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Journal of Intelligent & Fuzzy Systems 41 (2021) 4109–4126
DOI:10.3233/JIFS-210506
IOS Press
4109
Novel q-rung orthopair fuzzy interaction
aggregation operators and their application
to low-carbon green supply chain
management
Muhammad Riaza, Harish Gargb,∗, Hafiz Muhammad Athar Faridaand Muhammad Aslamc
aDepartment of Mathematics, University of the Punjab, Lahore, Pakistan
bSchool of Mathematics, Thapar Institute of Engineering and Technology, Deemed University, Patiala, Punjab,
India
cCollege of Sciences, King Khalid University Abha, Saudi Arabia
Abstract. The low-carbon supply chain management is big a challenge for the researchers due to the rapid increase in global
warming and environmental concerns. With the advancement of the environmental concerns and social economy, it is an
unavoidable choice for a business to achieve sustainable growth for low-carbon supply chain management. Since the root
of the chain depends upon the supplier selection and choosing an excellent low-carbon supply. Green supplier selection is
one of the most crucial activities in low-carbon supply chain management, it is critical to develop rigorous requirements
and a system for selection in low-carbon green supply chain management (LCGSCM). A q-rung orthopair fuzzy number
(q-ROFN) is pair of membership degree (MD) and non-membership degrees (NMD) which is reliable to address uncertainties
in the various real-life problems. This article sets out a decision analysis approach for interactions between MDs and NMDs
with the help of q-ROFNs. For this objective, we develop new aggregation operators (AOs) named as, q-rung orthopair
fuzzy interaction weighted averaging (q-ROFIWA) operator, q-rung orthopair fuzzy interaction ordered weighted averaging
(q-ROFIOWA) operator, q-rung orthopair fuzzy interaction hybrid averaging (q-ROFIHA) operator, q-rung orthopair fuzzy
interaction weighted geometric (q-ROFIWG) operator, q-rung orthopair fuzzy interaction ordered weighted geometric (q-
ROFIOWG) operator and q-rung orthopair fuzzy interaction hybrid geometric (q-ROFIHG) operator. These AOs define an
advanced approach for information fusion and modeling uncertainties in multi-criteria decision-making (MCDM). At the
end, a robust MCDM approach based on newly developed AOs is developed. Some significant properties of these AOS are
analyzed and the efficiency of the developed approach is assessed with a practical application towards sustainable low-carbon
green supply chain management.
Keywords: MCDM, Aggregation operators, interaction relation, low-carbon green supply chain management
1. Introduction
The complex problems involving vague informa-
tion have become a major issue for the decades.
∗Corresponding author. Harish Garg, School of Mathe-
matics, Thapar Institute of Engineering and Technology,
Deemed University, Patiala - 147004, Punjab, India. E-mails:
harish.garg@thapar.edu, harishg58iitr@gmail.com.
Information fusion and data analysis have gain the
attention of the researchers working in the fields
such as decision-making, medical diagnosis, pattern
recognition, computational intelligence and artifi-
cial intelligence. Traditionally the alternatives are
analyzed by a crisp number or linguistic number.
However, due to uncertainties, the data cannot easily
be aggregated. MCDM is a commonly used cognitive
ISSN 1064-1246/$35.00 © 2021 – IOS Press. All rights reserved.
AUTHOR COPY
4110 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
activity tool, the main aim of which is to choose an
optimal alternative among a finite number of alter-
natives using the preference information provided by
decision makers (DMs). Zadeh [22] has pioneered
the fuzzy set theory to describe vague information.
After Zadeh, Atanassov [23] introduced intuitionis-
tic fuzzy set (IFS), and Yager [24, 25] introduced
Pythagorean fuzzy set (PFS) which is an extension of
IFS. Then Yager [26] further extended PFS to q-rung
orthopair fuzzy set (q-ROFS) which is a strong model
to describe vague information in the real-life prob-
lems. A q-rung orthopair fuzzy number (q-ROFN)
is a pair of membership degrees (MDs) and non-
membership degree (NMDs) with the property that
the sum of qth power of both MDs and NMDs may
be less than or identical to one. Clearly, the values
of qwith q≥1 will increase the valuation space to
choose more suitable q-ROFNs.
Aggregation operators such as averaging and geo-
metric operators for IFSs proposed by Xu et al. [27].
Many researches extended aggregation operators to
various fuzzy sets; Mahmood et al. [28], Riaz et al.
[55], Wei et al. [29], Hashmi et al. [30], Feng et
al. [33], Yang et al. [53], Chen et al. [24], Zhao
et al. [32], Garg [34]. Wang et al. [48, 49] developed
interactive Hamacher power AOs and interactive
Archimedean norm operations related to PFS. Wang
and Li introduced Pythagorean fuzzy interaction
power Bonferroni mean AOs [50]. Pythagorean
fuzzy interaction AOs introduced by Wei [51] and
Gao et al. [52]. The notion of linear Diophantine
fuzzy set (LDFS) introduced by Riaz and Hashmi
[35]. A LDFS is new extension of fuzzy sets and
new approach to decision making under vagueness.
Riaz et al. introduced the certain extensions of AOs
like q-ROFS Einstein [36], prioritized [37], Einstein
prioritized [39] and some hybrid AOs [38]. They
developed some interesting applications of these
operators towards MCDM. Liu and Liu [40] initiated
the idea of q-ROF bonferroni mean AOs. Garg
and Chen [41] presented the neutrality operators
for q-ROFSs and its based algorithm to solve the
MCDM problems. Liu et al. [43] proposed the
idea of q-ROF Heronian mean AOs and application
related to MCDM. Joshi and Gegov [44] established
the notion of q-ROF confidence based AOs and their
application to MCDM. Riaz et al. [42] introduced
novel concept of bipolar picture fuzzy. Garg [45] pre-
sented a novel concept of CN-qROFS to address the
decision making problems by utilizing the q-ROFSs.
The main objectives and advantages of the
manuscript are listed as follows.
1) q-ROF AOs are dependent on algebraic opera-
tional principles initiated by Liu and Wang [31]
and do not accept interaction between MD and
NMDs. For example, Assume ˘
αt=(μt,ν
t), t
varies from 1 to n is the assortment of q-ROFNs.
If ˘
αj=(μj,0) with μjis non-zero, then by
operational principles in [31], we get ν˘
αtע
αj=
0. that is NMD of product of all ˘
αtand ˘
αjis
zero if one of the NMD become zero and other
NMDs is not zero. In addition, all q-ROFN AOs
dependent on algebraic operating principles are
also inappropriate for all contingencies. For
example, if we take q-ROFWA(˘
α1,˘
α2,... ˘
αt)
in [31], we get νq−ROFWA(˘
α1,˘
α2,... ˘
αt)=0if
one of the NMD of (˘
α1,˘
α2,... ˘
αt) is zero
but others are non-zero. Therefore, there is
need to improve the operational principles of
q-ROFNs.
2) In many decision-making issues, certain
attributes are always associated in such a way
that inter-relationships between them should
be taken into consideration. We should also
pay close attention to aggregation techniques
that can account for inter-relationships between
multiple attributes. Apparently, by taking into
consideration the interactions between MD and
NMDs, the first problem listed above can be
resolved by He et al. [7] with new interaction
operational laws.
3) In view of the reasons discussed above, we have
developed various aggregation operators with
new interaction operational laws named as, q-
rung orthopair fuzzy interaction weighted aver-
aging (q-ROFIWA) operator, q-rung orthopair
fuzzy interaction ordered weighted averaging
(q-ROFIOWA) operator, q-rung orthopair fuzzy
interaction hybrid averaging (q-ROFIHA)
operator, q-rung orthopair fuzzy interaction
weighted geometric (q-ROFIWG) operator,
q-rung orthopair fuzzy interaction ordered
weighted geometric (q-ROFIOWG) operator
and q-rung orthopair fuzzy interaction hybrid
geometric (q-ROFIHG) operator.
4) Based on the proposed AOs an algorithm
is developed for modeling uncertainties in
MCDM. Some significant properties of sug-
gested operators are also defined.
5) The efficiency of the proposed MCDM
approach is analyzed by a practical application
towards sustainable low-carbon supply chain
management.
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M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4111
The remainder of this paper is set out as follows.
Section 2 consists of a variety of basic principles
related to q-ROFSs. The q-ROF interaction AOs were
addressed in Section 3. Based on new interaction
operations, In Section 4, we present an algorithm
to solve MCDM problems. Section 5 sets out an
application relating to the selection of the supplier of
low-carbon green supply chain management. Some
concluding remarks and future directions are made
in Section 6.
2. Some fundamental concepts
In this section of the paper, we discuss few basic
laws and operational principles of q-ROFSs and q-
ROFNs.
Definition 2.1. [26] A q-rung orthopair fuzzy set (q-
ROFS)
Ton a universe Qis defined as
T={ς, μ
T(ς),ν
T(ς):ς∈Q}
where μ
T,ν
T:Q→[0,1] defines the MD and
NMD of the alternative ς∈Qand ∀ςwe have
0≤μq
T(ς)+νq
T(ς)≤1.
Liu and Wang [31] proposed some basic operations
on q-ROFNs. In the next definition, we review these
operations.
Definition 2.2. [31] Let ˘
α1=μ1,ν
1and ˘
α2=
μ2,ν
2be q-ROFNs and σ>0, then
(1) ˘
αc
1=ν1,μ
1
(2) ˘
α1∨˘
α2=max{μ1,ν
1}, min{μ2,ν
2}
(3) ˘
α1∧˘
α2=min{μ1,ν
1},max{μ2,ν
2}
(4) ˘
α1⊕˘
α2=(μq
1+μq
2−μq
1μq
2)1/q,ν
1ν2
(5) ˘
α1⊗˘
α2=μ1μ2,(νq
1+νq
2−νq
1νq
2)1/q
(6) σ˘
α1=(1 −(1 −μq
1)σ)1/q,ν
σ
1
(7)˘
ασ
1=μσ
1,(1 −(1 −νq
1)σ)1/q
Definition 2.3. [31] Let ˘
α=μ, νbe the q-ROFN,
then its score function iis defined as
i(˘
α)=μq−νq
where i(˘
α)∈[−1,1]. The score function will decide
the ranking of a set of q-ROFNs. The values of
score function of q-ROFNs determine their priori-
ties/ranking. In certain situations, although, the score
function is not really beneficial in the ranking of
two q-ROFNs. If the values of score function of two
q-ROFNs become same then we must use another
function named as accuracy function to discuss their
ranking.
Definition 2.4. [31] Let ˘
α=μ, νbe the q-ROFN,
then an accuracy function ´
Qof ˘
αis defined as
´
Q(˘
α)=μq+νq
where ´
Q(˘
α)∈[0,1].
Definition 2.5. Let ˘
α=μ˘
α,ν˘
αand β=μβ,ν
βbe
two q-ROFNs, and i(˘
α),i(βbe the score function
of ˘
αand β, and ´
Q(˘
α),´
Q(β) be the accuracy function
of ˘
αand β, respectively. Then
a) If i(˘
α)>i(β), then ˘
α>β
b) If i(˘
α)=i(β), then
–if´
Q(˘
α)>´
Q(β) then ˘
α>β,
–if´
Q(˘
α)=´
Q(β), then ˘
α=β.
2.1. q-ROF aggregation operators
Liu and Wang [31] developed some basic aggre-
gation operators for q-ROFNs named as q-rung
orthopair fuzzy weighted averaging (q-ROFWA)
and q-rung orthopair fuzzy weighted geometric (q-
ROFWG) operators.
Definition 2.6. [31] Assume that ˘
αq=μq,ν
qis the
family of q-ROFNs, and q-ROFWA: ϒn→ϒ,if
q-ROFWA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
g˘
αg
=˘
1˘
α1⊕˘
2˘
α2⊕...,˘
r˘
αr
where ϒnis the set of all q-ROFNs and
(˘
1,˘
2,...˘
r) be the weight vector (WV) of con-
sidered q-ROFNs with the condition that ˘
g>0,
˘
g∈[0,1] and r
g=1˘
g=1. Then, the q-ROFWA is
called the ”q-rung orthopair fuzzy weighted average
operator”.
By using q-ROFNs operational principles, we can
evaluate q-ROFWA operator as given below.
Theorem 2.1. [31] Let ˘
αq=μq,ν
qbe the assort-
ment of q-ROFNs,we also evaluate q-ROFWA
operator by
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4112 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
Table 1
Comparative analysis of q-ROFSs with other models
Theory MD NMD Benefits Drawbacks
Fuzzy sets [22] ×Use a fuzzy interval for MD Fail to use NMD,
to deal with vagueness has less benefits
IFS [23] Can tackle vagueness by Unable to cope with the situations
using MDs and NMDs 0 ≤MD+NMD >1
PFS [24, 25] A wider valuation space & Unable to cope with the situations
superior than the IFSs 0 ≤MD2+NMD2>1
q-ROFS [26] A wider valuation space & Unable to cope with the situations
superior to both IFSs and PFSs when MD=1 and NMD=1
q-ROFWA(˘
α1,˘
α2,... ˘
αr)
=q
(1 −
r
g=1
(1 −μq
g)˘
g),
r
g=1
ν˘
g
g
Definition 2.7. [31] Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFN, and q-ROFWG: ϒn→ϒ,if
q-ROFWG(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
α˘
g
g
=˘
α˘
1
1⊗˘
α˘
2
2⊗..., ˘
α˘
r
r
where ϒnis the set of all q-ROFNs and
(˘
1,˘
2,...˘
r) be the WV of considered q-ROFNs
with the condition that ˘
g>0, ˘
g∈[0,1] and
r
g=1˘
g=1. Then, the q-ROFWG is called the "q-
rung orthopair fuzzy weighted geometric operator".
By using q-ROFNs operational principles, we can
evaluate q-ROFWG operator as given below.
Theorem 2.2. [31] Let ˘
αq=μq,ν
qbe the assort-
ment of q-ROFNs, we can evaluate q-ROFWG by
q-ROFWG(˘
α1,˘
α2,... ˘
αr)
=r
g=1
μ˘
g
g,q
(1 −
r
g=1
(1 −νq
g)˘
g)
2.2. Superiority and advantage of q-ROFNs
The superiority of q-ROFNs can analyzed by
their comparative analysis with existing fuzzy num-
bers (FNs), intuitionistic fuzzy numbers (IFNs) and
Pythagorean fuzzy numbers (PFNs). We can not
argue about the non-membership degrees (NMDs)
of the alternatives in the decision-making dilemma
with simple FNs. The decision makers (DMs) have
to face certain limitations while using IFNs and PFNs.
Fig. 1. Geometrical interpretation of q-ROFNs.
They can not choose MDs and NMDs with a freedom
of mind with IFNs and PFNs. For example if DMs
choose a pair of MDs and NMDs as (0.96,0.45),
then this numbers fails to be IFN since 0.96 +
0.45 =1.41 >1 and also fails to be PFN because
0.962+0.452=1.1241 >1. But if we choose q=
3, then 0.963+0.453=0.9759 <1. This shows that
(0.96,0.45) is q-ROFN with q=3. This discus-
sion demonstrate that a q-ROFN is superior to IFN
and PFN. Table 1 shows a concise summary of q-
ROFSs, their advantages and benefits when compared
with exiting theories. Geometrical interpretation of
q-ROFNs is given in Figure 1.
3. q-ROF interaction aggregation operators
In this section, we will initially interpret some basic
operations of the q-ROFNs that consider the inter-
action of MDs and NMDs. Then we define some
q-ROF interaction averaging and q-ROF geometric
interaction AOs.
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M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4113
3.1. q-ROF interaction operations
Let ˘
α,˘
α1and ˘
α2be the two q-ROFNs, then the
interaction operations for q-ROF environment are
defined as
(1) ˘
α1⊕˘
α2=q
μq
1+μq
2−μq
1μq
2,q
νq
1+νq
2−νq
1νq
2−νq
1μq
2−μq
1νq
2
(2) ˘
α1⊗˘
α2=q
μq
1+μq
2−μq
1μq
2−μq
1νq
2−νq
1μq
2,q
νq
1+νq
2−νq
1νq
2
(3) λ˘
α=q
1−(1 −μq)λ,q
(1 −μq)λ−(1 −(μq+νq))λ,λ>0
(4) ˘
αλ=q
(1 −νq)λ−(1 −(νq+μq))λ,q
1−(1 −νq)λ,λ>0
3.2. q-ROF interaction averaging aggregation
operators
Definition 3.1. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, and q-ROFIWA : ϒn→ϒ
is a mapping, if
q-ROFIWA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
g˘
αg(1)
then this mapping q-ROFIWA is called "q-rung
orthopair fuzzy interaction weighted averaging oper-
ator", where (˘
1,˘
2,...˘
r) be the WV of considered
q-ROFNs with the condition that ˘
j>0, ˘
g∈[0,1]
and r
g=1˘
g=1
Based on new q-ROF interaction operations we
have the following theorem.
Theorem 3.1. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, then
q-ROFIWA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
g˘
αg
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
1−
r
g=11−μq
g˘
g
,
q
r
g=11−μq
g˘
g
−
r
g=11−(μq
g+νq
g)˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠(2)
where (˘
1,˘
2,...˘
r)be the WV of considered q-
ROFNs with the condition that ˘
g>0,˘
g∈[0,1]
and r
g=1˘
g=1.
Proof. We complete this theorem by mathematical
induction.
For g=2
q-ROFIWA(˘
α1,˘
α2)=˘
1˘
α1⊕˘
2˘
α2
By interaction operational laws of q-ROF, we have
˘
1˘
α1=⎛
⎝q
1−1−μq
1˘
1
,
q
1−μq
1˘
1
−1−(μq
1+νq
1)˘
1⎞
⎠
˘
2˘
α2=⎛
⎝q
1−1−μq
2˘
2
,
q
1−μq
2˘
2
−1−(μq
2+νq
2)˘
2⎞
⎠
Then,
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4114 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
q-ROFIWA(˘
α1,˘
α2)=˘
1˘
α1⊕˘
2˘
α2
=⎛
⎝q
1−1−μq
1˘
1
,q
1−μq
1˘
1
−1−(μq
1+νq
1)˘
1
⊕q
1−1−μq
2˘
2
,q
1−μq
2˘
2
−1−(μq
2+νq
2)˘
2⎞
⎠
=⎛
⎜
⎜
⎜
⎜
⎜
⎝
q
1−1−μq
1˘
11−μq
2˘
2
,
q
1−μq
1˘
11−μq
2˘
2
−1−(μq
1+νq
1)˘
11−(μq
2+νq
2)˘
2
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
1−
2
g=11−μq
g˘
g
,
q
2
g=11−μq
g˘
g
−
2
g=11−(μq
g+νq
g)˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Now let it will be hold for g=d
q-ROFIWA(˘
α1,˘
α2,... ˘
αd)=
d
g=1
˘
g˘
αg
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
1−
d
g=11−μq
g˘
g
,
q
d
g=11−μq
g˘
g
−
d
g=11−(μq
g+νq
g)˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Here, we will prove it for g=d+1
q-ROFIWA(˘
α1,˘
α2,... ˘
αd,˘
αd+1)
=
d
g=1
˘
g˘
αg⊕˘
d+1˘
αd+1
=⎛
⎝q
1−
d
g=11−μq
g˘
g
,
q
d
g=11−μq
g˘
g
−
d
g=11−(μq
g+νq
g)˘
g⎞
⎠
⊕⎛
⎝q
1−1−μq
d+1˘
d+1
,
q
1−μq
d+1˘
d+1
−1−(μq
d+1+νq
d+1)˘
d+1⎞
⎠
=⎛
⎝q
1−
d+1
g=11−μq
g˘
g
,
q
d+1
g=11−μq
g˘
g
−
d+1
g=11−(μq
g+νq
g)˘
g⎞
⎠
In this way, proof is completed.
Example 3.1. Let ˘
α1=(0.44,0.23), ˘
α2=
(0.56,0.67), ˘
α3=(0.76,0.13), ˘
α4=(0.23,0.77)
and ˘
α5=(0.87,0.33) be the q-ROFNs, WV is
˘
=(0.25,0.10,0.15,0.25,0.25) and q=3, we
have
q
1−
5
g=11−μq
g˘
g
=0.690788
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M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4115
q
5
g=11−μq
g˘
g
−
5
g=11−(μq
g+νq
g)˘
g
=0.517874
then by Equation (2)
q-ROFIWA(˘
α1,˘
α2,... ˘
α5)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
1−
5
g=11−μq
g˘
g
,
q
5
g=11−μq
g˘
g
−
5
g=11−(μq
g+νq
g)˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=(0.690788,0.517874)
This would easily be proven that the q-ROFIWA
operator has satisfied the following features.
Theorem 3.2. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and all ˘
αqare equal, i.e
˘
αq=˘
α, ∀qthen,
q-ROFIWA(˘
α1,˘
α2,... ˘
αq)=˘
α(3)
Theorem 3.3. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and let
˘
α−=min ˘
αq,˘
α+=max ˘
αq
Then
˘
α−≤q-ROFIWA(˘
α1,˘
α2,... ˘
αq)≤˘
α+
Theorem 3.4. Assume that ˘
αq=μq,ν
qand ˘
α
q=
μq,ν
qare the assortment of q-ROFNs and ˘
αq≤˘
α
q
for all q, then
q-ROFIWA(˘
α1,˘
α2,... ˘
αq)=q-ROFIWA(˘
α
1,˘
α
2,... ˘
α
q)
(4)
Further, we define the q-ROFIOWA operator as
given below.
Definition 3.2. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, and q-ROFIOWA : ϒn→
ϒ, is a mapping. if
q-ROFIOWA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
g˘
α(g)(5)
then the mapping q-ROFIOWA is called ”q-rung
orthopair fuzzy interaction ordered weighted aver-
aging operator”, where (˘
1,˘
2,...˘
r) be the WV of
considered q-ROFNs with the condition that ˘
j>0
and r
g=1˘
g=1. (1),(2),...(r) is a permu-
tation of (1,2,...r), s.t ˘
α(j−1) ≥˘
α(j).
Same as Theorem 3.10, we have the following
result.
Theorem 3.5. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, then
q-ROFIOWA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
g˘
α(g)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
1−
r
g=11−(μ(g))q˘
g
,
q
r
g=11−(μ(g))q˘
g
−
r
g=11−((μ(g))q+(ν(g))q)˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(6)
where (˘
1,˘
2,...˘
r)be the WV of considered q-
ROFNs with the condition that ˘
g>0,˘
g∈[0,1]
and r
g=1˘
g=1.(1),(2),...(r)is a permu-
tation of (1,2,...r), s.t ˘
α(j−1) ≥˘
α(j).
It can be easily proved that the q-ROFIOWA oper-
ator fulfil the following properties.
Theorem 3.6. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and all ˘
αqare equal, i.e
˘
αq=˘
α, ∀qthen,
q-ROFIOWA(˘
α1,˘
α2,... ˘
αq)=˘
α(7)
Theorem 3.7. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and let
AUTHOR COPY
4116 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
˘
α−=min ˘
αq,˘
α+=max ˘
αq
Then
˘
α−≤q-ROFIOWA(˘
α1,˘
α2,... ˘
αq)≤˘
α+
Theorem 3.8. Assume that ˘
αq=μq,ν
qand ˘
α
q=
μq,ν
qare the assortment of q-ROFNs and ˘
αq≤˘
α
q
for all q, then
q-ROFIOWA(˘
α1,˘
α2,... ˘
αq)=q-ROFIOWA(˘
α
1,˘
α
2,... ˘
α
q)
(8)
Theorem 3.9. Assume that ˘
αq=μq,ν
qthe assort-
ment of q-ROFNs and then
q-ROFIOWA(˘
α1,˘
α2,... ˘
αq)=q-ROFIOWA(˘
α
1,˘
α
2,... ˘
α
q)
(9)
where ˘
α
qis any permutation of ˘
αq.
The q-ROFIWA operators weight only the q-
ROFNs themselves, while the q-ROFIOWA operators
weight the ordered positions of the q-ROFNs instead
of weighting the arguments themselves. Conse-
quently, the weights represent two different issues for
both q-ROFIWA and q-ROFIOWA operators. Both
AOs, however, accept only one of them. In order
to resolve this drawback, we shall recommend the
following to the q-ROFIHA operator.
Definition 3.3. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, and q-ROFIHA : ϒn→ϒ,
is a mapping. if
q-ROFIHA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
g˘
˘
α(g) (10)
then the mapping q-ROFIHA is called ”q-rung
orthopair fuzzy interaction hybrid averaging opera-
tor”, where (˘
1,˘
2,...˘
r) be the WV of considered
q-ROFNs with the condition that ˘
j>0, ˘
g∈[0,1]
and r
g=1˘
g=1. ˘
˘
α(g)is the largest q-ROFN ˘
˘
α(g)=
n˘
˘
αj.(1,
2,...
r) is the associated WV with
j>0, g∈[0,1] and r
g=1g=1
If g=(1
n,1
n,... 1
n)T, then q-ROFIHA
operator changed into q-ROFIWA operator. If
˘
g=(1
n,1
n,... 1
n)T, then q-ROFIHA operator
changed into q-ROFIOWA operator.
Let ˘
˘
α(g)=(˘μ(g),˘
ν(g)). Same as Theorem
3.10 we have,
q-ROFHWA(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
g˘
˘
α(g)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
1−
r
g=11−(˘μ(g))q˘
g
,
q
r
g=11−(˘μ(g))q˘
g
−
r
g=11−(( ˘μ(g))q+(˘
ν(g))q)˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(11)
3.3. q-ROF interaction geometric aggregation
operators
Definition 3.4. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, and q-ROFIWG : ϒn→ϒ
is a mapping, if
q-ROFIWG(˘
α1,˘
α2,... ˘
αr)=
r
g=1
(˘
αg)˘
g(12)
then this mapping q-ROFIWG is called ”q-rung
orthopair fuzzy interaction weighted geometric oper-
ator”, where (˘
1,˘
2,...˘
r) be the WV of considered
q-ROFNs with the condition that ˘
j>0, ˘
g∈[0,1]
and r
g=1˘
g=1
AUTHOR COPY
M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4117
Based on q-ROF interaction operations we have
the following theorem.
Theorem 3.10. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, then
q-ROFIWG(˘
α1,˘
α2,... ˘
αr)=
r
g=1
(˘
αg)˘
g
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
r
g=11−νq
g˘
g
−
r
g=11−(νq
g+μq
g)˘
g
,
q
1−
r
g=11−νq
g˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(13)
where (˘
1,˘
2,...˘
r)be the WV of considered q-
ROFNs with the condition that ˘
g>0,˘
g∈[0,1]
and r
g=1˘
g=1.
Proof. Proof is same as Theorem 3.1.
Example 3.2. Let ˘
α1=(0.44,0.23), ˘
α2=
(0.56,0.67), ˘
α3=(0.76,0.13), ˘
α4=(0.23,0.77)
and ˘
α5=(0.87,0.33) be the q-ROFNs, WV is
˘
=(0.25,0.10,0.15,0.25,0.25) and q=3, we
have
q
5
g=11−νq
g˘
g
−
5
g=11−(νq
g+μq
g)˘
g
=0.659338
q
1−
5
g=11−νq
g˘
g
=0.566595
then by Theorem 3.10
q-ROFIWG(˘
α1,˘
α2,... ˘
α5)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
5
g=11−νq
g˘
g
−
5
g=11−(νq
g+μq
g)˘
g
,
q
1−
5
g=11−νq
g˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=(0.659338,0.566595)
This would be easily proven that the q-ROFIWG
operator has the following features.
Theorem 3.11. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and all ˘
αqare equal, i.e
˘
αq=˘
α, ∀qthen,
q-ROFIWG(˘
α1,˘
α2,... ˘
αq)=˘
α(14)
Theorem 3.12. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and let
˘
α−=min ˘
αq,˘
α+=max ˘
αq
Then
˘
α−≤q-ROFIWG(˘
α1,˘
α2,... ˘
αq)≤˘
α+
Theorem 3.13. Assume that ˘
αq=μq,ν
qand ˘
α
q=
νq,μ
qare the assortment of q-ROFNs and ˘
αq≤˘
α
q
for all q, then
q-ROFIWG(˘
α1,˘
α2,... ˘
αq)=q-ROFIWG(˘
α
1,˘
α
2,... ˘
α
q)
(15)
Further, we define the q-ROFIOWG operator as
follows.
Definition 3.5. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, and q-ROFIOWG : ϒn→
ϒis a mapping, if
q-ROFIOWG(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
α˘
g
(g)(16)
then the mapping q-ROFIOWG is called ”q-rung
orthopair fuzzy interaction ordered weighted geomet-
ric operator”, where (˘
1,˘
2,...˘
r) be the WV of
considered q-ROFNs with the condition that ˘
j>0
and r
g=1˘
g=1. (1),(2),...(r) is a permu-
tation of (1,2,...r), s.t ˘
α(j−1) ≥˘
α(j).
Same as Theorem 3.10, we have the following
result.
Theorem 3.14. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, then
AUTHOR COPY
4118 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
q-ROFIOWG(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
α˘
g
(g)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
r
g=11−(ν(g))q˘
g
−
r
g=11−((ν(g))q+(μ(g))q)˘
g
,
q
1−
r
g=11−(ν(g))q˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(17)
where (˘
1,˘
2,...˘
r)be the WV of considered q-
ROFNs with the condition that ˘
g>0,˘
g∈[0,1]
and r
g=1˘
g=1.(1),(2),...(r)is a permu-
tation of (1,2,...r), s.t ˘
α(j−1) ≥˘
α(j).
It can be easily proved that the q-ROFIOWG oper-
ator fulfil following properties.
Theorem 3.15. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and all ˘
αqare equal, i.e
˘
αq=˘
α, ∀qthen,
q-ROFIOWG(˘
α1,˘
α2,... ˘
αq)=˘
α(18)
Theorem 3.16. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs and let
˘
α−=min ˘
αq,˘
α+=max ˘
αq
Then
˘
α−≤q-ROFIOWG(˘
α1,˘
α2,... ˘
αq)≤˘
α+
Theorem 3.17. Assume that ˘
αq=μq,ν
qand ˘
α
q=
νq,μ
qare the assortment of q-ROFNs and ˘
αq≤˘
α
q
for all q, then
q-ROFIOWG(˘
α1,˘
α2,... ˘
αq)=q-ROFIOWG(˘
α
1,˘
α
2,... ˘
α
q)
(19)
Theorem 3.18. Assume that ˘
αq=μq,ν
qthe
assortment of q-ROFNs and then
q-ROFIOWG(˘
α1,˘
α2,... ˘
αq)=q-ROFIOWG(˘
α
1,˘
α
2,... ˘
α
q)
(20)
where ˘
α
qis any permutation of ˘
αq.
The q-ROFIWG operators weight only the
q-ROFNs themselves, while the q-ROFIOWG oper-
ators weight the ordered positions of the q-ROFNs
instead of weighting the arguments themselves. Con-
sequently, the weights represent two different issues
for both q-ROFIWG and q-ROFIOWG operators.
Both operators, however, accept only one of them. In
order to resolve this drawback, we shall recommend
the following to the q-ROFIHG operator.
Definition 3.6. Assume that ˘
αq=μq,ν
qis the
assortment of q-ROFNs, and q-ROFIHG : ϒn→ϒ
is a mapping, if
q-ROFIHG(˘
α1,˘
α2,... ˘
αr)=
r
g=1
˘
˘
αg
(g)(21)
then this mapping q-ROFIHG is called ”q-rung
orthopair fuzzy interaction hybrid geometric opera-
tor”, where (˘
1,˘
2,...˘
r) be the WV of considered
q-ROFNs with the condition that ˘
j>0, ˘
g∈[0,1]
and r
g=1˘
g=1. ˘
˘
α(g)is the largest q-ROFN ˘
˘
α(g)=
n˘
˘
αj.(1,
2,...
r) is the associated WV with
j>0, g∈[0,1] and r
g=1g=1
If g=(1
n,1
n,... 1
n)T, then q-ROFIHG opera-
tor changed into q-ROFIWG operator. If ˘
g=
(1
n,1
n,... 1
n)T, then q-ROFIHG operator changed
into q-ROFIOWG operator.
Let ˘
˘
α(g)=(˘
ν(g),˘μ(g)). Same as Theorem 3.10 we
have,
q-ROFHWG(˘
α1,˘
α2,... ˘
αr)=r
g=1˘
g˘
˘
α(g)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
q
r
g=11−(˘
ν(g))q˘
g
−
r
g=11−((˘
ν(g))q+(˘μ(g))q)˘
g
,
q
1−
r
g=11−(˘
ν(g))q˘
g
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(22)
AUTHOR COPY
M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4119
Fig. 2. Pictorial view of Algorithm.
4. MCDM with interaction aggregation
operators
MCDM method using the interaction aggregation
operators for q-ROFNs is presented in this section.
Suppose that ˘
={˘
1,˘
2,..., ˘
p}is the set of
alternatives and k={
k1,k2,...,kq}is the set
of criterion. Let ˘
be the WV, s.t ˘
j∈[0,1]
and n
j=1˘
j=1, (j=1,2,...,n) and ˘
jshow
the weight of kj. Alternatives on an attribute are
reviewed by the decision-maker (DM) and the assess-
ment measurements has to be in the q-ROFN. Assume
that ˘
f=(˘
αij )p×qis the decision matrix provided by
DM.(˘
αij ) represent a q-ROFN for alternative ˘
iasso-
ciated with the criterion kj.
Indeed an algorithm is being developed to resolve
an issue. Figure 2 shows the pictorial view of this
developed Algorithm.
Algorithm
Step 1.
The DM has given its personal opinion in the form
of q-ROFNs. ˘
αij =μij ,ν
ij towards the alternative
˘
iand hence construct a q-ROF decision matrix ˘
f=
(˘
αij )p×qas
˘
f=
k1k2kq
⎡
⎢
⎢
⎣⎤
⎥
⎥
⎦
˘
1(μ11,ν
11)(μ12,ν
12)······ (μ1q,ν
1q)
˘
2(μ21,ν
21)(μ22,ν
22)······ (μ2q,ν
2q)
.
.
..
.
........
.
.
˘
p(μp1,ν
p1)(μp2,ν
p2)······ (μpq,ν
pq)
Step 2.
Normalize the decision matrix. If there are different
types of criteria or attributes like cost (τc) and ben-
efit (τb). By normalize the decision matrix we deal
all criteria or attributes in the same way. Otherwise,
different criterion or attributes should be aggregate in
different ways.
jij = ˘
αc
ij ;j∈τc
˘
αij ;j∈τb.
(23)
where ˘
αc
ij show the compliment of ˘
αij .
Step 3.
Based on decision matrix acquired from step 2, the
aggregated value of the alternative ˘
iunder various
parameter kjis obtained using either q-ROFIWA, q-
ROFIWG operators etc and hence get the collective
value jifor each alternative ˘
i(i=1,2, ...m).
Step 4.
Calculate the score functions for all jifor q-ROFNs
by using the Definition 2.3.
Step 5.
Rank all jias per the score values to choose the most
desirable option.
5. An application for the choice of low-carbon
GSCM
As a result of the growing effect on the reduc-
tion of greenhouse gas (GHG) emissions, several
changes have indeed been recorded in the discourse
to the conventional functioning of supply chains.
Such reforms have also been triggered by increas-
ing pressure from interested parties, including such
state agencies, non-governmental organizations and
industry. While the environmental impact of the com-
panies has been tracked since the late 19th century,
the emphasis has changed from individual organi-
zations to supply chains since the advent of supply
chain management (SCM) in the 1990s. At the same
time, Green Supply Chain Management (GSCM) also
drawn attention by academics and organizations [1].
However, the GSCM process is very detailed and
also covers environmentalregulations, environmental
safety and decontamination. Late entrepreneurs have
AUTHOR COPY
4120 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
focused primarily on decarbonization. The majority
of these induced by human activity GHG concentra-
tions are the result of an increase in industrialization,
population and thus distribution networks. As a result,
regulating GHG emissions from supply chains would
not only resolve climate change issues, but could also
comply with regulatory requirements. As a conse-
quence, removing emissions from its supply chains
is a crucial capability for any organisation [2]. The
idea of LCSCM is therefore attracting the interest of
academic and commercial. Changing climate was a
global threat. And per the United Nations, environ-
mental conditions are becoming even more extreme
and GHG emissions are now at their highest in record
[3]. As regions around the world have positioned
climate change policies at the centre of their eco-
nomic and social policies, demand for SCM to reduce
carbon emissions is growing. The logistics industry,
characterised by high energy usage and carbon con-
sumption, is under immense strain in the fight against
environmental warming. LCSCM is described as "a
tactic that incorporates CO2or CO2alternative or
GHG emissions, whether as a constraint or as an
objective in the design and planning of the sup-
ply chain" [4]. It is understood to be a combination
of low-carbon operations management and SCM in
regard to environmental change, focusing on activi-
ties that reduce carbon foot printing in supply chains
[5], such as low-carbon products, production, pro-
cesses and logistics. A considerable amount of work
on low-carbon SCM has been done, concentrating
mostly on the realistic methods and processes used
to achieve SCM in the areas of carbon efficiency and
pollution problems, as well as on the issues of car-
bon footprint measurement. Significant public health
concerns and environmental degradation of its cur-
rent patterns of economic development have been
raised by increasingly frequent red alerts of dan-
gerous toxic smoke in Asian countries. In recent
decades, major increases in carbon emissions have
contributed to climate change and global warming.
Several governments and policy makers have been
inspired by this challenge to take measures to min-
imis pollution. China’s manufacturing industry, as the
world ’s leading producer, only achieved a cumula-
tive value of $2.9 trillion in 2014. In China today,
pressure on manufacturers to adopt green technol-
ogy and reduce carbon emissions is increasing [20].
The selection of carbon-free suppliers is an important
aspect of this process. Generally speaking, the choice
of suppliers requires input from multiple divisions
within the company, and decisions or preferences
about the comparison of different suppliers are often
ambiguous and unclear. In order to resolve this impor-
tant problem, this article also suggests the following
framework for dealing with MCDM, where q-ROFNs
have preferences for DMs. The suggested methodol-
ogy is then applied to a case study on issues relevant
to the selection of carbon-free suppliers. Due to the
recent emergence of competition between supply
chains, the important impact of the efficiency of the
supply chain on the output of each participant has led
a number of supplier selection frameworks to con-
sider the cumulative efficiency of the supply chain
Much of the research recently presented has assumed
a centralized supply chain, taking into account the
supply chain perspective. Supply chain members are
disproportionately based on their own goals, and this
usually leads to low overall supply chain produc-
tion.The optimal performance of the entire supply
chain is achieved by centralized decision-making or
by the introduction of a set of communication mech-
anisms. Suppliers also have a major role to play in
the corporate climate and in the global supply chain.
There is a significant quantity of research findings
on this subject in accordance with the importance
of the selection of suppliers. In addition to models
that indicate the choice of goods and services based
on the supply chain perspective rather than on the
customer’s knowledge, the advancement in decision-
making in the selection of suppliers was initiated
mainly from models that only viewed the cost fac-
tor as multi-objective models with qualitative and
quantitative criteria. However, the dynamics of the
decentralized supply chains and the conflicting inter-
ests of its members have prompted many scholars to
develop models that address the overall integration of
the supply chain as a final aim in a centralized sense,
although this situation is not true in many cases. When
suppliers have capacity constraints, fragmented sup-
ply chains become even more complicated and they
need to create coalitions to meet total demand In this
situation, the question is what providers are going to
make a coalition and how the resulting shares will be
distributed.
Every supply chain is made up of a complex
mixture of nodes, links and activities and processes
that make it possible for all to function. The use
of oil, environmental assets, raw ingredients and
scrap production will be powered by these kinds
of supply chain components. This, in fact, has
an effect on the carbon footprint of the atmo-
sphere. Chain Management nodes are some of the
premises associated with the production, transport
AUTHOR COPY
M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4121
and consumption of end-to-end goods. Suppliers
and Company Manufactured goods, Stores, Retail
Outlets, Exchange and Replace damaged Operations
and Integration Points are all supply chain nodes that
use resources to carry out their production process.
The ties between some of these nodes are container,
transport and logistics players that transfer goods
via the Supply Chain. Vehicles, aircraft and railway
all use energy in the provision of their services.
The processes and decisions taken to manage such
activities include selection of components, selec-
tion of suppliers, network development, location
planning of nodes, distribution requirements and
inventory levels, among many other elements [6].
These decisions would definitely decide the rate of
carbon footprint formation and environmental effect
in the supply of products and services. Overall, key
prevailing considerations need to be addressed when
identifying and maximizing any end-to-end supply
chain approach from an environmental perspective:
1. Reduce pollution and boost quality
2. Start reducing emissions of carbon
3. Significantly reduce use
4. Conservation of environmental resources
5. Reduce-Reuse-Recycle
6. Promote the use of renewable, natural resources
Improving the environmental effect of a business
must be a core part of the management strategy and
the corporate goal set for it to thrive. What are the
fundamental principles of SCM? In 1997, the Supply
Chain Management Review published a manuscript
entitled ”The Seven Principles of Supply Chain Man-
agement” by Anderson et al. [6]. SCM was a fairly
new idea at the time, but this manuscript did an excel-
lent job of explaining the core concepts of SCM in one
round. About 20 years have passed and this article is
known as the "classic" manuscript and was published
again in 2010. From now on, both academic articles
and business journals have received more than 300
citations for this manuscript. Pictorial view of these
7 principles is given in Figure 3.
The two main types of descriptive and analytical
models can be classified in the Supplier Selection
studies. The descriptive studies examine the main cri-
teria of the supplier selection and evaluation process.
23 parameters were specified by Dickson [8], which
were considered by contractors to be involved in var-
ious vendor selection issues. He found that the most
important parameters were performance, time deliv-
ery and cost, and Wind et al. [9] found that many
factors had been involved in several other vendor
Fig. 3. Fundamental Principles of SCM.
selection processes. In a multi-criteria selection of
international journals from 2000 to 2008, Ho et al.
[10] analyzed all methods and concluded that the
most common parameters used to measure the out-
put of vendors were quality, followed by distribution,
cost or price, and so on. Weber et al. reviewed 74
papers on the selection of suppliers in empiric lit-
erature models and identified a range of techniques
that have appeared in studies over the last 25 years.
Most of the methods were linear weighting, regres-
sion models, and some optimization algorithms, they
concluded. For a more recent review of supplier
evaluation and selection methods. Amid et al. [12]
considered fuzzy criteria in the vendor selection
framework. In order to obtain aggregate ratings from
various suppliers, Jolai et al. [13] proposed a fuzzy
MCDM mechanism and then recommended the most
relevant ones using the second-level objective pro-
gramming (GP) methodology. Sevkli et al. [14] used
an empirical hierarchical approach to test the weights
of their fuzzy linear programming model for the
selection of suppliers. Due to global warming and
climate change, environmental issues have become
more important across the different industries and
regions [15]. Growing attention has been given in
recent decades to the study of the LCGSCM in an
attempt to minimis environmental pollution, aware-
ness and environmental protection [16]. Sustainable
supply chain development and the resulting reduction
of environmental constraints are highly dependent on
the identification and selection of appropriate low-
carbon suppliers. Simply put, the MCDM problem
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4122 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
Table 2
q-ROF decision matrix taking by decision maker
k1k2k3k4k5
˘
1(0.74, 0.62) (0.34, 0.64) (0.31, 0.81) (0.74, 0.21) (0.68, 0.24)
˘
2(0.82, 0.21) (0.72, 0.31) (0.64, 0.24) (0.34, 0.82) (0.94, 0.16)
˘
3(0.68, 0.41) (0.25, 0.54) (0.31, 0.61) (0.24, 0.68) (0.38, 0.45)
˘
4(0.36, 0.72) (0.36, 0.74) (0.32, 0.82) (0.34, 0.21) (0.54, 0.23)
˘
5(0.85, 0.21) (0.32, 0.75) (0.11, 0.65) (0.78, 0.40) (0.82, 0.31)
Table 3
Normalized q-ROF decision matrix
k1k2k3k4k5
˘
1(0.74, 0.62) (0.64, 0.34) (0.81, 0.31) (0.74, 0.21) (0.68, 0.24)
˘
2(0.82, 0.21) (0.31, 0.72) (0.24, 0.64) (0.34, 0.82) (0.94, 0.16)
˘
3(0.68, 0.41) (0.54, 0.25) (0.61, 0.31) (0.24, 0.68) (0.38, 0.45)
˘
4(0.36, 0.72) (0.74, 0.36) (0.82, 0.32) (0.34, 0.21) (0.54, 0.23)
˘
5(0.85, 0.21) (0.75, 0.32) (0.65, 0.11) (0.78, 0.40) (0.82, 0.31)
would represent the vendor selection mechanism
since, in the decision-making phase [17], cacophonic
and multiple parameters should be evaluated and
checked. To date, the assessment and procurement of
low carbon suppliers has been carried out using the
MCDM methods, but it has also been presumed that
the attribute details is certain and accurate [18]. Fortu-
nately, due to the inconsistency of human reasoning
involved, the rapid economic growth and dynamic
commercial climate find it harder for decision-makers
to provide reliable analysis or preference details.
Tong and Wang [19] have recently used induced IF
operator to solve the low-carbon vendor selection
problem. To address low-carbon supplier selection,
Zeng et al. [21] implemented PF self-confidence AOs.
The relaxing limitations on the MD and NMD of q-
ROFSs allow an expanded range to render q-ROFS
superior to IFS and PFS in the definition of unreliable
and unclear details, as described in the introduction.
In the q-ROF context, it is therefore important and
appropriate to thoroughly investigate the low carbon
vendor selection issue.
To illustrate the possibilities for solving the low
carbon supplier problem by means of reasoning based
on the q-ROFSs, There are five alternatives ˘
i(i=
1,2,3,4,5) and we consider k1=low carbon tech-
nology, k2=cost, k3=risk factor, k4=capacity
and k5=economic efficiency as attributes. In this
example we use q-ROFNs as input data for ranking
the given alternatives under the given attributes. Also
the WV ˘
is (0.25,0.10,0.15,0.25,0.25). We take
q=3.
Using q-ROFIWA operator
Step 1.
Construct the decision matrix given by the decision
maker in Table 2 consist on q-ROF information.
Step 2.
Normalize the rating of Table 2 because the attribute
k2=cost and k3=risk factor and hence the result
is summarized in Table 3.
Step 3.
Evaluate ji=q-ROFIWA(ji1,ji2,...,jip) and we
get
j1=(0.732124,0.441151)
j2=(0.784507,0.521041)
j3=(0.536483,0.495753)
j4=(0.595338,0.478843)
j5=(0.796099,0.309708)
Step 4.
Compute the score functions for all ji, and get
i(j1)=0.306569
i(j2)=0.341371
i(j3)=0.0325654
i(j4)=0.101239
i(j5)=0.474839
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M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4123
Table 4
q-ROF decision matrix taking by decision maker
k1k2k3k4k5
˘
1(0.74, 0.62) (0.34, 0.64) (0.31, 0.81) (0.74, 0.21) (0.68, 0.24)
˘
2(0.82, 0.21) (0.72, 0.31) (0.64, 0.24) (0.34, 0.82) (0.94, 0.16)
˘
3(0.68, 0.41) (0.25, 0.54) (0.31, 0.61) (0.24, 0.68) (0.38, 0.45)
˘
4(0.36, 0.72) (0.36, 0.74) (0.32, 0.82) (0.34, 0.21) (0.54, 0.23)
˘
5(0.85, 0.21) (0.32, 0.75) (0.11, 0.65) (0.78, 0.40) (0.82, 0.31)
Table 5
Normalized q-ROF decision matrix
k1k2k3k4k5
˘
1(0.74, 0.62) (0.64, 0.34) (0.81, 0.31) (0.74, 0.21) (0.68, 0.24)
˘
2(0.82, 0.21) (0.31, 0.72) (0.24, 0.64) (0.34, 0.82) (0.94, 0.16)
˘
3(0.68, 0.41) (0.54, 0.25) (0.61, 0.31) (0.24, 0.68) (0.38, 0.45)
˘
4(0.36, 0.72) (0.74, 0.36) (0.82, 0.32) (0.34, 0.21) (0.54, 0.23)
˘
5(0.85, 0.21) (0.75, 0.32) (0.65, 0.11) (0.78, 0.40) (0.82, 0.31)
Step 5.
Rank all the ji(i=1,2,...,p) according to the
score values,
j5j2j1j4j3
From this rating, we get j5corresponds to ˘
5,so ˘
5
is the best alternative.
Using q-ROFIWG operator
Step 1.
Construct the decision matrix given by the decision
maker in Table 4 consist on q-ROF information.
Step 2.
The normalized values of the rating by converting
the cost attribute into benefit types are given in
Table 5.
Step 3.
Evaluate ji=q-ROFIWG(ji1,ji2,...,jip).
j1=(0.736296,0.429279)
j2=(0.716623,0.635174)
j3=(0.523751,0.509905)
j4=(0.582005,0.498162)
j5=(0.796203,0.30902)
Step 4.
Calculate the score functions for all ji.
i(j1)=0.320062
i(j2)=0.111762
i(j3)=0.011096
i(j4)=0.0735154
i(j4)=0.475235
Step 5.
Rank all the ji(i=1,2,...,p) according to the
score values,
j5j2j1j4j3
j5corresponds to ˘
5,so ˘
5is the best alternative.
5.1. Comparison analysis and advantages of the
proposed AOs
We present a comparative review of recommended
operators with some current AOs in this section. That
both achieve the same final result is the excellence
of our suggested AOs By resolving the information
data with some existing AOs, we compare our results
and get the same optimal decision. This illustrates
our proposed model’s strength and consistency. The
comparison can be made of the AOs presented with
some current AOs is given in the Table 6. Basic AOs
are dependent on algebraic operational principles and
do not accept interaction between MD and NMDs.
All q-ROFN AOs dependent on algebraic operating
principles are also inappropriate for all contingencies.
For example, if we take q-ROFWA(˘
α1,˘
α2,... ˘
αt)
in [31], we get νq−ROFWA(˘
α1,˘
α2,... ˘
αt)=0 if one of
the NMD of (˘
α1,˘
α2,... ˘
αt) is zero but others are
non-zero. Therefore, there is need to improve the
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4124 M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application
Table 6
Comparison of proposed operators with some exiting operators
Method Ranking of alternatives The optimal alternative
q-ROFEWA (Riaz et al. [36]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFEOWA (Riaz et al. [36]) ˘
5˘
2˘
1˘
4˘
3˘
5
ST-q-ROFWA (Garg [46]) ˘
5˘
2˘
3˘
4˘
1˘
5
ST-q-ROFWG (Garg [46]) ˘
5˘
2˘
3˘
4˘
1˘
5
q-ROFPWA (Riaz et al. [37]) ˘
5˘
2˘
3˘
4˘
1˘
5
q-ROFPWG (Riaz et al. [37]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFWA (Liu & Wang [31]) ˘
5˘
2˘
1˘
3˘
4˘
5
q-ROFWG (Liu & Wang [31]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFWBM (Liu & Liu [40]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFWGBM (Liu & Liu [40]) ˘
5˘
2˘
1˘
3˘
4˘
5
q-ROFHM (Zhao et al. [32]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFWHM (Zhao et al. [32]) ˘
5˘
2˘
3˘
4˘
1˘
5
q-ROFHWAGA (Riaz et al. [38]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFHOWAGA (Riaz et al. [38]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFHM (Liu et al. [43]) ˘
5˘
3˘
1˘
4˘
2˘
5
q-ROFWHM (Liu et al. [43]) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFEPWA (Riaz et al. [39]) ˘
5˘
2˘
1˘
3˘
4˘
5
q-ROFEPWG (Liu et al. [39]) ˘
5˘
2˘
1˘
4˘
3˘
5
CQROFWA (Joshi & Gegov [44]) ˘
5˘
2˘
4˘
1˘
3˘
5
CQROFWG (Joshi & Gegov [44]) ˘
5˘
2˘
4˘
1˘
3˘
5
q-ROFIWA (Proposed) ˘
5˘
2˘
1˘
4˘
3˘
5
q-ROFIWG (Proposed) ˘
5˘
2˘
1˘
4˘
3˘
5
operational principles of q-ROFNs. In many other
decision-making issues, certain attributes are always
associated in such a way that inter-relationships
between them should be taken into consideration.
We should also pay close attention to aggregation
techniques that can account for inter-relationships
between multiple attributes. In this way our proposed
AOs are more reliable, technically sound and sus-
tainable from existing q-ROF AOs which are already
exist. In our proposed AOs there is a parameter q
involved, this increase in value of qprovides an
opportunity to select MDs and NMDs from a larger
area. Initially, when qis increased from 1 to 2, there
is a rapid increase in the area bounded between
coordinate axes and the curve μq+νq=1. As the
value of qincreases (say after q= 2) increase in
area gets slower and slower. Therefore for practi-
cal purposes, values from 2 to 10 are more useful.
It can be seen that about 99% area of the unit square
[0,1] ×[0,1] is available for the selection of MD and
NMDs when q=10. However no restriction other
than q≥1 can be imposed on q. Although qis a real
number, yet for an integral value it is bit easy to know
about the area from where MD and NMDs are being
selected (see [47]). Table 6 shows the comparative
analysis of proposed operators with some existing
operators.
6. Conclusion
Since AOs serve a significant part in decision-
making, therefore, in this paper we developed several
AOs for interaction of membership degrees and
non-membership degrees of q-ROFNs, named as ”q-
rung orthopair fuzzy interaction weighted averaging
(q-ROFIWA) operator, q-rung orthopair fuzzy inter-
action ordered weighted averaging (q-ROFIOWA)
operator, q-rung orthopair fuzzy interaction hybrid
averaging (q-ROFIHA) operator, q-rung orthopair
fuzzy interaction weighted geometric (q-ROFIWG)
operator, q-rung orthopair fuzzy interaction ordered
weighted geometric (q-ROFIOWG) operator and q-
rung orthopair fuzzy interaction hybrid geometric
(q-ROFIHG) operator”. We discussed some certain
fundamental properties of these developed operators.
Finally, a descriptive example was given concern-
ing the selection of low-carbon suppliers to highlight
the possibilities for applying the proposed approach
to green supply chain management. Based on new
interaction aggregation operators for q-ROFNs an
algorithm is developed for modeling uncertainties in
MCDM. Some significant properties of these AOS
are analyzed and the efficiency of the developed
MCDM approach is assessed with a practical appli-
cation towards sustainable low-carbon green supply
AUTHOR COPY
M. Riaz et al. / Novel q-rung orthopair fuzzy interaction aggregation operators and their application 4125
chain management. In future studies, we shall study
applications of the proposed methodology under the
different fuzzy environment and solve some more
practical problems [49, 50, 53, 54, 56–58].
Acknowledgment
The authors extend their appreciation to the Dean-
ship of Scientific Research at King Khalid University,
Abha 61413, Saudi Arabia for funding this work
through research groups program under grant number
R.G. P-1/23/42.
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