Multi-criteria group decision making based on Archimedean power partitioned Muirhead mean operators of q-rung orthopair fuzzy numbers

Two critical tasks in multi-criteria group decision making (MCGDM) are to describe criterion values and to aggregate the described information to generate a ranking of alternatives. A flexible and superior tool for the first task is q-rung orthopair fuzzy number (qROFN) and an effective tool for the second task is aggregation operator. So far, nearly thirty different aggregation operators of qROFNs have been presented. Each operator has its distinctive characteristics and can work well for specific purpose. However, there is not yet an operator which can provide desirable generality and flexibility in aggregating criterion values, dealing with the heterogeneous interrelationships among criteria, and reducing the influence of extreme criterion values. To provide such an aggregation operator, Muirhead mean operator, power average operator, partitioned average operator, and Archimedean T-norm and T-conorm operations are concurrently introduced into q-rung orthopair fuzzy sets, and an Archimedean power partitioned Muirhead mean operator of qROFNs and its weighted form are presented and a MCGDM method based on the weighted operator is proposed in this paper. The generalised expressions of the two operators are firstly defined. Their properties are explored and proved and their specific expressions are constructed. On the basis of the specific expressions, a method for solving the MCGDM problems based on qROFNs is then designed. Finally, the feasibility and effectiveness of the method is demonstrated via a numerical example, a set of experiments, and qualitative and quantitative comparisons.


description,
here are many different kinds of available tools, where fuzzy set is a well-known kind [2][3][4][5][6][7][8][9][10][11][12].To date, nearly thirty different types of fuzzy sets have been presented [13].Yager's generalised orthopair fuzzy set [14], commonly known as q-rung orthopair fuzzy set (qROFS), is one of the most important and popular types among them.

A qROFS consists of an element and a q-rung orthopair membership grade (commonly known as q-rung orthopair fuzzy number (qROFN), which is used to quantify the degrees of membership and non-membership of the element to the qROFS.In a qROFS, both the degrees of membership and non-membership and the sum of the q-th (q = 1, 2, 3, . ..) power of the degree of membership and the q-th power of the degree of non-membership are restricted to [0,1].In other words, the rung q in a qROFS is adjustable under the premise of satisfying this condition.Because of this characteristic, qROFS can be regarded as the generalisation of Zadeh's fuzzy set (FS) [15], Atanassov's intuitionistic fuzzy set (IFS) [16], and Yager's Pythagorean fuzzy set (PFS) [17].This is because qROFS will reduce to FS when q = 1 and the sum of the degrees of membership and non-membership is equal to 1, will reduce to IFS when q = 1, and will reduce to PFS when q = 2.In addition, the expressiveness of a qROFS will continue to increase as q increases, which provides enough freedom for the description of fuzzy information.Due to such advantage, qROFSs have received extensive attention in the field of MCGDM during the past few years.Various research topics regarding qROFSs for MCGDM, such as correlation and correlation coefficient of qROFSs [18], distance measures of qROFSs [19], similarity measures of qROFSs [20], application of qROFSs in practical MCGDM problems [21,22], operational rules of qROFNs [23][24][25], and aggregation operators of qROFNs [26 −38], are gaining importance and popularity within academia.

The other critical task in MCGDM is to fuse the described criterion information to generate a ranking of all alternatives.For such fusion, aggregation operator is regarded as an effective tool.So far, over twenty different aggregation operators of qROFNs have been presented.They are the weighted averaging (WA) operator and the weighted geometric (WG) operator presented by Liu and Wang [26], the weighted Bonferron mean (WBM) operator and the weighted geometric Bonferroni mean (WGBM) operator presented by Liu and Liu [27], the weighted Archimedean Bonferroni mean (WABM) operators presented by Liu and Wang [28], the weighted partitioned Bonferroni mean (WPBM) operator and the weighted partitioned geometric Bonferroni mean (WPGBM) operator presented by Yang and Pang [29], the weighted Heronian mean (WHM) operator and the weighted geometric Heronian mean (WGHM) operator presented by Wei et al. [30], the WHM � operator (This operator is different from the WHM operator presented by Wei et al. [30], though they have the same names) and the weighted partitioned Heronian mean (WPHM) operator presented by Liu et al. [31], the weighted Maclaurin symmetric mean (WMSM) operator and the weighted geometric Maclaurin symmetric mean (WGMSM) operator presented by Wei et al. [32], the weighted power Maclaurin symmetric mean (WPMSM) operator presented by Liu et al. [33], the weighted power partitioned Maclaurin symmetric mean (WPPMSM) operator presented by Bai et al. [34], the weighted Muirhead mean (WMM) operator and the weighted geometric Muirhead mean (WGMM) operator presented by Wang et al. [35], the weighted extended Bonferroni mean (WEBM) operator presented by Liu et al. [36], the weighted exponential (WE) operator presented by Peng et al. [37], and the weighted point (WP) operators presented by Xing et al. [38].Each operator has its own characteristics and can work well for its specific purpose.But there is not yet an operator that has the following three characteristics at the same time: (1) Provide satisfying generality and flexibility in the aggregation of qROFNs; (2) Deal with the situation in which the criteria are divided into several parts and there are interrelationships among different criteria in each part whereas the criteria in different parts are independent of each other; (3) Reduce the negative effect of the unduly high or unduly low criterion values on the aggregation results.

In practical MCGDM problems, aggregation of criterion values is a complex process, in which the preferences of decision makers may change frequently.An ideal aggregation operator should be general and flexible enough to adapt to such change.Moreover, there are usually complex relationships among the different criteria considered in the problems.It is also of importance for an aggregation operator to capture the comple interrelationships of different criteria to generate more reasonable aggregation results [28].Further, the values of criteria are generally assessed by domain experts.It is often difficult to ensure the absolute objectivity, which means that a few biased experts will give biased assessment values [33].To obtain reasonable aggregation results, it is of necessity to reduce the negative influence of biased criterion values in the aggregation.Based on these considerations, the motivations of the present paper are explained as follows:

1. To develop an aggregation operator of qROFNs which can capture the complex interrelationships among criteria, the Muirhead mean (MM) operator [39] and partitioned average operator are introduced.The MM operator, which is a generalisation of the generalised arithmetic average operator, Bonferroni mean (BM) operator [40,41], Maclaurin symmetric mean (MSM) operator [42,43], and generalised geometric average operator, is an a l-inone aggregation operator for capturing the interrelationships of criteria.It is applicable in the cases where all criteria are independent of each other, where there are interrelationships between any two criteria, and where there are interrelationships between any multiple (three or more) criteria [44][45][46][47][48][49].The partitioned average operator is an aggregation operator that has the capability of aggregating the parameters in different partitions using the same aggregation operator and aggregating the aggregation results of different partitions using the arithmetic average operator [50][51][52][53][54][55].

2. To enable the aggregation operator to reduce the negative effect of extreme criterion values on the aggregation results, the power average (PA) operator [56] is combined into the partitioned MM operator.The PA operator is an aggregation operator that can assign weights to the aggregated parameters by calculating the support degrees between these parameters, which makes it capable of reducing the negative influence of unreaso able parameter values on the aggregation results [57][58][59][60][61][62].

3. To improve the generality and flexibility of the combined aggregation operator, the operational rules of qROFNs based on the Archimedean T-norm and T-conorm (ATT) are leveraged to perform the operations in the operator.The ATT are operations for generalising the logical conjunction and disjunction to fuzzy logic.They are important tools that can generate versatile and flexible operational rules for fuzzy numbers and the aggre ation operators based on them are rather general and flexible for aggregating fuzzy information [28,[63][64][65][66][67][68].

As can be summarised from the motivations above, this paper aims to present a set of Archimedean power partitioned MM operators of qROFNs and propose a MCGDM method based on them.This aim is achieved through combining the MM operator, the partitioned average operator, the PA operator, and the ATT.As a result, the presented aggregation operators combine all of their characteristics.

The remainder of the paper is organised as follo s.A brief introduction of some basic concepts is provided Section 2. Sections 3 explains the details of the presented Archimedean power partitioned MM operators.The specific process of the proposed MCGDM method is described in Section 4. Section 5 demonstrates and evaluates the presented operators and proposed method via example, experiments, and comparisons.Section 6 ends the paper w th a conclusion.


Preliminaries

In this section, some prerequisites in qROFS theory, operational rules of qROFNs based on ATT, PA operator, and MM operator are briefly introduced to facilitate the understanding of the present paper.


qROFS theory

qROFS [14] is the generalisation of FS [15], IFS [16], and PFS [17].Its formal definition is as follow:

Definition 1 [14].A qROFS S in a finite universe of discourse X is S = {<x, μ S (x), ν S (x)> | x2X},

D(a i , a j
(i, j = 1, 2, . .., n and j 6 ¼ i; D(a i , a j ) is the distance between a i and a j ) be the degree of support for a i from a j which has the following properties: (1)
0 � Sup(a i , a j ) � 1; (2) Sup(a i , a j ) = Sup(a j , a i ); (3) Sup(a i , a j ) � Sup(a r , a s ) if |a i −a j | � |a r −a s |,

and
Tða i Þ ¼ X n j¼1;j6 ¼i Supða i ; a j Þð5Þ
Then the aggregation function
PA a 1 ; a 2 ; . . . ; a n ð Þ ¼ X n i¼1 ðð1 þ Tða i ÞÞa i Þ X n i¼1 ð1 þ Tða i ÞÞð6Þ
is called the PA operator.


Partitioned average operator

The partitioned average operator can aggregate the arguments in different partitions using the same (a 1 , a 2 , . .., a n ) be a collection of crisp numbers, S = ., a n , S k = {a 1 , a 2 ,

k |), Q i �Q j and
i �Q j (i, j = 1, 2, . .., n) be respectively the sum and product operations of Q i and Q j based on ATT, aQ r and Q s b (r, s = 1, 2, . .., n; a, b > 0) be respectively the multiplication operation of Q r and the power operation of Q s based on ATT,
Definition 10. Let Q 1 , Q 2 , . . ., Q n (Q i = <μ i , ν i >, i = 1, 2, . . ., n) be n qROFNs (q = 1, 2, 3, . . .), (Q 1 , Q 2 , . . ., Q n ) be a collection of Q 1 , Q 2 , . . ., Q n , S = {Q 1 , Q 2 , . . ., Q n } be an ordered set of Q 1 , Q 2 , . . ., Q n , S k = {Q 1 , Q 2 , . . ., Q |Sk| } (k = 1,Sup(Q r , Q s ) = 1 − D(Q r , Q s ) (r, s = 1, 2, . . ., n and s 6 ¼ r; D(Q r , Q s ) is the distance between Q r and Q s ) be the degree of support for Q r from Q s which satisfy 0 � Sup(Q r , Q s ) � 1, Sup(Q r , Q s ) = Sup(Q s , Q r ),andSup(Q r , Q s ) � Sup(Q u , Q v ) if |Q r −Q s | � |Q u −Q v |,andTðQ r Þ ¼ X n s¼1;s6 ¼r SupðQ r ; Q s Þð9Þ
Then the aggregation function According to Eq (1)-( 4) and (10), the following theorem is obtained:
qROFAPPMM D Q 1 ; Q 2 ; . . . ; Q n ð Þ ¼ 1 N � N k¼1 1 jS k j! � p2P jS k j � jS k j i k ¼1 nð1 þ TðQ pði k Þ ÞÞ X n j¼1 ð1 þ TðQ j ÞÞ Q pði k Þ 0 B B B B @ 1 C C C C A d i k 0 B B B B @ 1 C C C C A 1 X jS k j , . . ., Q n ) be a collection of Q 1 , Q 2 , . . ., Q n . Then qROFAPPMM D ðQ 1 ; Q 2 ; . . . ; Q n Þ ¼ hm; nið11Þ
and it is still a qROFN, where
m ¼ g À 1 1 N X N k¼1 g f À 1 1 X jS k j i k ¼1 d i k f g À 1 1 jS k j! X p2P jS k j g � f À 1 � X jS k j i k ¼1 � d i k f ðg À 1 ððnx pði k Þ Þgðm pði k Þ ÞÞÞ ��� 0 @ 1 A 0 @ 1 A 0 B B B B @ 1 C C C C A 0 B B B B B @ 1 C C C C C A 0 B B B B B @ 1 C C C C C Að12Þn ¼ f À 1 1 N X N k¼1 f g À 1 1 X jS k j i k ¼1 d i k g f À 1 1 jS k j! X p2P jS k j f � g À 1 � X jS k j i k ¼1 � d i k gðf À 1 ððnx pði k Þ Þf ðn pði k Þ ÞÞÞ ��� 0 @ 1 A 0 @ 1 A 0 B B B B @ 1 C C C C A 0 B B B B B @ 1 C C C C C A 0 B B B B B @ 1 C C C C C Að13Þ
and ξ p(ik) is a PA factor which can b or:
Theorem 2. Let Q 1 , Q 2 , . . ., Q n (Q i = <μ i , ν i >, i = 1, 2, . . ., n) be n qROFNs (q = 1, 2, 3, . . .) and 2 , . . ., Q n . If Q i = Q = ely.

From Eq (11), it is not difficult to see that the qROFAPPMM operator has ve influence of unreasonable aggregated qROFNs, because the operator is constructed via combining the MM, partitioned average, and PA operators.In addition, it can also be seen that the qRO-FAPPMM operator has desirable generality, since different specific qROFAPPMM operators will be obtained if different if specific functions are assigned to f.For example, n ð Þ ¼ * 1 À Y N k¼1 1 À 1 À Y p2P jS k j � 1 À Y jS k j i k ¼1 ð1 À ð1 À m q pði k Þ Þ nx pði k Þ Þ d i k � ! 1 jS k j!
0 B B @ 1 C C A 1 X jS k j i k ¼1 d i k 0 B B B B B B B B @ 1 C C C C C C C C A 0 B B B B B B B B @ 1 C C C C C C C C A 1 N 0 B B B B B B B B B B B @ 1 C C C C C C C C C C C A 1=q ; Y N k¼1 1 À 1 À Y p2P jS k j � 1 À Y jS k j i k ¼1 ð1 À n qnx pði k Þ pði k Þ Þ d i k � ! 1 jS k j! 0 B B @ 1 C C A 1 X jS k j i k ¼1 d i k 0 B B B B B B B B @ 1 C C C C C C C C A 0 B B B B B B B B @ 1 C C C C C C C C A 1 N 0 B B B B B B B B B B B @ 1 C C C C C C C C C C C A 1=q +ð15Þ
where ξ p(ik) is a PA factor which can be calculated via Eq (14).


If
f(t) = In[(2−t q )/t q ], then g(t) = In[(1+t q )/(1−t q )], f −1 (t) = [2/(e t +1
)] 1/q , and g −1 (t) = [(e t −1)/(e t +1)] 1/q .A q-rung orthopair fuzzy Archimedean Einstein pow * Y N k¼1 ðm 0 þ 3Þ 1= X jS k j i k ¼1 d i k þ 3ðm 0 À 1Þ 1= X jS k j i k ¼1 d i k ðm 0 þ 3Þ 1= X jS k j i k ¼1 C C A 0 B B B B B B @ 1 C C C C C C A 1 N þ 1 0 B B B B B B B B B pði k Þ Þ nx pði k Þ þ 3ð1 À m q pði k Þ Þ nx pði k Þ ð1 þ m q pði k Þ Þ nx pði k Þ À ð1 À m q pði k Þ Þ nx pði k Þ ! d i k þ 3 Y jS s cons = X jS k j i k ¼1 d i k À ðn 0 À 1Þ 1= X jS k j i k ¼1 d i k 0 B =q +ð19Þ
where
m 0 ¼ Y p2P jS k j Y jS k j i k ¼1 ðl þ ð1 À lÞð1 À m q pði k Þ ÞÞ nx pði k Þ þ ðl 2 À 1Þð1 À m q pði k Þ Þ nx À n 0 À 1ÞÞ d i k 0 @ 1 Að28Þm 0 ¼ 1 À log ε 1 þ ε 1À m q pði k a Eq (14).


qROFWAPPMM operator

The qROFAPPMM operator has advantages in having satisfying generality and flexibility, ca are the same as the functions of δ ik in the qROFAPPMM operator

see Eq (11)).

Accor
ing to Eqs (1)-( 4) and ( 31), the following theorem is obtained:
Theorem 4. Let Q 1 , Q 2 , . . ., Q n (Q i = <μ i , ν i >, i = 1, 2, . . ., n) be n qROFNs (q = 1, 2, 3, . . .) and (Q 1 , Q 2 , . . ., Q n ) be a collection of Q 1 , Q 2 , . . ., Q n . Then qROFWAPPMM D ðQ 1 ; Q 2 ; . . . ; Q n Þ ¼ hm; nið32Þ
and it is still a qROFN, where
m ¼ g À 1 1 N X N k¼1 g f À 1 1 X jS k j i k ¼1 d i k f g À 1 1 jS k j! X p2P jS k j g f À 1 X jS k j i k ¼1 d i k f g À 1 n � w pði k Þ x pði k Þ � , X n t¼1 ðw t x t ÞÞgðm pði k Þ Þ !!!!! 0 @ 1 A 0 @ 1 A 0 B B B B @ 1 C C C C A 0 B B B B B @ 1 C C C C C A 0 B B B B B @ 1 C C C C C Að33Þn ¼ f À 1 1 N X N k¼1 f g À 1 1 X jS k j i nd ξ t are two PA factors which can be calculated via Eq (14).

The proof of Theorem 4 is similar to the proof of Theorem 1 (see Appendix A in S1 File) and is omitted here.It is worth nothing that the qROFWAPPMM operator no longer has the properties of ed as follows:
1. If f(t) = −Int Y N k¼1 1 À 1 À Y p2P jS k j 1 À Y jS k j i k ¼1 1 À n qnðw pði k Þ x pði k Þ Þ � X n t¼1 ðw t x t Þ pði k Þ ! d i k !! 1 jS k j! 0 B B B @ 1 C C C A 1 X jS k j i k ¼1 d i k 0 B B B B B B B B @ 1 C C C C C C C C A 0 B B B B B B B B @ 1 C C C C C C C C A 1 N 0 B B B B B B B B B B B @ 1 C C C C C C C C C C C A 1=q +ð35Þ
where ξ p(ik) and ξ t are two PA factors which can be calculated via Eq (14).
2. If f(t) = In[(2−t q )/t q ], then g(t) = In[(1+t q )/(1−t q )], f −1 (t) = [2/(e t +1
)] 1/q , and g −1 (t) = [(e t −1)/(e t ucted according to Eq (32):
qROFWAEPPMM D Q 1 ; Q 2 ; . . . ; Q n ð Þ ¼ * Y N k¼1 ðm 0 þ 3Þ 1= X jS k j i k ¼1 d i k þ 3ðm A 0 B B B B B B @ 1 C C C C C C A 1 N þ 1 0 B B B B B B B B B @ 1 C C C C C C C C C A q pði A 0 B B B B B B @ 1 C C C C C C A 1 N þ l À 1 0 B B B B B B B B B B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C C C C C C C A 1=q ; l , Y N k¼1 ðn 0 pði k Þ , ððε À 1Þ=ðε m 000 À 1ÞÞ 1= X jS k j i k ¼1 d i k 0 B B B B @ 1 C C C C A . ., C |Ck| (k = 1, riteria in each C k whereas the criteria in different C k are independent of

ach other, a
ector of weights of criteria w = [w 1 , w 2 , . .., w n ] such that 0 � w 1 , w 2 , . .., w n � 1, w 1 +w 2 +. ..+w n = 1, and each element respectively denotes the relative importance of C 1 , C 2 , . .., C n , a set of experts
E = {E 1 , E 2 , . . ., E M }, a vector of weights of experts $ = [$ 1 , $ 2 , . . ., $ M ] such that 0 � $ 1 , $ 2 , . . ., $ M � 1, $ 1 +$ 2 +. . .+$ M = 1,
and each element respectively denotes the relative importance of E 1 , E 2 , . .., E M , and M q-rung orthopair fuzzy decision matrices M h = [Q h,i,j ] m×n (h = 1, 2,. .., M; i = 1, 2,. .., m; j = 1, 2,. .., n) such that Q h,i,j = <μ h,i,j , ν h,i,j > is a qROFN which denotes the evaluation value of C j with respect to A i provided by E k .Based on these components, the MCGDM problem can be described as: Determining the optimal alternative with the help of a ranking of the elements of A based on C, M h , w, and $.Using the qROF-WAPPMM operator, the problem is solved according to the following steps:

1. Normalise the q-rung orthopair fuzzy decision matrices M h .Generally, a MCGDM problem may contain two types of criteria, i.e. benefit and cost criteria, which affect the results of decision making positively and negatively, respectively.To eliminate the negative effect, the qrung orthopair fuzzy decision matrices
M h = [Q h,i,j ] m×n = <μ h,i,j , ν h,i,j > are normalised as M N;h ¼ ½Q h;i;j � m�n ¼ ½hm h;i;j ; n h;i;j i� m�n ; if C j is a benefit criterion ½hn h;i;j ; m h;i;j i� m�n ; if C j is a cost criterionð51Þ
( 2. Calculate the power weights of Q h,i,j .The power weights of Q h,i,j ar computed using
W h;i;j ¼ ð$ h x h Þ . X M z¼1 ð$ z x z Þ ¼ $ h 1 þ X M x¼1;x6 ¼h ð1 À DðQ h;i;j ; Q x i;j ÞÞ ! !. X M z¼1 $ z 1 þ X M y¼1;y6 ¼z ð1 À DðQ z;i;j ; Q y;i;j ÞÞ ! !ð52Þ
where
D(Q h N,h and the expert weight vector $ as input, the collective values of Q h,i,j are computed using
Q i;j ¼ hm i;j ; n i;j i ¼ qROFWAPPMM D ðQ 1;i;j ; Q 2;i;j ; . . . ; Q M;i;j Þð53Þ
where qROFWAPPMM is an arbitrary specific qROFWAPPMM operator, such as the qROFWAAPPMM operator in Eq (35), the qROFWAEPPMM operator in Eq (36), the qROFWAHPPMM operator in Eq (39), and the qROFWAFPPMM operator in Eq (42), and the values of the elements in Δ = (δ 1 , δ 2 , . .., δ M ) (It is worth nothing that there is only one partition) are determined via identifying the ould be mutually independent.Thus δ 1 > 0 and δ 2 = δ 3 = . . .= δ M = 0.

4. Calculate the power weights of Q i,j .The power weights of Q i,j are computed using
W i;j ¼ ðw j x j Þ

X n t¼1 ðw t x t Þ ¼ w j 1 þ X n r¼1;
6 ¼j ð1 À DðQ i;j ; Q i;r ÞÞ ! !. X n t¼1 w t 1 þ X n s¼1;s6 ¼t ð1 À DðQ i;t ; Q i;s ÞÞ ! !ð54Þ
where
D(Q i,j , Q i,r ) (D(Q i,r , Q i,s
)) is the Minkowski-type distance between Q i,j and Q i,r (Q i,t and Q i,s ) that can be computed via the Equation in Definition 5.


5.

Calculate the collective values of Q i,j .On the basis of the N partitions
C k = {C 1 , C 2 , . . ., C | Ck| }, suppose S i = {Q i,1 , Q i,2 , . .

, Q i,n
is an ordered set of Q i,1 , Q i,2 , . . ., Q i,n , S i,k = {Q i,1 , Q i,2 , . . ., Q i,|Sk| } are N partitions of S that correspond to C k , δ 1 , δ 2 , . . ., δ |Si,k| (k = 1, 2, . . ., N and δ 1 , 2 , . . ., δ |Si,k| � 0 but not at the same time δ 1 = δ 2 = . . . = δ |Si,k| = 0) are |S i,k | real numbers that respectively correspond to Q i,1 , Q i,2 , . . ., Q i,|Sk| , Δ k = (δ 1 , δ 2 , . . ., δ |Si,k| ) is a col- lection of δ 1 , δ 2 , . . ., δ |Si,k| , Δ = (Δ 1 , Δ 2 , . . ., Δ N ) is a collection of Δ 1 , Δ 2 , . . ., Δ N , p(i k ) is a per- mutation of (Q i ¼ hm i ; n i i ¼ qROFWAPPMM D ðQ i;1 ; Q i;2 ; . . . ; Q i;n Þð55Þ
where qROFWAPPMM is the same specific qROFWAPPMM operator used in Eq (53), and the values of the elements in 6. Calculate the scores and accuracies of Q i .The scores and accuracies of Q i can be respectively computed using the Equations in Definitions 2 and 3.
Δ = (Δ 1 , Δ 2 , . . ., Δ N ) (Δ k = (δ 1 , δ 2 , . . ., δ |Si,
7. Generate a ranking of A i .On the basis of the scores and accuracies of Q i , a ranking of A i can be generated according to the comparison rules in Definition 4.

8. Determine the optimal alternative.The optimal alternative is determined with the help of the ranking.


Example, experiments, and comparisons

In this section, a numerical example is firstly used to illustrate the working process of the proposed MCGDM method.Then a set of experiments are carried out to explore the influence of different specific operators and parameter values on the aggregation results.Finally, qualitative and quantitative comparisons to the existing methods are made to demonstrate the feasibility and effectiveness of the proposed method.


Example

A numerical example about the determination of the best industry for investment from five possible industries (adapted on the basis of Reference [31]) is used to demonstrate the proposed MCGDM method.

To make full use of idle capital, the board of directors of a company decided to invest ew industry.Five industries were identified as possible industries for investment after preliminary research.The five alternative industries are medical industry (A 1 ), real estate development industry (A 2 ), Internet industry (A 3 ), education and training industry (A 4 ), and manufacturing industry (A 5 ).To select the best industry for investment, the board of directors appointed an expert panel, which consists of four different experts E 1 , E 2 , E 3 , and E 4 .The relative importance of these experts is quantified by the weight vector $ = [0.30,0.22, 0.28, 0.20].The four experts were asked to evaluate the five alternative industries on the basis of five criteria, which are the amount of capital profit (C 1 ), the market potential (C 2 ), the risk of capital loss (C 3 ), the growth potential (C 4 ), and the stability of policy (C 5 ).The relative importance of these criteria is measured by the weight v cture of interrelationships, the five criteria are divided into two partitions C 1 = {C 1 , C 3 , C 5 } and C

ustry for investment is d
termined as medical industry (A 1 ).


Experiments

To explore the effect of using different specific operators and assigning different parameter values on the aggregation results, the following three experiments were carried out:

(1) Experiment 1 aims to show the influence of using different specific operators on the aggregation results.In this experiment, the presented qROFWAAPPMM (see Eq (35)), qROF-WAEPPMM (see Eq (36)), qROFWAHPPMM (see Eq (39)), and qROFWAFPPMM (see Eq (42)) operators are respectively used to calculate the collective values of Q h,i,j and the collective values of Q i,j in the numerical example (When calculating the collective values of Q h,i,j , q = 3, λ = 3, ε = 2, and Δ = (δ 1 , δ 2 , δ 3 , δ 4 ) = (1, 0, 0, 0); When calculating the collective values of Q i,j , q = 3, λ = 3, ε = 2, and Δ = (Δ 1 , Δ 2 ) = ((δ 1 , δ 2 , δ 3 ), (δ 1 , δ 2 )) = ((1, 2, 3), (1, 2))).The results of the experiment are the calculated scores of Q i and the generated rankings of A i , which are listed in Table 1.As can be seen from the table, there is slight difference among the scores of the same Q i calculated by the four pairs of specific operators, and the rankings of A i also indicate small difference with respect to the four pairs of specific operators.These indicate that the use of different sp cific operators has no obvious influence on the aggregation results.

(2) Experiment 2 aims to show the influence of assigning different q values on the aggregation results.In this experiment, the presented qROFWAHPPMM (see Eq (39) and qROFWAFPPMM (see Eq (42)) operators (Since the qROFWAHPPMM operator is the generalisation of the qROFWAAPPMM and qROFWAEPPMM operators, they are not included in this experiment and will not be included in he subsequent experiments and comparisons for the sake of simplicity) are respectively used to calculate the collective values of Q h,i,j and the collective values of Q i,j in the numerical example (When calculating the collective values of Q h,i,j , λ = 3, ε = 2, and Δ = (δ 1 , δ 2 , δ 3 , δ 4 ) = (1, 0, 0, 0); When calculating the collective values of Q i,j , λ = 3, ε = 2, and Δ = (Δ 1 , Δ 2 ) = ((δ 1 , δ 2 , δ 3 ), (δ 1 , δ 2 )) = ((1, 2, 3), (1, 2))).The results of the experiment are the calculated scores of Q i and the generated rankings of A i , which are depicted in Fig 1 .From the figure, it can be seen that the ranking will change as q changes.For the pair of qROFWAHPPMMs, the ranking is A 1 � A 3 � A 5 � A 4 � A 2 when q = 3, changes to A 1 � A 3 � A 4 � A 5 � A 2 when q = 4, 5, 6, and becomes A 3 � A 1 � A 4 � A 5 � A 2 when q = 7, 8, 9, 10; The best alternative changes from A 1 to A 3 from q = 7.For the pair of qROFWAFPPMMs, the ranking is
A 1 � A 3 � A 4 � A 5 � A 2 when q = 3, changes to A 3 � A 1 � A 4 � A 5
� A 2 when q = 4, 5, 6, 7, 8, 9, and becomes A 3 � A 4 � A 1 � A 5 � A 2 when q = 10; The best alternative changes from A 1 to A 3 from q = 4. Based on these results, it is recommended that the smallest q which can meet 0 � μ q + ν q � 1 is assigned in practical applications.For example, since M N,3 contains <0.9, 0.6>, q is assigned 3 since 0.9 2 + 0.6 2 > 1 and 0.9 3 + 0.6 3 < 1.

(3) Experiment 3 aims to show the influence of assigning different λ (ε) values on the aggregation results.In this experiment, the presented qROFWAHPPMM (see Eq (39) and qROFWAFPPMM (see Eq (42)) operators are respectively used to calculate the collective values of Q h,i,j and the collective values of Q i,j in the numerical example (When calculating the collective values of Q h,i,j , q = 3 and Δ = (δ 1 , δ 2 , δ 3 , δ 4 ) = (1, 0, 0, 0); When calculating the collective values of Q i,j , q = 3 and Δ = (Δ 1 , Δ 2 ) = ((δ 1 , δ 2 , δ 3 ), (δ 1 , δ 2 )) = ((1, 2, 3), (1, 2))).The results of the experiment are the calculated scores of Q i and the generated rankings of A i , which are depicted in Fig 2 .It can be seen from the figure that the scores computed by the pair of qROFWAHPPMMs (qROFWAFPPMMs) gradually decrease as λ (ε) gradually increases.Therefore, the parameter λ (ε) can be seen as a pessimistic factor for MCGDM problems.Generally, if the attitude of a decision maker is neutral, a small λ (ε) (e.g.λ = 1, 2, 3; ε = 2, 3, 4) is recommended.If the attitude is pessimistic enough, a bigger λ (ε) can be assigned when the pair of qROFWAHPPMMs (qROFWAFPPMMs) is used.Otherwise, a smaller λ (ε) is recommended.


Comparisons

As mentioned in the introduction, more than twenty different aggregation operators of qROFNs have been presented within academia.Representative examples are the WA and WG [26], WBM and WGBM [27], WABM [28], WPBM and WPGBM [29], WHM and WGHM [30], WHM � and WPHM [31], WMSM and WGMSM [32], WPMSM [33], WPPMSM [34], WMM and WGMM [35], WEBM [36], WE [37], and WP [38] operators.In this subsection, qualitative and quantitative comparisons between the MCGDM methods based on these operators and the proposed MCGDM method are carried out to demonstrate its feasibility and

ffectivenes
.

5.3.1.Qualitative comparison.Generally, a qualitative comparison among different MCGDM methods can be carried out by comparing their characteristics.For the twenty existing methods and the proposed method, the generality and flexibility in the aggregation of qROFNs, the capability to deal with the interrelationships among different criteria, and the capability to reduce the negative influence of the unduly high or unduly low criterion values on the aggregation results are selected as the comparison characteristics.The results of the comparison are shown in Table 2.The details of the comparison are explained as follows:

1. Generality and flexibility: For the WP method, any one of the twenty different WP operators can be used in the aggregation.Therefore, its generality and flexibility can be seen as moderate.The generality and flexibility of the WABM method and the proposed (WAPPMM) method are desirable since the aggregations are based on the operations of any family of ATTs.The aggregations in the remaining methods are based on the operation of a specific family of ATT.Relatively, they have limited generality and flexibility.

2. When all criteria are independent of each other: It is no doubt that all of the l sted methods can deal with this case.3. When there are interrelationships between any two criteria: The WE, WP, WA, and WG methods are only suitable for the independent case.All other methods have the capability of dealing with the case in which there are interrelationships between any two criteria.

4. When there are interrelationships among any multiple criteria: The WMSM, WGMSM, WPMSM, WPPMSM, WMM, and WGMM methods and t