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http://www.aimspress.com/journal/Math
AIMS Mathematics, 7(9): 17529–17569.
DOI: 10.3934/math.2022966
Received: 25 June 2022
Revised: 14 July 2022
Accepted: 25 July 2022
Published: 29 July 2022
Research article
2-tuple linguistic q-rung orthopair fuzzy CODAS approach and its
application in arc welding robot selection
Sumera Naz1, Muhammad Akram2,∗, Afia Sattar1and Mohammed M. Ali Al-Shamiri3,4
1Department of Mathematics, Division of Science and Technology, University of Education, Lahore,
Pakistan
2Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan
3Department of Mathematics, Faculty of science and arts, Mahayl Assir, King Khalid University,
Saudi Arabia
4Department of Mathematics and Computer, Faculty of Science, Ibb University, Ibb, Yemen
*Correspondence: Email: m.akram@pucit.edu.pk.
Abstract: Industrial robots enable manufacturers to produce high-quality products at low cost, so
they are a key component of advanced production technology. Welding, assembly, disassembly,
painting of printed circuit boards, pick-and-place mass production of consumer products, laboratory
research, surgery, product inspection and testing are just some of the applications of industrial robots.
All functions are done with a high level of endurance, speed and accuracy. Many competing attributes
must be evaluated simultaneously in a comprehensive selection method to determine the performance
of industrial robots. In this research article, we introduce the 2TLq-ROFS as a new advancement in
fuzzy set theory to communicate complexities in data and presents a decision algorithm for selecting
an arc welding robot utilizing the 2-tuple linguistic q-rung orthopair fuzzy (2TLq-ROF) set, which
can dynamically delineate the space of ambiguous information. We propose the q-ROF Hamy mean
(q-ROFHM) and the q-ROF weighted Hamy mean (q-ROFWHM) operators by combining the
q-ROFS with Hamy mean operator. We investigate the properties of some of the proposed operators.
Then based on the proposed maximization bias, a new optimization model is built, which is able to
exploit the DM preference to find the best objective weights among attributes. Next, we extend the
COmbinative Distance-Based ASsessment (CODAS) method to 2TLq-ROF-CODAS version which
not only covers the uncertainty of human cognition but also gives DMs a larger space to represent
their decisions. To validate our strategy, we present a case study of arc welding robot selection.
Finally, the effectiveness and accuracy of the method are proved by parameter analysis and
comparative analysis results. The results show that our method effectively addresses the evaluation
and selection of arc welding robots and captures the relationship between an arbitrary number of
attributes.
17530
Keywords: 2-tuple linguistic q-rung orthopair fuzzy set; MAGDM; CODAS method; arc welding
robot
Mathematics Subject Classification: 03E72, 90B50
1. Introduction
Industrial robots are machines used for manufacturing. In material handling, spot welding,
material removal, arc welding, inspection and testing, handling, assembly, finishing and painting,
robots are used to perform repetitive, difficult and dangerous with greater precision, accuracy and
precision task speed. The main reasons for industrial use of industrial robots are to reduce operating
costs and increase manufacturing efficiency. An industrial robot has several parameters, including
mechanical weight, payload capacity, repeatability, etc. [1]. These parameters make it a MAGDM
problem. Welding is the most sought-after skill in any industrial business. Since the invention of
industrial robots, there has been a high demand for industrial robots for welding applications. Arc
welding, metal arc welding, carbon arc welding, metal inert gas welding, plasma arc welding,
tungsten inert gas welding, electro-slag welding and submerged arc welding [2] are some of several
types of arc welding. Arc welding robots are programmed to perform all forms of arc welding tasks.
In arc welding [3], electricity is used to form an arc between an electrode and a conductive base
metal. Arc welding is widely used in most manufacturing companies. However, as technology
advances and product demand increases, manufacturing companies are turning to robot-assisted
manufacturing [4]. To provide manufacturers a common configuration to let them choose between a
variety of arc welding robots. The objective of this study is to investigate using MAGDM approaches
to prioritize industrial arc welding robots. To choose the best robot for multiple objectives, different
MAGDM techniques such as VIKOR, ELECTRE, and compromise ranking techniques were used.
For robot selection, researchers used a decision model based on fuzzy linear regression. Four criteria
were used to evaluate twenty-seven industrial robots [5]. The Analytic Hierarchy Process (AHP) and
TOPSIS MADM techniques were used to compare and assess seven industrial robot choices based on
two criterion and six sub-criteria [6]. Using a set of objective data, a PROMETHEE II approach was
used to select the robot. Fourteen and seven distinct industrial robots were compared based on four
and five criteria, respectively, in two numerical illustrations [7]. For solving robot selection problems
with incomplete weight information, an integrated model based on hesitant 2-tuple linguistic term sets
and an expanded QUALIFLEX technique was developed [8]. For robot selection, the VIKOR method
was introduced, which used a type-2 fuzzy sets methodology to evaluate eight industrial robot
alternatives using seven criteria [9]. The application of the COPRAS method’s multi-criteria approach
to solve an industrial robot selection problem was demonstrated. Seven different industrial robot
models were chosen and compared based on five alternatives [10]. The WASPAS approach was
proposed as an MADM tool for picking the best robot among seven different real-time industrial robot
models that were assessed using five criteria [11]. To evaluate mobile robot selection for a hospital
pharmacy, a fuzzy extended VIKOR method was created by combining fuzzy AHP and VIKOR-based
techniques. On the basis of seven parameters, three different mobile robots were compared [12].
MAGDM is a fascinating research topic that has attracted widespread attention from scholars and
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
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scientists all over the world [13–24]. Decision-makers (DMs) use some tools in the MAGDM
framework to effectively and appropriately articulate their evaluation values. Following that, several
approaches or strategies are used to identify the ranking order of viable choices and make the ultimate
selection. The q-ROFS, developed by Yager [25], is an effective tool for representing DM assessment
data. As an extension of the intuitionistic fuzzy set [26] and Pythagorean fuzzy set [27], q-ROFS
successfully models DMs’ reluctance when presenting their assessment information, as it allows the
membership degree (MD) and non-membership degree (NMD) sets of some values in the interval [0,
1]. The qth power of the MD and the qth power of the NMD must be ≤1 to satisfy the q-ROFS
constraint. The preceding data can be represented as Q=(0.6,0.9), which is a q-rung orthopair fuzzy
number (since (0.6)q+(0.9)q≤1). As a result, q-ROFS has been widely used in MAGDM, and
several new decision-making methods have been proposed. An MAGDM technique based on the
q-ROF geometric-arithmetic weighted averaging operator was proposed by Liu and Wang [28]. Liu
and Liu [29] extended the traditional Bonferroni mean operator to the q-ROF set and developed an
MAGDM technique based on the q-ROF Bonferroni mean operator, recognizing the correlation
between many attributes may affect the decision results. The MAGDM technique proposed by Wei
et al. [30] is subject to the q-ROF Maclaurin symmetric mean (MSM) operator. In light of the fact that
the association between q-ROF numbers may be heterogeneous, Liu et al. [31] suggested an MAGDM
approach based on the q-ROF distributed Heronian mean operator. Yang et al. [32] developed a deep
learning and q-ROF interactive weighted Heronian averaging operator-based online shopping
assistance model. The q-ROF power MSM operators were provided by Liu et al. [33] to develop a
new MAGDM technique from the expert group’s viewpoint. The MAGDM technique was developed
by Hussain et al. [34] using the group-based generalized q-ROF average aggregation operations. To
solve the MAGDM difficulties, He et al. [35] proposed the q-ROF power Bonferroni mean operator.
The complex q-ROF MSM operators were further established by Ali and Mahmood [36]. The works
cited above demonstrate the effectiveness of q-ROFSs in dealing with the difficult assessment values
of DMs in the MAGDM technique.
Since Zadeh [37] proposed linguistic variation (LV) theory, in particular to solve the ensemble of
linguistic MAGDM challenges, many advances have been made in the study of linguistic MAGDM
challenges. Fuzzy linguistic techniques have been proven effective in various fields and applications.
Several researchers have studied the problem of group decision-making, where both attributes and
decision expert weights are represented as linguistic words in the recent literature. They suggested
an MAGDM-based method that focuses on actual language knowledge, defined linguistic assessment
operational principles, established a few new operators, and defined linguistic assessment operational
principles. Originally, Herrera and Martinez [38] proposed the 2TL representation approach. It is
comprised of a linguistic term and a number and represents linguistic information with a pair of values
known as a 2-tuple. In linguistic information processing, the 2TL model has precise characteristics.
It prevented information loss and distortion, which previously occurred during linguistic information
processing. This strategy has been increasingly popular in recent years for group DM [39, 40]. They
also proposed the 2TL computational model and 2TL aggregation operators, as well as DM techniques.
Wang [41] provided a model for determining which agile manufacturing system is best for you. Deng
et al. [42] investigated novel complex T-SF 2TL Muirhead mean aggregation operators. Wei and
Gao [43] developed several Pythagorean fuzzy 2TL power AOs using the power average and power
geometric operations with Pythagorean fuzzy 2TL information to tackle the MAGDM challenges. To
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
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tackle the MAGDM problem using 2TLq-ROF information, Ju et al. [44] developed the 2TLq-ROF
weighted AO and the 2TLq-ROF weighted geometric operator. They also propose the 2TLq-ROF
Muirhead mean operator and the 2TLq-ROF dual Muirhead mean operator.
Many wide assortments of studies have been undertaken to learn more about the correlation between
arguments, which is a crucial feature of aggregated data. The Hamy mean (HM) operator is one of the
more comprehensive, adaptable, and dominant concepts used to operate troublesome and contradictory
information in real-life challenges, and certain scholars have implemented it in the environment of
numerous domains to find the relation between any number of attributes. Liang [45] also initiated
the HM operators for IFSs, Li et al. [46] proposed the Dombi HM operators for IFSs, Wu et al. [47]
initiated the Dombi HM operators for interval-valued IFSs, and developed the Dombi HM operators
for interval-valued IFSs, Li et al. [48] investigated the HM operators for PFS, and Wang et al. [49]
investigated the HM operators under the q-ROFSs. Ghorabaee et al. [50] established the CODAS
technique, which is an efficient and up-to-date decision-making methodology. It is a distance-based
method that employs Euclidean distance (ED) and Hamming distance (HM) measures. As a primary
comparison measure, this method employs the ED. Whenever the EDs between two alternatives are
relatively close, HDs are employed to compare them. A threshold parameter determines the degree
of closeness of EDs. On the basis of the AHP and CODAS methods, Panchal et al. [51] developed
an integrated MAGDM architecture. Badi et al. [52] used the CODAS technique to determine the
ideal location for a desalination facility on Libya’s northwest coast. Ghorabaee et al. [53] applied the
CODAS approach to picking the most attractive providers in a fuzzy environment. Pamucar et al. [54]
proposed a novel CODAS approach based on linguistic neutrosophic numbers. However, no one has
utilized the HM operators’ idea in the domain of 2TL-q-ROFS in terms of CODAS approach yet.
In this study, we use 2TLq-ROFS as it provides a stronger definition of fuzziness and thus more
accurate evaluation of the decision making process by permitting DMs to assess a wider range due to
the uncertainties in the addressed problems and the lack of information and inconsistencies among
expert groups. So, we developed the 2TLq-ROFS as a new evolution in FS theory for communicating
data complexities. The 2TLq-ROFS involves the integration of 2TL and q-ROF sets and expands the
q-ROFS adaptability. When making a collective choice, the DMs may only have a hazy idea of how
much they like one alternative over another and are unable to measure their preferences with exact
numerical numbers. Rather than numerical variables, it is more appropriate to communicate their
preferences through linguistic variables. We devised a technique called the maximizing deviation
approach to discover the ideal relative weights of qualities under linguistic context, based on the
premise that the attribute with a greater deviation value among alternatives should be considered with
a greater weight. The development has the notable feature of being able to reduce the influence of
DMs’ subjectivity and make adequate use of decision information. Then, using the HM operator, we
suggested a generic strategy for grouping multi-attribute DM issues with linguistic information, in
which preference values are expressed as linguistic variables. Furthermore, we use CODAS method
which is a powerful technique to solve a group DM challenge and selecting the best alternative for
selection of the best arc welding robot. It has several advantages that aren’t considered by other
MAGDM approaches [55]. These are the main contributions of this study:
(i) We introduce 2TLq-ROFS as a new advance in FS theory to communicate the complexity of the
data. 2TLq-ROFS combines the advantages of 2TL and q-ROF sets, increasing the versatility of
q-ROFS.
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
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(ii) We introduce a family of HM aggregation operators for 2TLq-ROFS, such as 2TLq-ROFHM
operator, 2TLq-ROFDHM operator, 2TLq-ROFWHM operator and 2TLq-ROFWDHM. The
2TLq-ROFWDHM operator is used to deal with group decision-making problems with
interrelated attributes.
(iii) Some theorems, properties, and formal definitions of the proposed information aggregation
operators are inferred from existing situations.
(iv) Based on the 2TLq-ROFWHM and 2TLq-ROFWDHM operators, a 2TLq-ROF-CODAS method
is proposed to rank the alternatives. A novel MAGDM model is used to fuse the evaluation
preferences of DMs.
(v) A decision-making system based on 2TLq-ROF-CODAS method for evaluating and selecting arc
welding robots is designed.
The following is the structure of the paper: Section 2 covers various key ideas, including the 2TL
representation model, the description of q-ROFS, the HM operator, and the dual HM operator.
Section 3 introduces the concept of 2TLq-ROFSs and how it works. The 2TLq-ROFHM,
2TLq-ROFDHM, 2TLq-ROFWHM and 2TLq-ROFWDHM aggregation operators with optimal
properties are developed in section 4. In the section 5, the MAGDM policy is constructed by using the
2TLq-ROFWHM and 2TLq-ROFWDHM operators in the 2TLq-ROFS environment. The section 6
provides numerical examples, parameter effects, comparative analysis, and benefits to illustrate the
usefulness and superiority of the established method. Finally, Section 7 summarizes the research and
suggests future directions.
2. Preliminaries
Definition 2.1. [56] Let there exists a linguistic term set (LTS) S={st|t=0,1, . . . , τ}with odd
cardinality, where stindicates a possible linguistic term for a linguistic variable. If st,s∈S, then the
LTS meets the following characteristics:
(i) The set is ordered: st>s, if and only if t> .
(ii) Max operator: max(st,s)=st,if and only if t≥.
(iii) Min operator: min(st,s)=st,if and only if t≤.
(iv) Negative operator: Neg(st)=ssuch that =τ−t.
The 2TL representation model based on the idea of symbolic translation, introduced by Herrera and
Martinez [57], is useful for representing the linguistic assessment information by means of a 2-tuple
(st, υt), where stis a linguistic label from predefined LTS Sand υtis the value of symbolic translation,
and υt∈[−0.5,0.5).
Definition 2.2. [57] Let %be the result of an aggregation of the indices of a set of labels assessed in a
LTS S, i.e., the result of a symbolic aggregation operation, %∈[0, τ], where τis the cardinality of S.
Let t=round(%) and υ=%−tbe two values, such that, t∈[0, τ] and υ∈[−0.5,0.5) then υis called a
symbolic translation.
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
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Definition 2.3. [57] Let S={st|t=1, . . . , τ}be a LTS and %∈[0, τ] is a number value representing
the aggregation result of linguistic symbolic. Then the function ∆used to obtain the 2TL information
equivalent to %is defined as:
∆: [0, τ]→S×[−0.5,0.5),
∆(%)=
st,t=round(%)
υ=%−t, υ ∈[−0.5,0.5).(2.1)
Definition 2.4. [57] Let S={st|t=1, . . . , τ}be a LTS and (st, υt) be a 2-tuple, there exists a function
∆−1that restore the 2-tuple to its equivalent numerical value %∈[0, τ]⊂R,where
∆−1:S×[−0.5,0.5) →[0, τ],
∆−1(st, υ)=t+υ=%. (2.2)
Yager [25] defined the q-rung orthopair fuzzy set as an extension of intuitionistic fuzzy set and
Pythagorean fuzzy set as follows.
Definition 2.5. [25] For any universal set X, a q-ROFS is of the form
T={hx,p(x),l(x)i|x∈X},
where p,l:X→[0,1] represent the MD and NMD, respectively, with the condition 0 ≤pq(`)+lq(`)≤
1 for positive number q≥1 and r(`)=q
p1−(pq(`)+lq(`)) is known as the degree of refusal of `in
T. To express information conveniently, the pair (p,l) is known as a q-rung orthopair fuzzy number
(q-ROFN).
Aq-ROFN is a generalized form of existing fuzzy framework and it reduces to:
(i) Pythagorean fuzzy number (PFN); by taking qas 2.
(ii) Intuitionistic fuzzy number (IFN); by taking qas 1.
(iii) Fuzzy number (FN); by taking las zero and qas 1.
Definition 2.6. Let a(=1,2,...,n) be a set of non-negative real numbers. Some HM aggregation
operators are defined as follows:
(1) Hamy mean [58]: HM(κ)(a1,a2,...,an)=P
1≤t1<...<tκ≤n
κ
Q
=1
at
1
κ
Cκ
n;
(2) Weighted Hamy mean [58]: WHM(κ)
$(a1,a2,...,an)=P
1≤t1<...<tκ≤n
κ
Q
=1
(at)$t
1
κ
Cκ
n;
(3) Dual Hamy mean [59]: DHM(κ)(a1,a2,...,an)=
Q
1≤t1<...<tκ≤n
κ
P
=1
at
κ
1
Cκ
n
;
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17535
(4) Weighted dual Hamy mean [59]: WDHM(κ)
$(a1,a2,...,an)=
Q
1≤t1<...<tκ≤n
κ
P
=1
$tat
κ
1
Cκ
n
,
where κis a parameter and κ=1,2,...,n,t1,t2,...,tκare κinteger values taken from the set
{1,2,...,n}of tinteger values, Cκ
ndenotes the binomial coefficient, and Cκ
n=n!/(κ!(n−κ)!).
For other concepts and applications, the readers are refer to [60–63].
3. 2-Tuple linguistic q-rung orthopair fuzzy set
We introduce the 2TLq-ROFS with its operational laws as a new advancement of FS theory, in this
section. Inspired by the ideas of 2TL and q-ROF sets, we develop the new concept of 2TLq-ROFS
by combining both the advantages of 2TL and q-ROF sets, as an extension of 2TLIFSs and 2TLPFSs.
The newly proposed set has flexibility due to the qth power of MD and NMD. The mathematical
representation of 2TLq-ROFS is described as follows.
Definition 3.1. Let S={st|t=0,1, . . . , τ}be a LTS with odd cardinality. If (sp(x), ℘(x)),(sr(x), ζ(x))
is defined for sp(x),sr(x)∈S, ℘(x), ζ (x)∈[−0.5,0.5), where (sp(x), ℘(x)) and (sr(x), ζ (x)) represent
the MD and NMD by 2TLSs, respectively. A 2TL q-rung orthopair fuzzy set is defined as:
ℵ={hx,((sp(x), ℘(x)),(sr(x), ζ(x)))i|x∈X},(3.1)
where 0 ≤∆−1(sp(x), ℘(x)) ≤τ, 0≤∆−1(sr(x), ζ(x)) ≤τ, and
0≤(∆−1(sp(x), ℘(x)))q+(∆−1(sr(x), ζ(x)))q≤τq.
To compare any two 2TLq-ROFNs, their score value and accuracy value are defined as follows.
Definition 3.2. Let η=((sp, ℘),(sr, ζ )) be a 2TLq-ROFN. Then the score function Sof a 2TLq-ROFN
η, can be represented as:
S(η)= ∆ τ
21+∆−1(sp,℘)
τq
−∆−1(sr,ζ)
τq,S(η)∈[0, τ],(3.2)
and its accuracy function His defined as:
H(η)= ∆ τ∆−1(sp,℘)
τq
+∆−1(sr,ζ)
τq,H(η)∈[0, τ].(3.3)
Definition 3.3. Let η1=((sp1, ℘1),(sr1, ζ1)) and η2=((sp2, ℘2),(sr2, ζ2)) be two 2TLq-ROFNs, then
these two 2TLq-ROFNs can be compared according to the following rules:
(1) If S(η1)>S(η2), then η1> η2;
(2) If S(η1)=S(η2), then
•If H(η1)>H(η2), then η1> η2;
•If H(η1)=H(η2), then η1∼η2.
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
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Definition 3.4. Let η1=((sp1, ℘1),(sl1, ζ1)) and η2=((sp2, ℘2),(sl2, ζ2)) be two 2TLq-ROFNs. We
define the 2TLq-ROF normalized ED and HD as:
ED(η1, η2)= ∆
τ
2∆−1(sp1,℘1)
τq
−∆−1(sp2,℘2)
τq
q
+∆−1(sη1,ζ1)
τq
−∆−1(sη2,ζ2)
τq
q1
q
.(3.4)
HD(η1, η2)= ∆ τ
2∆−1(sp1,℘1)
τq
−∆−1(sp2,℘2)
τq
+∆−1(sr1,ζ1)
τq
−∆−1(sr2,ζ2)
τq.(3.5)
We now put forward the novel operational laws based on 2TLq-ROFNs, including addition,
multiplication, scalar multiplication, power and ranking rules.
Definition 3.5. Let η=((sp, ℘),(sr, ζ)), η1=((sp1, ℘1),(sr1, ζ1)), and η2=((sp2, ℘2),(sr2, ζ2)) be three
2TLq-ROFNs, q≥1, then
(1) η1⊕η2=
∆
τq
r1−1−∆−1(sp1,℘1)
τq1−∆−1(sp2,℘2)
τq
,∆τ∆−1(sr1,ζ1)
τ∆−1(sr2,ζ2)
τ
;
(2) η1⊗η2=
∆τ∆−1(sp1,℘1)
τ∆−1(sp2,℘2)
τ,∆
τq
r1−1−∆−1(sr1,ζ1)
τq1−∆−1(sr2,ζ2)
τq
;
(3) λη =
∆
τq
r1−1−∆−1(sp,℘)
τqλ
,∆τ∆−1(sr,ζ)
τλ
, λ > 0;
(4) ηλ=
∆ τ∆−1(sp,℘)
τλ!,∆
τq
r1−1−∆−1(sr,ζ)
τqλ
, λ > 0.
4. Some 2TLq-ROF Hamy mean aggregation operators
Hara et al. [58] proposed the concept of Hamy mean operator. In this Section, the 2TLq-ROFHM,
2TLq-ROFWHM, 2TLq-ROFDHM, and 2TLq-ROFWDHM operators for aggregating the
2TLq-ROFNs are proposed to extend the HM aggregation operators to the 2TLq-ROFS environment.
Since 2TLq-ROFS is a useful technique for expressing ambiguous data in a real-world
decision-making context. Core features of aggregation operators are idempotency, monotonicity, and
boundedness.
4.1. 2TLq-ROFHM aggregation operator
This subsection introduces the new concept of the 2TLq-ROFHM operator for aggregating 2TLq-
ROFNs and examines its distinctive and preferred features.
Definition 4.1. Let η=(sp, ℘),(sr, ζ )(=1,2,...,n) be a collection of 2TLq-ROFNs. The
2TLq-ROFHM operator is a mapping Tn→Tsuch that
2TLq-ROFHM(κ)(η1, η2, . . . , ηn)=
⊕1≤t1<...<tκ≤n⊗κ
=1ηt1
κ
Cκ
n
.(4.1)
Theorem 4.1. Utilizing the 2TLq-ROFHM operator, the aggregated value is likewise a 2TLq-ROFN
value, where
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17537
2TLq-ROFHM(κ)(η1, η2, . . . , ηn)
=
∆
τq
v
u
t1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1
∆−1(sp,℘ )
τ!q
κ
1
Cκ
n
,
∆
τ
Q
1≤t1<...<tκ≤n
q
s1−
κ
Q
=11−∆−1(sr,ζ)
τq1
κ
1
Cκ
n
.(4.2)
Proof. By utilizing Definition 3.5, we get
⊗κ
=1ηt=
∆ τ
κ
Q
=1
∆−1(sp,℘ )
τ!,∆
τq
s1−
κ
Q
=11−∆−1(sr,ζ)
τq
.
Thus,
⊗κ
=1ηt1
κ=
∆
τ κ
Q
=1
∆−1(sp,℘ )
τ!1
κ
,∆
τq
s1−
κ
Q
=11−∆−1(sr,ζ)
τq1
κ
.
Therefore,
⊕1≤t1<...<tκ≤n⊗κ
=1ηt1
κ=
∆
τq
s1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1
∆−1(sp,℘ )
τ!q
κ
,
∆
τQ
1≤t1<...<tκ≤n
q
s1−
κ
Q
=11−∆−1(sr,ζ)
τq1
κ
.
Furthermore,
2TLq-ROFHM(κ)(η1, η2, . . . , ηn)=
∆
τq
v
u
t1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1
∆−1(sp,℘)
τ!q
κ
1
Cκ
n
,
∆
τ
Q
1≤t1<...<tκ≤n
q
s1−
κ
Q
=11−∆−1(sr,ζ)
τq1
κ
1
Cκ
n
.
The desirable properties of the 2TLq-ROFHM operator, such as idempotency, monotonicity, and
boundedness, are also described below.
Property 4.1. (Idempotency). If all η=(sp, ℘ ),(sr, ζ)(=1,2,...,n)are equal, for all , then
2TLq-ROFFHM(κ)(η1, η2, . . . , ηn)=η.
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17538
Proof.
2TLq-ROFHM(κ)(η1, η2, . . . , ηn)=
∆
τq
v
u
t1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1
∆−1(sp,℘)
τ!q
κ
1
Cκ
n
,
∆
τ
Q
1≤t1<...<tκ≤n
q
s1−
κ
Q
=11−∆−1(sr,ζ)
τq1
κ
1
Cκ
n
=
∆
τq
v
u
t1−
1−∆−1(sp,℘)
τκ
q
κ!Cκ
n
1
Cκ
n
,
∆
τ
q
r1−1−∆−1(sr,ζ)
τqκ1
κ
Cκ
n
1
Cκ
n
=((sp, ℘),(sr, ζ )) =η.
Property 4.2. (Monotonicity). Let η=(sp, ℘ ),(sr, ζ)and η0
=(s0
p, ℘0
),(s0
r, ζ0
)(=1,2,...,n)
be two sets of 2TLq-ROFNs, if η≤η0
, for all , then
2TLq-ROFHM(κ)(η1, η2, . . . , ηn)≤2TLq-ROFHM(κ)(η0
1, η0
2, . . . , η0
n).
Proof. Let η=(sp, ℘),(sr, ζ )and η0
=(s0
p, ℘0
),(s0
r, ζ0
)(=1,2,...,n) be two sets of 2TLq-
ROFNs, let
(sp, ℘)= ∆
τ
q
v
u
u
u
u
t1−Y
1≤t1<...<tκ≤n
1−
κ
Y
=1
∆−1(sp, ℘ )
τ
q
κ
1
Cκ
n
,
(sr, ζ)= ∆
τ
Y
1≤t1<...<tκ≤n
q
v
u
u
t1−
κ
Y
=1
1− ∆−1(sr, ζ )
τ!q
1
κ
1
Cκ
n
,
given that (sp, ℘ )≤(s0
p, ℘0
); then
κ
Y
=1
∆−1(sp, ℘ )
τ
q
κ
≤
κ
Y
=1
∆−1(s0
p, ℘0
)
τ
q
κ
.
Moreover,
Y
1≤t1<...<tκ≤n
1−
κ
Y
=1
∆−1(sp, ℘ )
τ
q
κ
1
Cκ
n
≥Y
1≤t1<...<tκ≤n
1−
t
Y
=1
∆−1(s0
p, ℘0
)
τ
q
κ
1
Cκ
n
.
Furthermore,
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17539
∆
τ
q
v
u
u
u
u
t1−Y
1≤t1<...<tκ≤n
1−
κ
Y
=1
∆−1(sp, ℘ )
τ
q
κ
1
Cκ
n
≤∆
τ
q
v
u
u
u
u
t1−Y
1≤t1<...<tκ≤n
1−
κ
Y
=1
∆−1(s0
p, ℘0
)
τ
q
κ
1
Cκ
n
.
Therefore, (sp, ℘)≤(s0
p, ℘0). Similarly, we can show that (sr, ζ)≥(s0
r, ζ0).
Hence, 2TLq-ROFHM(κ)(η1, η2, . . . , ηn)≤2TLq-ROFHM(κ)(η0
1, η0
2, . . . , η0
n).
Property 4.3. (Boundedness). Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n)be a collection of 2TLq-
ROFNs, and let η−=min((sp, ℘ ),(sr, ζ)) and η+=max ((sp, ℘ ),(sr, ζ)); then
η−≤2TLq-ROFHM(κ)(η1, η2, . . . , ηn)≤η+.
From Property 4.1,
2TLq-ROFHM(κ)(η−
1, η−
2, . . . , η−
n)=η−,
2TLq-ROFHM(κ)(η+
1, η+
2, . . . , η+
n)=η+.
From Property 4.2,
η−≤2TLq-ROFHM(κ)(η1, η2, . . . , ηn)≤η+.
4.2. 2TLq-ROFWHM aggregation operator
The 2TLq-ROFHM aggregation operator does not show the weighting values of attributes in
Theorem 4.1. To overcome the constraints of the 2TLq-ROFHM operator, we shall introduce the
2TLq-ROFWHM operator with certain preferred features.
Definition 4.2. Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n) be a collection of 2TLq-ROFNs with
weighting vector $=($1, $2, . . . , $n)T, thereby satisfying $∈[0,1] and
n
P
=1
$=1. The
2TLq-ROFWHM operator is a mapping Tn→Tsuch that
2TLq-ROFWHM(κ)
$(η1, η2, . . . , ηn)=
⊕1≤t1<...<tκ≤n⊗κ
=1(ηt)$t1
κ
Cκ
n
.(4.3)
Theorem 4.2. Using the 2TLq-ROFWHM operator, the aggregated value is likewise a 2TLq-ROFN
value, where
2TLq-ROFWHM(κ)
$(η1, η2, . . . , ηn)
=
∆
τq
v
u
t1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1∆−1(sp,℘)
τ$t!q
κ
1
Cκ
n
,
∆
τ
Q
1≤t1<...<tκ≤n
q
s1− κ
Q
=11−∆−1(sr,ζ)
τq$t!1
κ
1
Cκ
n
.(4.4)
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17540
Proof. By utilizing Definition 3.5, we get
(ηt)$t= ∆τ∆−1(sp,℘ )
τ$t,∆
τq
r1−1−∆−1(sr,ζ )
τq$t
!.
Then,
⊗κ
=1(ηt)$t=
∆ τ
κ
Q
=1∆−1(sp,℘)
τ$t!,∆
τq
s1−
κ
Q
=11−∆−1(sr,ζ)
τq$t
.
Thus,
⊗κ
=1ηt$t1
κ=
∆
τ κ
Q
=1∆−1(sp,℘)
τ$t!1
κ
,∆
τq
s1− t
Q
=11−∆−1(sr,ζ)
τq$t!1
κ
.
Therefore,
⊕1≤t1<...<tκ≤n⊗κ
=1ηtx$t1
κ
=
∆
τq
s1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1∆−1(sp,℘)
τ$t!q
κ
,
∆
τQ
1≤t1<...<tκ≤n
q
s1− κ
Q
=11−∆−1(sr,ζ)
τq$t!1
κ
.
Furthermore,
2TLq-ROFWHM(κ)
$(η1, η2, . . . , ηn)
=
∆
τq
v
u
t1−Q
1≤t1<...<tκ≤n
1− κ
Q
=1∆−1(sp,℘)
τ$t!q
κ
1
Cκ
n
,
∆
τ
Q
1≤t1<...<tκ≤n
q
s1− κ
Q
=11−∆−1(sr,ζ)
τq$t!1
κ
1
Cκ
n
.
Property 4.4. (Monotonicity). Let η=((sp, ℘ ),(sr, ζ)) and η0
=((s0
p, ℘0
),(s0
r, ζ0
))( =1,2,...,n)
be two sets of 2TLq-ROFNs, if η≤η0
, for all , then
2TLq-ROFWHM(κ)
$(η1, η2, . . . , ηn)≤2TLq-ROFWHM(κ)
$(η0
1, η0
2, . . . , η0
n).
Property 4.5. (Boundedness). Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n)be a collection of 2TLq-
ROFNs, and let η−=min((sp, ℘ ),(sr, ζ)) and η+=max ((sp, ℘ ),(sr, ζ)); then
η−≤2TLq-ROFWHM(κ)
$(η1, η2, . . . , ηn)≤η+.
Idempotency is obviously not a feature of the 2TLq-ROFWHM operator.
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17541
4.3. 2TLq-ROFDHM aggregation operator
In this subsection, we will augment the DHM operator with 2TLq-ROFS to propose the 2TLq-
ROFDHM operator for aggregating 2TLq-ROFNs, and also examine its desirable features.
Definition 4.3. Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n) be a collection of 2TLq-ROFNs. The
2TLq-ROFDHM operator is a mapping Tn→Tsuch that
2TLq-ROFDHM(κ)(η1, η2, . . . , ηn)= ⊗1≤t1<...<tκ≤n ⊕κ
=1ηt
κ!!1
Cκ
n
.(4.5)
Theorem 4.3. The aggregated value by utilizing 2TLq-ROFDHM operator is also a 2TLq-ROFN,
where
2TLq-ROFDHM(κ)(η1, η2, . . . , ηn)
=
∆
τ
Q
1≤t1<...<tκ≤n
q
s1−
κ
Q
=11−∆−1(sp,℘)
τq1
κ
1
Cκ
n
,
∆
τq
v
u
t1−Q
1≤t1<...<tκ<n
1− κ
Q
=1
∆−1(sr,ζ )
τ!q
κ
1
Cκ
n
.(4.6)
Property 4.6. (Idempotency). If all η=((sp, ℘ ),(sr, ζ))( =1,2,...,n)are equal, for all , then
2TLq-ROFDHM(κ)(η1, η2, . . . , ηn)=η.
Property 4.7. (Monotonicity). Let η=((sp, ℘ ),(sr, ζ)) and η0
=((s0
p, ℘0
),(s0
r, ζ0
))( =1,2,...,n)
be two sets of 2TLq-ROFNs, if η≤η0
, for all , then
2TLq-ROFDHM(κ)(η1, η2, . . . , ηn)≤2TLq-ROFDHM(κ)(η0
1, η0
2, . . . , η0
n).
Property 4.8. (Boundedness). Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n)be a collection of 2TLq-
ROFNs, and let η−=min((sp, ℘ ),(sr, ζ)) and η+=max ((sp, ℘ ),(sr, ζ)); then
η−≤2TLq-ROFDHM(κ)(η1, η2, . . . , ηn)≤η+.
4.4. 2TLq-ROFWDHM aggregation operator
The value of the aggregated arguments is not taken into account by the 2TLq-ROFDHM operator,
as demonstrated in Theorem 4.3. However, in many real-life circumstances, particularly in MAGDM,
attribute weights play an important role in the aggregation process. The attributes’ values are omitted
by the 2TLq-ROFDHM operator. The 2TLq-ROFWDHM operator is proposed to overcome the
constraints of 2TLq-ROFDHM.
Definition 4.4. Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n) be a collection of 2TLq-ROFNs with
weighting vector $=($1, $2, . . . , $n)T, thereby satisfying $∈[0,1] and
n
P
=1
$=1. The
2TLq-ROFWDHM operator is a mapping Tn→Tsuch that
2TLq-ROFWDHM(κ)
$(η1, η2, . . . , ηn)= ⊗1≤t1<...<tκ≤n ⊕κ
=1$tηt
κ!!1
Cκ
n
.(4.7)
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17542
Theorem 4.4. Using the 2TLq-ROFWDHM operator, the aggregated value is likewise a 2TLq-ROFN
value, where
2TLq-ROFWDHM(κ)
$(η1, η2, . . . , ηn)
=
∆
τ
Q
1≤t1<...<tκ≤n
q
s1− κ
Q
=11−∆−1(sp,℘)
τq$t!1
κ
1
Cκ
n
,
∆
τq
v
u
t1−Q
1≤t1<...<tκ<n
1− κ
Q
=1∆−1(sr,ζ)
τ$t!q
κ
1
Cκ
n
.(4.8)
Property 4.9. (Monotonicity). Let η=((sp, ℘),(sr, ζ )) and η0
=((s0
p, ℘0
),(s0
r, ζ0
)),(=1,2,...,n)
be two sets of 2TLq-ROFNs, if η≤η0
, for all , then
2TLq-ROFWDHM(κ)
$(η1, η2, . . . , ηn)≤2TLq-ROFWDHM(κ)
$(η0
1, η0
2, . . . , η0
n).
Property 4.10. (Boundedness). Let η=((sp, ℘ ),(sr, ζ))( =1,2,...,n)be a collection of 2TLq-
ROFNs, and let η−=min((sp, ℘ ),(sr, ζ)) and η+=max ((sp, ℘ ),(sr, ζ)); then
η−≤2TLq-ROFWDHM(κ)
$(η1, η2, . . . , ηn)≤η+.
Idempotency is obviously not a feature of the 2TLq-ROFWDHM operator.
5. MAGDM based on the maximizing deviation and CODAS method
This section gives a framework for calculating attribute weights and the ranking orders for all the
alternatives with incomplete weight information under 2TLq-ROF environment.
Suppose there are ealternatives R={R1,R2,...,Re},nattributes G={G1,G2,...,Gn}, and g
experts E={Θ1,Θ2,...,Θg}, and let $=($1, $2, . . . , $n)Tand $0=($0
1, $0
2, . . . , $0
g)Tbe the
weighting vector of the attributes and weighting vector of the experts satisfying $∈[0,1], $0
`∈[0,1],
Pn
=1$=1, and
g
P
`=1
$0
`=1, respectively.
5.1. Calculation of optimal weights utilizing maximizing deviation method
Case 1: Completely unknown information on attribute weights
To find the best relative weights for attributes G∈G, we build an optimization model based on the
maximizing deviation method in a 2TLq-ROF environment. The deviation of the alternative Rtfrom
all other alternatives for the attribute can be expressed as:
Dt($)=
e
X
k=1
dηt, ηk$,t=1,2,...,e, =1,2,...,n(5.1)
where,
d(ηt,hk)= ∆
τ
2
∆−1spt, ℘t
τ
q
−
∆−1spk, ℘k
τ
q
q
+
∆−1srt, ζt
τ
q
−
∆−1srk, ζk
τ
q
q
1
q
(5.2)
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17543
denotes the 2TLq-ROF ED between the 2TLq-ROFEs htand hk.
Let
D($)=
e
X
t=1
Dt($)=
e
X
t=1
e
X
k=1
$d(ηt,hk), =1,2,...,n.(5.3)
D($) represents the deviation value of all alternatives to other alternatives for the attribute G∈G.
(M−1)
max D($)=
n
P
=1
e
P
t=1
e
P
k=1
$d(ηt,hk)
s.t. $≥0, =1,2,...,n,
n
P
=1
$2
=1
.
In order to solve the above model, we consider
L($, k)=
n
X
=1
e
X
t=1
e
X
k=1
$d(ηt,hk)+k
2
n
X
=1
$2
−1
(5.4)
which represents the Lagrange function of the constrained optimization problem (M-1), where kis a
real number, denoting the Lagrange multiplier variable. Then the partial derivatives of L are calculated
as:
∂L
∂$
=
e
X
t=1
e
X
k=1
d(ηt,hk)+k$=0,(5.5)
∂L
∂k=1
2
n
X
=1
$2
−1
=0.(5.6)
It follows from Eq (5.5) that
$=
−
e
P
t=1
e
P
k=1
d(ηt,hk)
k, =1,2,...,n.(5.7)
Putting Eq (5.7) into Eq (5.6), we get
k=−v
u
tn
X
=1
e
X
t=1
e
X
k=1
d(ηt,hk)
2
.(5.8)
Obviously, k<0,
e
P
t=1
e
P
k=1
d(ηt,hk) denotes the sum of all the alternatives’ deviations from the jth
attribute, and sn
P
=1 e
P
t=1
e
P
k=1
d(ηt,hk)!2
denotes the sum of all of the alternatives’ deviations for all the
attributes. Then utilizing Eqs (5.7) and (5.8), we get
$=
e
P
t=1
e
P
k=1
d(ηt,hk)
sn
P
=1 e
P
t=1
e
P
k=1
d(ηt,hk)!2
.(5.9)
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17544
For the sake of simplicity,
χ=
e
X
t=1
e
X
k=1
d(ηt,hk)=1,2,...,n.(5.10)
Then the Eq (5.9) becomes
$=χ
sn
P
=1
χ2
, =1,2,...,n.(5.11)
It is simple to verify that $(=1,2,...,n) are positive and fulfill the constrained conditions in the
model (M-1) and that the solution is unique using Eq (5.11).
By normalizing $(=1,2,...,n), to let the sum of $into a unit, we have
$∗
=$
n
P
=1
$
=χ
n
P
=1
χ
, =1,2,...,n.(5.12)
Case 2: Partly known information on attribute weights
In some cases, the weighting vectors’ information is only partially known rather than completely
unknown. In these cases, the constrained optimization model can be designed as follows, based on the
set of weight’s information that is known, Ψ
(M−2)
max D($)=
n
P
=1
e
P
t=1
e
P
k=1
$d(ηt,hk)
s.t. $ ∈Ψ, $ ≥0, =1,2,...,n,
n
P
=1
$=1
where Ψalso refers to a collection of restriction constraints that the weight value $should satisfy in
order to fulfil the requirements in real-world scenarios. A linear programming model (M−2) is used.
We acquire the best solution $=($1, $2, . . . , $n)T, by solving this model, which can be used as the
weighting vector for the attributes.
5.2. CODAS approach for MAGDM under 2TLq-ROF environment
In this subsection, we present a new approch to deal with MAGDM problems, known as 2TLq-
ROF-CODAS model based on 2TLq-ROFWHM and 2TLq-ROFWDHM operators by considering the
flexibility of 2TLq-ROFNs. The preference of alternatives is calculated using two measures in this
method. The largest and the most important measurement is the ED between alternatives and the
negative-ideal solution (NIS), and the second measure is the HD. It is clear that the alternative which
has greater distance from the NIS is more preferable. The ED and HD measures are used for the relative
assessment (RA) of alternatives in order to construct the RA based matrix to fuse the information. The
technique of implementing the 2TLq-ROF-CODAS approach is described in the following steps:
Step 1. Switch the linguistic information into 2TLq-ROFNs η`
t=((sp`
t, ℘`
t),(sr`
t, ζ`
t))(`=1,2, . . . g).
Step 2. According to 2TLq-ROFNs η`
t=((sp`
t, ℘`
t),(sr`
t, ζ`
t))(`=1,2, . . . g) and by utilizing Eqs (4.3)
and (4.7), independent panel evaluations can be combined to form the fused 2TLq-ROFNs matrix
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17545
ηt=((spt, ℘t),(srt, ζt)).
z=[ηt]e×n=
η11 η12 . . . η1n
η21 η22 . . . η2n
.
.
..
.
..
.
..
.
.
ηe1ηe2. . . ηen
.(5.13)
Step 3. Calculate the weighted 2TLq-ROFNs matrix as follows:
tt=$⊗ηt,(5.14)
where $is the attribute weight of G, and 0 ≤$≤1, Pn
=1$=1.
Step 4. Calculate the NIS by using 2TLq-ROFNs’ score function. If the score function is similar, the
accuracy function is used to rank the 2TLq-ROFNs:
NIS =[NIS ]1×n; (5.15)
NIS =min
tS(tt).(5.16)
Step 5. Calculate the weighted E Dtand H Dtas follows:
EDt=
n
X
=1
ED(tt,NIS ); (5.17)
HDt=
n
X
=1
HD(tt,NIS ).(5.18)
Step 6. In the following equations, build the relative assessment matrix RA:
RA =[ht`]e×e; (5.19)
ht`=(EDt−ED`)+(g(EDt−ED`)×(HDt−H D`)),(5.20)
where `∈ {1,2,3,...,g}and gdenotes a significant function that could be designed:
g(θ)=(1 if |θ|≥=
0 if |θ|<=,(5.21)
where = ∈ [0.01,0.05] specified by DMs. Here, ==0.02.
Step 7. Derive the average solution (£t) by using:
£t=
g
X
`=1
ht`.(5.22)
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17546
Step 8. On the basis of computing outcomes of £t, all the alternatives can be ranked. The best option
has the highest evaluation score.
The scheme of the developed approach for MAGDM problems is shown in Figure 1.
Figure 1. The scheme of the developed approach for MAGDM.
6. Numerical example: Case study
Robotic welding is the most visible manifestation of current welding technology. The first
generation robotic welding systems used a two-pass weld method, with the first pass committed to
learn the seam geometry and the second pass committed to track and weld the seam. The second
generation of robotic welding systems came with the technological advancements , which tracked the
seam in real-time while learning and seam-tracking at the same time. Third-generation robotic
welding systems are the most advanced in robotic welding technology, as they not only function in
real-time but also understand the quickly changing geometry of the seam while operating in
unorganized situations. Higher product quality criteria should drive it at a lower cost and generate a
dependable weld, according to the selection of industrial arc welding robots. Weight density,
replicability, freight capacity, maximum reach, Average power consumption, and Motion of a robot
are some of the characteristics that can be used to characterize robots. All of these aspects must be
taken into account when choosing robots for a certain application. The most prevalent type of robot in
industrial robotic arc welding is one with a revolute (or jointed arm) arrangement, which is based on
the workspace geometry. The CODAS approach is used to investigate the selection of industrial
robots for arc welding operations in this study. The data for arc welding robots was gathered to apply
nine distinct robots with six controllable axes and varied controllers from their manufacturers. Six
AIMS Mathematics Volume 7, Issue 9, 17529–17569.
17547
attributes are assigned to these nine robots. After looking over several datasheets offered by robot
manufacturers to describe their goods, the selection criteria were evaluated. The opinions of industry
professionals are also taken into account. The selection criteria were decided after a discussion
between the research group and an industry specialist. The final decision matrix was reviewed using
the joint decision of both groups, and the significant traits possessed by each robot were used as
criteria for evaluation.The following are the six important attributes shown in Table 1 to consider
while choosing an arc welding robot:
Table 1. Description of evaluation attributes.
Criterion Explanation
Weight density This criterion takes into account the
physical weight of the robot. In general,
consumer chooses a lighter robot. The
weight density is usually expressed in kg
(G1).
Replicability This refers to a robot’s ability to
repeat a task over and over again.
More replicability is often preferred.
Replicability is usually measured in
millimeters (G2).
Freight capacity The highest total weight a robot can lift in
one turn is referred to as freight capacity.
Being more is often preferred. The weight
density is usually expressed in kg (G3).
Maximum reach This is the average of the maximum
vertical and horizontal distances a robot’s
arm can extend to complete a task. It is
common to want to be more. The robot’s
maximum reach is typically measured in
millimeters (G4).
Average power consumption It refers to the robot’s average power
consumption in units of electricity. It is
often desirable that a robot consumes less
energy. The robot’s power consumption
is usually measured in kilowatts (G5).
Motion of a robot The motion of a robot at a reference
point near the end effector’s tip is referred
to as robot motion. Trajectory, speed,
acceleration, and acceleration derivative
are commonly used to describe robot
motion. The robot’s motion is usually
expressed in ms−1or ms−2(G6)
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Comprehensive above, the set of nine alternatives R={R1,R2, . . . R9}is evaluated by four experts
E={Θ1,Θ2,Θ3,Θ4}which consists of experienced engineers and customers in evaluation stage having
weights $0=(0.19,0.31,0.17,0.33)T. The four experts use the six attributes shown in Table 2 to select
the best alternatives for additive manufacturing of linear delta robot.
The linguistic variables of 2TLq-ROFNs are recorded in Table 3.
Establish the 2TLq-ROF evaluation matrix z`=[η`
t]9×6(`=1,2,3,4,5,6) in Table 4 based on
linguistic variables listed in Table 3, which are the assessments of four DMs.
Transformation of the linguistic decision matrix given in Table 4 into 2TLq-ROF decision matrix
shown in Table 5.
Table 2. Attributes their symbols and units.
Sr. No. Criteria Units Symbol
1 Weight density Kg G1
2 Replicability (+/−)mm G2
3 Freight capacity Kg G3
4 Maximum Reach mm G4
5 Average Power Consumption KW G5
6 Motion of a robot ms−2G6
Table 3. Linguistic variables and 2TLq-ROFNs.
Linguistic variables 2TLq-ROFNs
Certainly high value (CHV) ((s8,0),(s0,0))
Very high value (VHV) ((s7,0),(s1,0))
High value(HV) ((s6,0),(s2,0))
Above average value (AAV) ((s5,0),(s3,0))
Average vlaue (AV) ((s4,0),(s4,0))
Under average value (UAV) ((s3,0),(s5,0))
Low value (LV) ((s2,0),(s6,0))
Very low value (VLV) ((s1,0),(s7,0))
Certainly low value (CLV) ((s0,0),(s8,0))
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Table 4. Linguistic assessing matrix by four decision makers.
Experts Alternatives Attributes
G1G2G3G4G5G6
Θ1
R1AV LV VHV VLV CLV CHV
R2HV UAV CLV AAV VHV LV
R3VHV VLV CHV HV UAV CLV
R4CLV AAV LV VHV VLV AAV
R5CHV HV UAV CLV LV AV
R6LV VHV VLV CHV AV HV
R7UAV CLV AAV AV HV VLV
R8AAV AV HV UAV CHV VHV
R9VLV CHV AV LV AAV UAV
Θ2
R1VLV AV HV UAV LV CHV
R2LV VHV VLV CHV CLV AAV
R3AV HV UAV CLV AAV VHV
R4CLV AAV LV VHV VLV UAV
R5VHV VLV CHV HV UAV AV
R6HV UAV CLV AAV AV LV
R7AAV LV VHV AV CHV CLV
R8UAV CHV AV LV VHV VLV
R9CHV CLV AAV VLV HV AV
Θ3
R1AV UAV LV AAV HV VLV
R2LV VHV AV HV UAV AAV
R3VHV VLV CHV AV LV UAV
R4CHV AV UAV VLV AAV HV
R5HV CHV VLV UAV CLV VHV
R6AAV LV CLV CHV VLV AV
R7CLV AAV HV VHV AV CHV
R8VLV HV VHV CLV CHV LV
R9UAV CLV AAV LV VHV CLV
Θ4
R1CHV LV UAV AV AAV HV
R2LV HV CLV UAV CHV AAV
R3UAV VHV CHV HV VLV CHV
R4AV AAV VLV CLV HV LV
R5VLV CLV AV LV VHV UAV
R6AAV CHV LV VLV CLV AV
R7VHV VLV AAV CHV AV VHV
R8HV AV VHV AAV UAV VLV
R9CLV UAV HV VHV LV CLV
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Table 5. The assessing matrix with 2TLq-ROFNs.
Experts Alternatives Attributes
G1G2G3G4G5G6
Θ1
R1((s4,0),(s4,0)) ((s2,0),(s6,0)) ((s7,0),(s1,0)) (( s1,0),(s7,0)) ((s0,0),(s8,0)) (( s8,0),(s0,0))
R2((s6,0),(s2,0)) ((s3,0),(s5,0)) ((s0,0),(s8,0)) (( s5,0),(s3,0)) ((s7,0),(s1,0)) (( s2,0),(s6,0))
R3((s7,0),(s1,0)) ((s1,0),(s7,0)) ((s8,0),(s0,0)) (( s6,0),(s2,0)) ((s3,0),(s5,0)) (( s0,0),(s8,0))
R4((s0,0),(s8,0)) ((s5,0),(s3,0)) ((s2,0),(s6,0)) (( s7,0),(s1,0)) ((s1,0),(s7,0)) (( s5,0),(s3,0))
R5((s8,0),(s0,0)) ((s6,0),(s2,0)) ((s3,0),(s5,0)) (( s0,0),(s8,0)) ((s2,0),(s6,0)) (( s4,0),(s4,0))
R6((s2,0),(s6,0)) ((s7,0),(s1,0)) ((s1,0),(s7,0)) (( s8,0),(s0,0)) ((s4,0),(s4,0)) (( s6,0),(s2,0))
R7((s3,0),(s5,0)) ((s0,0),(s8,0)) ((s5,0),(s3,0)) (( s4,0),(s4,0)) ((s6,0),(s2,0)) (( s1,0),(s7,0))
R8((s5,0),(s3,0)) ((s4,0),(s4,0)) ((s6,0),(s2,0)) (( s3,0),(s5,0)) ((s8,0),(s0,0)) (( s7,0),(s1,0))
R9((s1,0),(s7,0)) ((s8,0),(s0,0)) ((s4,0),(s4,0)) (( s2,0),(s6,0)) ((s5,0),(s3,0)) (( s3,0),(s5,0))
Θ2
R1((s1,0),(s7,0)) ((s4,0),(s4,0)) ((s6,0),(s2,0)) (( s3,0),(s5,0)) ((s2,0),(s6,0)) (( s8,0),(s0,0))
R2((s2,0),(s6,0)) ((s7,0),(s1,0)) ((s1,0),(s7,0)) (( s8,0),(s0,0)) ((s0,0),(s8,0)) (( s5,0),(s3,0))
R3((s4,0),(s4,0)) ((s6,0),(s2,0)) ((s3,0),(s5,0)) (( s0,0),(s8,0)) ((s5,0),(s3,0)) (( s7,0),(s1,0))
R4((s0,0),(s8,0)) ((s5,0),(s3,0)) ((s2,0),(s6,0)) (( s7,0),(s1,0)) ((s1,0),(s7,0)) (( s3,0),(s5,0))
R5((s7,0),(s1,0)) ((s1,0),(s7,0)) ((s8,0),(s0,0)) (( s6,0),(s2,0)) ((s3,0),(s5,0)) (( s4,0),(s4,0))
R6((s6,0),(s2,0)) ((s3,0),(s5,0)) ((s0,0),(s8,0)) (( s5,0),(s3,0)) ((s4,0),(s4,0)) (( s2,0),(s6,0))
R7((s5,0),(s3,0)) ((s2,0),(s6,0)) ((s7,0),(s1,0)) (( s4,0),(s4,0)) ((s8,0),(s0,0)) (( s0,0),(s8,0))
R8((s3,0),(s5,0)) ((s8,0),(s0,0)) ((s4,0),(s4,0)) (( s2,0),(s6,0)) ((s7,0),(s1,0)) (( s1,0),(s7,0))
R9((s8,0),(s0,0)) ((s0,0),(s8,0)) ((s5,0),(s3,0)) (( s1,0),(s7,0)) ((s6,0),(s2,0)) (( s4,0),(s4,0))
Θ3
R1((s4,0),(s4,0)) ((s3,0),(s5,0)) ((s2,0),(s6,0)) (( s5,0),(s3,0)) ((s6,0),(s2,0)) (( s1,0),(s7,0))
R2((s2,0),(s6,0)) ((s7,0),(s1,0)) ((s4,0),(s4,0)) (( s6,0),(s2,0)) ((s3,0),(s5,0)) (( s5,0),(s3,0))
R3((s7,0),(s1,0)) ((s1,0),(s7,0)) ((s8,0),(s0,0)) (( s4,0),(s4,0)) ((s2,0),(s6,0)) (( s3,0),(s5,0))
R4((s8,0),(s0,0)) ((s4,0),(s4,0)) ((s3,0),(s5,0)) (( s1,0),(s7,0)) ((s5,0),(s3,0)) (( s6,0),(s2,0))
R5((s6,0),(s2,0)) ((s8,0),(s0,0)) ((s1,0),(s7,0)) (( s3,0),(s5,0)) ((s0,0),(s8,0)) (( s7,0),(s1,0))
R6((s5,0),(s3,0)) ((s2,0),(s6,0)) ((s0,0),(s8,0)) (( s8,0),(s0,0)) ((s1,0),(s7,0)) (( s4,0),(s4,0))
R7((s0,0),(s8,0)) ((s5,0),(s3,0)) ((s6,0),(s2,0)) (( s7,0),(s1,0)) ((s4,0),(s4,0)) (( s8,0),(s0,0))
R8((s1,0),(s7,0)) ((s6,0),(s2,0)) ((s7,0),(s1,0)) (( s0,0),(s8,0)) ((s8,0),(s0,0)) (( s2,0),(s6,0))
R9((s3,0),(s5,0)) ((s0,0),(s8,0)) ((s5,0),(s3,0)) (( s2,0),(s6,0)) ((s7,0),(s1,0)) (( s0,0),(s8,0))
Θ4
R1((s8,0),(s0,0)) ((s2,0),(s6,0)) ((s3,0),(s5,0)) (( s4,0),(s4,0)) ((s5,0),(s3,0)) (( s6,0),(s2,0))
R2((s2,0),(s6,0)) ((s6,0),(s2,0)) ((s0,0),(s8,0)) (( s3,0),(s5,0)) ((s8,0),(s0,0)) (( s5,0),(s3,0))
R3((s3,0),(s5,0)) ((s7,0),(s1,0)) ((s8,0),(s0,0)) (( s6,0),(s2,0)) ((s1,0),(s7,0)) (( s8,0),(s0,0))
R4((s4,0),(s4,0)) ((s5,0),(s3,0)) ((s1,0),(s7,0)) (( s0,0),(s8,0)) ((s6,0),(s2,0)) (( s2,0),(s6,0))
R5((s1,0),(s7,0)) ((s0,0),(s8,0)) ((s4,0),(s4,0)) (( s2,0),(s6,0)) ((s7,0),(s1,0)) (( s3,0),(s5,0))
R6((s5,0),(s3,0)) ((s8,0),(s0,0)) ((s2,0),(s6,0)) (( s1,0),(s7,0)) ((s0,0),(s8,0)) (( s4,0),(s4,0))
R7((s7,0),(s1,0)) ((s1,0),(s7,0)) ((s5,0),(s3,0)) (( s8,0),(s0,0)) ((s4,0),(s4,0)) (( s7,0),(s1,0))
R8((s6,0),(s2,0)) ((s4,0),(s4,0)) ((s7,0),(s1,0)) (( s5,0),(s3,0)) ((s3,0),(s5,0)) (( s1,0),(s7,0))
R9((s0,0),(s8,0)) ((s3,0),(s5,0)) ((s6,0),(s2,0)) (( s7,0),(s1,0)) ((s2,0),(s6,0)) (( s0,0),(s8,0))
6.1. Results of the case study
6.1.1. Decision-making procedure based on the 2TLq-ROFWHM operator
The MAGDM technique to select best arc welding robot involves the following cases:
Case 1: Assume that the information about the attribute weights is completely unknown: Utilize the
Eq (5.12) to get the optimal weight vector $=(0.1574,0.1881,0.2079,0.1398,0.1449,0.1619)T.
Step 1. Individual expert assessments can be integrated into the collective assessing matrix with 2TLq-
ROFNs, according to Tables 4 and 5 and Eq (4.3) (q=4 and κ=3) (see Table 6).
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Table 6. Combined assessing matrix with 2TLq-ROFNs utilizing 2TLq-ROFWHM operator.
G1G2G3
R1((s7,−0.3891),(s4,−0.2277)) ((s6,0.0997),(s4,−0.0756)) ((s7,−0.1965),(s3,0.2730))
R2((s6,0.0081),(s4,0.1537)) ((s7,0.3642),(s2,0.1177)) ((s0,0),(s8,0))
R3((s7,−0.0292),(s3,−0.1356)) ((s6,0.4762),(s4,0.1480)) ((s8,0),(s0,0))
R4((s0,0),(s8,0)) ((s7,0.0485),(s2,0.2979)) ((s5,0.4993),(s5,−0.3450))
R5((s7,0.0101),(s3,0.0473)) ((s5,−0.3156),(s7,−0.1305)) ((s7,−0.3557),(s4,−0.3383))
R6((s7,−0.0726),(s3,−0.1504)) ((s7,0.0101),(s3,0.2390)) ((s0,0),(s8,0))
R7((s5,0.3996),(s6,0.0833)) ((s4,−0.1232),(s7,0.0370)) ((s7,0.3708),(s2,−0.1898))
R8((s7,−0.4816),(s4,−0.3182)) ((s7,0.2618),(s2,0.3874)) ((s7,0.3997),(s2,−0.1125))
R9((s5,−0.0390),(s7,−0.2683)) ((s0,0),(s8,0)) ((s7,0.1479),(s2,0.1867))
G4G5G6
R1((s6,0.2466),(s4,−0.1424)) ((s5,−0.1877),(s7,−0.4491)) ((s7,0.2954),(s3,−0.1998))
R2((s7,0.2201),(s3,−0.4669)) ((s6,−0.0280),(s6,−0.0258)) ((s7,−0.1752),(s3,−0.0620))
R3((s6,−0.2876),(s6,−0.3069)) ((s6,−0.0466),(s4,−0.2966)) ((s6,−0.0874),(s6,−0.0255))
R4((s5,0.2866),(s7,−0.3760)) ((s6,0.0168),(s4,0.4795)) ((s6,0.4588),(s4,−0.4168))
R5((s5,−0.3423),(s7,−0.3634)) ((s5,−0.1427),(s7,−0.4539)) ((s7,−0.2352),(s3,−0.0154))
R6((s7,−0.1643),(s3,0.4871)) ((s5,−0.3171),(s7,−0.2974)) ((s7,−0.4421),(s3,0.3799))
R7((s7,0.3384),(s2,0.3524)) ((s7,0.2766),(s2,−0.3623)) ((s5,0.2401),(s7,−0.3311))
R8((s5,−0.4488),(s7,−0.3605)) ((s7,0.4941,(s2,−0.0552)) ((s5,0.4849),(s5,−0.1040))
R9((s6,0.0488),(s4,0.3846)) ((s7,0.0848),(s3,−0.0375)) ((s0,0),(s8,0))
Step 2. Determine the weighted assessing matrix with 2TLq-ROFNs using Eq (5.14) (see Table 7).
Table 7. Combined weighted assessing matrix with 2TLq-ROFNs.
G1G2G3
R1((s6,0.0539),(s5,−0.0285)) ((s6,−0.2441),(s5,−0.3317)) ((s7,−0.4061),(s4,−0.2107))
R2((s5,0.4486),(s5,0.2840)) ((s7,0.0878),(s3,−0.0722)) ((s0,0),(s8,0))
R3((s6,0.4378),(s4,0.1766)) ((s6,0.1349),(s5,−0.1320)) ((s8,0),(s0,0))
R4((s0,0),(s8,0)) ((s7,−0.2648),(s3,0.1142)) ((s5,0.2848),(s5,0.0870))
R5((s6,0.4812),(s4,0.3434)) ((s4,0.3848),(s7,0.1293)) ((s6,0.4298),(s4,0.1621))
R6((s6,0.3903),(s4,0.1630)) ((s7,−0.3064),(s4,0.0374)) ((s0,0),(s8,0))
R7((s5,−0.1340),(s7,0.2730)) ((s4,−0.3785),(s7,0.2604)) ((s7,0.1988),(s2,0.3094))
R8((s6,−0.0415),(s5,−0.1043)) ((s7,−0.0291),(s3,0.2056)) ((s7,0.2309),(s2,0.3916))
R9((s4,0.4567),(s7,0.1722)) ((s0,0),(s8,0)) ((s7,−0.0436),(s3,−0.2953))
G4G5G6
R1((s6,−0.4600),(s5,0.3085)) ((s4,0.2348),(s7,0.1206)) ((s7,−0.1581),(s4,0.0392))
R2((s7,−0.4190),(s4,0.1904)) ((s5,0.3148),(s7,−0.2517)) ((s6,0.3139),(s4,0.1675))
R3((s5,0.0255),(s7,−0.3929)) ((s5,0.2969),(s6,−0.4309)) ((s5,0.3897),(s7,−0.3847))
R4((s5,−0.3703),(s7,0.1945)) ((s5,0.3582),(s6,−0.2940)) ((s6,−0.0676),(s,−0.2575))
R5((s4,0.0599),(s7,0.2022)) ((s4,0.2756),(s7,0.1176)) ((s6,0.2501),(s4,0.2387))
R6((s6,0.1459),(s5,0.0155)) ((s4,0.1177),(s7,0.2162)) ((s6,0.0339),(s5,−0.4345))
R7((s7,−0.2744),(s4,0.0196)) ((s7,−0.3088),(s4,−0.0700)) ((s5,−0.2523),(s7,0.1062))
R8((s4,−0.0354),(s7,0.2039)) ((s7,−0.0325),(s4,−0.4910)) ((s5,−0.0211),(s6,−0.1888))
R9((s5,0.3467),(s6,−0.2952)) ((s6,0.2762),(s4,0.4843)) ((s0,0),(s8,0))
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Step 3. Calculate the NIS by Eq (5.16).
NIS ={((s0,0),(s8,0)),((s0,0),(s8,0)),((s0,0),(s8,0)),((s4,−0.0354),(s7,0.2039))
((s4,0.1177),(s7,0.2162)),((s0,0),(s8,0))}.
Step 4. Calculate the H Dtand E Dt:
HD1=2.9440,HD2=2.5988,HD3=3.0646,HD4=2.1374,HD5=2.2474,
HD6=2.3838,HD7=2.6034,HD8=3.1897,HD9=1.7095.
ED1=3.5009,ED2=2.9797,ED3=3.4908,ED4=2.6627,E D5=2.6899,
ED6=2.7760,ED7=2.9501,ED8=3.6246,ED8=1.9711.
Step 5. Determine the RA matrix (see Table 8).
Table 8. Relative assessment matrix (RA).
R1R2R3R4R5R6R7R8R9
R10 0.8665 0.0101 1.6448 1.5077 1.2851 0.8915 −0.3694 2.7643
R2−0.8665 0 −0.9769 0.7783 0.6413 0.4187 0.0251 −1.2359 1.8979
R3−0.0101 0.9769 0 1.7553 1.6182 1.3956 1.0020 −0.2589 2.8748
R4−1.6448 −0.7783 −1.7553 0 −0.1371 −0.3597 −0.7533 −2.0142 1.1195
R5−1.5077 −0.6413 −1.6182 0.1371 0 −0.2226 −0.6162 −1.8771 1.2566
R6−1.2851 −0.4187 −1.3956 0.3597 0.2226 0 −0.3936 −1.6545 1.4792
R7−0.8915 −0.0251 −1.0020 0.7533 0.6162 0.3936 0 −1.2609 1.8728
R80.3694 1.2359 0.2589 2.0142 1.8771 1.6545 1.2609 0 3.1337
R9−2.7643 −1.8979 −2.8748 −1.1195 −1.2566 −1.4792 −1.8728 −3.1337 0
Step 6. Derive the £tby using Eq (5.22). The results of £tare as follows:
£1=8.6007,£2=0.6820,£3=9.3537,£4=−6.3231,£5=−5.0894,
£6=−3.0861,£7=0.4563,£8=11.8046,£9=−16.39873.
Step 7. On the basis of computing results of £t, all the alternatives can be ranked. The ranking of
alternatives is as follows:
R8>R3>R1>R2>R7>R6>R5>R4>R9.
So, R8is the best alternative.
Case 2: The weights of attributes are partly known, and the information of known weights is as follows:
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Ψ = {0.15 ≤$1≤0.2,0.16 ≤$2≤0.18,0.05 ≤$3≤0.15,0.25 ≤$4≤0.35,
0.3≤$5≤0.45,0.09 ≤$6≤0.13,
6
X
=1
$=1}.
To construct the single-objective model, utilize the model (M-2) as follows:
(M−2)
max D($)=17.5771$1+21.0079$2+23.2248$3+15.6182$4+16.1851$5+18.0836$6
s.t.w∈Ψ,w≥0,j=1,2,...,6,
6
P
=1
w=1.
We obtain the optimal weighting vector by solving this model
$=(0.1500,0.1600,0.0500,0.2500,0.3000,0.0900)T.
Step 1. Determine the weighted assessing matrix with 2TLq-ROFNs using Eq (5.14) (see Table 9).
Table 9. Combined weighted assessing matrix with 2TLq-ROFNs.
G1G2G3
R1((s4,0.3805),(s7,0.1469)) ((s4,0.0216),(s7,0.1384)) ((s3,0.4930),(s8,−0.3496))
R2((s4,−0.1115),(s7,0.2509)) ((s5,0.2350),(s6,0.4675)) ((s0,0),(s8,0))
R3((s5,−0.2826),(s7,−0.1423)) ((s4,0.3307),(s7,0.2019)) ((s8,0),(s0,0))
R4((s0,0),(s8,0)) ((s5,−0.1305),(s7,−0.4475)) ((s3,−0.3221),(s8,−0.2137))
R5((s5,−0.2428),(s7,−0.0783)) ((s3,0.0013),(s8,−0.1927)) ((s3,0.3777),(s8,−0.3066))
R6((s5,−0.3257),(s7,−0.1476)) ((s5,−0.1706),(s7,−0.0775)) ((s0,0),(s8,0))
R7((s3,0.4426),(s8,−0.3220)) ((s2,0.4665),(s8,−0.1625)) ((s4,0.0118),(s7,0.4271))
R8((s4,0.3003),(s7,0.1209)) ((s5,0.1080),(s7,−0.4073)) ((s4,0.0193),(s7,0.4427))
R9((s3,0.1395),(s8,0.2045)) ((s0,0),(s8,0)) ((s4,−0.2276),(s7,0.4976))
G4G5G6
R1((s5,−0.3960),(s7,−0.3335)) ((s4,−0.3952),(s8,−0.4655)) ((s5,−0.4962),(s7,0.2788))
R2((s6,−0.4103),(s6,0.0011)) ((s5,−0.4401),(s7,0.3290)) ((s4,0.0492),(s7,0.3103))
R3((s4,0.1512),(s7,0.3477)) ((s5,−0.4562),(s7,−0.3610)) ((s3,0.3673),(s8,−0.2075))
R4((s4,−0.1886),(s8,−0.3687)) ((s5,−0.4008),(s7,−0.2775)) ((s4,−0.2445),(s7,0.4421))
R5((s3,0.3309),(s8,−0.3651)) ((s4,−0.3597),(s8,−0.4672)) ((s4,−0.0015),(s7,0.3274))
R6((s5,0.1621),(s7,−0.4997)) ((s4,−0.4967),(s8,−0.4136)) ((s4,−0.1684),(s7,0.4031))
R7((s6,−0.2606),(s6,0.1089)) ((s6,−0.1166),(s6,−0.4517)) ((s3,−0.0639),(s8,−0.1300))
R8((s3,0.2513),(s8,−0.3643)) ((s6,0.1821),(s5,0.2339)) ((s3,0.0888),(s8,−0.3458))
R9((s4,0.4321),(s7,−0.1166)) ((s5,0.4627),(s6,0.0617)) ((s0,0),(s8,0))
Step 3. Calculate the NIS by Eq (5.16).
NIS ={((s0,0),(s8,0)),((s0,0),(s8,0)),((s0,0),(s8,0)),((s4,−0.0354),(s7,0.2039))
((s4,0.1177),(s7,0.2162)),((s0,0),(s8,0))}.
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Step 4. Calculate the H Dtand E Dt.
HD1=2.9440,HD2=2.5988,HD3=3.0646,HD4=2.1374,HD5=2.2474,
HD6=2.3838,HD7=2.6034,HD8=3.1897,HD9=1.7095.
ED1=3.5009,ED2=2.9797,ED3=3.4908,ED4=2.6627,E D5=2.6899,
ED6=2.7760,ED7=2.9501,ED8=3.6246,ED9=1.9711.
Step 5. Determine the RA matrix (see Table 10).
Table 10. Relative assessment matrix (RA).
R1R2R3R4R5R6R7R8R9
R10 0.8665 0.0101 1.6448 1.5077 1.2851 0.8915 −0.3694 2.7643
R2−0.8665 0 −0.9769 0.7783 0.6413 0.4187 0.0251 −1.2359 1.8979
R3−0.0101 0.9769 0 1.7553 1.6182 1.3956 1.0020 −0.2589 2.8748
R4−1.6448 −0.7783 −1.7553 0 −0.1371 −0.3597 −0.7533 −2.0142 1.1195
R5−1.5077 −0.6413 −1.6182 0.1371 0 −0.2226 −0.6162 −1.8771 1.2566
R6−1.2851 −0.4187 −1.3956 0.3597 0.2226 0 −0.3936 −1.6545 1.4792
R7−0.8915 −0.0251 −1.0020 0.7533 0.6162 0.3936 0 −1.2609 1.8728
R80.3694 1.2359 0.2589 2.0142 1.8771 1.6545 1.2609 0 3.1337
R9−2.7643 −1.8979 −2.8748 −1.1195 −1.2566 −1.4792 −1.8728 −3.1337 0
Step 6. Derive the £tby using Eq (5.22). The results of £tare as follows:
£1=8.6007,£2=0.6820,£3=9.3537,£4=−6.3231,£5=−5.0894,
£6=−3.0861,£7=0.4563,£8=11.8046,£9=−16.3987.
Step 7. On the basis of computing results of £t, all the alternatives can be ranked. The ranking of
alternatives is as follows:
R8>R3>R1>R2>R7>R6>R5>R4>R9.
So, R8is the best alternative.
6.1.2. Decision-making procedure based on the 2TLq-ROFWDHM operator
The MAGDM technique to select best arc welding robot involves the following cases:
Case 1: Assume that the information about the attribute weights is completely unknown: Utilize the
Eq (5.12) to get the optimal weight vector $=(0.1574,0.1881,0.2079,0.1398,0.1449,0.1619)T.
Step 1. Individual expert assessments can be integrated into the collective assessing matrix with 2TLq-
ROFNs, according to Tables 4 and 5 and Eq (4.7) (q=4 and κ=3) (see Table 11).
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Table 11. Combined assessing matrix with 2TLq-ROFNs utilizing 2TLq-ROFWDHM
operator.
G1G2G3
R1((s6,−0.0961),(s6,−0.3603)) ((s2,0.1718),(s7,0.1783)) ((s4,0.0624),(s6,0.2396))
R2((s3,−0.3998),(s7,0.0990)) ((s5,−0.2385),(s5,0.4972)) ((s1,0.4422),(s8,−0.2737))
R3((s4,0.2422),(s6,0.0676)) ((s4,0.3482),(s6,0.1491)) ((s8,0),(s0,0))
R4((s6,−0.1292),(s6,−0.1601)) ((s3,0.4843),(s6,0.3538)) ((s2,−0.4878),(s7,0.4843))
R5((s7,−0.0444),(s4,0.2884)) ((s6,0.3337),(s6,−0.4017)) ((s6,−0.0684),(s5,−0.4954))
R6((s4,−0.2807),(s6,0.3164)) ((s7,−0.3151),(s5,−0.0977)) ((s1,−0.0599),(s8,−0.2005))
R7((s4,0.0689),(s6,0.2841)) ((s2,0.0635,(s7,0.4490)) ((s4,0.4322),(s6,−0.2882))
R8((s3,0.4526),(s7,−0.3928)) ((s6,0.4542),(s5,−0.110)) ((s5,−0.2896),(s6,−0.4666))
R9((s5,0.2397),(s6,0.3559)) ((s5,0.3566),(s6,0.1203)) ((s4,−0.2063),(s6,0.1743))
G4G5G6
R1((s3,−0.2832),(s7,−0.0381)) ((s3,0.2726),(s7,−0.1462)) ((s8,0),(s0,0))
R2((s6,0.4719),(s5,−0.0714)) ((s7,−0.3232),(s5,−0.1548)) ((s3,0.3950),(s6,0.4988))
R3((s4,−0.2593),(s7,−0.4324)) ((s3,−0.4913),(s7,0.1456)) ((s7,−0.1266),(s5,−0.3632))
R4((s4,0.4649),(s6,0.3020)) ((s3,0.3650),(s7,−0.1510)) ((s3,0.1467),(s7,−0.2130))
R5((s3,−0.0851),(s7,0.0082)) ((s4,−0.4871),(s7,−0.3113)) ((s4,−0.4392),(s6,0.4802))
R6((s8,0),(s0,0) ((s2,0.3524),(s7,0.3384)) ((s3,0.1466),(s7,−0.2670))
R7((s7,−0.2974),(s5,−0.3171)) ((s6,0.4789),(s5,−0.1396)) ((s7,−0.1059),(s5,−0.3389))
R8((s3,−0.4455),(s7,0.1303)) ((s8,0,(s0,0) ((s3,−0.2232),(s7,0.1423))
R9((s3,0.1835),(s7,0.1663)) ((s4,0.1105),(s6,0.1671)) ((s2,0.1042),(s7,0.4790))
Step 2. Determine the weighted assessing matrix with 2TLq-ROFNs (see Table 12).
Table 12. Weighted assessing matrix with 2TLq-ROFNs.
G1G2G3
R1((s7,−0.3993),(s5,0.0934)) ((s3,−0.0159),(s7,−0.1225)) ((s5,−0.4604),(s6,0.0193))
R2((s4,−0.0714),(s7,−0.4190)) ((s5,0.4034),(s5,0.1645)) ((s2,−0.0902),(s8,−0.3910))
R3((s5,0.3549),(s6,−0.4932)) ((s5,0.0447),(s6,−0.1950)) ((s8,0),(s0,0))
R4((s7,−0.4227),(s5,0.2854)) ((s4,0.2667),(s6,0.0106)) ((s2,−0.0130),(s7,0.3257))
R5((s7,0.3222),(s4,−0.1602)) ((s7,−0.2953),(s5,0.2628)) ((s6,0.2297),(s5,0.2811))
R6((s5,−0.0728),(s6,−0.2464)) ((s7,−0.0160),(s5,−0.4076)) ((s1,0.3354),(s8,−0.2997))
R7((s5,0.2154),(s6,−0.2788)) ((s3,−0.1290),(s7,0.1869)) ((s5,−0.1174),(s5,0.4940))
R8((s5,−0.2994),(s6,0.0500)) ((s7,−0.1991),(s5,−0.4203)) ((s5,0.1376),(s5,0.3183))
R9((s6,0.1205),(s6,−0.2067)) ((s6,−0.0101),(s6,−0.2236)) ((s4,0.2872),(s6,−0.0462))
G4G5G6
R1((s4,0.3587),(s6,0.2841)) ((s5,−0.2479),(s6,0.2093)) ((s8,0),(s0,0))
R2((s7,0.1011),(s4,0.3037)) ((s7,0.2000),(s5,−0.4528)) ((s5,−0.4211),(s6,−0.0267))
R3((s5,0.2174),(s6,−0.1365)) ((s4,0.0702),(s7,−0.4642)) ((s7,0.2473),(s4,0.0.1865))
R4((s6,−0.2367),(s6,0.4050)) ((s5,−0.1702),(s6,0.2042)) ((s4,0.3580),(s6,0.2737))
R5((s5,−0.4653),(s6,0.3357)) ((s5,−0.0475),(s6,0.0333)) ((s5,−0.2768),(s6,−0.0457))
R6((s8,0),(s0,0) ((s4,−0.0795),(s7,−0.2330)) ((s4,0.3579),(s6,0.2165))
R7((s7,0.2423),(s4,0.0825)) ((s7,0.0749),(s4,0.2785)) ((s7,0.2615),(s4,0.2088))
R8((s4,0.2103),(s6,0.4752)) ((s8,0),(s0,0)) ((s4,0.0172),(s7,−0.3362))
R9((s5,−0.2350),(s6,0.1438)) ((s5,0.4272),(s6,−0.4952)) ((s3,0.3536),(s7,0.0678))
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Step 3. Calculate the NIS by Eq (5.16).
NIS ={((s4,−0.0714),(s7,−0.4190),(s3,−0.1290)),((s7,0.1869),(s1,0.3354),(s8,−0.2997)),
((s4,0.2103),(s6,0.4752),(s4,−0.0795)),((s7,−0.2330),(s3,0.3536),(s7,0.0678))}.
Step 4. Calculate the H Dtand E Dt.
HD1=1.6508,HD2=1.4910,HD3=2.1233,HD4=1.0588,HD5=1.8788,
HD6=1.5187,HD7=2.1069,HD8=1.8443,HD9=1.1751.
ED1=1.9154,ED2=1.6925,ED3=2.2336,ED4=1.2755,E D5=2.0536,
ED6=1.7067,ED7=2.3246,ED8=2.1120,ED9=1.3759.
Step 5. Determine the RA matrix (see Table 13).
Table 13. Relative assessment matrix (RA).
.
R1R2R3R4R5R6R7R8R9
R10 0.3827 −0.7907 1.2319 −0.3662 0.3409 −0.8652 −0.3901 1.0152
R2−0.3827 0 −1.1735 0.8492 −0.7489 −0.0142 −1.2480 −0.7729 0.6325
R30.7907 1.1735 0 2.0226 0.4245 1.1316 −0.0745 0.4006 1.8059
R4−1.2319 −0.8492 −2.0226 0 −1.5981 −0.8910 −2.0972 −1.6220 −0.2167
R50.3662 0.7489 −0.4245 1.5981 0 0.7071 −0.4990 −0.0239 1.3814
R6−0.3409 0.0142 −1.1316 0.8910 −0.7071 0 −1.2061 −0.7310 0.6743
R70.8652 1.2480 0.0745 2.0972 0.4990 1.2061 0 0.4751 1.8804
R80.3901 0.7729 −0.4006 1.6220 0.0239 0.7310 −0.4751 0 1.4053
R9−1.0152 −0.6325 −1.8059 0.2167 −1.3814 −0.6743 −1.8804 −1.4053 0
Step 6. Derive the £tby using Eq (5.22). The results of £tare as follows:
£1=0.5583,£2=−2.8586,£3=7.6750,£4=−10.5288,£5=3.8543,
£6=−2.5370,£7=8.3456,£8=4.0695,£9=−8.5783.
Step 7. On the basis of computing results of £t, all the alternatives can be ranked. The ranking of
alternatives is as follows:
R7>R3>R8>R5>R1>R6>R2>R9>R4.
So, R7is the best alternative.
Case 2: The weights of attributes are partly known, and the information of known weights is as
follows:
Ψ = {0.15 ≤$1≤0.2,0.16 ≤$2≤0.18,0.05 ≤$3≤0.15,0.25 ≤$4≤0.35,
0.3≤$5≤0.45,0.09 ≤$6≤0.13,
6
X
=1
$=1}.
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To construct the single-objective model, utilize the model (M-2) as follows:
(M−2)
max D($)=17.5771$1+21.0079$2+23.2248$3+15.6182$4+16.1851$5+18.0836$6
s.t.w∈ =,w≥0,j=1,2,...,6,
6
P
=1
w=1.
We obtain the optimal weighting vector by solving this model
$=(0.1500,0.1600,0.0500,0.2500,0.3000,0.0900)T.
Step 1. Determine the weighted assessing matrix with 2TLq-ROFNs using Eq (5.14) (see Table 14).
Table 14. Combined weighted assessing matrix with 2TLq-ROFNs.
G1G2G3
R1((s8,−0.3564),(s4,−0.3858)) ((s6,0.4936),(s5,0.0110)) ((s8,−0.2665),(s3,0.1101))
R2((s7,−0.2411),(s5,−0.1498)) ((s7,0.3627),(s4,−0.4322)) ((s7,0.3432),(s4,0.4644))
R3((s7,0.2739),(s4,0.0657)) ((s7,0.2565),(s4,0.0607)) ((s8,0),(s0,0))
R4((s8,−0.3629),(s4,−0.2386)) ((s7,0.0038),(s4,0.2273)) ((s7,0.3607),(s4,0.1157))
R5((s8,−0.1661),(s3,−0.3069)) ((s8,−0.2934),(s4,−0.3587)) ((s8,−0.1188),(s3,−0.3242))
R6((s7,0.1317),(s4,0.1317)) ((s8,−0.2266),(s3,0.1496)) ((s7,0.1878),(s5,−0.3895))
R7((s7,−0.2285),(s4,0.1055)) ((s6,0.4407),(s5,0.3486)) ((s8,−0.2328),(s3,−0.2034))
R8((s7,0.0526),(s4,0.3772)) ((s8,−0.2702),(s3,0.1405)) ((s8,−0.2091),(s3,−0.3033))
R9((s8,−0.4920),(s4,0.1640)) ((s8,−0.4750),(s4,0.0379)) ((s8,−0.2929),(s3,0.0695))
G4G5G6
R1((s6,0.1071),(s5,0.2954)) ((s6,0.1183),(s5,0.3971)) ((s8,0),(s0,0))
R2((s8,−0.4129),(s4,0.4646)) ((s8,0.4224),(s4,−0.1227)) ((s7,0.4061),(s4,−0.2140))
R3((s7,0.3846),(s5,−0.1023)) ((s6,−0.3507),(s6,−0.2773)) ((s8,−0.1085),(s3,−0.4259))
R4((s7,−0.0854),(s5,0.3467)) ((s6,0.1696),(s5,0.3921)) ((s7,0.3556),(s4,0.0171))
R5((s6,0.2155),(s5,0.3451)) ((s6,0.2498),(s5,0.2271)) ((s7,0.4379),(s4,−0.2282))
R6((s8,0),(s0,0)) ((s6,−0.4586),(s6,−0.0364)) ((s7,0.3556),(s4,−0.0279))
R7((s8,−0.3462),(s3,0.3498)) ((s8,−0.4904),(s4,−0.3572)) ((s8,−0.1064),(s3,−0.4116))
R8((s6,0.0137),(s5,0.4829)) ((s8,0),(s0,0)) ((s7,0.2733),(s4,0.3421))
R9((s6,0.3539),(s5,0.1601)) ((s7,−0.4486),(s5,0.2670)) ((s7,0.0940),(s5,−0.2737))
Step 2. Calculate the NIS by Eq (5.16).
NIS ={((s7,−0.2411),(s5,−0.1498)),((s6,0.4407),(s5,0.3486)),((s7,0.1878),(s5,−0.3895)),
((s6,0.0137),(s5,0.4829))((s6,−0.4586),(s6,−0.0364)),((s7,0.0940),(s5,−0.2737))}.
Step 3. Calculate the H Dtand E Dt.
HD1=0.7762,HD2=1.1088,HD3=0.9165,HD4=0.7636,HD5=1.0274,
HD6=0.9459,HD7=1.2580,HD8=1.1456,HD9=0.8280.
ED1=0.9534,ED2=1.3068,ED3=1.1052,ED4=0.8774,E D5=1.2471,
ED6=1.1608,ED7=1.5152,ED8=1.3791,ED9=0.9814.
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Step 4. Determine the RA matrix (see Table 15).
Table 15. Relative assessment matrix (RA).
R1R2R3R4R5R6R7R8R9
R10−0.6860 −0.2921 0.0886 −0.5449 −0.3772 −1.0436 −0.7952 −0.0798
R20.6860 0 0.3939 0.7745 0.1410 0.3088 −0.3576 −0.1093 0.6062
R30.2921 −0.3939 0 0.3807 −0.2529 −0.0851 −0.7515 −0.5032 0.2123
R4−0.0886 −0.7745 −0.3807 0 −0.6335 −0.4658 −1.1322 −0.8838 −0.1684
R50.5449 −0.1410 0.2529 0.6335 0 0.1677 −0.4987 −0.2503 0.4652
R60.3772 −0.3088 0.0851 0.4658 −0.1677 0 −0.6664 −0.4180 0.2974
R71.0436 0.3576 0.7515 1.1322 0.4987 0.6664 0 0.2484 0.9638
R80.7952 0.1093 0.5032 0.8838 0.2503 0.4180 −0.2484 0 0.7155
R90.0798 −0.6062 −0.2123 0.1684 −0.4652 −0.2974 −0.9638 −0.7155 0
Step 5. Derive the £tby using Eq (5.22). The results of £tare as follows:
£1=−3.7303,£2=2.4435,£3=−1.1015,£4=−4.5274,£5=1.1742,
£6=−0.3353,£7=5.6622,£8=3.4269,£9=−3.0122.
Step 6. On the basis of computing results of £t, all the alternatives can be ranked. The ranking of
alternatives is as follows:
R7>R8>R2>R5>R6>R3>R9>R1>R4.
So, R7is the best alternative.
6.2. Parameter analysis
The impact of qand κon arc welding robot selection is investigated in this section. First, as indicated
in Tables 16 and 17, we find the average solutions of the arc welding robots as qvalues vary (from 1
to 8) (κ=3) in the 2TLq-ROFWHM operator. After altering q, Tables 18 and 19 show how the average
solutions of the alternatives differ and ranking on the basis of the average solutions shown in Tables 16
and 17. If the DM wants to make a judgement based on complicated data, just increase qto enlarge the
information representation space of 2TLq-ROFS. Effect of variation of qand κon the 2TLq-ROFWHM
operator is shown in Figures 2 and 3, respectively.
To investigate the effects of the parameter κon arc welding robot selection and decision outcomes
in depth. Tables 20 and 21 (q=4) show the average solutions obtained after adjusting the values of κ
in both the 2TLq-ROFWHM and 2TLq-ROFWDHM operators. The values of average solutions differ
when the parameter κin the 2TLq-ROFWDHM operator is changed, as shown in Tables 20 and 21,
although the ranking orders are essentially the same in most cases shown in Tables 22 and 23. Effect
of variation of qand κon the 2TLq-ROFWDHM is shown in Figures 4 and 5, respectively. When
the parameters based on the 2TLq-ROFWHM and 2TLq-ROFWDHM operators are changed, both the
score values and alternative ranking change, indicating that the parameter κinfluences the arc welding
robot selection assessment process.
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Table 16. Average solutions with different parameter qin 2TLq-ROFWHM operator.
Parameter £1£2£3£4£5£6£7£8£9
q=1 9.2419 0.4522 8.0325 −5.5686 −5.6983 −2.6495 −0.7889 12.5685 −15.5897
q=2 8.3757 0.8741 8.8384 −6.4119 −5.5053 −2.9184 0.4589 12.1618 −15.8735
q=3 8.6811 0.4570 9.9984 −6.2013 −4.6643 −3.2330 0.3786 11.4614 −16.8780
q=4 9.2018 0.1342 10.9091 −6.9945 −3.6352 −3.6145 0.4292 11.1241 −17.5541
q=5 9.0140 −0.0512 11.6425 −7.1200 −3.0797 −4.0100 0.5400 11.0921 −18.0277
q=6 9.4112 −0.4079 12.0041 −7.2024 −2.5031 −4.4406 0.7780 10.7475 −18.3869
q=7 9.0445 −0.8170 12.9281 −7.2637 −1.8192 −4.8432 0.9374 10.5033 −18.6703
q=8 9.0544 −0.9737 13.3726 −7.3163 −1.3976 −5.2068 1.1006 10.2916 −18.9248
Table 17. Average solutions with different parameter κin 2TLq-ROFWHM operator.
Parameter £1£2£3£4£5£6£7£8£9
κ=1−1.2161 −0.3893 1.6164 −6.7267 5.3976 −7.5968 8.8638 0.7522 −0.7012
κ=2 6.3461 −1.8035 8.3705 −7.0144 −3.2918 0.3788 2.6722 9.5008 −15.1586
κ=3 8.6007 0.6820 9.3537 −6.3231 −5.0894 −3.0861 0.4563 11.8046 −16.3987
κ=4 11.5855 2.7295 1.3611 −2.0245 −11.4463 0.5527 −6.1084 15.4832 −12.1329
Table 18. Alternative ranking with different parameter qin 2TLq-ROFWHM operator.
Parameter Ranking
q=1R8>R1>R3>R2>R7>R6>R5>R4>R9
q=2R8>R3>R1>R2>R7>R6>R5>R4>R9
q=3R8>R3>R1>R2>R7>R6>R5>R4>R9
q=4R8>R3>R1>R7>R2>R6>R5>R4>R9
q=5R8>R3>R1>R7>R2>R5>R6>R4>R9
q=6R8>R3>R1>R7>R2>R5>R6>R4>R9
q=7R8>R3>R1>R7>R2>R5>R6>R4>R9
q=8R8>R3>R1>R7>R2>R5>R6>R4>R9
Table 19. Alternative ranking with different parameter κin 2TLq-ROFWHM operator.
Parameter Ranking
κ=1R7>R5>R3>R8>R2>R9>R1>R4>R6
κ=2R8>R3>R1>R7>R6>R2>R5>R4>R9
κ=3R8>R3>R1>R2>R7>R6>R5>R4>R9
κ=4R8>R1>R2>R3>R6>R4>R7>R5>R9
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Figure 2. Variation of qin 2TLq-ROFWHM operator.
Figure 3. Variation of κin 2TLq-ROFWHM operator.
Table 20. Average solutions with different parameter qin 2TLq-ROFWDHM operator.
Parameter £1£2£3£4£5£6£7£8£9
q=1 0.1530 −3.1792 7.1507 −12.0965 5.5735 −1.8540 8.0698 3.1966 −7.0140
q=2 0.1057 −2.9132 7.8509 −11.0315 4.3727 −2.5697 8.5268 3.7767 −8.1183
q=3 1.0370 −2.8208 7.4580 −10.1699 3.4083 −2.4021 8.1027 4.3676 −8.9809
q=4 2.0292 −2.8761 6.8321 −9.6699 2.5850 −1.9598 7.6085 5.0272 −9.5762
q=5 2.6374 −2.9743 6.9033 −9.3430 2.0591 −1.4359 7.1377 4.9781 −9.9625
q=6 3.1882 −3.1053 6.7752 −9.1052 1.5023 −0.8554 6.6762 5.1364 −10.2125
q=7 3.9542 −3.2434 6.3435 −8.9285 1.0057 −0.2839 6.2217 5.2570 −10.3264
q=8 4.2588 −3.3906 6.2811 −8.8002 0.5206 0.2603 5.7110 5.4004 −10.2414
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Table 21. Average solutions with different parameter κin 2TLq-ROFWDHM operator.
Parameter £1£2£3£4£5£6£7£8£9
κ=1 3.4005 2.5863 1.4628 −5.4874 −8.1530 2.2088 −0.6670 12.0452 −7.3961
κ=2 3.7906 −0.9149 11.8515 −14.4767 1.8085 −0.3653 6.5394 9.4213 −17.6544
κ=3 0.5583 −2.8586 7.6750 −10.5288 3.8543 −2.5370 8.3456 4.0695 −8.5783
κ=4−2.6107 −3.5564 1.8935 −13.2726 12.1627 −7.5384 11.5950 0.5158 0.8112
Table 22. Alternative ranking with different parameter qin 2TLq-ROFWHM operator.
Parameter Ranking
q=1R7>R3>R5>R8>R1>R6>R2>R9>R4
q=2R7>R3>R5>R8>R1>R6>R2>R9>R4
q=3R7>R3>R8>R5>R1>R6>R2>R9>R4
q=4R7>R3>R8>R5>R1>R6>R2>R9>R4
q=5R7>R3>R8>R1>R5>R6>R2>R4>R9
q=6R3>R7>R8>R1>R5>R6>R2>R4>R9
q=7R3>R7>R8>R1>R5>R6>R2>R4>R9
q=8R3>R7>R8>R1>R5>R6>R2>R4>R9
Table 23. Alternative ranking with different parameter κin 2TLq-ROFWDHM operator.
Parameter Ranking
κ=1R8>R1>R2>R6>R3>R7>R4>R9>R5
κ=2R3>R8>R7>R1>R5>R6>R2>R4>R9
κ=3R7>R3>R8>R5>R1>R6>R2>R9>R4
κ=4R5>R7>R3>R9>R8>R1>R2>R6>R4
Figure 4. Variation of qin 2TLq-ROFWDHM operator.
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Figure 5. Variation of κin 2TLq-ROFWDHM operator.
6.3. Comparative analysis
In this subsection, we use certain validated approaches to cope with the proposed MAGDM problem
and analyze the outcomes with our developed framework to check its feasibility and effectiveness.
We carefully compute the evaluation outcomes for the selection of arc welding robots utilizing these
strategies. Tables 24 and 25 illustrated by Figures 6 and 7, respectively, summarize the output of the
comparisons among the developed CODAS method and existing EDAS and TOPSIS methods.
Table 24. Evaluation outcomes utilizing different methodologies based on 2TLq-ROFWHM
operator.
Alternatives EDAS Ranking CODAS Ranking TOPSIS Ranking
R10.6255 III 8.6007 III 0.6850 III
R20.7340 II 0.6820 IV 0.6278 IV
R30.5452 VI 9.3537 II 0.7280 II
R40.3315 VIII −6.3231 VIII 0.5059 VIII
R50.4210 VII −5.0894 VII 0.5399 VII
R60.5998 IV −3.0861 VI 0.5812 VI
R70.5728 V 0.4563 V 0.6136 V
R80.8143 I 11.8046 I 0.7400 I
R90.1015 IX −16.3987 IX 0.4010 IX
Table 25. Evaluation outcomes utilizing different methodologies based on 2TLq-
ROFWDHM operator.
Alternatives EDAS Ranking CODAS Ranking TOPSIS Ranking
R10.4897 VI 0.5583 V 0.3909 V
R20.2840 VII −2.8586 VII 0.3490 VII
R30.8716 I 7.6750 II 0.4678 II
R40.2296 IX −10.5288 IX 0.2593 IX
R50.6853 II 3.8543 IV 0.4197 IV
R60.2521 VIII −2.5370 VI 0.3634 VI
R70.5000 IV 8.3456 I 0.4727 I
R80.5000 III 4.0695 III 0.4274 III
R90.5000 V −8.5783 VIII 0.2784 VIII
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Figure 6. Comparison of CODAS method based on 2TLq-ROFWHM operator with different
approaches.
Figure 7. Comparison of CODAS method based on 2TLq-ROFWDHM operator with
different approaches.
There is some variation in the ranking order of the alternatives due to the basic behavior of the
various aggregation methods. However, in most cases, the most acceptable alternatives are the same
for the existing method and the proposed method, as given in Tables 24 and 25 and shown in Figures 6
and 7, respectively. Therefore, by comparing the results of EDAS and TOPSIS methods, we can
conclude that R8and R7are the best arc welding robots.
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7. Conclusions
The CODAS ranking method is very useful and efficient when dealing with complex MAGDM
difficulties. Some experts use it to evaluate a handful of alternatives by using various properties. In
this paper, we have proposed AO and extended the CODAS method to MAGDM with 2TLq-ROFS,
based on two different types of distance measurements. The main advantage of the proposed technique
compared to techniques already in use is that it not only addresses 2TLq-ROFS, but also has a strong
ability to identify the best alternatives. We have developed 2TLq-ROFS as a new advance in FS theory
for conveying data complexity. 2TLq-ROFS has involved the integration of 2TL terms and q-ROF sets,
increasing the adaptability of q-ROFS. Inspired by traditional AO, we have proposed two aggregations
(2TLq-ROFHM and 2TLq-ROFWHM operators) to aggregate 2TLq-ROFS, and further have explored
their basic features. We have devised a technique called the maximizing deviation method to discover
ideal relative weights for attributes in linguistic contexts, with the premise that attributes with larger
deviation values among the alternatives should be considered to have larger weights. The distinctive
feature of this development is that it can reduce the influence of the subjectivity of decision makers
and make full use of decision information. Furthermore, the CODAS method is extended to solve
the MAGDM challenge using 2TLq-ROFS, which can fully consider both ED and HD. Finally, a
practical example is given to demonstrate the suggested method for evaluating and selecting an arc
welding robot. We have also examined the influence of different parameters on the selection of the
arc welding robot. The proposed method is also compared with the EDAS and TOPSIS methods to
demonstrate their advantages and efficacy. The four main contributions of this study are as follows: (1)
The development of 2TLq-ROFS; (2) the extension of the classical CODAS method to 2TLq-ROFS;
(3) the CODAS of the 2TLq-ROFS MAGDM problem method design to provide DM with an effective
way to solve MAGDM problems; and (4) present a case study on the evaluation and selection of arc
welding robots to demonstrate the applicability, feasibility and effectiveness of the proposed MAGDM
method. We will continue to extend our proposed model to other ambiguous cases and application
domains for the next study.
Acknowledgements
The fourth author extends his appreciation to the Deanship of Scientific Research at King Khalid
University for funding this work through the General Research Project under grant number
(R.G.P.2/48/43).
Conflict of interest
The authors declare no conflicts of interest.
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