# Linguistic Interval-Valued Intuitionistic Fuzzy Archimedean Power Muirhead Mean Operators for Multiattribute Group Decision-Making

Authors: Yuchu Qin; Xiaolan Cui; Meifa Huang; Yanru Zhong; Zhemin Tang; Peizhi Shi
Journal: Complexity
 (2020)

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## Abstract

Two important tasks in multiattribute group decision-making (MAGDM) are to describe the attribute values and to generate a ranking of all alternatives. A superior tool for the first task is linguistic interval-valued intuitionistic fuzzy number (LIVIFN), and an effective tool for the second task is aggregation operator (AO). To date, nearly ten AOs of LIVIFNs have been presented. Each AO has its own features and can work well in its specific context. But there is not yet an AO of LIVIFNs that can offer desirable generality and flexibility in aggregating attribute values and capturing attribute interrelationships and concurrently reduce the influence of unreasonable attribute values. To this end, a linguistic interval-valued intuitionistic fuzzy Archimedean power Muirhead mean operator and its weighted form, which have such capabilities, are presented in this paper. Firstly, the generalised expressions of the AOs are established by a combination of the Muirhead mean operator and the power average operator under the Archimedean T-norm and T-conorm operations of LIVIFNs. Then the properties of the AOs are explored and proved, their specific expressions are constructed, and the special cases of the specific expressions are discussed. After that, a new method for solving the MAGDM problems based on LIVIFNs is designed on the basis of the weighted AO. Finally, the designed method is illustrated via a practical example, and the presented AOs are evaluated via experiments and comparisons.


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## Body

## 1. Introduction

Multiattribute group decision-making (MAGDM) or multiattribute group decision analysis refers to a process of finding the most desirable alternatives from a set of finite alternatives on the basis of a ranking or the collective attribute values of all alternatives, in which the value of each attribute is provided by a group of experts [1]. This process has two critical tasks. This first task is to describe the values of attributes, while the second task is to generate a ranking of all alternatives.For the description of attribute values, real number is a default tool. But it is usually difficult for an expert to provide the assessment results in the form of crisp values because of ambiguity and incomplete information. To this end, fuzzy set is naturally introduced in MAGDM and becomes a popular tool in this field [2, 3]. To date, more than twenty different types of fuzzy sets have been presented with academia [4], where Atanassov’s intuitionistic fuzzy set (IFS) [5] and interval-valued IFS (IVIFS) [6] are two representative examples. IFS and IVIFS are powerful extensions of Zadeh’s fuzzy set [7] for dealing with vagueness. Both of them have a membership degree (MD) and a nonmembership degree (NMD), which can quantify the degrees of satisfaction and dissatisfaction, respectively. Due to such strong expressiveness, IFSs and IVIFSs have been widely used to express the values of attributes in MAGDM [8]. Many research topics about IFSs and IVIFSs for MAGDM, such as operational rules for the sets [9–13], fuzzy calculus for the sets [14–17], score and accuracy functions of the sets [18–22], preference relations of the sets [23–26], similarity and distance measures of the sets [27–30], aggregation operators (AOs) for the sets [31–35], and decision-making approaches based on the sets [36–40], have received widespread attention during the past few decades.Although IFSs and IVIFSs have gained importance and popularity in MAGDM, they can only be leveraged to express the rating values in quantitative aspect (i.e., the numerical rating values). In practical decision-making problems, the rating values may be denoted by other kinds of variables, where linguistic variables are one of the most important types [41–45]. For example, selection of a proper 3D printer from a certain number of alternatives to print a specific component is a classic decision-making problem in manufacturing domain. In this problem, the performance parameters of a 3D printer, such as surface roughness, strength, elongation, and hardness, may be described by some of linguistic terms like “small”, “medium”, and “large”. Under such case, IFSs and IVIFSs are not applicable. To address this issue, Chen et al. [46] extended the IFS to the linguistic IFS (LIFS), in which the MD and NMD are represented by linguistic variables. As a result, the rating values in the form of single-valued linguistic terms can be expressed by LIFSs. Because of such expressiveness, LIFSs have been applied to solve decision-making problems by a number of researchers. For example, Li et al. [47] developed a set of new operational rules and entropy for LIFSs; Zhang et al. [48] designed an extended outranking approach for decision-making problems with LIFSs; Liu et al. [49], Garg and Kumar [50], Peng et al. [51], and Teng and Liu [52] presented some AOs for LIFSs; Jin et al. [53] established a decision support model for MAGDM with the preferences relations of LIFSs.To further improve the expressiveness of LIFS, Garg and Kumar [54, 55] extended it to the linguistic interval-valued IFS (LIVIFS), in which the MD and NMD are quantified by the intervals of linguistic variables. LIVIFS can provide more freedom to experts, because it allows them to describe their preferences using intervals of linguistic terms. Due to such characteristic, Garg and Kumar [55] presented a weighted average (WA) operator, an ordered WA (OWA) operator, a hybrid average (HA) operator, a weighted geometric (WG) operator, an ordered WG (OWG) operator, and a hybrid geometric (HG) operator of linguistic interval-valued intuitionistic fuzzy numbers (LIVIFNs) and applied these operators to solve MAGDM problems; Kumar and Garg [56] presented a prioritised weighted averaging (PWA) operator, a prioritised ordered weighted averaging (POWA) operator, a prioritised weighted geometric (PWG) operator, and a prioritised ordered weighted geometric (POWG) operator and study their applications in MAGDM problems; Liu and Qin [57] presented a weighted Maclaurin symmetric mean (WMSM) operator of LIVIFNs and proposed a new decision-making method based on it; Garg and Kumar presented [58] an extended TOPSIS group decision-making method under LIVIFS environment; Tang et al. [59] developed a procedure for MAGDM with LIVIFSs that can cope with inconsistent and incomplete preference relations of LIVIFSs.For the generation of a ranking of all alternatives, there are usually two ways. One way is to use conventional decision-making methods (e.g., TOPSIS, VIKOR, and ELECTRE), and the other way is to adopt AOs. Generally, AOs can resolve the MAGDM problems more effectively than conventional methods, because they can generate both the collective attribute values and a ranking of all alternatives, while conventional methods can only provide a ranking [60]. So far, over ten AOs of LIVIFNs have been presented. Representative examples are the WA, OWA, HA, WG, OWG, and HG operators presented by Garg and Kumar [54, 55], the PWA, POWA, PWG, and POWG operators presented by Kumar and Garg [56], and the WMSM operator presented by Liu and Qin [57]. Each of these AOs can work well under its specific circumstance, but none of them can provide desirable generality and flexibility in aggregating attribute values and capturing attribute interrelationships and concurrently reduce the negative influence of extreme attribute values on aggregation result.In practical decision-making problems, the aggregation of attribute values is a complicated process, in which a set of general and versatile AOs is needed. Further, the attributes considered in the problems are always not independent of each other, but are interrelated. Thus, it is important and useful to use a general, flexible, and effective AO to capture the interrelationships of different attributes for making a reasonable decision [60]. The seven existing AOs, however, are sometimes not versatile and flexible, since all of them are based on a specific type of T-norm and T-conorm (i.e., Algebraic T-norm and T-conorm). Except the WMSM operator, all of the AOs can only deal with the situation in which the attributes are independent of each other or have priority relationships. In addition, the values of attributes are generally evaluated by domain experts. It is usually difficult to ensure the absolute objectivity of this way, which means that some biased experts will provide extreme attribute values [61]. To get a reasonable aggregation result under such circumstance, it is of necessity to reduce the effect of unduly high or unduly low attribute values. But none of the seven existing operators can achieve this. Based on these considerations, the motivations and objectives of the present paper are as follows:(1)
To develop a versatile and flexible AO of LIVIFNs, the Archimedean T-norm and T-conorm (ATNTC) operations [62], which can generate versatile and flexible operational rules for fuzzy numbers, are introduced to establish a set of operational rules of LIVIFNs(2)
To make the AO have the capability to capture the interrelationships among attributes, the Muirhead mean (MM) operator [63], which is suitable for the situations where all aggregated arguments are independent of each other, where there are interrelationships between any two arguments, and where there are interrelationships among any multiple arguments [64, 65], is selected as the core component of the AO(3)
To make the AO capable to reduce the negative influence of biased attribute values on the aggregation result, the power average (PA) operator [66], which has the capability to reduce the negative effect of unreasonable argument values, is combined with the MM operatorBased on the analysis above, this paper aims to present a linguistic interval-valued intuitionistic fuzzy weighted Archimedean power MM operator for MAGDM. This aim is achieved via combining the ATNTC operations, the MM operator, and the PA operator with weights in the context of LIVIFSs. The major contributions of the paper are as follows:(1)
A set of general and flexible operational rules of LIVIFNs based on ATNTC operations is developed. The operational rules of LIVIFNs in the existing AOs of LIVIFNs are based on Algebraic T-norm and T-conorm, which are sometimes not versatile and flexible enough. The developed operational rules are based on any types of T-norm and T-conorm. They have satisfying generality and flexibility.(2)
A weighted Archimedean power MM operator of LIVIFNs is presented to solve the MAGDM problems based on LIVIFNs. Compared to the existing AOs of LIVIFNs, the presented AO has generality and flexibility in aggregating attribute values and capturing attribute interrelationships and concurrently can reduce the influence of extreme attribute values.The rest of the paper is organised as follows. A brief introduction of some prerequisites is provided Section2. Section 3 explains the details of the presented AO of LIVIFNs. A MAGDM method based on the AO is designed in Section 4. Section 5 demonstrates the method and evaluates the AOs. Section 6 ends the paper with a conclusion.


## 2. Preliminaries

In this section, some prerequisites in LIVIFS theory, operational rules for LIVIFNs, PA operator, and MM operator are briefly introduced to facilitate the understanding of the paper.


### 2.1. LIVIFS Theory

LIVIFS was extended from IVIFS and LIFS by Garg and Kumar [54, 55]. Its formal definition is as follows.Definition 1 (see [55]).
LetSo,h=stso≤sh be a continuous linguistic term set (where sh is a possible value for a variable and h is a positive integer, and for any sx,sy∈So,h,sx>syiffx>y). A LIVIFS A in a finite universe of discourse XisA=<x,sμx,svx>x∈X (in this paper, the symbol in sr denotes that the subscript of s is r), where sμx=sμLx,sμUx and svx=svLx,svUx are subsets of [s0, sh] and, respectively, stand for the linguistic MD and NMD of x to A, and sμUx+svUx≤sh i.e.,μUx+vUx≤h for any x ∈ X. The linguistic intuitionistic index of x to A is sπx=sπLx,sπUx=sh−μUx−vUx,sh−μLx−vLx.
A pair,sμLx+sμUx,svLx+svUx, is called a LIVIFN. For convenience, a LIVIFN is denoted as α=sa,sb,sc,sd, where sa,sb,sc,sd∈S0,h,sa,sb⊆s0,sh,sc,sd⊆s0,sh, and b + d ≤ h. To compare two LIVIFNs, their scores and accuracies are required, which can be calculated according to the following definitions.Definition 2 (see [55]).
Letα = ([sa, sb], [sc, sd]) be a LIVIFN. Then its score is(1)Sα=s2h+a+b−d4.Definition 3 (see [55]).
Letα = ([sa, sb], [sc, sd]) be a LIVIFN. Then its accuracy is(2)Aα=Sa+b+c+d2.
UsingS (α) and A (α), two LIVIFNs can be compared via the following definition.Definition 4 (see [55]).
Letα1=sa1,sb1,sc1,sd1, and α2=sa2,sb2,sc2,sd2 and be any two LIVIFNs, S (α1) and S (α2) be, respectively, the scores of α1 and α2, and A (α1) and A (α2) be, respectively, the accuracies of α1 and α2. Then, (1) if S (α1) > S (α2), then α1 > α2; (2) if S (α1) = S (α2) and A (α1) > A (α2), then α1 > α2; (3) if S (α1) = S (α2) and A (α1) = A (α2), then α1 = α2.
To calculate the distance between two LIVIFNs, a distance measure of LIVIFNs is required. On the basis of the distance measure of IVIFNs introduced by Xu [67] and the distance measure of LIFNs introduced by Liu and Liu [68], the following distance measure for LIVIFNs is defined.Definition 5.
Letα1=sa1,sb1,sc1,sd1, and α2=sa2,sb2,sc2,sd2 be any two LIVIFNs. Then, the distance of α1 and α2 is(3)Dα1,α2=a1−a2+b1−b2+c1−c2+d1−d24h.
Obviously,D(α1, α2) satisfies (1) Dα1,α2=Dα2,α1≥0; (2) if α3=sa3,sb3,sc3,sd3 is an arbitrary LIVIFN, then Dα1,α3≤Dα1,α2+Dα2,α3.


### 2.2. Operational Rules

Motivated by the concept of T-norm and T-conorm, a set of operational rules for LIVIFNs based on the Algebraic T-norm and T-conorm were presented by Garg and Kumar [54, 55], Kumar and Garg [56], and Liu and Qin [57]. These operational rules are sometimes not very versatile and flexible since they are just based on a specific type of ATNTC. Inspired by the general operational rules for IFNs [69] and q-rung orthopair fuzzy numbers [60] (which are based on any types of ATNTCs), a set of operational rules for LIVIFNs based on ATNTCs is developed. The definition of the operational rules is as follows.Definition 6.
Letα1=sa1,sb1,sc1,sd1, α2=sa2,sb2,sc2,sd2, and α=saα,sbα,scα,sdα be any three LIVIFNs, and r be a real number that satisfies r > 0, the sum, product, multiplication, and power operations of LIVIFNs based on the Archimedean T-norm Tx,y=f−1fx+fy and its T-conorm Cx,y=g−1gx+gy can be, respectively, defined as follows:(4)α1⊕α2=sCa1,a2,sCb1,b2,sTc1,c2,sTd1,d2=sg−1ga1+ga2,sg−1gb1+gb2,sf−1fc1+fc2,sf−1fd1+fd2,(5)α1⊗α2=sTa1,a2,sTb1,b2,sCc1,c2,sCd1,d2=sf−1fa1+fa2,sf−1fb1+fb2,sg−1gc1+gc2,sg−1gd1+gd2,(6)rα=sg−1rgaα,sg−1rgbα,sf−1rfcα,sf−1rfdα,(7)αr=sf−1rfaα,sf−1rfbα,sg−1rgcα,sg−1rgdα.
Equations (4)‒(7) are generalised form of the operational rules for LIVIFNs based on ATNTCs. If f and g are assigned specific functions, then specific operational rules can be achieved. The following are four examples:(1)
Ifft=−Int/h and gt=−In1−t/h, then f−1t=he−t and g−1t=h−he−t. Four operational rules for LIVIFNs based on Algebraic T-norm and T-conorm are obtained as follows:(8)α1⊕α2=sa1+a2−a1a2/h,sb1+b2−b1b2/h,sc1c2/h,sd1d2/h,(9)α1⊗α2=sa1a2/h,sb1b2/h,sc1+c2−c1c2/h,sd1+d2−d1d2/h,(10)rα=sh1−1−aα/hr,sh1−1−bα/hr,shcα/hr,shdα/hr,(11)αr=shaα/hr,shbα/hr,sh1−1−cα/hr,sh1−1−dα/hr.(2)
Ifft=−In2h−t/t and gt=Inh+t/h−t, then f−1t=2h/et+1 and g−1t=het−h/et+1. Four operational rules for LIVIFNs based on Einstein T-norm and T-conorm are obtained as follows:(12)α1⊕α2=sh2a1+a2/h2+a1a2,sh2b1+b2/h2+b1b2,shc1c2/2h2−hc1+c2+c1c2,shd1d2/2h2−hd1+d2+d1d2,(13)α1⊗α2=sha1a2/2h2−ha1+a2+a1a2,shb1b2/2h2−hb1+b2+b1b2,sh2c1+c2/h2+c1c2,sh2d1+d2/h2+d1d2,(14)rα=shh+aαr−hh−aαr/h+aαr+h−aαr,shh+bαr−hh−bαr/h+bαr+h−bαr,s2hcαr/2h−cαr+cαr,s2hdαr/2h−dαr+dαr,(15)αr=s2haαr/2h−aαr+aαr,s2hbαr/2h−bαr+bαr,shh+cαr−hh−cαr/h+cαr+h−cαr,shh+dαr−hh−dαr/h+dαr+h−dαr.(3)
Ifft=Inhλ+1−λt/tλ>0 and gt=Inh+λ−1t/h−t then f−1t=λh/et+λ−1 and g−1t=het−h/et+λ−1. Four operational rules for LIVIFNs based on Hamacher T-norm and T-conorm are obtained as follows:(16)α1⊕α2=sh2a1+a2+λ−2ha1a2/h2+λ−1a1a2,sh2b1+b2+λ−2hb1b2/h2+λ−1b1b2,shc1c2/λh2+h−λhc1+c2+λ−1c1c2,shd1d2/λh2+h−λhd1+d2+λ−1d1d2,(17)α1⊗α2=sha1a2/λh2+h−λha1+a2+λ−1a1a2,shb1b2/λh2+h−λhb1+b2+λ−1b1b2,sh2c1+c2+λ−2hc1c2/h2+λ−1c1c2,sh2d1+d2+λ−2hd1d2/h2+λ−1d1d2,(18)rα=shh+λ−1aαr−hh−aαr/h+λ−1aαr+λ−1h−aαr,shh+λ−1bαr−hh−bαr/h+λ−1bαr+λ−1h−bαr,sλhcαr/hλ+1−λcαr+λ−1cαr,sλhdαr/hλ+1−λdαr+λ−1dαr,(19)αr=sλhaαr/hλ+1−λaαr+λ−1aαr,sλhbαr/hλ+1−λbαr+λ−1bαr,shh+λ−1cαr−hh−cαr/h+λ−1cαr+λ−1h−cαr,shh+λ−1dαr−hh−dαr/h+λ−1dαr+λ−1h−dαr.(4)
Ifft=−Inε−1/εt/h−1ε>1 and gt=−Inε−1/ε1−t/h−1, then f−1t=logε1+ε−1/e−th and g−1t=h−logε1+ε−1/e−th. Four operational rules for LIVIFNs based on Frank T-norm and T-conorm are obtained as follows:(20)α1⊕α2=sh−logε1+ε1−a1/h−1ε1−a2/h−1/ε−1h,sh−logε1+ε1−b1/h−1ε1−b2/h−1/ε−1h,slogε1+εc1/h−1εc2/h−1/ε−1h,slogε1+εd1/h−1εd2/h−1/ε−1h,(21)α1⊗α2=slogε1+εa1/h−1εa2/h−1/ε−1h,slogε1+εb1/h−1εb2/h−1/ε−1h,sh−logε1+ε1−c1/h−1ε1−c2/h−1/ε−1h,sh−logε1+ε1−d1/h−1ε1−d2/h−1/ε−1h,(22)rα=sh−logε1+ε1−aα/h−1r/ε−1r−1h,sh−logε1+ε1−bα/h−1r/ε−1r−1h,slogε1+εcα/h−1r/ε−1r−1h,slogε1+εdα/h−1r/ε−1r−1h,(23)αr=slogε1+εaα/h−1r/ε−1r−1h,slogε1+εbα/h−1r/ε−1r−1h,sh−logε1+ε1−cα/h−1r/ε−1r−1h,sh−logε1+ε1−dα/h−1r/ε−1r−1h.
The operational rules in equations (8)‒(11) are the four operational rules in [54–57]. From these examples, it can be seen that the developed operational rules can be used to derive any operational rules based on ATNTCs and the operational rules in [54–57] are just one special case of the developed operational rules. Therefore, the developed operational rules are more general and flexible than the operational rules in [54–57].


### 2.3. PA Operator

The PA operator, introduced by Yager [66], has the capability to assign weights to arguments via computing their support degrees. This makes it possible to reduce the negative effect of the unduly high or unduly low argument values on the aggregation result. The formal definition of this operator is as follows.Definition 7 (see [66]).
Let (a1, a2, …, an) be a collection of crisp numbers, Sai,aj=1−Dai,aj (where, Dai,aj is the distance of ai and aj and i, j = 1, 2, …, n and j ≠ i) be the support degree for ai from aj which has the following properties: (1) 0≤Sai,aj≤1; (2) Sai,aj=Saj,ai; (3) Sai,aj≥Sap,ap if ai−aj≤ap−ap, and(24)Tai=∑j=1,j≠inSai,aj.
Then the aggregation function(25)PAa1,a2,...,an=∑i=1n1+Taiai∑i=1n1+Tai,is called the PA operator.


### 2.4. MM Operator

The MM operator was firstly introduced to aggregate crisp numbers by Muirhead [63]. It can capture the interrelationships of arguments and provide a generalised form of several other AOs. The formal definition of the MM operator is as follows.Definition 8 (see [63]).
Let (a1, a2, …, an) be a collection of crisp numbers, Q=Q1,Q2,…,Qn (where Q1,Q2,…,Qn≥0 but not at the same time Q1=Q2=⋯=Qn=0) be a collection of n real numbers, p(i) be a permutation of (1, 2, …, n), and Pn be the set of all permutations of (1, 2, …, n). Then the aggregation function(26)MMQa1,a2,…,an=1n!∑p∈Pn∏i=1napiQi1/Σi=1nQi,is called the MM operator.
In this operator, whether the interrelationships are considered depends on the values ofQii=1,2,…,n: (1) if Q1=Q>0 and Q2=Q3=⋯=Qn=0, then the interrelationships are not considered; (2) if Q1=Q2>0 and Q3=Q4=⋯=Qn=0, then the interrelationships between any two crisp numbers are considered; (3) if Q1,Q2,…Qk>0k=3,4,…,n and Qk+1=Qk+2=⋯=Qn=0, then the interrelationships among any k crisp numbers are considered. Further, different AOs can be obtained via assigning different values to Q1,Q2,…,Qn:(1)
IfQ1=Q>0 and Q2=Q3=⋯=Qn=0 then the MM operator will reduce to the generalised arithmetic average (GAA) operator. When Q = 1, it will become the arithmetic average (AA) operator.(2)
IfQ1,Q2>0 and Q3=Q4=⋯=Qn=0, then the MM operator will reduce to the Bonferroni mean (BM) operator.(3)
IfQ1=Q2=⋯=Qk=1 and Qk+1=Qk+2=⋯=Qn=0, then the MM operator will reduce to the Maclaurin symmetric mean (MSM) operator.(4)
IfQ1=Q2=⋯=Qn=Q>0, then the MM operator will reduce to the generalised geometric average (GGA) operator. When Q = 1, it will become the geometric average (GA) operator.


## 2.1. LIVIFS Theory

LIVIFS was extended from IVIFS and LIFS by Garg and Kumar [54, 55]. Its formal definition is as follows.Definition 1 (see [55]).
LetSo,h=stso≤sh be a continuous linguistic term set (where sh is a possible value for a variable and h is a positive integer, and for any sx,sy∈So,h,sx>syiffx>y). A LIVIFS A in a finite universe of discourse XisA=<x,sμx,svx>x∈X (in this paper, the symbol in sr denotes that the subscript of s is r), where sμx=sμLx,sμUx and svx=svLx,svUx are subsets of [s0, sh] and, respectively, stand for the linguistic MD and NMD of x to A, and sμUx+svUx≤sh i.e.,μUx+vUx≤h for any x ∈ X. The linguistic intuitionistic index of x to A is sπx=sπLx,sπUx=sh−μUx−vUx,sh−μLx−vLx.
A pair,sμLx+sμUx,svLx+svUx, is called a LIVIFN. For convenience, a LIVIFN is denoted as α=sa,sb,sc,sd, where sa,sb,sc,sd∈S0,h,sa,sb⊆s0,sh,sc,sd⊆s0,sh, and b + d ≤ h. To compare two LIVIFNs, their scores and accuracies are required, which can be calculated according to the following definitions.Definition 2 (see [55]).
Letα = ([sa, sb], [sc, sd]) be a LIVIFN. Then its score is(1)Sα=s2h+a+b−d4.Definition 3 (see [55]).
Letα = ([sa, sb], [sc, sd]) be a LIVIFN. Then its accuracy is(2)Aα=Sa+b+c+d2.
UsingS (α) and A (α), two LIVIFNs can be compared via the following definition.Definition 4 (see [55]).
Letα1=sa1,sb1,sc1,sd1, and α2=sa2,sb2,sc2,sd2 and be any two LIVIFNs, S (α1) and S (α2) be, respectively, the scores of α1 and α2, and A (α1) and A (α2) be, respectively, the accuracies of α1 and α2. Then, (1) if S (α1) > S (α2), then α1 > α2; (2) if S (α1) = S (α2) and A (α1) > A (α2), then α1 > α2; (3) if S (α1) = S (α2) and A (α1) = A (α2), then α1 = α2.
To calculate the distance between two LIVIFNs, a distance measure of LIVIFNs is required. On the basis of the distance measure of IVIFNs introduced by Xu [67] and the distance measure of LIFNs introduced by Liu and Liu [68], the following distance measure for LIVIFNs is defined.Definition 5.
Letα1=sa1,sb1,sc1,sd1, and α2=sa2,sb2,sc2,sd2 be any two LIVIFNs. Then, the distance of α1 and α2 is(3)Dα1,α2=a1−a2+b1−b2+c1−c2+d1−d24h.
Obviously,D(α1, α2) satisfies (1) Dα1,α2=Dα2,α1≥0; (2) if α3=sa3,sb3,sc3,sd3 is an arbitrary LIVIFN, then Dα1,α3≤Dα1,α2+Dα2,α3.


## 2.2. Operational Rules

Motivated by the concept of T-norm and T-conorm, a set of operational rules for LIVIFNs based on the Algebraic T-norm and T-conorm were presented by Garg and Kumar [54, 55], Kumar and Garg [56], and Liu and Qin [57]. These operational rules are sometimes not very versatile and flexible since they are just based on a specific type of ATNTC. Inspired by the general operational rules for IFNs [69] and q-rung orthopair fuzzy numbers [60] (which are based on any types of ATNTCs), a set of operational rules for LIVIFNs based on ATNTCs is developed. The definition of the operational rules is as follows.Definition 6.
Letα1=sa1,sb1,sc1,sd1, α2=sa2,sb2,sc2,sd2, and α=saα,sbα,scα,sdα be any three LIVIFNs, and r be a real number that satisfies r > 0, the sum, product, multiplication, and power operations of LIVIFNs based on the Archimedean T-norm Tx,y=f−1fx+fy and its T-conorm Cx,y=g−1gx+gy can be, respectively, defined as follows:(4)α1⊕α2=sCa1,a2,sCb1,b2,sTc1,c2,sTd1,d2=sg−1ga1+ga2,sg−1gb1+gb2,sf−1fc1+fc2,sf−1fd1+fd2,(5)α1⊗α2=sTa1,a2,sTb1,b2,sCc1,c2,sCd1,d2=sf−1fa1+fa2,sf−1fb1+fb2,sg−1gc1+gc2,sg−1gd1+gd2,(6)rα=sg−1rgaα,sg−1rgbα,sf−1rfcα,sf−1rfdα,(7)αr=sf−1rfaα,sf−1rfbα,sg−1rgcα,sg−1rgdα.
Equations (4)‒(7) are generalised form of the operational rules for LIVIFNs based on ATNTCs. If f and g are assigned specific functions, then specific operational rules can be achieved. The following are four examples:(1)
Ifft=−Int/h and gt=−In1−t/h, then f−1t=he−t and g−1t=h−he−t. Four operational rules for LIVIFNs based on Algebraic T-norm and T-conorm are obtained as follows:(8)α1⊕α2=sa1+a2−a1a2/h,sb1+b2−b1b2/h,sc1c2/h,sd1d2/h,(9)α1⊗α2=sa1a2/h,sb1b2/h,sc1+c2−c1c2/h,sd1+d2−d1d2/h,(10)rα=sh1−1−aα/hr,sh1−1−bα/hr,shcα/hr,shdα/hr,(11)αr=shaα/hr,shbα/hr,sh1−1−cα/hr,sh1−1−dα/hr.(2)
Ifft=−In2h−t/t and gt=Inh+t/h−t, then f−1t=2h/et+1 and g−1t=het−h/et+1. Four operational rules for LIVIFNs based on Einstein T-norm and T-conorm are obtained as follows:(12)α1⊕α2=sh2a1+a2/h2+a1a2,sh2b1+b2/h2+b1b2,shc1c2/2h2−hc1+c2+c1c2,shd1d2/2h2−hd1+d2+d1d2,(13)α1⊗α2=sha1a2/2h2−ha1+a2+a1a2,shb1b2/2h2−hb1+b2+b1b2,sh2c1+c2/h2+c1c2,sh2d1+d2/h2+d1d2,(14)rα=shh+aαr−hh−aαr/h+aαr+h−aαr,shh+bαr−hh−bαr/h+bαr+h−bαr,s2hcαr/2h−cαr+cαr,s2hdαr/2h−dαr+dαr,(15)αr=s2haαr/2h−aαr+aαr,s2hbαr/2h−bαr+bαr,shh+cαr−hh−cαr/h+cαr+h−cαr,shh+dαr−hh−dαr/h+dαr+h−dαr.(3)
Ifft=Inhλ+1−λt/tλ>0 and gt=Inh+λ−1t/h−t then f−1t=λh/et+λ−1 and g−1t=het−h/et+λ−1. Four operational rules for LIVIFNs based on Hamacher T-norm and T-conorm are obtained as follows:(16)α1⊕α2=sh2a1+a2+λ−2ha1a2/h2+λ−1a1a2,sh2b1+b2+λ−2hb1b2/h2+λ−1b1b2,shc1c2/λh2+h−λhc1+c2+λ−1c1c2,shd1d2/λh2+h−λhd1+d2+λ−1d1d2,(17)α1⊗α2=sha1a2/λh2+h−λha1+a2+λ−1a1a2,shb1b2/λh2+h−λhb1+b2+λ−1b1b2,sh2c1+c2+λ−2hc1c2/h2+λ−1c1c2,sh2d1+d2+λ−2hd1d2/h2+λ−1d1d2,(18)rα=shh+λ−1aαr−hh−aαr/h+λ−1aαr+λ−1h−aαr,shh+λ−1bαr−hh−bαr/h+λ−1bαr+λ−1h−bαr,sλhcαr/hλ+1−λcαr+λ−1cαr,sλhdαr/hλ+1−λdαr+λ−1dαr,(19)αr=sλhaαr/hλ+1−λaαr+λ−1aαr,sλhbαr/hλ+1−λbαr+λ−1bαr,shh+λ−1cαr−hh−cαr/h+λ−1cαr+λ−1h−cαr,shh+λ−1dαr−hh−dαr/h+λ−1dαr+λ−1h−dαr.(4)
Ifft=−Inε−1/εt/h−1ε>1 and gt=−Inε−1/ε1−t/h−1, then f−1t=logε1+ε−1/e−th and g−1t=h−logε1+ε−1/e−th. Four operational rules for LIVIFNs based on Frank T-norm and T-conorm are obtained as follows:(20)α1⊕α2=sh−logε1+ε1−a1/h−1ε1−a2/h−1/ε−1h,sh−logε1+ε1−b1/h−1ε1−b2/h−1/ε−1h,slogε1+εc1/h−1εc2/h−1/ε−1h,slogε1+εd1/h−1εd2/h−1/ε−1h,(21)α1⊗α2=slogε1+εa1/h−1εa2/h−1/ε−1h,slogε1+εb1/h−1εb2/h−1/ε−1h,sh−logε1+ε1−c1/h−1ε1−c2/h−1/ε−1h,sh−logε1+ε1−d1/h−1ε1−d2/h−1/ε−1h,(22)rα=sh−logε1+ε1−aα/h−1r/ε−1r−1h,sh−logε1+ε1−bα/h−1r/ε−1r−1h,slogε1+εcα/h−1r/ε−1r−1h,slogε1+εdα/h−1r/ε−1r−1h,(23)αr=slogε1+εaα/h−1r/ε−1r−1h,slogε1+εbα/h−1r/ε−1r−1h,sh−logε1+ε1−cα/h−1r/ε−1r−1h,sh−logε1+ε1−dα/h−1r/ε−1r−1h.
The operational rules in equations (8)‒(11) are the four operational rules in [54–57]. From these examples, it can be seen that the developed operational rules can be used to derive any operational rules based on ATNTCs and the operational rules in [54–57] are just one special case of the developed operational rules. Therefore, the developed operational rules are more general and flexible than the operational rules in [54–57].


## 2.3. PA Operator

The PA operator, introduced by Yager [66], has the capability to assign weights to arguments via computing their support degrees. This makes it possible to reduce the negative effect of the unduly high or unduly low argument values on the aggregation result. The formal definition of this operator is as follows.Definition 7 (see [66]).
Let (a1, a2, …, an) be a collection of crisp numbers, Sai,aj=1−Dai,aj (where, Dai,aj is the distance of ai and aj and i, j = 1, 2, …, n and j ≠ i) be the support degree for ai from aj which has the following properties: (1) 0≤Sai,aj≤1; (2) Sai,aj=Saj,ai; (3) Sai,aj≥Sap,ap if ai−aj≤ap−ap, and(24)Tai=∑j=1,j≠inSai,aj.
Then the aggregation function(25)PAa1,a2,...,an=∑i=1n1+Taiai∑i=1n1+Tai,is called the PA operator.


## 2.4. MM Operator

The MM operator was firstly introduced to aggregate crisp numbers by Muirhead [63]. It can capture the interrelationships of arguments and provide a generalised form of several other AOs. The formal definition of the MM operator is as follows.Definition 8 (see [63]).
Let (a1, a2, …, an) be a collection of crisp numbers, Q=Q1,Q2,…,Qn (where Q1,Q2,…,Qn≥0 but not at the same time Q1=Q2=⋯=Qn=0) be a collection of n real numbers, p(i) be a permutation of (1, 2, …, n), and Pn be the set of all permutations of (1, 2, …, n). Then the aggregation function(26)MMQa1,a2,…,an=1n!∑p∈Pn∏i=1napiQi1/Σi=1nQi,is called the MM operator.
In this operator, whether the interrelationships are considered depends on the values ofQii=1,2,…,n: (1) if Q1=Q>0 and Q2=Q3=⋯=Qn=0, then the interrelationships are not considered; (2) if Q1=Q2>0 and Q3=Q4=⋯=Qn=0, then the interrelationships between any two crisp numbers are considered; (3) if Q1,Q2,…Qk>0k=3,4,…,n and Qk+1=Qk+2=⋯=Qn=0, then the interrelationships among any k crisp numbers are considered. Further, different AOs can be obtained via assigning different values to Q1,Q2,…,Qn:(1)
IfQ1=Q>0 and Q2=Q3=⋯=Qn=0 then the MM operator will reduce to the generalised arithmetic average (GAA) operator. When Q = 1, it will become the arithmetic average (AA) operator.(2)
IfQ1,Q2>0 and Q3=Q4=⋯=Qn=0, then the MM operator will reduce to the Bonferroni mean (BM) operator.(3)
IfQ1=Q2=⋯=Qk=1 and Qk+1=Qk+2=⋯=Qn=0, then the MM operator will reduce to the Maclaurin symmetric mean (MSM) operator.(4)
IfQ1=Q2=⋯=Qn=Q>0, then the MM operator will reduce to the generalised geometric average (GGA) operator. When Q = 1, it will become the geometric average (GA) operator.


## 3. Aggregation Operators

In this section, a linguistic interval-valued intuitionistic fuzzy Archimedean power MM (LIVIFAPMM) operator and its weighted form, i.e., a linguistic interval-valued intuitionistic fuzzy Archimedean weighted power MM (LIVIFAWPMM) operator, are presented. The properties of these AOs are explored, and their specific cases are discussed.


### 3.1. LIVIFAPMM Operator

A LIVIFAPMM operator is an AO of LIVIFNs constructed via combining the PA operator and the MM operator under linguistic interval-valued intuitionistic fuzzy environment. The operations in this operator are based on ATNTCs. The formal definition of the operator is as follows.Definition 9.
Letα1,α2,…,αn (where αisai,sbi,sci,sdi,i=1,2,…,n) be a collection of n LIVIFNs, Q=Q1=Q2=⋯=Qn (where Q1=Q2=⋯=Qn≥0 but not at the same time Q1=Q2=⋯=Qn=0) be a collection of n real numbers that, respectively, correspond to α1,α2,…,αn,pi be a permutation of (1, 2, …, n), Pn be the set of all permutations of (1, 2, …, n), αi⊕αj and αi⊗αjj=1,2,…,n be, respectively, the sum and product operations of αi and αj based on ATNTCs, rαk and αrk (k = 1, 2, …, n; r > 0) be, respectively, the multiplication and power operations of αk based on ATNTCs, Sαi,αj=1−Dαi,αj (where Dαi,αj is the distance of αi and αj and i, j = 1, 2, …, n and j ≠ i) be the support degree for αi from αj which satisfies 0≤Sαi,αj≤1, Sαi,αj=Sαj,αi, and Sαi,αj≥Sαp,αp if Sαi,αj≤αp,αp, and(27)Tαi=∑j=1,j≠inSαi,αj.
Then the aggregation function(28)LIVIFAPMMQα1,α2,…,αn=1n!⊕p∈Pn⊗i=1nn1+Tαpi∑j=1n1+TαjαpiQi1/Σi=1nQi,is called the LIVIFAPMM operator.
In this operator, the values ofQi are used to capture the interrelationships of the aggregated LIVIFNs: (1) if Q1>0 and Q2=Q3=⋯=Qn=0, then the LIVIFNs are independent of each other; (2) if Q1,Q2>0 and Q3=Q4=⋯=Qn=0, then the interrelationships between any two LIVIFNs are considered; (3) if Q1=Q2=⋯=Qk>0k=3,4,…,n and Qk+1=Qk+2=⋯=Qn=0, then the interrelationships among any k LIVIFNs are considered.
According to equations (4)‒(7) and (28), the following theorem is obtained.Theorem 1.
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. Then(29)LIVIFAPMMQα1,α2,…,αn=sa,sb,sc,sd,and it is still a LIVIFN, where(30)sa=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nξpigapi,sb=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nξpigbpi,sc=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nξpifcpi,sd=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nξpifdpi,and ξp(i) is a PA factor that can be computed by(31)ξi=1+∑p=1,p≠in1−Dαi,αp∑j=1n1+∑q=1,q≠jn1−Dαj,αq.For the details regarding the proof of this theorem, please refer to Appendix A. The following three theorems respectively state the idempotency, commutativity, and boundedness of the LIVIFPMM operator.Theorem 2 (idempotency).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. If αi=α=saα,sbα,scα,sdα for all i=1,2,…,n, then LIVIFAPMMQα1,α2,…,αn=α.Theorem 3 (commutativity).
Letα1,α2,…,αn (where ai=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. If β1,β2,…,βn is any permutation of α1,α2,…,αn, then LIVIFAPMMQα1,α2,…,αn=LIVIFAPMMQβ1,β2,…,βn.Theorem 4 (boundedness).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs, α−=sminai,sminbi,smaxcici,smaxdi, and α+=smaxai,smaxbi,smincici,smindi, Then α−≤LIVIFAPMMQα1,α2,…,αn≤α+.For the details regarding the proofs of these three theorems, please refer to Appendices B–D, respectively.Equation (29) is a generalised form of the LIVIFAPMM operator. If specific functions are assigned to f and g, then specific operators can be constructed. For example, if the additive generators of Algebraic T-norm and T-conorm [60, 69] are, respectively, assigned to f and g, i.e., f(t) = −In(t/h) and gt = −In(1−t/h), then a linguistic interval-valued intuitionistic fuzzy power MM (LIVIFPMM) operator is constructed:(32)LIVIFPMMQα1,α2,…,αn=sh1−∏p∈Pn1−∏i=1n1−1−api/hnξpiQi1/n!1/Σi=1nQish1−∏p∈Pn1−∏i=1n1−1−bpi/hnξpiQi1/n!1/Σi=1nQi,sh1−1−∏p∈Pn1−∏i=1n1−cpi/hnξpiQi1/n!1/Σi=1nQish1−1−∏p∈Pn1−∏i=1n1−dpi/hnξpiQi1/n!1/Σi=1nQi.This operator has the following special cases:(1)
IfQ1=Q>0 and Q2=Q3=⋯=Qn=0, then the LIVIFPMM operator will reduce to(33)sh1−∏i=1n1−1−ai/hnξiQ1/n1/Qsh1−∏i=1n1−1−bi/hnξiQ1/n1/Q,sh1−1−∏i=1n1−1−ci/hnξiQ1/n1/Qsh1−1−∏i=1n1−1−di/hnξiQ1/n1/Q=LIVIFPGAAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power GAA (LIVIFPGAA) operator. WhenQ = 1, it will become(34)sh1−∏i=1n1−1−ai/hnξi1/n,sh1−∏i=1n1−1−bi/hnξi1/n,sh∏i=1nci/hnξi1/n,sh∏i=1ndi/hnξi1/n=LIVIFPAAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power AA (LIVIFPAA) operator.(2)
IfQ1,Q2>0 and Q3=Q4=⋯=Qn=0 then the LIVIFPMM operator will reduce to(35)sh1−∏i,j=1j≠in1−1−ai/hnξiQ11−1−aj/hnξjQ21/nn−11/Q1+Q2sh1−∏i,j=1j≠in1−1−bi/hnξiQ11−1−bj/hnξjQ21/nn−11/Q1+Q2,sh1−1−∏i,j=1j≠in1−1−ci/hnξiQ11−cj/hnξjQ21/nn−11/Q1+Q2sh1−1−∏i,j=1j≠in1−1−di/hnξiQ11−dj/hnξjQ21/nn−11/Q1+Q2=LIVIFPBMQ1,Q2α1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power BM (LIVIFPBM) operator.(3)
IfQ1=Q2=⋯=Qk=1 and Qk+1=Qk+2=⋯=Qn=0, then the LIVIFPMM operator will reduce to(36)sh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−aij/hnξijk!n−k!/n!1/ksh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−bij/hnξijk!n−k!/n!1/k,sh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−cij/hnξijk!n−k!/n!1/ksh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−dij/hnξijk!n−k!/n!1/k=LIVIFPMSMkα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power MSM (LIVIFPMSM) operator.(4)
IfQ1=Q2=⋯=Qn=Q>0, then the LIVIFPMM operator will reduce to(37)sh1−1−∏i=1n1−1−ai/hnξiQ1/n1/Qsh1−1−∏i=1n1−1−bi/hnξiQ1/n1/Q,sh1−∏i=1n1−1−ci/hnξiQ1/n1/Qsh1−∏i=1n1−1−di/hnξiQ1/n1/Q=LIVIFPGGAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power GGA (LIVIFPGGA) operator. WhenQ = 1, it will become(38)sh∏i=1nai/hnξi1/n,sh∏i=1nbi/hnξi1/n,sh1−∏i=1n1−1−ci/hnξi1/n,sh1−∏i=1n1−1−di/hnξi1/n=LIVIFPGAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power GA (LIVIFPGA) operator.Similarly, if the additive generators of other (e.g., Einstein, Hamacher, Frank) ATNTCs [60, 69] are, respectively, assigned to f and g, then other specific operators can be constructed according to equation (29).


### 3.2. LIVIFAWPMM Operator

The LIVIFAPMM operator has advantages in having desirable generality and flexibility, capturing the complex interrelationships of LIVIFNs, and reducing the negative effect of unreasonable LIVIFNs on the aggregation result. But it does not consider the relative importance of each aggregated LIVIFN. To this end, weights are introduced and a LIVIFAWPMM operator is presented. The formal definition of this operator is as follows.Definition 10.
On the basis of Definition9, let w1,w2,…,wn be, respectively, the weights of α1,α2,…,αn such that 0≤w1,w2,…,wn≤1 and w1+w2+⋯+wn=1. Then the aggregation function(39)LIVIFAWPMMQα1,α2,…,αn=1n!⊕p∈Pn⊗i=1nnwpi1+Tαpi∑j=1n1+TαjαpiQi1/Σi=1nQi,is called the LIVIFAWPMM operator.
In this operator, the function ofQi is the same as the function of Qi in the LIVIFAPMM operator (see equation (28)).
According to equations (4)‒(7) and (39), the following theorem is obtained.Theorem 5.
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. Then(40)LIVIFAWPMMQα1,α2,…,αn=sa,sb,sc,sd,and it is still a LIVIFN, where(41)sa=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nwpiξpi/∑t=1nwtξtgapi,sb=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nwpiξpi/∑t=1nwtξtgbpi,sc=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nwpiξpi/∑t=1nwtξtfcpi,sd=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nwpiξpi/∑t=1nwtξtfdpi,and ξp(i) is a PA factor that can be computed by equation (31).The proof of Theorem5 is similar to the proof of Theorem 1 (see Appendix A) and is omitted here. The following two theorems, respectively, state the commutativity and boundedness of the LIVIFAWPMM operator (it is worth nothing that the LIVIFAWPMM operator no longer has the property of idempotency due to the introduce of weights).Theorem 6 (commutativity).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. If β1,β2,…,βn is any permutation of α1,α2,…,αn, then LIVIFAWPMMQ α1,α2,…,αn = LIVIFAWPMMQ β1,β2,…,βn.Theorem 7 (boundedness).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs, α−=sminai,sminbi,smaxcici,smaxdi, and α+=smaxai,smaxbi,smincici,smindi. Then α−≤LIVIFAWPMMQα1,α2,…,αn≤α+.The proofs of these theorems are, respectively, similar to the proofs of Theorems3 (see Appendix C) and 4 (see Appendix D) and are omitted here.Equation (40) is a generalised form of the LIVIFAWPMM operator. If specific functions are assigned to f and g, then specific operators can be constructed. For example, if the additive generators of Algebraic T-norm and T-conorm [60, 69] are, respectively, assigned to f and g, i.e., ft=−Int/h and gt=−In1−t/h, then a linguistic interval-valued intuitionistic fuzzy weighted power MM (LIVIFWPMM) operator is constructed:(42)LIVIFWPMMQα1,α2,…,αn=sh1−∏p∈Pn1−∏i=1n1−1−api/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQish1−∏p∈Pn1−∏i=1n1−1−bpi/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQi,sh1−1−∏p∈Pn1−∏i=1n1−cpi/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQish1−1−∏p∈Pn1−∏i=1n1−dpi/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQi.This operator has the following special cases:(1)
IfQ1=Q>0 and Q2=Q3=⋯=Qn=0, then the LIVIFWPMM operator will reduce to(43)sh1−∏i=1n1−1−ai/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−∏i=1n1−1−bi/hnwiξi/∑t=1nwtξtQ1/n1/Q,sh1−1−∏i=1n1−1−ci/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−1−∏i=1n1−1−di/hnwiξi/∑t=1nwtξtQ1/n1/Q=LIVIFWPGAAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power GAA (LIVIFWPGAA) operator. WhenQ = 1, it will become(44)sh1−∏i=1n1−1−ai/hnwiξi/∑t=1nwtξt1/n,sh1−∏i=1n1−1−bi/hnwiξi/∑t=1nwtξt1/n,sh∏i=1nci/hnwiξi/∑t=1nwtξt1/n,sh∏i=1ndi/hnwiξi/∑t=1nwtξt1/n=LIVIFWPAAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power AA (LIVIFWPAA) operator.(2)
IfQ1,Q2>0 and Q3=Q4=⋯=Qn=0, then the LIVIFWPMM operator will reduce to(45)sh1−∏i,j=1j≠in1−1−ai/hnwiξi/∑t=1nwtξtQ11−1−aj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2sh1−∏i,j=1j≠in1−1−bi/hnwiξi/∑t=1nwtξtQ11−1−bj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2,sh1−1−∏i,j=1j≠in1−1−ci/hnwiξi/∑t=1nwtξtQ11−cj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2sh1−1−∏i,j=1j≠in1−1−di/hnwiξi/∑t=1nwtξtQ11−dj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2=LIVIFWPBMQ1,Q2α1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power BM (LIVIFWPBM) operator.(3)
IfQ1=Q2=⋯=Qk=1 and Qk+1=Qk+2=⋯=Qn=0, then the LIVIFWPMM operator will reduce to(46)sh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−aij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/ksh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−bij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/k,sh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−cij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/ksh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−dij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/k=LIVIFWPMSMkα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power MSM (LIVIFWPMSM) operator.(4)
IfQ1=Q2=⋯=Qn=Q>0, then the LIVIFWPMM operator will reduce to(47)sh1−1−∏i=1n1−1−ai/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−1−∏i=1n1−1−bi/hnwiξi/∑t=1nwtξtQ1/n1/Q,sh1−∏i=1n1−1−ci/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−∏i=1n1−1−di/hnwiξi/∑t=1nwtξtQ1/n1/Q=LIVIFWPGGAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power GGA (LIVIFWPGGA) operator. WhenQ = 1, it will become(48)sh∏i=1nai/hnwiξi/∑t=1nwtξt1/n,sh∏i=1nbi/hnwiξi/∑t=1nwtξt1/n,sh1−∏i=1n1−1−ci/hnwiξi/∑t=1nwtξt1/n,sh1−∏i=1n1−1−di/hnwiξi/∑t=1nwtξt1/n=LIVIFWPGAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power GA (LIVIFWPGA) operator.Similarly, if the additive generators of other (e.g., Einstein, Hamacher, and Frank) ATNTCs [60, 69] are, respectively, assigned to f and g, then other specific operators can be constructed according to equation (40).


## 3.1. LIVIFAPMM Operator

A LIVIFAPMM operator is an AO of LIVIFNs constructed via combining the PA operator and the MM operator under linguistic interval-valued intuitionistic fuzzy environment. The operations in this operator are based on ATNTCs. The formal definition of the operator is as follows.Definition 9.
Letα1,α2,…,αn (where αisai,sbi,sci,sdi,i=1,2,…,n) be a collection of n LIVIFNs, Q=Q1=Q2=⋯=Qn (where Q1=Q2=⋯=Qn≥0 but not at the same time Q1=Q2=⋯=Qn=0) be a collection of n real numbers that, respectively, correspond to α1,α2,…,αn,pi be a permutation of (1, 2, …, n), Pn be the set of all permutations of (1, 2, …, n), αi⊕αj and αi⊗αjj=1,2,…,n be, respectively, the sum and product operations of αi and αj based on ATNTCs, rαk and αrk (k = 1, 2, …, n; r > 0) be, respectively, the multiplication and power operations of αk based on ATNTCs, Sαi,αj=1−Dαi,αj (where Dαi,αj is the distance of αi and αj and i, j = 1, 2, …, n and j ≠ i) be the support degree for αi from αj which satisfies 0≤Sαi,αj≤1, Sαi,αj=Sαj,αi, and Sαi,αj≥Sαp,αp if Sαi,αj≤αp,αp, and(27)Tαi=∑j=1,j≠inSαi,αj.
Then the aggregation function(28)LIVIFAPMMQα1,α2,…,αn=1n!⊕p∈Pn⊗i=1nn1+Tαpi∑j=1n1+TαjαpiQi1/Σi=1nQi,is called the LIVIFAPMM operator.
In this operator, the values ofQi are used to capture the interrelationships of the aggregated LIVIFNs: (1) if Q1>0 and Q2=Q3=⋯=Qn=0, then the LIVIFNs are independent of each other; (2) if Q1,Q2>0 and Q3=Q4=⋯=Qn=0, then the interrelationships between any two LIVIFNs are considered; (3) if Q1=Q2=⋯=Qk>0k=3,4,…,n and Qk+1=Qk+2=⋯=Qn=0, then the interrelationships among any k LIVIFNs are considered.
According to equations (4)‒(7) and (28), the following theorem is obtained.Theorem 1.
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. Then(29)LIVIFAPMMQα1,α2,…,αn=sa,sb,sc,sd,and it is still a LIVIFN, where(30)sa=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nξpigapi,sb=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nξpigbpi,sc=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nξpifcpi,sd=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nξpifdpi,and ξp(i) is a PA factor that can be computed by(31)ξi=1+∑p=1,p≠in1−Dαi,αp∑j=1n1+∑q=1,q≠jn1−Dαj,αq.For the details regarding the proof of this theorem, please refer to Appendix A. The following three theorems respectively state the idempotency, commutativity, and boundedness of the LIVIFPMM operator.Theorem 2 (idempotency).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. If αi=α=saα,sbα,scα,sdα for all i=1,2,…,n, then LIVIFAPMMQα1,α2,…,αn=α.Theorem 3 (commutativity).
Letα1,α2,…,αn (where ai=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. If β1,β2,…,βn is any permutation of α1,α2,…,αn, then LIVIFAPMMQα1,α2,…,αn=LIVIFAPMMQβ1,β2,…,βn.Theorem 4 (boundedness).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs, α−=sminai,sminbi,smaxcici,smaxdi, and α+=smaxai,smaxbi,smincici,smindi, Then α−≤LIVIFAPMMQα1,α2,…,αn≤α+.For the details regarding the proofs of these three theorems, please refer to Appendices B–D, respectively.Equation (29) is a generalised form of the LIVIFAPMM operator. If specific functions are assigned to f and g, then specific operators can be constructed. For example, if the additive generators of Algebraic T-norm and T-conorm [60, 69] are, respectively, assigned to f and g, i.e., f(t) = −In(t/h) and gt = −In(1−t/h), then a linguistic interval-valued intuitionistic fuzzy power MM (LIVIFPMM) operator is constructed:(32)LIVIFPMMQα1,α2,…,αn=sh1−∏p∈Pn1−∏i=1n1−1−api/hnξpiQi1/n!1/Σi=1nQish1−∏p∈Pn1−∏i=1n1−1−bpi/hnξpiQi1/n!1/Σi=1nQi,sh1−1−∏p∈Pn1−∏i=1n1−cpi/hnξpiQi1/n!1/Σi=1nQish1−1−∏p∈Pn1−∏i=1n1−dpi/hnξpiQi1/n!1/Σi=1nQi.This operator has the following special cases:(1)
IfQ1=Q>0 and Q2=Q3=⋯=Qn=0, then the LIVIFPMM operator will reduce to(33)sh1−∏i=1n1−1−ai/hnξiQ1/n1/Qsh1−∏i=1n1−1−bi/hnξiQ1/n1/Q,sh1−1−∏i=1n1−1−ci/hnξiQ1/n1/Qsh1−1−∏i=1n1−1−di/hnξiQ1/n1/Q=LIVIFPGAAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power GAA (LIVIFPGAA) operator. WhenQ = 1, it will become(34)sh1−∏i=1n1−1−ai/hnξi1/n,sh1−∏i=1n1−1−bi/hnξi1/n,sh∏i=1nci/hnξi1/n,sh∏i=1ndi/hnξi1/n=LIVIFPAAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power AA (LIVIFPAA) operator.(2)
IfQ1,Q2>0 and Q3=Q4=⋯=Qn=0 then the LIVIFPMM operator will reduce to(35)sh1−∏i,j=1j≠in1−1−ai/hnξiQ11−1−aj/hnξjQ21/nn−11/Q1+Q2sh1−∏i,j=1j≠in1−1−bi/hnξiQ11−1−bj/hnξjQ21/nn−11/Q1+Q2,sh1−1−∏i,j=1j≠in1−1−ci/hnξiQ11−cj/hnξjQ21/nn−11/Q1+Q2sh1−1−∏i,j=1j≠in1−1−di/hnξiQ11−dj/hnξjQ21/nn−11/Q1+Q2=LIVIFPBMQ1,Q2α1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power BM (LIVIFPBM) operator.(3)
IfQ1=Q2=⋯=Qk=1 and Qk+1=Qk+2=⋯=Qn=0, then the LIVIFPMM operator will reduce to(36)sh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−aij/hnξijk!n−k!/n!1/ksh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−bij/hnξijk!n−k!/n!1/k,sh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−cij/hnξijk!n−k!/n!1/ksh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−dij/hnξijk!n−k!/n!1/k=LIVIFPMSMkα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power MSM (LIVIFPMSM) operator.(4)
IfQ1=Q2=⋯=Qn=Q>0, then the LIVIFPMM operator will reduce to(37)sh1−1−∏i=1n1−1−ai/hnξiQ1/n1/Qsh1−1−∏i=1n1−1−bi/hnξiQ1/n1/Q,sh1−∏i=1n1−1−ci/hnξiQ1/n1/Qsh1−∏i=1n1−1−di/hnξiQ1/n1/Q=LIVIFPGGAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power GGA (LIVIFPGGA) operator. WhenQ = 1, it will become(38)sh∏i=1nai/hnξi1/n,sh∏i=1nbi/hnξi1/n,sh1−∏i=1n1−1−ci/hnξi1/n,sh1−∏i=1n1−1−di/hnξi1/n=LIVIFPGAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy power GA (LIVIFPGA) operator.Similarly, if the additive generators of other (e.g., Einstein, Hamacher, Frank) ATNTCs [60, 69] are, respectively, assigned to f and g, then other specific operators can be constructed according to equation (29).


## 3.2. LIVIFAWPMM Operator

The LIVIFAPMM operator has advantages in having desirable generality and flexibility, capturing the complex interrelationships of LIVIFNs, and reducing the negative effect of unreasonable LIVIFNs on the aggregation result. But it does not consider the relative importance of each aggregated LIVIFN. To this end, weights are introduced and a LIVIFAWPMM operator is presented. The formal definition of this operator is as follows.Definition 10.
On the basis of Definition9, let w1,w2,…,wn be, respectively, the weights of α1,α2,…,αn such that 0≤w1,w2,…,wn≤1 and w1+w2+⋯+wn=1. Then the aggregation function(39)LIVIFAWPMMQα1,α2,…,αn=1n!⊕p∈Pn⊗i=1nnwpi1+Tαpi∑j=1n1+TαjαpiQi1/Σi=1nQi,is called the LIVIFAWPMM operator.
In this operator, the function ofQi is the same as the function of Qi in the LIVIFAPMM operator (see equation (28)).
According to equations (4)‒(7) and (39), the following theorem is obtained.Theorem 5.
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. Then(40)LIVIFAWPMMQα1,α2,…,αn=sa,sb,sc,sd,and it is still a LIVIFN, where(41)sa=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nwpiξpi/∑t=1nwtξtgapi,sb=sf−11/∑i=1nQifg−11/n!∑p∈Pngf−1∑i=1nQifg−1nwpiξpi/∑t=1nwtξtgbpi,sc=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nwpiξpi/∑t=1nwtξtfcpi,sd=sg−11/∑i=1nQigf−11/n!∑p∈Pnfg−1∑i=1nQigf−1nwpiξpi/∑t=1nwtξtfdpi,and ξp(i) is a PA factor that can be computed by equation (31).The proof of Theorem5 is similar to the proof of Theorem 1 (see Appendix A) and is omitted here. The following two theorems, respectively, state the commutativity and boundedness of the LIVIFAWPMM operator (it is worth nothing that the LIVIFAWPMM operator no longer has the property of idempotency due to the introduce of weights).Theorem 6 (commutativity).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs. If β1,β2,…,βn is any permutation of α1,α2,…,αn, then LIVIFAWPMMQ α1,α2,…,αn = LIVIFAWPMMQ β1,β2,…,βn.Theorem 7 (boundedness).
Letα1,α2,…,αn (where αi=sai,sbi,sci,sdi,i=1,2,3,…,n) be a collection of n LIVIFNs, α−=sminai,sminbi,smaxcici,smaxdi, and α+=smaxai,smaxbi,smincici,smindi. Then α−≤LIVIFAWPMMQα1,α2,…,αn≤α+.The proofs of these theorems are, respectively, similar to the proofs of Theorems3 (see Appendix C) and 4 (see Appendix D) and are omitted here.Equation (40) is a generalised form of the LIVIFAWPMM operator. If specific functions are assigned to f and g, then specific operators can be constructed. For example, if the additive generators of Algebraic T-norm and T-conorm [60, 69] are, respectively, assigned to f and g, i.e., ft=−Int/h and gt=−In1−t/h, then a linguistic interval-valued intuitionistic fuzzy weighted power MM (LIVIFWPMM) operator is constructed:(42)LIVIFWPMMQα1,α2,…,αn=sh1−∏p∈Pn1−∏i=1n1−1−api/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQish1−∏p∈Pn1−∏i=1n1−1−bpi/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQi,sh1−1−∏p∈Pn1−∏i=1n1−cpi/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQish1−1−∏p∈Pn1−∏i=1n1−dpi/hnwpiξpi/∑t=1nwtξtQi1/n!1/Σi=1nQi.This operator has the following special cases:(1)
IfQ1=Q>0 and Q2=Q3=⋯=Qn=0, then the LIVIFWPMM operator will reduce to(43)sh1−∏i=1n1−1−ai/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−∏i=1n1−1−bi/hnwiξi/∑t=1nwtξtQ1/n1/Q,sh1−1−∏i=1n1−1−ci/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−1−∏i=1n1−1−di/hnwiξi/∑t=1nwtξtQ1/n1/Q=LIVIFWPGAAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power GAA (LIVIFWPGAA) operator. WhenQ = 1, it will become(44)sh1−∏i=1n1−1−ai/hnwiξi/∑t=1nwtξt1/n,sh1−∏i=1n1−1−bi/hnwiξi/∑t=1nwtξt1/n,sh∏i=1nci/hnwiξi/∑t=1nwtξt1/n,sh∏i=1ndi/hnwiξi/∑t=1nwtξt1/n=LIVIFWPAAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power AA (LIVIFWPAA) operator.(2)
IfQ1,Q2>0 and Q3=Q4=⋯=Qn=0, then the LIVIFWPMM operator will reduce to(45)sh1−∏i,j=1j≠in1−1−ai/hnwiξi/∑t=1nwtξtQ11−1−aj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2sh1−∏i,j=1j≠in1−1−bi/hnwiξi/∑t=1nwtξtQ11−1−bj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2,sh1−1−∏i,j=1j≠in1−1−ci/hnwiξi/∑t=1nwtξtQ11−cj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2sh1−1−∏i,j=1j≠in1−1−di/hnwiξi/∑t=1nwtξtQ11−dj/hnwjξj/∑t=1nwtξtQ21/nn−11/Q1+Q2=LIVIFWPBMQ1,Q2α1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power BM (LIVIFWPBM) operator.(3)
IfQ1=Q2=⋯=Qk=1 and Qk+1=Qk+2=⋯=Qn=0, then the LIVIFWPMM operator will reduce to(46)sh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−aij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/ksh1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−1−bij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/k,sh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−cij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/ksh1−1−∏1≤i1<⋯<ik≤nn1−∏j=1k1−dij/hnwijξij/∑t=1nwtξtk!n−k!/n!1/k=LIVIFWPMSMkα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power MSM (LIVIFWPMSM) operator.(4)
IfQ1=Q2=⋯=Qn=Q>0, then the LIVIFWPMM operator will reduce to(47)sh1−1−∏i=1n1−1−ai/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−1−∏i=1n1−1−bi/hnwiξi/∑t=1nwtξtQ1/n1/Q,sh1−∏i=1n1−1−ci/hnwiξi/∑t=1nwtξtQ1/n1/Qsh1−∏i=1n1−1−di/hnwiξi/∑t=1nwtξtQ1/n1/Q=LIVIFWPGGAQα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power GGA (LIVIFWPGGA) operator. WhenQ = 1, it will become(48)sh∏i=1nai/hnwiξi/∑t=1nwtξt1/n,sh∏i=1nbi/hnwiξi/∑t=1nwtξt1/n,sh1−∏i=1n1−1−ci/hnwiξi/∑t=1nwtξt1/n,sh1−∏i=1n1−1−di/hnwiξi/∑t=1nwtξt1/n=LIVIFWPGAα1,α2,…,αn,which is a linguistic interval-valued intuitionistic fuzzy weighted power GA (LIVIFWPGA) operator.Similarly, if the additive generators of other (e.g., Einstein, Hamacher, and Frank) ATNTCs [60, 69] are, respectively, assigned to f and g, then other specific operators can be constructed according to equation (40).


## 4. MAGDM Method

In this section, an MAGDM method based on the LIVIFAWPMM operator is designed to resolve the MAGDM problems based on LIVIFNs.In general, a MAGDM problem based on LIVIFNs can be formalised by a set of optionsO=O1,O2,…,Om, a set of attributes A=A1,A2,…,An, a vector of weights of attributes w=w1,w2,…,wn such that 0≤w1,w2,…,wn≤1 and w1+w2+…+wn=1, a set of experts E=E1,E2,…,EnL, a vector of weights of experts ϖ=ϖ1,ϖ2,…,ϖn such that 0≤ϖ1,ϖ2,…,ϖn≤1 and ϖ1+ϖ2+…+ϖn=1, and L linguistic interval-valued intuitionistic fuzzy decision matrices Mh=αh,i,jm×nh=1,2,…,L;i=1,2,…,m;j=1,2,…,n such that sah,i,j,sbh,i,j,sch,i,j,sdh,i,j is a LIVIFN that denotes the evaluation value of Aj with respect to Oi provided by Eh. Based on these components, the MAGDM problem can be formalised as determining the optimal option according to a ranking of all options in O based on A, Mh, w, and ϖ. Using the LIVIFAWPMM operator, the problem is solved according to the following steps:(1)
Normalise the linguistic interval-valued intuitionistic fuzzy decision matricesMh. To balance the physical dimensions of the rating values in Mh, they are normalised as(49)Nh=sah,i,j,sbh,i,j,sch,i,j,sdh,i,jm×n,C1,sch,i,j,sdh,i,j,sah,i,j,sbh,i,jm×n,C2,whereC1 denotes “if Cj is a benefit attribute” and C2 denotes “if Cj is a cost attribute.”(2)
Compute the power weights ofαh,i,j The power weights of αh,i,j are calculated using(50)Wh,i,j=ϖhξh∑z=1Lϖzξz=ϖh1+∑x=1,x≠hL1−Dαh,i,j,αx,i,j∑z=1Lϖz1+∑y=1,y≠zL1−Dαz,i,j,αy,i,j,whereDαh,i,j,αx,i,jDαz,i,j,αy,i,j is the distance of αh,i,j and αx,i,j(αz,i,j and αy,i,j) that can be calculated via equation (3).(3)
Compute the collective values ofαh,i,j. Taking the matrices Nh and the set ϖ as input, the collective values of αh,i,j are calculated using(51)αi,j=sai,j,sbi,j,sci,j,sdi,j=LIVIFAWPMMQα1,i,j,α2,i,j,…,αL,i,j,where LIVIFAWPMM is an arbitrary specific LIVIFAWPMM operator (for example, it can be the LIVIFWPMM operator in equation (42)), and the values of Q=Q1,Q2,…,QL are assigned as Q1>0 and Q1=Q3=⋯=QL=0 since the rating values of different experts should generally be mutually independent.(4)
Compute the power weights ofαi,j. The power weights of αi,j are calculated using(52)Wi,j=wjξj∑t=1nwtξt=wj1+∑r=1,r≠jn1−Dαi,j,αi,r∑t=1nwt1+∑s=1,s≠tn1−Dαi,t,αi,s,whereDαi,j,αi,rDαi,t,αi,s is the distance of αi,j and αi,r αi,tandαi,s that can be calculated via equation (3).(5)
Calculate the collective values ofαi,j. The collective values of αi,j are computed using(53)αi=sai,sbi,sci,sdi=LIVIFAWPMMQαi,1,αi,2,…,αi,n,where LIVIFAWPMM is the same specific operator used in equation (51), and the values of Q=Q1,Q2,…,Qn are determined by identifying the interrelationships of the n attributes in A. If n attributes are mutually independent, then Q1>0 and Q2=Q2=⋯=Qn=0. If there are interrelationships between any two attributes, then Q1,Q2>0 and Q3=Q4=⋯=Qn=0. If there are interrelationships among any kk=3,4,…,n, Q1,Q2,…,Qk>0 and Qk+1=Qk+2=⋯=Qn=0.(6)
Compute the scores and accuracies ofαi. The scores and accuracies of αi are calculated using equations (1) and (2), respectively.(7)
Generate a ranking ofOi. According to the scores and accuracies of αi and the comparison rules in Definition 4, a ranking of Oi is generated.(8)
Determine the optimal option. The optimal option is determined according to the ranking.


## 5. Example, Experiments, and Comparisons

In this section, a practical example is firstly used to illustrate the designed MAGDM method. Then a set of test experiments are carried out to validate the method and explore the effects of different values ofQ on the aggregation and ranking results. Finally, qualitative and quantitative comparisons between the presented AO and the existing AOs are reported to show the characteristics and advantages of the presented AO.


### 5.1. Example

Three-dimensional (3D) printing refers to a series of emerging manufacturing technologies that build 3D physical objects from 3D model data, in which materials are stacked layer by layer through a specific process like sintering, melting, jetting, and lamination. A well-known characteristic of 3D printing technologies compared with traditional manufacturing technologies is that they can be used to fabricate 3D objects with complicated geometric structures and heterogeneous materials without additional cost. Due to such characteristic, the study and application of 3D printing technologies have received extensive attention from the academia and industry. Some even believed that 3D printing technologies would trigger a new round of manufacturing revolution.The existing 3D printing technologies can be divided into vat photopolymerisation, material jetting, binder jetting, powder bed fusion, material extrusion, directed energy deposition, and sheet lamination. Based on these technologies, over one thousand different industrial 3D printers have been developed and identified in the market so far. A controversy regarding which printer is better than the others is meaningless, as each printer has its own features and application range. However, research on the selection of a suitable printer from a specific number of alternative printers for printing a specific component is nontrivial. This is because such selection needs a comprehensive understanding of the features of all alternative printers and a dynamic interaction with the quality of the built component, while most users lack such knowledge and experience. In addition, different printers could have similarities or overlaps at features, which brings certain difficulties to the selection objectively.To offer an effective tool for selection of 3D printers, many different kinds of methods have been presented, where methods based on multiattribute decision-making (MADM) are one of the most important kinds. This kind of methods determine appropriate printers by synthetically assessing the values of multiple interrelated attributes of all alternative printers, which are usually achieved from experiments or simulations or provided by domain experts. It is generally difficult to ensure the absolute objectivity of these ways. This means that there could be some attributes values having “noise.” To achieve reasonable selection result in this case, it is of necessity to capture the interrelationships of the attributes and concurrently reduce the influence of the attributes values having “noise.” But there is yet no evidence that any of the existing MADM-based methods for 3D printer selection have such capabilities. The proposed MAGDM method can meet this requirement when the values of attributes of all alternative printers are quantified by LIVIFNs. The following is an illustrative example about the application of the proposed method in the 3D printer selection.In this example, a user needs to select a proper printer from four alternative printers, denoted asP1, P2, P3, and P4, to print a component using a specific material. The user invited three experienced domain experts, denoted as E1, E2, and E3, to evaluate the four alternative printers based on four attributes, which are the surface roughness (A1), strength (A2), elongation (A3), and hardness (A4) of the built component. The relative importance of the three experts is quantified by ϖ = [0.4, 0.3, 0.3]. The relative importance of the four attributes is measured by w = [0.1, 0.3, 0.3, 0.3]. In the evaluation, the three experts were asked to use LIVIFNs. The available linguistic variables are extremely small (s0), very small (s1), small (s2), slightly small (s3), medium (s4), slightly large (s5), large (s6), very large (s7), and extremely large (s8). The evaluation results of the three experts are, respectively, listed in the following three matrices:(54)M1=s2,s4,s3,s4s6,s6,s1,s2s5,s6,s1,s1s6,s6,s1,s1s3,s5,s2,s3s5,s6,s1,s2s4,s5,s1,s1s4,s6,s1,s2s1,s1,s6,s7s6,s6,s1,s2s3,s4,s3,s3s5,s6,s1,s2s1,s1,s7,s7s3,s4,s2,s3s3,s5,s2,s3s2,s3,s3,s4,M2=s3,s4,s3,s4s5,s6,s1,s2s6,s6,s1,s2s5,s6,s1,s2s3,s5,s1,s3s6,s6,s1,s1s5,s6,s1,s2s3,s5,s1,s2s1,s2,s6,s6s5,s6,s1,s1s3,s5,s3,s3s5,s6,s1,s2s1,s2,s6,s6s3,s3,s3,s3s3,s4,s2,s3s3,s4,s3,s4,M3=s3,s3,s3,s4s6,s6,s2,s2s5,s6,s1,s1s6,s7,s1,s1s3,s4,s2,s3s6,s7,s1,s1s5,s5,s1,s3s5,s5,s1,s2s2,s3,s5,s5s4,s5,s1,s2s3,s4,s3,s4s4,s5,s2,s2s1,s1,s6,s7s3,s4,s2,s3s3,s5,s1,s2s4,s5,s3,s3.According to the conditions above, the determination can be carried out leveraging the designed MAGDM method. Its process includes the following steps:(1)
Normalise the fuzzy decision matricesMh (h = 1, 2, 3). Since surface roughness is cost attribute and strength, elongation, and hardness are benefit attributes, the fuzzy decision matrices M1, M2, and M3 are, respectively, normalised as follows:(55)N1=s3,s4,s2,s4s6,s6,s1,s2s5,s6,s1,s1s6,s6,s1,s1s2,s3,s3,s5s5,s6,s1,s2s4,s5,s1,s1s4,s6,s1,s2s6,s7,s1,s1s6,s6,s1,s2s3,s4,s3,s3s5,s6,s1,s2s7,s7,s1,s1s3,s4,s2,s3s3,s5,s2,s3s2,s3,s3,s4,N2=s3,s4,s3,s4s5,s6,s1,s2s6,s6,s1,s2s5,s6,s1,s2s1,s3,s3,s5s6,s6,s1,s1s5,s6,s1,s2s3,s5,s1,s2s6,s6,s1,s2s5,s6,s1,s1s3,s5,s3,s3s5,s6,s1,s2s6,s6,s1,s2s3,s3,s3,s3s3,s4,s2,s3s3,s4,s3,s4,N3=s3,s4,s3,s3s6,s6,s2,s2s5,s6,s1,s1s6,s7,s1,s1s2,s3,s3,s4s6,s7,s1,s1s5,s5,s1,s3s5,s5,s1,s2s5,s5,s2,s3s4,s5,s1,s2s3,s4,s3,s4s4,s5,s2,s2s6,s7,s1,s1s3,s4,s2,s3s3,s5,s1,s2s4,s5,s3,s3,(2)
Compute the power weights ofαh,i,ji=1,2,3,4;j=1,2,3,4. According to equation (50), the power weights of αh,i,j are calculated and listed in the following matrices:(56)W1,i,j4×4=0.39870.40260.40260.40390.40260.39610.39740.40000.40000.40130.40260.40390.40000.40260.40390.3987,W2,i,j4×4=0.30230.29870.29550.29640.29870.30360.30130.30000.30680.30100.29870.30290.29670.29550.29970.3057,W3,i,j4×4=0.29900.29870.30190.29970.29870.30030.30130.30000.29320.29770.29870.29320.30330.30190.29640.2956.(3)
Compute the collective values ofαh,i,j. Taking Nh and ϖ as input, the collective values of αh,i,j can be calculated using equation (51). Here, the LIVIFWPMM operator in equation (42) (when adapting the operator, Q = (1, 0, 0)) is used in the calculation. The calculated results are listed in the following matrix:(57)αi,j4×4=s3.0000,s4.0000,s2.5522,s3.6703s5.7425,s6.0000,s1.2300,s2.0000s5.3387,s6.0000,s1.0000,s1.2273s5.7446,s6.3751,s1.0000,s1.2281s1.7173,s3.0000,s3.0000,s4.6776s5.6516,s6.3759,s1.0000,s1.3159s4.6367,s5.3450,s1.0000,s1.7158s4.0767,s5.4492,s1.0000,s2.0000s5.7475,s6.2930,s1.2253,s1.7070s5.2226,s5.7434,s1.0000,s1.6234s3.0000,s4.3294,s3.0000,s3.2692s4.7360,s5.7476,s1.2253,s2.0000s6.4843,s6.7716,s1.0000,s1.2284s3.0000,s3.7274,s2.2546,s3.0000s3.0000,s4.7299,s1.6285,s2.6603s2.9663,s3.9843,s3.0000,s3.6739.(4)
Compute the power weights ofαi,j. According to equation (52), the power weights of αi,j are calculated and listed in the following matrix:(58)Wi,j4×4=0.08880.30440.30440.30240.08610.29620.30890.30890.10170.30910.27840.31080.08880.30540.30680.2990.(5)
Calculate the collective values ofαi,j. The collective values of αi,j can be computed according to equation (53). Since the LIVIFWPMM operator in equation (42) has been leveraged in the third step, this operator (when adapting the operator, Q = (1, 2, 3, 4), i.e., there are interrelationships among the four attributes) is also used in this step to complete the calculation. The calculated results are listed as follows:(59)α1=S4.6635,S5.2066,S1.9801,S2.4910,α2=S3.8050,S4.7172,S2.0661,S2.8436,α3=S4.3660,S5.2202,S1.9944,S2.4892,α4=S3.4932,S4.4878,S2.3124,S2.9583.(6)
Compute the scores and accuracies ofαi. According to equations (1) and (2), the scores and accuracies of αi are calculated and, respectively, listed as follows:(60)Sα1=S5.3498,Sα2=S4.9031,Sα3=S5.2757,Sα4=S4.6776,Aα1=S7.1706,Aα2=S6.7160,Aα3=S7.0349,Aα3=S6.6258.(7)
Generate a ranking ofPi. According to the scores and accuracies of αi and the comparison rules in Definition 4, a ranking of Pi is generated as P1≻P3≻P2≻P4.(8)
Determine the best printer. According to the ranking, the best 3D printer is determined as printerP1.


### 5.2. Experiments

#### 5.2.1. Validation Experiments

To demonstrate the effectiveness of an MADM method, Wang and Triantaphyllou [70] presented the following three test criteria:(1)
Criterion 1: “An effective MADM method should not change the place of the best option in the generated ranking when a nonoptimal option is replaced by a new worse option under the condition that the weight of each attribute remains unchanged”(2)
Criterion 2: “The rankings of options generated by an effective MADM method should be transitive”(3)
Criterion 3: “If a MADM problem is decomposed into several sub MADM problems and the same MADM method is used in the problem and its sub problems, then a collective ranking of all sub MADM problems should be the same as the ranking of the undecomposed MADM problem.”According to these three criteria, thirteen test experiments using the practical example were designed and carried out. The options included in each of the thirteen experiments are listed in Table1. In Experiments 1, 2, and 3, the nonoptimal options P2, P3, and P4 are, respectively, replaced by the new worse options P2′,P3′,andP4′, whose evaluation results are listed in Table 2. Using the designed method, the rankings of Experiments 1, 2, and 3 can be generated and are shown in Table 1. From the generated rankings, it can be seen that the best options in them remain unchanged. Thus, the designed method can meet Criterion 1.Table 1
The details and results of the validation experiments.
Experiment
Included options
Generated ranking
Experiment 1
P
1, P2′, P3, P4
P
1 ≻ P3 ≻ P4 ≻ P2′
Experiment 2
P
1, P2, P3′, P4
P
1 ≻ P2 ≻ P3′ ≻ P4
Experiment 3
P
1, P2, P3, P4′
P
1 ≻ P3 ≻ P2 ≻ P4′
Experiment 4
P
1, P2
P
1 ≻ P2
Experiment 5
P
1, P3
P
1 ≻ P3
Experiment 6
P
1, P4
P
1 ≻ P4
Experiment 7
P
2, P3
P
3 ≻ P2
Experiment 8
P
2, P4
P
2 ≻ P4
Experiment 9
P
3, P4
P
3 ≻ P4
Experiment 10
P
1, P2, P3
P
1 ≻ P3 ≻ P2
Experiment 11
P
1, P2, P4
P
1 ≻ P2 ≻ P4
Experiment 12
P
1, P3, P4
P
1 ≻ P3 ≻ P4
Experiment 13
P
2, P3, P4
P
3 ≻ P2 ≻ P4Table 2
The evaluation results of the new worse options.
Option
AttributeA1
AttributeA2
AttributeA3
AttributeA4
P
2′ in M1
([s3, s5], [s1, s2])
([s4, s5], [s1, s2])
([s3, s4], [s1, s1])
([s3, s5], [s1, s2])
P
2′ in M2
([s3, s5], [s1, s3])
([s5, s5], [s1, s1])
([s4, s5], [s1, s2])
([s2, s4], [s1, s2])
P
2′ in M3
([s3, s4], [s1, s2])
([s5, s6], [s1, s1])
([s4, s4], [s1, s3])
([s4, s4], [s1, s2])
P
3′ in M1
([s1, s1], [s5, s6])
([s5, s5], [s1, s2])
([s2, s3], [s3, s3])
([s4, s5], [s1, s2])
P
3′ in M2
([s5, s5], [s1, s2])
([s4, s5], [s1, s1])
([s2, s4], [s3, s3])
([s4, s5], [s1, s2])
P
3′ in M3
([s4, s4], [s2, s3])
([s3, s4], [s1, s2])
([s2, s3], [s3, s4])
([s3, s4], [s2, s2])
P
4′ in M1
([s1, s1], [s6, s6])
([s2, s3], [s2, s3])
([s2, s4], [s2, s3])
([s1, s2], [s3, s4])
P
4′ in M2
([s1, s2], [s5, s5])
([s2, s2], [s3, s3])
([s2, s3], [s2, s3])
([s2, s3], [s3, s4])
P
4′ in M3
([s1, s1], [s5, s6])
([s2, s3], [s2, s3])
([s2, s4], [s1, s2])
([s3, s4], [s3, s3])The remaining ten test experiments (i.e., Experiments 4−13) aim to solve all sub MAGDM problems of the MAGDM problem in the practical example. Using the designed method, the rankings of these experiments can be generated and are also shown in Table1. From the rankings of Experiments 2, 4, 7, and 10, Experiments 4, 6, 8, and 11, Experiments 5, 6, 9, and 12, and Experiments 7, 8, 9, and 13, it can be seen that transitivity is satisfied. Therefore, the designed method can meet Criterion 2. In addition, if the rankings of Experiments 4−9 or Experiments 10−13 are combined, then the same collective ranking P1≻P3≻P2≻P4 will be obtained, which is exactly the ranking of the undecomposed MAGDM problem. From this point of view, the designed method can also meet Criterion 3.In summary, the designed method is effective for resolving the practical MAGDM problems since it can meet all of the three test criteria.


#### 5.2.2. Exploration Experiment

To show the influence of different values ofQ on the aggregation and ranking results, a test experiment using the practical example was designed and carried out. In this experiment, Q was respectively assigned (1, 0, 0, 0), (2, 0, 0, 0), (3, 0, 0, 0), (4, 0, 0, 0), (1, 1, 0, 0), (1, 2, 0, 0), (1, 3, 0, 0), (1, 4, 0, 0), (1, 1, 1, 0), (1, 2, 3, 0), (1, 1, 1, 1), (2, 2, 2, 2), (3, 3, 3, 3), (4, 4, 4, 4), and (1, 2, 3, 4) when calculating the collective values of αi,j in Step (5) in the example. Based on these different values of Q, the LIVIFWPMM operator will reduce to different operators, which are, respectively, listed in Table 3. The results of the experiment are the computed scores of αi and the generated ranking of Pi under each Q, which are also listed in Table 3. As can be seen from the table, the rankings of the four 3D printers generated by the proposed method may have difference, in the situations where (1) all of the four attributes are independent of each other, (2) there are interrelationships between any two attributes, (3) there are interrelationships among any three attributes, and (4) there are interrelationships among the four attributes. This indicates that the presented LIVIFWPMM operator has the capability and generality to capture the interrelationships of attributes. In practical application of the operator, Q is recommended to take (1, 0, 0, 0), (1, 2, 0, 0), (1, 2, 3, 0), and (1, 2, 3, 4) under the four situations, respectively.Table 3
The details and results of the exploration experiment.
Value ofQ
Actual operator
Interrelationships
The computed scores of all printers
The generated ranking
S (α1)
S (α2)
S (α3)
S (α4)
(1, 0, 0, 0)
LIVIFWPAA (equation (44))
Independent of each other
6.1822
5.8369
5.6327
4.7823
P
1
≻
P
2
≻
P
3
≻
P
4
(2, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.3032
5.9960
5.7047
4.8070
P
1
≻
P
2
≻
P
3
≻
P
4
(3, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.3892
6.1016
5.7753
4.8327
P
1
≻
P
2
≻
P
3
≻
P
4
(4, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.4509
6.1753
5.8420
4.8588
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
5.7939
5.4072
5.3790
4.7050
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 2, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
5.9428
5.5941
5.4509
4.7281
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 3, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
6.0762
5.7504
5.5429
4.7589
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 4, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
6.1780
5.8670
5.6315
4.7902
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 1, 0)
LIVIFWPMSM (equation (46))
Among any 3 attributes
5.4135
4.9825
5.2503
4.6695
P
1
≻
P
3
≻
P
2
≻
P
4
(1, 2, 3, 0)
LIVIFWPMM (equation (42))
Among any 3 attributes
5.7674
5.4217
5.3606
4.7043
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 1, 1)
LIVIFWPGA (equation (48))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(2, 2, 2, 2)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(3, 3, 3, 3)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(4, 4, 4, 4)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(1, 2, 3, 4)
LIVIFWPMM (equation (42))
Among the 4 attributes
5.3498
4.9031
5.2757
4.6776
P
1
≻
P
3
≻
P
2
≻
P
4


### 5.3. Comparisons

As mentioned in the introduction, representative AOs of LIVIFNs are the WA, OWA, HA, WG, OWG, and HG operators presented by Garg and Kumar [54, 55], the PWA, POWA, PWG, and POWG operators presented by Kumar and Garg [56], and the WMSM operator presented by Liu and Qin [57]. In this subsection, qualitative and quantitative comparisons between these AOs and the presented AO of LIVIFNs are carried out to evaluate the presented AO:(1)
Qualitative Comparison. This comparison was carried out via comparing the characteristics of the AOs. For the twelve AOs above, the generality and flexibility in aggregating the values of attributes, the generality in capturing the interrelationships of attributes, and the capability to reduce the effect of biased attribute values are selected as the comparison characteristics. The results of the comparison, as shown in Table 4, are explained as follows.(a)
Generality and Flexibility in Aggregating the Values of Attributes. Among the twelve AOs, the WA, OWA, HA, WG, OWG, HG, PWA, POWA, PWG, POWG, and WMSM operators perform the aggregation using the Algebraic T-norm and T-conorm, their generality and flexibility are relatively limited. Such characteristics of the presented Archimedean weighted power Muirhead mean (AWPMM) operator are satisfying since the aggregation in it can be performed by any types of ATNTCs (e.g., Algebraic, Einstein, Hamacher, and Frank T-norms and T-conorms).(b)
Generality in Capturing the Relationships of Attributes. The WA, OWA, HA, WG, OWG, HG, WMSM, and AWPMM operators are suitable for the case where all aggregated attributes are independent of each other. In addition, the WMSM and AWPMM operators can also be applicable in the situations where there are interrelationships between any two attributes and where there are relationships among any multiple attributes, because they, respectively, use the all-in-one MM and MSM operators for capturing interrelationships. It should be noted that MM is more generalised than MSM since MSM is just a special case of MM. From this point of view, the AWPMM operator is more generalised than the WMSM operator. The PWA, POWA, PWG, and POWG operators are specifically presented for dealing with the situation where the attributes are in different priority levels. They are totally different from other AOs from the perspective of application range.(c)
Capability to Reduce the Effect of Biased Attribute Values. Among the twelve AOs, only the AWPMM operator has this capability due to the combination of the PA operator.(2)
Quantitative Comparison. This comparison was carried out using the practical examples in the present paper and in [55, 57] as benchmarks. In the comparison, the WA and WG operators were selected as the AOs for the method of Garg and Kumar [55]. The PWA and PWG operators were chosen as the AOs for the method of Kumar and Garg [56]. The given weights in the three examples were directly used as the priority weights of the PWA and PWG operators to make the input of all comparison operators the same (it should be pointed out that the PWA and PWG operators will, respectively, reduce to the WA and WG operators in this case). The method of Liu and Qin [57] and the proposed method respectively used the WMSM operator and the weighted power MM (WPMM) operator (see equation (42)). Moreover, the same score function (i.e., the function in equation (1)) was used in all of these operators for easy comparison. The details and results of the comparison are listed in Table 5.Table 4
The results of the qualitative comparison.
AOs presented in the method
Generality and flexibility in aggregation
Generality in capturing interrelationships of attributes
Capability to reduce the effect
Independent of each other
Between any two
Among any multiple
In different priority levels
WA [55]
Limited
Yes
No
No
No
No
OWA [55]
Limited
Yes
No
No
No
No
HA [55]
Limited
Yes
No
No
No
No
WG [55]
Limited
Yes
No
No
No
No
OWG [55]
Limited
Yes
No
No
No
No
HG [55]
Limited
Yes
No
No
No
No
PWA [56]
Limited
No
No
No
Yes
No
POWA [56]
Limited
No
No
No
Yes
No
PWG [56]
Limited
No
No
No
Yes
No
POWG [56]
Limited
No
No
No
Yes
No
WMSM [57]
Limited
Yes
Yes
Yes
No
No
AWPMM
Satisfying
Yes
Yes
Yes
No
YesTable 5
The details and results of the quantitative comparison.
Benchmark
Decision-making method
Used AOs
Value of arguments
The calculated scores of all options
The generated ranking
S (α1)
S (α2)
S (α3)
S (α4)
Example in this paper
Garg and Kumar [55]
WA, WA
—
6.1654
5.8171
5.5985
4.8181
O
1
≻
O
2
≻
O
3
≻
O
4
Garg and Kumar [55]
WG, WG
—
5.9921
5.5132
5.3441
4.5370
O
1
≻
O
2
≻
O
3
≻
O
4
Kumar and Garg [56]
PWA, PWA
—
6.1654
5.8171
5.5985
4.8181
O
1
≻
O
2
≻
O
3
≻
O
4
Kumar and Garg [56]
PWG, PWG
—
5.9921
5.5132
5.3441
4.5370
O
1
≻
O
2
≻
O
3
≻
O
4
Liu and Qin [57]
WMSM, WMSM
k = 1, k = 3
5.4523
5.0127
5.2577
4.7343
O
1
≻
O
3
≻
O
2
≻
O
4
The designed method
WPMM, WPMM
Q = (1, 0, 0), Q = (1, 2, 3, 0)
5.7674
5.4217
5.3606
4.7043
O
1
≻
O
2
≻
O
3
≻
O
4
Example in [55]
Garg and Kumar [55]
WA, WA
—
5.3457
4.6145
5.1125
4.9687
O
1
≻
O
3
≻
O
4
≻
O
2
Garg and Kumar [55]
WG, WG
—
5.1069
4.3577
4.9218
4.5868
O
1
≻
O
3
≻
O
4
≻
O
2
Kumar and Garg [56]
PWA, PWA
—
5.3457
4.6145
5.1125
4.9687
O
1
≻
O
3
≻
O
4
≻
O
2
Kumar and Garg [56]
PWG, PWG
—
5.1069
4.3577
4.9218
4.5868
O
1
≻
O
3
≻
O
4
≻
O
2
Liu and Qin [57]
WMSM, WMSM
k = 1, k = 3
5.1768
4.2902
4.7499
4.6151
O
1
≻
O
3
≻
O
4
≻
O
2
The designed method
WPMM, WPMM
Q = (1, 0, 0), Q = (1, 2, 3, 0)
5.2054
4.3988
4.8582
4.6997
O
1
≻
O
3
≻
O
4
≻
O
2
Example in [57]
Garg and Kumar [55]
WA
—
6.2621
6.2709
5.6111
5.5823
O
2
≻
O
1
≻
O
3
≻
O
4
Garg and Kumar [55]
WG
—
6.1407
6.1620
5.3326
5.2541
O
2
≻
O
1
≻
O
3
≻
O
4
Kumar and Garg [56]
PWA, PWA
—
6.2621
6.2709
5.6111
5.5823
O
2
≻
O
1
≻
O
3
≻
O
4
Kumar and Garg [56]
PWG, PWG
—
6.1407
6.1620
5.3326
5.2541
O
2
≻
O
1
≻
O
3
≻
O
4
Liu and Qin [57]
WMSM
k = 3
6.1034
6.0611
5.5196
5.1819
O
1
≻
O
2
≻
O
3
≻
O
4
The designed method
WPMM
Q = (1, 2, 3, 0, 0)
6.1584
6.1157
5.5637
5.3056
O
1
≻
O
2
≻
O
3
≻
O
4As can be seen from Table5, the best option of the designed method is exactly the same as the best option of the method of Liu and Qin for all examples. This is mainly because the two methods are the most similar in nature (see Table 4). From this result, it can be observed that the designed method is feasible and effective for solving practical decision-making problems based on LIVIFNs. In addition, the rankings of the methods of Garg and Kumar and Kumar and Garg are different from that of the method of Liu and Qin and the designed method at the first and second places for the example in [57]. The reason is that the interrelationships in the former methods are set as “all attributes are independent of each other” and those in the latter two methods are set as “there are interrelationships among any three attributes.”From the qualitative comparison above, the advantage of combination of the PA and MM operators is providing generality in capturing the interrelationships of attributes and capability to reduce the effect of biased attribute values. Such advantage cannot be intuitively seen from the comparison results in Table5. To explicitly show the advantage, two additional quantitative comparison experiments were carried out. The first experiment aims to show the generality in capturing attribute relationships. In this experiment, the practical examples in the present paper and in [55, 57] were used to compare the proposed method and the methods of Garg and Kumar, Kumar and Garg, and Liu and Qin. Given the following are five assumptions: (a) all attributes are independent of each other; (b) there are interrelationships between any two attributes; (c) there are interrelationships among any three attributes; (d) there are interrelationships among any four attributes; (e) there are interrelationships among any five attributes (only for the example in [57]). The applicability of the four comparison methods and the results of the experiment are shown in Table 6. It can be intuitively seen from the table that the methods of Garg and Kumar and Kumar and Garg are applicable only under assumption a, while both the method of Liu and Qin and the designed method can be applied to generate ranking results under all assumptions. This indicates that both of the latter two methods have the generality in capturing the interrelationships of attributes.Table 6
The details and results of the first additional quantitative comparison experiment.
Benchmark
Decision-making method
Generated ranking under assumption (a)
Generated ranking under assumption (b)
Generated ranking under assumption (c)
Generated ranking under assumption (d)
Generated ranking under assumption (e)
Example in this paper
Garg and Kumar [55]
O
1
≻
O
2
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
—
Kumar and Garg [56]
O
1
≻
O
2
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
—
Liu and Qin [57]
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
3
≻
O
2
≻
O
4
O
3
≻
O
1
≻
O
4
≻
O
2
—
The designed method
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
3
≻
O
2
≻
O
4
—
Example in [55]
Garg and Kumar [55]
O
1
≻
O
3
≻
O
4
≻
O
2
Cannot be applied
Cannot be applied
Cannot be applied
—
Kumar and Garg [56]
O
1
≻
O
3
≻
O
4
≻
O
2
Cannot be applied
Cannot be applied
Cannot be applied
—
Liu and Qin [57]
O
1
≻
O
4
≻
O
3
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
—
The designed method
O
1
≻
O
4
≻
O
3
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
—
Example in [57]
Garg and Kumar [55]
O
2
≻
O
1
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
Cannot be applied
Kumar and Garg [56]
O
2
≻
O
1
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
Cannot be applied
Liu and Qin [57]
O
2
≻
O
1
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
The designed method
O
2
≻
O
1
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4The second experiment aims to show the capability to reduce the influence of extreme attribute values. In this experiment, the practical example in the present paper is used to compare the proposed method and the method of Liu and Qin, which have difference only in whether combining the PA operator whenQ = (1, 0, 0, 0) and k = 1 and Q = (1, 1, 1, 0) and k = 3 for the example. It is assumed that the value of attribute A2 of printer P1 evaluated by expert E1 (i.e., α1,1,2) is a biased attribute value. This value was constantly adjusted from high to low, as listed in the first column of Table 7. The change of the relative importance (i.e., weight) of α1,1,2 with respect to the adjusted value is also shown in Table 7. As can be intuitively seen from the table, the weight of α1,1,2 becomes smaller and smaller as the value of α1,1,2 changes from high to low in the proposed method, while it remains the same in this process in the method of Liu and Qin. The greater the change of the value of α1,1,2, the greater the bias. The relative importance of the value should be dynamically decreased to reduce the effect of this extreme value. From the comparison results, only the proposed method has such capability. That is, the method can reduce the effect of biased attribute values.Table 7
The details and results of the second additional quantitative comparison experiment.
Value ofα1,1,2
Change of weights in the designed method
Change of weights in the method of Liu and Qin [57]
Weights of expertE1 ([W1,i,j]4×4)
Weights of attributes ([Wi,j]4×4)
Weights of expertE1 ([W1,i,j]4×4)
Weights of attributes ([Wi,j]4×4)
([s6, s6], [s1, s2])
0
.
3987
0
.4026
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0888
0
.3044
0
.
3044
0
.
3024
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s5, s5], [s2, s3])
0
.
3987
0
.3974
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0906
0
.3044
0
.
3045
0
.
3005
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s4, s4], [s3, s4])
0
.
3987
0
.3859
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0923
0
.3039
0
.
3039
0
.
2999
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s3, s3], [s4, s5])
0
.
3987
0
.3728
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0936
0
.3035
0
.
3035
0
.
2994
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s2, s2], [s5, s6])
0
.
3987
0
.3577
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0945
0
.3032
0
.
3032
0
.
2991
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s1, s1], [s6, s7])
0
.
3987
0
.3402
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0952
0
.3030
0
.
3030
0
.
2988
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s0, s0], [s7, s8])
0
.
3987
0
.3195
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0955
0
.3029
0
.
3029
0
.
2987
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000On the basis of the comparisons above, the advantages of the designed method over the methods of Garg and Kumar, Kumar and Garg, and Liu and Qin are summarised as follows:(1)
Compared to the methods of Garg and Kumar and Kumar and Garg, the designed method is generalised and flexible for aggregation of attribute values and handling of attribute relationships and concurrently has the capability to reduce the influence of the distortion of attribute values.(2)
Compared to the method of Liu and Qin, the designed method has desirable generality and flexibility in aggregation of attribute values and can reduce the influence of the biased attribute values on aggregation result.


## 5.1. Example

Three-dimensional (3D) printing refers to a series of emerging manufacturing technologies that build 3D physical objects from 3D model data, in which materials are stacked layer by layer through a specific process like sintering, melting, jetting, and lamination. A well-known characteristic of 3D printing technologies compared with traditional manufacturing technologies is that they can be used to fabricate 3D objects with complicated geometric structures and heterogeneous materials without additional cost. Due to such characteristic, the study and application of 3D printing technologies have received extensive attention from the academia and industry. Some even believed that 3D printing technologies would trigger a new round of manufacturing revolution.The existing 3D printing technologies can be divided into vat photopolymerisation, material jetting, binder jetting, powder bed fusion, material extrusion, directed energy deposition, and sheet lamination. Based on these technologies, over one thousand different industrial 3D printers have been developed and identified in the market so far. A controversy regarding which printer is better than the others is meaningless, as each printer has its own features and application range. However, research on the selection of a suitable printer from a specific number of alternative printers for printing a specific component is nontrivial. This is because such selection needs a comprehensive understanding of the features of all alternative printers and a dynamic interaction with the quality of the built component, while most users lack such knowledge and experience. In addition, different printers could have similarities or overlaps at features, which brings certain difficulties to the selection objectively.To offer an effective tool for selection of 3D printers, many different kinds of methods have been presented, where methods based on multiattribute decision-making (MADM) are one of the most important kinds. This kind of methods determine appropriate printers by synthetically assessing the values of multiple interrelated attributes of all alternative printers, which are usually achieved from experiments or simulations or provided by domain experts. It is generally difficult to ensure the absolute objectivity of these ways. This means that there could be some attributes values having “noise.” To achieve reasonable selection result in this case, it is of necessity to capture the interrelationships of the attributes and concurrently reduce the influence of the attributes values having “noise.” But there is yet no evidence that any of the existing MADM-based methods for 3D printer selection have such capabilities. The proposed MAGDM method can meet this requirement when the values of attributes of all alternative printers are quantified by LIVIFNs. The following is an illustrative example about the application of the proposed method in the 3D printer selection.In this example, a user needs to select a proper printer from four alternative printers, denoted asP1, P2, P3, and P4, to print a component using a specific material. The user invited three experienced domain experts, denoted as E1, E2, and E3, to evaluate the four alternative printers based on four attributes, which are the surface roughness (A1), strength (A2), elongation (A3), and hardness (A4) of the built component. The relative importance of the three experts is quantified by ϖ = [0.4, 0.3, 0.3]. The relative importance of the four attributes is measured by w = [0.1, 0.3, 0.3, 0.3]. In the evaluation, the three experts were asked to use LIVIFNs. The available linguistic variables are extremely small (s0), very small (s1), small (s2), slightly small (s3), medium (s4), slightly large (s5), large (s6), very large (s7), and extremely large (s8). The evaluation results of the three experts are, respectively, listed in the following three matrices:(54)M1=s2,s4,s3,s4s6,s6,s1,s2s5,s6,s1,s1s6,s6,s1,s1s3,s5,s2,s3s5,s6,s1,s2s4,s5,s1,s1s4,s6,s1,s2s1,s1,s6,s7s6,s6,s1,s2s3,s4,s3,s3s5,s6,s1,s2s1,s1,s7,s7s3,s4,s2,s3s3,s5,s2,s3s2,s3,s3,s4,M2=s3,s4,s3,s4s5,s6,s1,s2s6,s6,s1,s2s5,s6,s1,s2s3,s5,s1,s3s6,s6,s1,s1s5,s6,s1,s2s3,s5,s1,s2s1,s2,s6,s6s5,s6,s1,s1s3,s5,s3,s3s5,s6,s1,s2s1,s2,s6,s6s3,s3,s3,s3s3,s4,s2,s3s3,s4,s3,s4,M3=s3,s3,s3,s4s6,s6,s2,s2s5,s6,s1,s1s6,s7,s1,s1s3,s4,s2,s3s6,s7,s1,s1s5,s5,s1,s3s5,s5,s1,s2s2,s3,s5,s5s4,s5,s1,s2s3,s4,s3,s4s4,s5,s2,s2s1,s1,s6,s7s3,s4,s2,s3s3,s5,s1,s2s4,s5,s3,s3.According to the conditions above, the determination can be carried out leveraging the designed MAGDM method. Its process includes the following steps:(1)
Normalise the fuzzy decision matricesMh (h = 1, 2, 3). Since surface roughness is cost attribute and strength, elongation, and hardness are benefit attributes, the fuzzy decision matrices M1, M2, and M3 are, respectively, normalised as follows:(55)N1=s3,s4,s2,s4s6,s6,s1,s2s5,s6,s1,s1s6,s6,s1,s1s2,s3,s3,s5s5,s6,s1,s2s4,s5,s1,s1s4,s6,s1,s2s6,s7,s1,s1s6,s6,s1,s2s3,s4,s3,s3s5,s6,s1,s2s7,s7,s1,s1s3,s4,s2,s3s3,s5,s2,s3s2,s3,s3,s4,N2=s3,s4,s3,s4s5,s6,s1,s2s6,s6,s1,s2s5,s6,s1,s2s1,s3,s3,s5s6,s6,s1,s1s5,s6,s1,s2s3,s5,s1,s2s6,s6,s1,s2s5,s6,s1,s1s3,s5,s3,s3s5,s6,s1,s2s6,s6,s1,s2s3,s3,s3,s3s3,s4,s2,s3s3,s4,s3,s4,N3=s3,s4,s3,s3s6,s6,s2,s2s5,s6,s1,s1s6,s7,s1,s1s2,s3,s3,s4s6,s7,s1,s1s5,s5,s1,s3s5,s5,s1,s2s5,s5,s2,s3s4,s5,s1,s2s3,s4,s3,s4s4,s5,s2,s2s6,s7,s1,s1s3,s4,s2,s3s3,s5,s1,s2s4,s5,s3,s3,(2)
Compute the power weights ofαh,i,ji=1,2,3,4;j=1,2,3,4. According to equation (50), the power weights of αh,i,j are calculated and listed in the following matrices:(56)W1,i,j4×4=0.39870.40260.40260.40390.40260.39610.39740.40000.40000.40130.40260.40390.40000.40260.40390.3987,W2,i,j4×4=0.30230.29870.29550.29640.29870.30360.30130.30000.30680.30100.29870.30290.29670.29550.29970.3057,W3,i,j4×4=0.29900.29870.30190.29970.29870.30030.30130.30000.29320.29770.29870.29320.30330.30190.29640.2956.(3)
Compute the collective values ofαh,i,j. Taking Nh and ϖ as input, the collective values of αh,i,j can be calculated using equation (51). Here, the LIVIFWPMM operator in equation (42) (when adapting the operator, Q = (1, 0, 0)) is used in the calculation. The calculated results are listed in the following matrix:(57)αi,j4×4=s3.0000,s4.0000,s2.5522,s3.6703s5.7425,s6.0000,s1.2300,s2.0000s5.3387,s6.0000,s1.0000,s1.2273s5.7446,s6.3751,s1.0000,s1.2281s1.7173,s3.0000,s3.0000,s4.6776s5.6516,s6.3759,s1.0000,s1.3159s4.6367,s5.3450,s1.0000,s1.7158s4.0767,s5.4492,s1.0000,s2.0000s5.7475,s6.2930,s1.2253,s1.7070s5.2226,s5.7434,s1.0000,s1.6234s3.0000,s4.3294,s3.0000,s3.2692s4.7360,s5.7476,s1.2253,s2.0000s6.4843,s6.7716,s1.0000,s1.2284s3.0000,s3.7274,s2.2546,s3.0000s3.0000,s4.7299,s1.6285,s2.6603s2.9663,s3.9843,s3.0000,s3.6739.(4)
Compute the power weights ofαi,j. According to equation (52), the power weights of αi,j are calculated and listed in the following matrix:(58)Wi,j4×4=0.08880.30440.30440.30240.08610.29620.30890.30890.10170.30910.27840.31080.08880.30540.30680.2990.(5)
Calculate the collective values ofαi,j. The collective values of αi,j can be computed according to equation (53). Since the LIVIFWPMM operator in equation (42) has been leveraged in the third step, this operator (when adapting the operator, Q = (1, 2, 3, 4), i.e., there are interrelationships among the four attributes) is also used in this step to complete the calculation. The calculated results are listed as follows:(59)α1=S4.6635,S5.2066,S1.9801,S2.4910,α2=S3.8050,S4.7172,S2.0661,S2.8436,α3=S4.3660,S5.2202,S1.9944,S2.4892,α4=S3.4932,S4.4878,S2.3124,S2.9583.(6)
Compute the scores and accuracies ofαi. According to equations (1) and (2), the scores and accuracies of αi are calculated and, respectively, listed as follows:(60)Sα1=S5.3498,Sα2=S4.9031,Sα3=S5.2757,Sα4=S4.6776,Aα1=S7.1706,Aα2=S6.7160,Aα3=S7.0349,Aα3=S6.6258.(7)
Generate a ranking ofPi. According to the scores and accuracies of αi and the comparison rules in Definition 4, a ranking of Pi is generated as P1≻P3≻P2≻P4.(8)
Determine the best printer. According to the ranking, the best 3D printer is determined as printerP1.


## 5.2. Experiments

### 5.2.1. Validation Experiments

To demonstrate the effectiveness of an MADM method, Wang and Triantaphyllou [70] presented the following three test criteria:(1)
Criterion 1: “An effective MADM method should not change the place of the best option in the generated ranking when a nonoptimal option is replaced by a new worse option under the condition that the weight of each attribute remains unchanged”(2)
Criterion 2: “The rankings of options generated by an effective MADM method should be transitive”(3)
Criterion 3: “If a MADM problem is decomposed into several sub MADM problems and the same MADM method is used in the problem and its sub problems, then a collective ranking of all sub MADM problems should be the same as the ranking of the undecomposed MADM problem.”According to these three criteria, thirteen test experiments using the practical example were designed and carried out. The options included in each of the thirteen experiments are listed in Table1. In Experiments 1, 2, and 3, the nonoptimal options P2, P3, and P4 are, respectively, replaced by the new worse options P2′,P3′,andP4′, whose evaluation results are listed in Table 2. Using the designed method, the rankings of Experiments 1, 2, and 3 can be generated and are shown in Table 1. From the generated rankings, it can be seen that the best options in them remain unchanged. Thus, the designed method can meet Criterion 1.Table 1
The details and results of the validation experiments.
Experiment
Included options
Generated ranking
Experiment 1
P
1, P2′, P3, P4
P
1 ≻ P3 ≻ P4 ≻ P2′
Experiment 2
P
1, P2, P3′, P4
P
1 ≻ P2 ≻ P3′ ≻ P4
Experiment 3
P
1, P2, P3, P4′
P
1 ≻ P3 ≻ P2 ≻ P4′
Experiment 4
P
1, P2
P
1 ≻ P2
Experiment 5
P
1, P3
P
1 ≻ P3
Experiment 6
P
1, P4
P
1 ≻ P4
Experiment 7
P
2, P3
P
3 ≻ P2
Experiment 8
P
2, P4
P
2 ≻ P4
Experiment 9
P
3, P4
P
3 ≻ P4
Experiment 10
P
1, P2, P3
P
1 ≻ P3 ≻ P2
Experiment 11
P
1, P2, P4
P
1 ≻ P2 ≻ P4
Experiment 12
P
1, P3, P4
P
1 ≻ P3 ≻ P4
Experiment 13
P
2, P3, P4
P
3 ≻ P2 ≻ P4Table 2
The evaluation results of the new worse options.
Option
AttributeA1
AttributeA2
AttributeA3
AttributeA4
P
2′ in M1
([s3, s5], [s1, s2])
([s4, s5], [s1, s2])
([s3, s4], [s1, s1])
([s3, s5], [s1, s2])
P
2′ in M2
([s3, s5], [s1, s3])
([s5, s5], [s1, s1])
([s4, s5], [s1, s2])
([s2, s4], [s1, s2])
P
2′ in M3
([s3, s4], [s1, s2])
([s5, s6], [s1, s1])
([s4, s4], [s1, s3])
([s4, s4], [s1, s2])
P
3′ in M1
([s1, s1], [s5, s6])
([s5, s5], [s1, s2])
([s2, s3], [s3, s3])
([s4, s5], [s1, s2])
P
3′ in M2
([s5, s5], [s1, s2])
([s4, s5], [s1, s1])
([s2, s4], [s3, s3])
([s4, s5], [s1, s2])
P
3′ in M3
([s4, s4], [s2, s3])
([s3, s4], [s1, s2])
([s2, s3], [s3, s4])
([s3, s4], [s2, s2])
P
4′ in M1
([s1, s1], [s6, s6])
([s2, s3], [s2, s3])
([s2, s4], [s2, s3])
([s1, s2], [s3, s4])
P
4′ in M2
([s1, s2], [s5, s5])
([s2, s2], [s3, s3])
([s2, s3], [s2, s3])
([s2, s3], [s3, s4])
P
4′ in M3
([s1, s1], [s5, s6])
([s2, s3], [s2, s3])
([s2, s4], [s1, s2])
([s3, s4], [s3, s3])The remaining ten test experiments (i.e., Experiments 4−13) aim to solve all sub MAGDM problems of the MAGDM problem in the practical example. Using the designed method, the rankings of these experiments can be generated and are also shown in Table1. From the rankings of Experiments 2, 4, 7, and 10, Experiments 4, 6, 8, and 11, Experiments 5, 6, 9, and 12, and Experiments 7, 8, 9, and 13, it can be seen that transitivity is satisfied. Therefore, the designed method can meet Criterion 2. In addition, if the rankings of Experiments 4−9 or Experiments 10−13 are combined, then the same collective ranking P1≻P3≻P2≻P4 will be obtained, which is exactly the ranking of the undecomposed MAGDM problem. From this point of view, the designed method can also meet Criterion 3.In summary, the designed method is effective for resolving the practical MAGDM problems since it can meet all of the three test criteria.


### 5.2.2. Exploration Experiment

To show the influence of different values ofQ on the aggregation and ranking results, a test experiment using the practical example was designed and carried out. In this experiment, Q was respectively assigned (1, 0, 0, 0), (2, 0, 0, 0), (3, 0, 0, 0), (4, 0, 0, 0), (1, 1, 0, 0), (1, 2, 0, 0), (1, 3, 0, 0), (1, 4, 0, 0), (1, 1, 1, 0), (1, 2, 3, 0), (1, 1, 1, 1), (2, 2, 2, 2), (3, 3, 3, 3), (4, 4, 4, 4), and (1, 2, 3, 4) when calculating the collective values of αi,j in Step (5) in the example. Based on these different values of Q, the LIVIFWPMM operator will reduce to different operators, which are, respectively, listed in Table 3. The results of the experiment are the computed scores of αi and the generated ranking of Pi under each Q, which are also listed in Table 3. As can be seen from the table, the rankings of the four 3D printers generated by the proposed method may have difference, in the situations where (1) all of the four attributes are independent of each other, (2) there are interrelationships between any two attributes, (3) there are interrelationships among any three attributes, and (4) there are interrelationships among the four attributes. This indicates that the presented LIVIFWPMM operator has the capability and generality to capture the interrelationships of attributes. In practical application of the operator, Q is recommended to take (1, 0, 0, 0), (1, 2, 0, 0), (1, 2, 3, 0), and (1, 2, 3, 4) under the four situations, respectively.Table 3
The details and results of the exploration experiment.
Value ofQ
Actual operator
Interrelationships
The computed scores of all printers
The generated ranking
S (α1)
S (α2)
S (α3)
S (α4)
(1, 0, 0, 0)
LIVIFWPAA (equation (44))
Independent of each other
6.1822
5.8369
5.6327
4.7823
P
1
≻
P
2
≻
P
3
≻
P
4
(2, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.3032
5.9960
5.7047
4.8070
P
1
≻
P
2
≻
P
3
≻
P
4
(3, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.3892
6.1016
5.7753
4.8327
P
1
≻
P
2
≻
P
3
≻
P
4
(4, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.4509
6.1753
5.8420
4.8588
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
5.7939
5.4072
5.3790
4.7050
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 2, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
5.9428
5.5941
5.4509
4.7281
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 3, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
6.0762
5.7504
5.5429
4.7589
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 4, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
6.1780
5.8670
5.6315
4.7902
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 1, 0)
LIVIFWPMSM (equation (46))
Among any 3 attributes
5.4135
4.9825
5.2503
4.6695
P
1
≻
P
3
≻
P
2
≻
P
4
(1, 2, 3, 0)
LIVIFWPMM (equation (42))
Among any 3 attributes
5.7674
5.4217
5.3606
4.7043
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 1, 1)
LIVIFWPGA (equation (48))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(2, 2, 2, 2)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(3, 3, 3, 3)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(4, 4, 4, 4)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(1, 2, 3, 4)
LIVIFWPMM (equation (42))
Among the 4 attributes
5.3498
4.9031
5.2757
4.6776
P
1
≻
P
3
≻
P
2
≻
P
4


## 5.2.1. Validation Experiments

To demonstrate the effectiveness of an MADM method, Wang and Triantaphyllou [70] presented the following three test criteria:(1)
Criterion 1: “An effective MADM method should not change the place of the best option in the generated ranking when a nonoptimal option is replaced by a new worse option under the condition that the weight of each attribute remains unchanged”(2)
Criterion 2: “The rankings of options generated by an effective MADM method should be transitive”(3)
Criterion 3: “If a MADM problem is decomposed into several sub MADM problems and the same MADM method is used in the problem and its sub problems, then a collective ranking of all sub MADM problems should be the same as the ranking of the undecomposed MADM problem.”According to these three criteria, thirteen test experiments using the practical example were designed and carried out. The options included in each of the thirteen experiments are listed in Table1. In Experiments 1, 2, and 3, the nonoptimal options P2, P3, and P4 are, respectively, replaced by the new worse options P2′,P3′,andP4′, whose evaluation results are listed in Table 2. Using the designed method, the rankings of Experiments 1, 2, and 3 can be generated and are shown in Table 1. From the generated rankings, it can be seen that the best options in them remain unchanged. Thus, the designed method can meet Criterion 1.Table 1
The details and results of the validation experiments.
Experiment
Included options
Generated ranking
Experiment 1
P
1, P2′, P3, P4
P
1 ≻ P3 ≻ P4 ≻ P2′
Experiment 2
P
1, P2, P3′, P4
P
1 ≻ P2 ≻ P3′ ≻ P4
Experiment 3
P
1, P2, P3, P4′
P
1 ≻ P3 ≻ P2 ≻ P4′
Experiment 4
P
1, P2
P
1 ≻ P2
Experiment 5
P
1, P3
P
1 ≻ P3
Experiment 6
P
1, P4
P
1 ≻ P4
Experiment 7
P
2, P3
P
3 ≻ P2
Experiment 8
P
2, P4
P
2 ≻ P4
Experiment 9
P
3, P4
P
3 ≻ P4
Experiment 10
P
1, P2, P3
P
1 ≻ P3 ≻ P2
Experiment 11
P
1, P2, P4
P
1 ≻ P2 ≻ P4
Experiment 12
P
1, P3, P4
P
1 ≻ P3 ≻ P4
Experiment 13
P
2, P3, P4
P
3 ≻ P2 ≻ P4Table 2
The evaluation results of the new worse options.
Option
AttributeA1
AttributeA2
AttributeA3
AttributeA4
P
2′ in M1
([s3, s5], [s1, s2])
([s4, s5], [s1, s2])
([s3, s4], [s1, s1])
([s3, s5], [s1, s2])
P
2′ in M2
([s3, s5], [s1, s3])
([s5, s5], [s1, s1])
([s4, s5], [s1, s2])
([s2, s4], [s1, s2])
P
2′ in M3
([s3, s4], [s1, s2])
([s5, s6], [s1, s1])
([s4, s4], [s1, s3])
([s4, s4], [s1, s2])
P
3′ in M1
([s1, s1], [s5, s6])
([s5, s5], [s1, s2])
([s2, s3], [s3, s3])
([s4, s5], [s1, s2])
P
3′ in M2
([s5, s5], [s1, s2])
([s4, s5], [s1, s1])
([s2, s4], [s3, s3])
([s4, s5], [s1, s2])
P
3′ in M3
([s4, s4], [s2, s3])
([s3, s4], [s1, s2])
([s2, s3], [s3, s4])
([s3, s4], [s2, s2])
P
4′ in M1
([s1, s1], [s6, s6])
([s2, s3], [s2, s3])
([s2, s4], [s2, s3])
([s1, s2], [s3, s4])
P
4′ in M2
([s1, s2], [s5, s5])
([s2, s2], [s3, s3])
([s2, s3], [s2, s3])
([s2, s3], [s3, s4])
P
4′ in M3
([s1, s1], [s5, s6])
([s2, s3], [s2, s3])
([s2, s4], [s1, s2])
([s3, s4], [s3, s3])The remaining ten test experiments (i.e., Experiments 4−13) aim to solve all sub MAGDM problems of the MAGDM problem in the practical example. Using the designed method, the rankings of these experiments can be generated and are also shown in Table1. From the rankings of Experiments 2, 4, 7, and 10, Experiments 4, 6, 8, and 11, Experiments 5, 6, 9, and 12, and Experiments 7, 8, 9, and 13, it can be seen that transitivity is satisfied. Therefore, the designed method can meet Criterion 2. In addition, if the rankings of Experiments 4−9 or Experiments 10−13 are combined, then the same collective ranking P1≻P3≻P2≻P4 will be obtained, which is exactly the ranking of the undecomposed MAGDM problem. From this point of view, the designed method can also meet Criterion 3.In summary, the designed method is effective for resolving the practical MAGDM problems since it can meet all of the three test criteria.


## 5.2.2. Exploration Experiment

To show the influence of different values ofQ on the aggregation and ranking results, a test experiment using the practical example was designed and carried out. In this experiment, Q was respectively assigned (1, 0, 0, 0), (2, 0, 0, 0), (3, 0, 0, 0), (4, 0, 0, 0), (1, 1, 0, 0), (1, 2, 0, 0), (1, 3, 0, 0), (1, 4, 0, 0), (1, 1, 1, 0), (1, 2, 3, 0), (1, 1, 1, 1), (2, 2, 2, 2), (3, 3, 3, 3), (4, 4, 4, 4), and (1, 2, 3, 4) when calculating the collective values of αi,j in Step (5) in the example. Based on these different values of Q, the LIVIFWPMM operator will reduce to different operators, which are, respectively, listed in Table 3. The results of the experiment are the computed scores of αi and the generated ranking of Pi under each Q, which are also listed in Table 3. As can be seen from the table, the rankings of the four 3D printers generated by the proposed method may have difference, in the situations where (1) all of the four attributes are independent of each other, (2) there are interrelationships between any two attributes, (3) there are interrelationships among any three attributes, and (4) there are interrelationships among the four attributes. This indicates that the presented LIVIFWPMM operator has the capability and generality to capture the interrelationships of attributes. In practical application of the operator, Q is recommended to take (1, 0, 0, 0), (1, 2, 0, 0), (1, 2, 3, 0), and (1, 2, 3, 4) under the four situations, respectively.Table 3
The details and results of the exploration experiment.
Value ofQ
Actual operator
Interrelationships
The computed scores of all printers
The generated ranking
S (α1)
S (α2)
S (α3)
S (α4)
(1, 0, 0, 0)
LIVIFWPAA (equation (44))
Independent of each other
6.1822
5.8369
5.6327
4.7823
P
1
≻
P
2
≻
P
3
≻
P
4
(2, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.3032
5.9960
5.7047
4.8070
P
1
≻
P
2
≻
P
3
≻
P
4
(3, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.3892
6.1016
5.7753
4.8327
P
1
≻
P
2
≻
P
3
≻
P
4
(4, 0, 0, 0)
LIVIFWPGAA (equation (43))
Independent of each other
6.4509
6.1753
5.8420
4.8588
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
5.7939
5.4072
5.3790
4.7050
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 2, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
5.9428
5.5941
5.4509
4.7281
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 3, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
6.0762
5.7504
5.5429
4.7589
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 4, 0, 0)
LIVIFWPBM (equation (45))
Between any 2 attributes
6.1780
5.8670
5.6315
4.7902
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 1, 0)
LIVIFWPMSM (equation (46))
Among any 3 attributes
5.4135
4.9825
5.2503
4.6695
P
1
≻
P
3
≻
P
2
≻
P
4
(1, 2, 3, 0)
LIVIFWPMM (equation (42))
Among any 3 attributes
5.7674
5.4217
5.3606
4.7043
P
1
≻
P
2
≻
P
3
≻
P
4
(1, 1, 1, 1)
LIVIFWPGA (equation (48))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(2, 2, 2, 2)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(3, 3, 3, 3)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(4, 4, 4, 4)
LIVIFWPGGA (equation (47))
Among the 4 attributes
4.8961
4.3272
5.1750
4.6449
P
3
≻
P
1
≻
P
4
≻
P
2
(1, 2, 3, 4)
LIVIFWPMM (equation (42))
Among the 4 attributes
5.3498
4.9031
5.2757
4.6776
P
1
≻
P
3
≻
P
2
≻
P
4


## 5.3. Comparisons

As mentioned in the introduction, representative AOs of LIVIFNs are the WA, OWA, HA, WG, OWG, and HG operators presented by Garg and Kumar [54, 55], the PWA, POWA, PWG, and POWG operators presented by Kumar and Garg [56], and the WMSM operator presented by Liu and Qin [57]. In this subsection, qualitative and quantitative comparisons between these AOs and the presented AO of LIVIFNs are carried out to evaluate the presented AO:(1)
Qualitative Comparison. This comparison was carried out via comparing the characteristics of the AOs. For the twelve AOs above, the generality and flexibility in aggregating the values of attributes, the generality in capturing the interrelationships of attributes, and the capability to reduce the effect of biased attribute values are selected as the comparison characteristics. The results of the comparison, as shown in Table 4, are explained as follows.(a)
Generality and Flexibility in Aggregating the Values of Attributes. Among the twelve AOs, the WA, OWA, HA, WG, OWG, HG, PWA, POWA, PWG, POWG, and WMSM operators perform the aggregation using the Algebraic T-norm and T-conorm, their generality and flexibility are relatively limited. Such characteristics of the presented Archimedean weighted power Muirhead mean (AWPMM) operator are satisfying since the aggregation in it can be performed by any types of ATNTCs (e.g., Algebraic, Einstein, Hamacher, and Frank T-norms and T-conorms).(b)
Generality in Capturing the Relationships of Attributes. The WA, OWA, HA, WG, OWG, HG, WMSM, and AWPMM operators are suitable for the case where all aggregated attributes are independent of each other. In addition, the WMSM and AWPMM operators can also be applicable in the situations where there are interrelationships between any two attributes and where there are relationships among any multiple attributes, because they, respectively, use the all-in-one MM and MSM operators for capturing interrelationships. It should be noted that MM is more generalised than MSM since MSM is just a special case of MM. From this point of view, the AWPMM operator is more generalised than the WMSM operator. The PWA, POWA, PWG, and POWG operators are specifically presented for dealing with the situation where the attributes are in different priority levels. They are totally different from other AOs from the perspective of application range.(c)
Capability to Reduce the Effect of Biased Attribute Values. Among the twelve AOs, only the AWPMM operator has this capability due to the combination of the PA operator.(2)
Quantitative Comparison. This comparison was carried out using the practical examples in the present paper and in [55, 57] as benchmarks. In the comparison, the WA and WG operators were selected as the AOs for the method of Garg and Kumar [55]. The PWA and PWG operators were chosen as the AOs for the method of Kumar and Garg [56]. The given weights in the three examples were directly used as the priority weights of the PWA and PWG operators to make the input of all comparison operators the same (it should be pointed out that the PWA and PWG operators will, respectively, reduce to the WA and WG operators in this case). The method of Liu and Qin [57] and the proposed method respectively used the WMSM operator and the weighted power MM (WPMM) operator (see equation (42)). Moreover, the same score function (i.e., the function in equation (1)) was used in all of these operators for easy comparison. The details and results of the comparison are listed in Table 5.Table 4
The results of the qualitative comparison.
AOs presented in the method
Generality and flexibility in aggregation
Generality in capturing interrelationships of attributes
Capability to reduce the effect
Independent of each other
Between any two
Among any multiple
In different priority levels
WA [55]
Limited
Yes
No
No
No
No
OWA [55]
Limited
Yes
No
No
No
No
HA [55]
Limited
Yes
No
No
No
No
WG [55]
Limited
Yes
No
No
No
No
OWG [55]
Limited
Yes
No
No
No
No
HG [55]
Limited
Yes
No
No
No
No
PWA [56]
Limited
No
No
No
Yes
No
POWA [56]
Limited
No
No
No
Yes
No
PWG [56]
Limited
No
No
No
Yes
No
POWG [56]
Limited
No
No
No
Yes
No
WMSM [57]
Limited
Yes
Yes
Yes
No
No
AWPMM
Satisfying
Yes
Yes
Yes
No
YesTable 5
The details and results of the quantitative comparison.
Benchmark
Decision-making method
Used AOs
Value of arguments
The calculated scores of all options
The generated ranking
S (α1)
S (α2)
S (α3)
S (α4)
Example in this paper
Garg and Kumar [55]
WA, WA
—
6.1654
5.8171
5.5985
4.8181
O
1
≻
O
2
≻
O
3
≻
O
4
Garg and Kumar [55]
WG, WG
—
5.9921
5.5132
5.3441
4.5370
O
1
≻
O
2
≻
O
3
≻
O
4
Kumar and Garg [56]
PWA, PWA
—
6.1654
5.8171
5.5985
4.8181
O
1
≻
O
2
≻
O
3
≻
O
4
Kumar and Garg [56]
PWG, PWG
—
5.9921
5.5132
5.3441
4.5370
O
1
≻
O
2
≻
O
3
≻
O
4
Liu and Qin [57]
WMSM, WMSM
k = 1, k = 3
5.4523
5.0127
5.2577
4.7343
O
1
≻
O
3
≻
O
2
≻
O
4
The designed method
WPMM, WPMM
Q = (1, 0, 0), Q = (1, 2, 3, 0)
5.7674
5.4217
5.3606
4.7043
O
1
≻
O
2
≻
O
3
≻
O
4
Example in [55]
Garg and Kumar [55]
WA, WA
—
5.3457
4.6145
5.1125
4.9687
O
1
≻
O
3
≻
O
4
≻
O
2
Garg and Kumar [55]
WG, WG
—
5.1069
4.3577
4.9218
4.5868
O
1
≻
O
3
≻
O
4
≻
O
2
Kumar and Garg [56]
PWA, PWA
—
5.3457
4.6145
5.1125
4.9687
O
1
≻
O
3
≻
O
4
≻
O
2
Kumar and Garg [56]
PWG, PWG
—
5.1069
4.3577
4.9218
4.5868
O
1
≻
O
3
≻
O
4
≻
O
2
Liu and Qin [57]
WMSM, WMSM
k = 1, k = 3
5.1768
4.2902
4.7499
4.6151
O
1
≻
O
3
≻
O
4
≻
O
2
The designed method
WPMM, WPMM
Q = (1, 0, 0), Q = (1, 2, 3, 0)
5.2054
4.3988
4.8582
4.6997
O
1
≻
O
3
≻
O
4
≻
O
2
Example in [57]
Garg and Kumar [55]
WA
—
6.2621
6.2709
5.6111
5.5823
O
2
≻
O
1
≻
O
3
≻
O
4
Garg and Kumar [55]
WG
—
6.1407
6.1620
5.3326
5.2541
O
2
≻
O
1
≻
O
3
≻
O
4
Kumar and Garg [56]
PWA, PWA
—
6.2621
6.2709
5.6111
5.5823
O
2
≻
O
1
≻
O
3
≻
O
4
Kumar and Garg [56]
PWG, PWG
—
6.1407
6.1620
5.3326
5.2541
O
2
≻
O
1
≻
O
3
≻
O
4
Liu and Qin [57]
WMSM
k = 3
6.1034
6.0611
5.5196
5.1819
O
1
≻
O
2
≻
O
3
≻
O
4
The designed method
WPMM
Q = (1, 2, 3, 0, 0)
6.1584
6.1157
5.5637
5.3056
O
1
≻
O
2
≻
O
3
≻
O
4As can be seen from Table5, the best option of the designed method is exactly the same as the best option of the method of Liu and Qin for all examples. This is mainly because the two methods are the most similar in nature (see Table 4). From this result, it can be observed that the designed method is feasible and effective for solving practical decision-making problems based on LIVIFNs. In addition, the rankings of the methods of Garg and Kumar and Kumar and Garg are different from that of the method of Liu and Qin and the designed method at the first and second places for the example in [57]. The reason is that the interrelationships in the former methods are set as “all attributes are independent of each other” and those in the latter two methods are set as “there are interrelationships among any three attributes.”From the qualitative comparison above, the advantage of combination of the PA and MM operators is providing generality in capturing the interrelationships of attributes and capability to reduce the effect of biased attribute values. Such advantage cannot be intuitively seen from the comparison results in Table5. To explicitly show the advantage, two additional quantitative comparison experiments were carried out. The first experiment aims to show the generality in capturing attribute relationships. In this experiment, the practical examples in the present paper and in [55, 57] were used to compare the proposed method and the methods of Garg and Kumar, Kumar and Garg, and Liu and Qin. Given the following are five assumptions: (a) all attributes are independent of each other; (b) there are interrelationships between any two attributes; (c) there are interrelationships among any three attributes; (d) there are interrelationships among any four attributes; (e) there are interrelationships among any five attributes (only for the example in [57]). The applicability of the four comparison methods and the results of the experiment are shown in Table 6. It can be intuitively seen from the table that the methods of Garg and Kumar and Kumar and Garg are applicable only under assumption a, while both the method of Liu and Qin and the designed method can be applied to generate ranking results under all assumptions. This indicates that both of the latter two methods have the generality in capturing the interrelationships of attributes.Table 6
The details and results of the first additional quantitative comparison experiment.
Benchmark
Decision-making method
Generated ranking under assumption (a)
Generated ranking under assumption (b)
Generated ranking under assumption (c)
Generated ranking under assumption (d)
Generated ranking under assumption (e)
Example in this paper
Garg and Kumar [55]
O
1
≻
O
2
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
—
Kumar and Garg [56]
O
1
≻
O
2
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
—
Liu and Qin [57]
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
3
≻
O
2
≻
O
4
O
3
≻
O
1
≻
O
4
≻
O
2
—
The designed method
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
3
≻
O
2
≻
O
4
—
Example in [55]
Garg and Kumar [55]
O
1
≻
O
3
≻
O
4
≻
O
2
Cannot be applied
Cannot be applied
Cannot be applied
—
Kumar and Garg [56]
O
1
≻
O
3
≻
O
4
≻
O
2
Cannot be applied
Cannot be applied
Cannot be applied
—
Liu and Qin [57]
O
1
≻
O
4
≻
O
3
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
—
The designed method
O
1
≻
O
4
≻
O
3
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
O
1
≻
O
3
≻
O
4
≻
O
2
—
Example in [57]
Garg and Kumar [55]
O
2
≻
O
1
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
Cannot be applied
Kumar and Garg [56]
O
2
≻
O
1
≻
O
3
≻
O
4
Cannot be applied
Cannot be applied
Cannot be applied
Cannot be applied
Liu and Qin [57]
O
2
≻
O
1
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
The designed method
O
2
≻
O
1
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4
O
1
≻
O
2
≻
O
3
≻
O
4The second experiment aims to show the capability to reduce the influence of extreme attribute values. In this experiment, the practical example in the present paper is used to compare the proposed method and the method of Liu and Qin, which have difference only in whether combining the PA operator whenQ = (1, 0, 0, 0) and k = 1 and Q = (1, 1, 1, 0) and k = 3 for the example. It is assumed that the value of attribute A2 of printer P1 evaluated by expert E1 (i.e., α1,1,2) is a biased attribute value. This value was constantly adjusted from high to low, as listed in the first column of Table 7. The change of the relative importance (i.e., weight) of α1,1,2 with respect to the adjusted value is also shown in Table 7. As can be intuitively seen from the table, the weight of α1,1,2 becomes smaller and smaller as the value of α1,1,2 changes from high to low in the proposed method, while it remains the same in this process in the method of Liu and Qin. The greater the change of the value of α1,1,2, the greater the bias. The relative importance of the value should be dynamically decreased to reduce the effect of this extreme value. From the comparison results, only the proposed method has such capability. That is, the method can reduce the effect of biased attribute values.Table 7
The details and results of the second additional quantitative comparison experiment.
Value ofα1,1,2
Change of weights in the designed method
Change of weights in the method of Liu and Qin [57]
Weights of expertE1 ([W1,i,j]4×4)
Weights of attributes ([Wi,j]4×4)
Weights of expertE1 ([W1,i,j]4×4)
Weights of attributes ([Wi,j]4×4)
([s6, s6], [s1, s2])
0
.
3987
0
.4026
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0888
0
.3044
0
.
3044
0
.
3024
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s5, s5], [s2, s3])
0
.
3987
0
.3974
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0906
0
.3044
0
.
3045
0
.
3005
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s4, s4], [s3, s4])
0
.
3987
0
.3859
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0923
0
.3039
0
.
3039
0
.
2999
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s3, s3], [s4, s5])
0
.
3987
0
.3728
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0936
0
.3035
0
.
3035
0
.
2994
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s2, s2], [s5, s6])
0
.
3987
0
.3577
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0945
0
.3032
0
.
3032
0
.
2991
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s1, s1], [s6, s7])
0
.
3987
0
.3402
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0952
0
.3030
0
.
3030
0
.
2988
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
([s0, s0], [s7, s8])
0
.
3987
0
.3195
0
.
4026
0
.
4039
0
.
4026
0
.
3961
0
.
3974
0
.
4000
0
.
4000
0
.
4013
0
.
4026
0
.
4039
0
.
4000
0
.
4026
0
.
4039
0
.
3987
0
.
0955
0
.3029
0
.
3029
0
.
2987
0
.
0861
0
.
2962
0
.
3089
0
.
3089
0
.
1017
0
.
3091
0
.
2784
0
.
3108
0
.
0888
0
.
3054
0
.
3068
0
.
2990
0
.
4000
0
.4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
4000
0
.
1000
0
.3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000
0
.
1000
0
.
3000
0
.
3000
0
.
3000On the basis of the comparisons above, the advantages of the designed method over the methods of Garg and Kumar, Kumar and Garg, and Liu and Qin are summarised as follows:(1)
Compared to the methods of Garg and Kumar and Kumar and Garg, the designed method is generalised and flexible for aggregation of attribute values and handling of attribute relationships and concurrently has the capability to reduce the influence of the distortion of attribute values.(2)
Compared to the method of Liu and Qin, the designed method has desirable generality and flexibility in aggregation of attribute values and can reduce the influence of the biased attribute values on aggregation result.


## 6. Conclusion

In this paper, a LIVIFAPMM operator and a LIVIFAWPMM operator have been presented to solve the MAGDM problems based on LIVIFNs. The generalised expressions of the two operators have been established. Their properties have been explored and proved and specific expressions have been constructed using the operational rules of LIVIFNs based on the Algebraic T-norm and T-conorm. On the basis of the presented LIVIFAWPMM operator, a method for resolving the LIVIFN-based MAGDM problems has been proposed. The paper has also introduced a practical example to illustrate the proposed method and reported a set of experiments and comparisons to evaluate it. The results of the experiments and comparisons suggest that the method is feasible and effective which has advantages in providing the generality and flexibility in aggregation of attribute values and capturing of attribute interrelationships and the capability to reduce the effect of the deviation of attribute values. Two main limitations of the proposed method are that the method has not captured the risk attitudes of decision makers and it cannot work properly under incomplete attribute information. Future work will focus especially on addressing these limitations. In addition, the application of the method in real MAGDM problems may also be studied.



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*Source: 2373762-2020-01-08.xml*