S2.9382,(0.3018ei2π0.0,0.6743ei2π0.0))	(S2.9832,(0.2703ei2π0.0,0.6020ei2π0.0))	(S3.8820,(0.3199ei2π0.0,0.5710ei2π0.0))

Table 14

The decision matrix from Example 9

Methods	Score Function	Ranking
CIFPA operator proposed by Rani and Garg [35]	Cannot be calculated	Cannot be calculated
CIFWA operator proposed by Garg and Rani [33]	Cannot be calculated	Cannot be calculated
WDM for PYLS proposed in [29]	SA1=0.9283,SA2=1.09, SA3=0.9210,SA4=0.9176	A2>A1>A4>A3
Method based on Cq-ROLS in this paper	SA1=0.8472,SA2=0.9982, SA3=0.8356,SA4=0.8355	A2>A1>A3>A4

cq-ROLS, complex q-rung orthopair linguistic set.

Table 15

Ranking results from different complex fuzzy sets for Example 8

From Table 15, we can know the Cq-ROLS is more generalized than existing CPYLS, CILS, q-ROLS, PYLS, ILS and CFSs, so we easily find that our proposed method is more superior and more reliable than existing methods. Therefore, the proposed method is more generalized than existing to cope with uncertain and complicated types of information easily.

6.3. The Qualitative Comparison with the Existing Methods

In this sub-section, we give some comparisons with some existing methods from a qualitative point of view. We compare our method with the work proposed by Ullah et al. [36] based on the similarity measures for complex PFS, the method proposed by Rani and Garg [35,37] based on the distance measures and power aggregation operators for CIFS, the method proposed by Garg and Rani [38,33] based on some robust correlation coefficient and generalized CIFS and their aggregation operators. The characteristic comparison of the proposed method with existing works is shown in Table 16.

Methods	Ability to Integrate Information	Generalized Operators Based on t-norm and t-conorm	Ability to Capture Information Using Complex Numbers	Ability to Handle Two-dimensional Information	Flexible According to Decision maker’s Preferences	Superior Characteristic of the Ideas
Zhang [39]	No	Yes	No	No	Yes	No
Liu [40]	No	Yes	No	No	Yes	No
Garg and Kumar [34]	No	Yes	No	No	Yes	No
Garg [41]	No	Yes	No	No	Yes	No
Garg and Rani [38,33]	Yes	Yes	Yes	Yes	Yes	No
Ullah et al. [36]	Yes	Yes	Yes	Yes	Yes	No
Rani and Garg [35,37]	Yes	Yes	Yes	Yes	Yes	No
Li et al. [29]	No	Yes	No	No	Yes	No
The proposed method for q= 1	Yes	Yes	Yes	Yes	Yes	No
The proposed method for q= 2	Yes	Yes	Yes	Yes	Yes	No
The proposed method for q= 3	Yes	Yes	Yes	Yes	Yes	Yes

Table 16

Comparison between existing methods and the proposed method

From Table 16, it is clear that our proposed method is more superior than existing works because the CILS and CPLS are only special cases of Cq-ROLS, which is the generalization of ILS and PLS.

The idea of Cq-ROFLS is more powerful and more general than existing methods, from the above analysis we have clear, if we take the values of parameter q=1, the proposed approach is converted to CIFLS and similarly, if we take the values of parameter q=2, the proposed approach is converted to CPFLS. We discussed two numerical examples for existing methods and solved by proposed approach. The comparison between proposed methods and existing methods are discussed in Table 16, to show the reliability and effectiveness of the proposed methods. Hence, the introduced methods in this manuscript is more powerful and more general than existing methods.

7. CONCLUSION

The notions of Cq-ROFS and LV are two different tools to describe uncertain and unpredictable information in MAGDM problems. Motivation of this paper is to propose a new concept, called Cq-ROLS to cope with unreliable and difficult information in real decision problems, which takes full benefits of Cq-ROFS and LV. Futher, we generalize the HM operator to Cq-ROLS and propose Cq-ROLHM, Cq-ROLWHM, Cq-ROLGHM, Cq-ROLWGHM operators and discuss their properties in detail. Moreover, we develop a novel approach to MAGDM using proposed operators. We also use a numerical example to describe the flexibility and explicitly of the proposed method. In last, the comparisons between proposed method and existing methods are also discussed in detail.

In the future, we will use the proposed method to solve some real decision problems [42], such as Efficiency evaluation [43], ecological environment quality assessment [44], supplier selection problems [45], and so on. We can also extend the IFSs [46], interval-valued intuitionistic fuzzy sets [47], to their complex types.

CONFLICT OF INTEREST

The authors declare no conflict of interest.

AUTHORS' CONTRIBUTIONS

Conceptualization, P.L., Z.A., T.M. (Peide Liu, Zeeshan Ali and Tahir Mahmood); methodology, P.L., Z.A., T.M.; software, Z.A.; validation, P.L., Z.A., T.M.; formal analysis, P.L., Z.A., T.M.; investigation, P.L., Z.A., T.M.; writing–original draft preparation, Z.A.; writing–review and editing, Z.A.; visualization, P.L., Z.A., T.M.; supervision, T.M.; funding acquisition, P.L.

ACKNOWLEDGMENT

This paper is supported by the National Natural Science Foundation of China (Nos. 71771140, 71471172), 文化名家暨“四个一批”人才项目(Project of cultural masters and “the four kinds of a batch” talents), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

REFERENCES

1.D. Ramot, R. Milo, M. Friedman, and A. Kandel, Complex fuzzy sets, IEEE Trans. Fuzzy Syst., Vol. 10, 2002, pp. 171-86.

2.L.A. Zadeh, Fuzzy sets, Inf. Control, Vol. 8, 1965, pp. 338-53.

3.A.U.M. Alkouri and A.R. Salleh, Linguistic variable, hedges and several distances on complex fuzzy sets, J. Intell. Fuzzy Syst., Vol. 26, 2014, pp. 2527-2535.

4.O. Yazdanbakhsh and S. Dick, Time-series forecasting via complex fuzzy logic, Frontiers of Higher Order Fuzzy Sets, Springer, New York, 2015, pp. 147-165.

5.L. Bi, S. Dai, and B. Hu, Complex fuzzy geometric aggregation operators, Symmetry, Vol. 10, 2018, pp. 251.

6.D. Moses, O. Degani, H.N. Teodorescu, M. Friedman, and A. Kandel, Linguistic coordinate transformations for complex fuzzy sets, IEEE, in IEEE International Fuzzy Systems. Conference Proceedings FUZZ-IEEE'99. 1999 (Cat. No. 99CH36315), Vol. 3, pp. 1340-1345.

7.Z. Chen, S. Aghakhani, J. Man, and S. Dick, A neurofuzzy architecture employing complex fuzzy sets, IEEE Trans. Fuzzy Syst., Vol. 19, 2011, pp. 305-322.

8.K.T. Atanassov, Intuitionistic Fuzzy Sets, Physica, Heidelberg, 1999, pp. 1-137.

9.H. Bustince and P. Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Set Syst., Vol. 79, 1996, pp. 403-405.

10.H. Garg and G. Kaur, Extended TOPSIS method for multi-criteria group decision-making problems under cubic intuitionistic fuzzy environment, Sci. Iran., Vol. 25, 2018, pp. 1-18.

11.A.M.D.J.S. Alkouri and A.R. Salleh, Complex intuitionistic fuzzy sets, in AIP Conference Proceedings (Kuala Lumpur), Vol. 1482, 2012, pp. 464-470.

12.R.R. Yager, Pythagorean fuzzy subsets, IEEE, in 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) (Edmonton), 2013, pp. 57-61.

13.H. Garg, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision‐making processes, Int. J. Intell. Syst., Vol. 31, 2016, pp. 1234-52.

14.H. Garg, A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making, Int. J. Intell. Syst., Vol. 31, 2016, pp. 886-920.

15.S. Dick, R.R. Yager, and O. Yazdanbakhsh, On Pythagorean and complex fuzzy set operations, IEEE Trans. Fuzzy Syst., Vol. 24, 2016, pp. 1009-1021.

16.H. Garg, A novel accuracy function under interval-valued pythagorean fuzzy environment for solving multicriteria decision making problem, J. Intell. Fuzzy Syst., Vol. 31, 2016, pp. 529-540.

17.P. Ren, Z. Xu, and X. Gou, Pythagorean fuzzy TODIM approach to multi-criteria decision making, Appl. Soft Comput., Vol. 42, 2016, pp. 246-59.

18.H. Garg, A new improved score function of an interval-valued pythagorean fuzzy set based TOPSIS method, Int. J. Uncertain. Quan., Vol. 7, 2017, pp. 463-74.

19.G. Wei and Y. Wei, Similarity measures of pythagorean fuzzy sets based on the cosine function and their applications, Int. J. Intell. Syst., Vol. 33, 2018, pp. 634-52.

20.H. Garg, Generalized pythagorean fuzzy geometric aggregation operators using Einstein t‐norm and t‐conorm for multicriteria decision‐making process, Int. J. Intell. Syst., Vol. 32, 2017, pp. 597-630.

21.G. Wei, Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making, J. Intell. Fuzzy Syst., Vol. 33, 2017, pp. 2119-2132.

22.R.R. Yager, Generalized orthopair fuzzy sets, IEEE Trans. Fuzzy Syst., Vol. 25, 2017, pp. 1222-30.

23.P. Liu and J. Liu, Some q‐rung orthopai fuzzy Bonferroni mean operators and their application to multi‐attribute group decision making, Int. J. Intell. Syst., Vol. 33, 2018, pp. 315-47.

24.G. Wei, H. Gao, and Y. Wei, Some q‐rung orthopair fuzzy Heronian mean operators in multiple attribute decision making, Int. J. Intell. Syst., Vol. 33, 2018, pp. 1426-58.

25.P. Liu and P. Wang, Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making, Int. J. Intell. Syst., Vol. 33, 2018, pp. 259-280.

26.P. Liu and P. Wang, Multiple-attribute decision making based on Archimedean Bonferroni operators of q-rung orthopair fuzzy numbers, IEEE Trans. Fuzzy Syst., Vol. 27, 2018, pp. 834-48.

27.P. Liu, S.M. Chen, and P. Wang, Multiple-attribute group decision-making based on q-rung orthopair fuzzy power maclaurin symmetric mean operators, IEEE Trans. Syst., Man, Cybern: Syst., 2018. in Press

28.K. Bai, X. Zhu, J. Wang, and R. Zhang, Some partitioned Maclaurin symmetric mean based on q-rung orthopair fuzzy information for dealing with multi-attribute group decision making, Symmetry, Vol. 10, 2018, pp. 383.

29.L. Li, R. Zhang, J. Wang, and X. Shang, Some q-rung orthopair linguistic Heronian mean operators with their application to multi-attribute group decision making, Arch. Control Sci., Vol. 8, 2018, pp. 551-583.

30.L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning—I, Info. Sci., Vol. 8, 1975, pp. 199-249.

31.J.Q. Wang and H.B. Li, Multi-criteria decision-making method based on aggregation operators for intuitionistic linguistic fuzzy numbers, Control Decis., Vol. 25, 2010, pp. 1571-1574.

32.H. Garg, Linguistic pythagorean fuzzy sets and its applications in multiattribute decision-making process, Int. J. Intell. Syst., Vol. 33, 2018, pp. 1234-1263.

33.H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arab. J. Sci. Eng., Vol. 44, 2019, pp. 2679-2698.

34.H. Garg and K. Kumar, Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis, Arab. J. Sci. Eng., Vol. 43, 2018, pp. 3213-27.

35.D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making, Expert Syst., Vol. 35, 2018, pp. e12325.

36.K. Ullah, T. Mahmood, Z. Ali, and N. Jan, On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition, Complex Intell. Syst., Vol. 1, 2019, pp. 1-13.

37.D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and their applications to the decision-making process, Int. J. Uncertain Quan., Vol. 7, 2017, pp. 423-439.

38.H. Garg and D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Appl. Intell., Vol. 49, 2019, pp. 496-512.

39.H. Zhang, Linguistic intuitionistic fuzzy sets and application in MAGDM, J. Appl. Math., Vol. 2014, 2014. Article ID 432092, 11 pages

40.P. Liu, Some generalized dependent aggregation operators with intuitionistic linguistic numbers and their application to group decision making, J. Comput. Syst. Sci., Vol. 79, 2013, pp. 131-43.

41.Y. Zhang, D. Huang, W. Gao, and V.G. Kaburlasos, A decision making approach with linguistic weight and unavoidable incomparable ranking, Int. J. Comput. Int. Syst., 2019. in Press

42.P. Li and N. Ding, A study of the determinants of performance of China’s OFDI firms: based on the perspective of institutional environment of host countries, Rev. Econ. Manag., Vol. 34, 2019, pp. 18-30.

43.Z. Fang, B. Zhang, and J. Qin, Evaluation of technological innovation efficiency of listed military enterprises in China from the perspective of innovation value chain, Rev. Econ. Manag., Vol. 35, 2019, pp. 37-48.

44.B. Ren and C. Lv, Changing trend and spatial distribution pattern of ecological environment quality in China, Rev. Econ. Manag., Vol. 35, 2019, pp. 120-134.

45.P. Liu and P. Wang, Some interval-valued intuitionistic fuzzy Schweizer–Sklar power aggregation operators and their application to supplier selection, Int. J. Syst. Sci., Vol. 49, 2018, pp. 1188-1211.

46.H. Garg and R. Arora, Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, J. Oper. Res. Soc., Vol. 69, 2018, pp. 1711-24.

47.S.K. Khan and M. Pal, Interval-valued intuitionistic fuzzy matrices, Notes on Intuitionistic Fuzzy Sets, Vol. 11, 2005, pp. 16-27.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 12 - 2
Pages: 1465 - 1496
Publication Date: 2019/11/25
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.191030.002 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Peide Liu
AU  - Zeeshan Ali
AU  - Tahir Mahmood
PY  - 2019
DA  - 2019/11/25
TI  - A Method to Multi-Attribute Group Decision-Making Problem with Complex q-Rung Orthopair Linguistic Information Based on Heronian Mean Operators
JO  - International Journal of Computational Intelligence Systems
SP  - 1465
EP  - 1496
VL  - 12
IS  - 2
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.191030.002
DO  - 10.2991/ijcis.d.191030.002
ID  - Liu2019
ER  -

download .riscopy to clipboard

Cq−ROLNs	Score Function	Ranking
A1	SA1=3.02	first
A2	SA2=2.76	second
A3	SA3=1.99	fourth
A4	SA4=2.16	thrid