Neu osophic Se s and Sys ems, Vol. 97, 2026
Uni e si y o New Mexico
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-Valued
In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
Mul i-Valued In e al Neu osophic So Se s and Thei
Agg ega ion Ope a o s
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and
Khalid Naeem5
1, College o Compu ing, In o ma ics and Ma hema ics, Uni e si i Teknologi MARA, Nege i Sembilan
B anch, Rembau Campus, 71300 Rembau, Nege i Sembilan, Malaysia; liyanakamal@ui m.edu.my
2Facul y o Compu e Science and Ma hema ics, Uni e si i Malaysia Te engganu,
21030 Kuala Te engganu, Te engganu, Malaysia; lazi[email p o ec ed]
3Facul y o Compu e Science and Ma hema ics, Uni e si i Malaysia Te engganu,
21030 Kuala Te engganu, Te engganu, Malaysia; ilya[email p o ec ed]
4Depa men o Ma hema ics, Gazian ep Uni e si y, Gazian ep 27310, Tu key;
[email p o ec ed]
5Depa men o Ma hema ics & S a is ics, Uni e si y o Laho e, Pakis an;
[email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : Recen s udies ha e inc easingly ocused on agg ega ion wi hin he neu osophic
en i onmen due o i s capabili y o handle ambigui y and unce ain y. Howe e , o he au ho s'
in o ma ion, no cu en s udy has in es iga ed agg ega ion ope a o s o mul i- alued in e al
neu osophic so numbe s (MVINSSs) in he con ex o al e na i e anking o decision-making
p oblems. This pape p oposes wo no el agg ega ion ope a o s: he mul i- alued in e al
neu osophic so -weigh ed geome ic a e aging (MVINSWGA) and he mul i- alued in e al
neu osophic so -weigh ed a i hme ic a e aging (MVINSWAA) ope a o s unde he MVINSS
amewo k. The undamen al p ope ies o he p oposed ope a o s, including idempo ency,
mono onici y, and boundedness, a e es ablished. In addi ion, a s uc u ed mul i-c i e ia g oup
decision-making (MCGDM) p ocedu e inco po a ing he p oposed ope a o s is in oduced. A
nume ical example in ol ing so wa e selec ion is p o ided o illus a e he applicabili y o he
sugges ed app oach. Compa a i e analysis con i ms he consis ency o anking esul s, indica ing
ha he MVINSWGA and MVINSWAA ope a o s a e obus and e ec i e in add essing MCGDM
p oblems wi hin he MVINSS en i onmen .
Keywo ds: a i hme ic agg ega ion; geome ic agg ega ion; mul i- alued neu osophic se ; so se ;
decision-making
1. In oduc ion
Classical se heo y e ec i ely models p oblems cha ac e ized by de e minacy and p ecision.
Howe e , i lacks he capabili y o manage he unce ain y and imp ecision equen ly encoun e ed
Neu osophic Se s and Sys ems, Vol. 97, 2026 661
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
in eal-wo ld si ua ions. To add ess his sho coming, se e al ma hema ical models ha e been
in oduced, including as uzzy se s [1], in e al- alued uzzy se s [2], in ui ionis ic uzzy se s (IFS)
[3], ague se s [4] and ough se s [5]. Despi e hei con ibu ions, hese models a e o en limi ed by
insu icien pa ame e iza ion [6].
To o e come his, Molod so p oposed he so se (SS) heo y [6], which emphasizes
pa ame e iza ion in decision app oxima ion a he han elying solely on membe ship unc ions.
Since i s incep ion, SS heo y has been applied o di e se a eas such as in eg a ion [7], op imiza ion
[8], game heo y [9], [10], la ice heo y [11-13], algeb aic s uc u es [14], [15], opology [16-18], da a
analysis and ope a ions esea ch [19-22], medical diagnosis [23], and decision-making unde
unce ain y [24-28].
In pa allel, Zadeh's uzzy se heo y [1] in oduced he concep o uzziness, enabling he handling
o imp ecise in o ma ion. Maji e al. [29] la e in eg a ed FS wi h SS o p opose he uzzy so se (FSS),
which o e s a amewo k o ep esen uzzy in o ma ion wi h pa ame e iza ion. FSS has been widely
explo ed [30-32] and applied in a eas such as o ecas ing [33], medicine [34], and lood p edic ion
[35].
To enhance FS u he , A anasso in oduced IFS [3], which inco po a es dual membe ship
unc ions— u h and alsi y—allowing simul aneous ep esen a ion o membe ship and non-
membe ship deg ees. This led o he de elopmen o he in ui ionis ic uzzy so se (IFSS) by
in eg a ing IFS and SS [36], wi h se e al s udies ollowing [37-41]. Howe e , IFS is cons ained by
he dependency be ween membe ship alues, whe e he sum o u h and alsi y is less han 1.
To add ess his, Sma andache [42] in oduced neu osophic se (NS) heo y, which he symbols
,
and
a e used o ep esen u h-membe ship unc ion, inde e minacy-membe ship unc ion and
alsi y-membe ship unc ion espec i ely, wi h each membe ship unc ion anging wi hin he non-
s anda d in e al ]⁻0, 1⁺[. This gene aliza ion allows NS o be e in e p e he ambiguous and
con using da a ha equen ly a ises in ac ual decision-making.
I can be said ha he NS is a new se ha o e comes he limi a ion o IFS. The dual membe ships o
IFSs a e unable o ca e o he inde ini e and ambiguous in o ma ion in which his kind o
in o ma ion always exis s in belie sys ems and decision-making p ocesses. The NS which consis s o
h ee independen membe ships o u h, inde e minacy and alsi y become an enhancemen o he
dual membe ships o IFSs. Fundamen ally, i is he gene aliza ion o he ypical in e al in IFS [3]
which is
[0,1].
Recen yea s ha e seen ac i e de elopmen in he s udy o neu osophic se (NS) heo y [43-47].
Recognizing he limi a ions o classical SS in unce ain con ex s, esea che s in eg a ed NS wi h SS,
o ming he neu osophic so se (NSS) [48]. Nume ous schola s ha e since wo ked on his concep .
[49-52]. This amewo k was la e ex ended in o he in e al- alued neu osophic so se (IVNSS)
[53], enabling in e al-based unce ain y modeling. The in e al- alued neu osophic se (IVNS)
Neu osophic Se s and Sys ems, Vol. 97, 2026 662
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
p oposed by Wang e al. [54] suppo s mo e exp essi e modeling o imp ecise, inadequa e, and
inconsis en da a and has gained a en ion in a ious s udies [55-57].
Meanwhile, Wang and Li ex ended NS in o he mul i- alued neu osophic se (MVNS) [58], whe e
he T, I, and F membe ships a e no limi ed o single alues [59-63]. Alkhazaleh [64] u he combined
MVNS wi h SS o o m he mul i- alued neu osophic so se (MVNSS), sui able o p oblems
in ol ing mul iple unce ain alues [65-68].
Despi e hese de elopmen s, challenges emain when decision-make s (DMs) a e aced wi h complex
p oblems and a e hesi an o p o ide single- alued o non-in e al assessmen s. To accommoda e
such scena ios, B oumi e al. [69] p oposed he mul i- alued in e al neu osophic se (MVINS),
which allows DMs o p o ide e alua ions in he o m o mul i- alued in e al membe ships. This
model has been discussed in se e al wo ks [69-72].
Building on his, Mohd Kamal e al. [73] in oduced he mul i- alued in e al neu osophic so se
(MVIN-SS) by in eg a ing SS and MVINS. This model is able o be used o mul i-c i e ia g oup
decision-making (MCGDM) cases and de ines undamen al ope a ions like in e sec ion, union,
complemen , AND, and OR.
In MCGDM, agg ega ion is a c i ical s ep, whe e e alua ions om mul iple DMs a e combined in o
a consensus decision. The weigh ed a i hme ic a e age [74] and weigh ed geome ic a e age [75] a e
ounda ional agg ega ion ope a o s, widely applied ac oss a ious domains. Ex ensions and a ian s
include he apezoidal in ui ionis ic uzzy p io i ized weigh ed a e aging and geome ic ope a o s
[76], agg ega ion unde iangula in ui ionis ic uzzy en i onmen s [77], single- alued neu osophic
weigh ed a e aging (SVNWA) [78], and in e al neu osophic weigh ed ope a o s [79-82].
Peng and Wang [62] ocused on agg ega ion in mul i- alued neu osophic en i onmen s, while Ye
[83] in oduced apezoidal neu osophic numbe -based ope a o s. Khan e al. [84] explo ed hesi an
uzzy agg ega ion using loga i hmic sphe ical unc ions. Gao e al. [85] de eloped a linguis ic
agg ega ion amewo k, and Cagman e al. [86] p oposed uzzy so agg ega ion ope a o s. Saqlain
e al. [49] and Jana and Pal [87] in oduced agg ega ion echniques o neu osophic hype so se s
and single- alued neu osophic so se s, espec i ely.
Despi e his p og ess, mos exis ing agg ega ion echniques a e es ic ed o he IFS, SVNS, IVNS,
and SVNSS domains. The e is a clea gap in explo ing agg ega ion ope a o s wi hin he MVIN-SS
amewo k, especially in MCGDM con ex s in ol ing in e al-based and mul i- alued e alua ions.
To add ess his, we p opose wo no el agg ega ion ope a o s o MVIN-SS: he mul i- alued in e al
neu osophic so -weigh ed a i hme ic a e aging (MVINSWAA) and geome ic a e aging
(MVINSWGA) ope a o s. These ope a o s e ec i ely agg ega e in o ma ion cha ac e ized by
unce ain y, agueness, and inde e minacy, and accommoda e in e al and mul i- alued inpu s.
Neu osophic Se s and Sys ems, Vol. 97, 2026 663
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
To ensu e he ma hema ical igo and eliabili y o he p oposed ope a o s, key p ope ies such as
idempo ency, mono onici y, and boundedness a e es ablished h ough algeb aic p oo s. A nume ical
example ocused on so wa e selec ion demons a es he p ac ical applica ion o hese ope a o s
wi hin an MCGDM amewo k.
The con ibu ions o his pape a e h ee old: (1) we p opose wo no el agg ega ion ope a o s—
MVINSWAA and MVINSWGA—wi hin he MVIN-SS amewo k; (2) we ma hema ically p o e
essen ial agg ega ion p ope ies including idempo ency, mono onici y, and boundedness; (3) we
demons a e he e ec i eness o he p oposed app oach h ough a eal-wo ld case s udy in ol ing
so wa e selec ion, using sco e unc ions o ank al e na i es.
This pape has he ollowing s uc u e: The undamen al e ms and ideas associa ed wi h MVIN-SS
a e e iewed in Sec ion 2. Sec ion 3 in oduces MVINSWAA and MVINSWGA ope a o s along wi h
hei ma hema ical p ope ies. Sec ion 4 p esen s an MCGDM amewo k inco po a ing he p oposed
ope a o s. Sec ion 5 p o ides an illus a i e example. Sec ion 6 o e s a compa a i e analysis wi h
exis ing me hods. Sec ion 7 w aps up he wo k and sugges s a eas o u he esea ch.
2. P elimina ies
In his sec ion, we p esen some de ini ions and p ope ies which a e ela ed o NS and MVIN-
SS.
2.1. Neu osophic Se
De ini ion 2.1 [42]
Le
U
be a uni e se o discou se, hen NS
A
can be de ined as
{ ( ), ( ), ( ) / , }
A A A
A y y y y y U
=
whe e
, , : ] 0, 1 [U
−+
→
de ine he deg ee o u h-membe ship
( ),
Ay
deg ee o inde e minacy
()
Ay
and deg ee o alsi y
()
Ay
espec i ely and he e is no es ic ion on he sum o
( ), ( )
AA
yy
and
( ),
Ay
so
0 ( ) ( ) ( ) 3 .
A A A
y y y
−+
+ +
Acco ding o philosophical pe spec i e, he NS de i es i s alue om ac ual s anda d o non-
s anda d subse s o
] 0, 1 [
−+
. Howe e , in eal implemen a ions, pa icula ly in scien i ic and
enginee ing domains, i is mo e app op ia e o adop he closed in e al
[ 0, 1]
, as he use o
] 0, 1 [
−+
p esen s di icul ies in eal-wo ld implemen a ions.
2.2. Mul i-Valued In e al Neu osophic Se
De ini ion 2.2 [69]
Le
U
be a space o poin s (objec s), wi h a gene ic elemen in
U
deno ed by
.y
An MVINS
A
o e
U
can be de ined as
{ ( ), ( ), ( ) / , }
l m n
A A A
A y y y y y U
=
whe e
1 1 2 2 1 1 2 2
( ) [ ( ), ( )], [ ( ), ( )], , [ ( ), ( )], ( ) [ ( ), ( )], [ ( ), ( )], , [ ( ),
l q q m
A A A A A A A A A A A A A
y y y y y y y y y y y y y
− + − + − + − + − + −
==( )],
Ay
+
1 1 2 2
( ) [ ( ), ( )], [ ( ), ( )], , [ ( ), ( )] }
n s s
A A A A A A A
y y y y y y y U
− + − + − +
=
such ha
0 ( ), ( ), ( ) 3,
l m n
A A A
y y y
+ + +
o all
1, 2, , ,lq=
1, 2, , ,m =
1, 2, , .ns=
Neu osophic Se s and Sys ems, Vol. 97, 2026 664
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
In his esea ch, he in e al u h-membe ship sequence
( ),
l
Ay
in e al inde e minacy-membe ship
sequence
()
m
Ay
, and in e al alsi y-membe ship sequence
()
n
Ay
o an elemen
y
a e assumed o
be equal, whe e
,q s==
espec i ely. The symbols
,,l m n
ep esen he dimensions o he MVINS
.A
. Clea ly, upon equalizing he lowe and uppe bounds o
( ), ( ), ( )
l m n
A A A
y y y
, he MVINS educes o
a MVNS.
De ini ion 2.3 [69]
Le
A
and
B
be wo MVINS. Then some ope a ions o MVINS a e gi en as ollows:
Neu osophic Se s and Sys ems, Vol. 97, 2026 665
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
1) Di e ence
1 1 1 1 2 2 2 2
1
{ ([ ( ) ( ), ( ) ( )],[ ( ) ( ), ( ) ( )], ,
[ ( ) ( ), ( ) ( )]),([
B B B B
A A A A
q q q q
BB
AA
A B y y y y y y y y
y y y y
− − + + − − + +
− − + +
=
1 1 1
2 2 2 2
1
( ) (1 ( )), ( ) (1 ( ))],[
( ) (1 ( )), ( ) (1 ( ))], , [ ( ) (1 ( )),
(y) (1 (y))]), ([
BB
AA
B B B
A A A
B
A
y y y y
y y y y y y
− + + −
− + + − − +
+−
− −
− − −
− 1 1 1 2 2 2 2
( ) ( ), ( ) ( )],[ ( ) ( ), ( ) ( )], ,
[ ( ) ( ), ( ) ( )]) / , .
B B B B
A A A A
s s s s
BB
AA
y y y y y y y y
y y y y y y U
− − + + − − + +
− − + +
2) Addi ion
1 1 1 1 2 2 2 2
{ ([( ( ) ( )) 1, ( ( ) ( )) 1],[( ( ) ( )) 1, ( ( ) ( )) 1], ,
[( ( ) ( )) 1,( ( )
B B B B
A A A A
q q q q
B
AA
A B y y y y y y y y
y y y
− − + + − − + +
− − +
+ = + + + +
+ + 1 1 1 1
2 2 2 2
( )) 1]),([( ( ) ( )) 1, ( ( ) ( )) 1],
[( ( ) ( )) 1, ( ( ) ( )) 1], [( ( ) ( )) 1, ( ( )
B B B
AA
B B B
A A A A
y y y y y
y y y y y y y
+ − − + +
− − + + − − +
+ +
+ + + +
1 1 1 1 2 2 2 2
( )) 1]),
,([( ( ) ( )) 1, ( ( ) ( )) 1],[( ( ) ( )) 1, ( ( ) ( )) 1],
, [( ( ) ( )) 1, ( ( )
B
B B B B
A A A A
s s s
B
AA
y
y y y y y y y y
y y y
+
− − + + − − + +
− − +
+ + + +
+ + ( )) 1]) / , .
s
By y y U
+
3) Scala Mul iplica ion
1 1 2 2
11
2
{ ([( ( )) 1,( ( )) 1)], [( ( )) 1,( ( )) 1)],
,[( ( )) 1,( ( )) 1)]), ([( ( )) 1,( ( )) 1)],
[(
A A A A
qq
A A A A
A
A y y y y
y y y y
− + − +
− + − +
−
=
2
1 1 2 2
( )) 1,( ( )) 1)], ,[( ( )) 1,( ( )) 1)]),
([( ( )) 1,( ( )) 1)], [( ( )) 1,( ( )) 1)],
,[( ( )) 1,( (
A A A
A A A A
ss
AA
y y y y
y y y y
yy
+ − +
− + − +
−+
)) 1)]) / , , }.y y U R
+
4) Scala Di ision
1 1 2 2
11
2
/ { ([( ( ) / ) 1,( ( ) / ) 1)], [ ( ) / ) 1,( ( ) / ) 1)],
,[ ( ) / ) 1,( ( ) / ) 1)]),([( ( ) / ) 1,( ( ) / ) 1)],
[ ( ) /
A A A A
qq
A A A A
A
A y y y y
y y y y
y
− + − +
− + − +
−
=
2
1 1 2 2
) 1,( ( ) / ) 1)], ,[ ( ) / ) 1,( ( ) / ) 1)]),
([( ( ) / ) 1,( ( ) / ) 1)], [( ( ) / ) 1,( ( ) / ) 1)],
,[ ( ) / ) 1,( ( ) / ) 1)
A A A
A A A A
ss
AA
y y y
y y y y
yy
+ − +
− + − +
−+
]) / , , }.y y U R
+
De ini ion 2.4 [72]
Le
( ), ( ), ( ) / ;
q s
L y y y y y U
=
be an MVINS. Then,
1 1 1
1 1 1 1
( ) ( ) (2 ) (2 )
3 2 2 2
l l m m n n
q s
A A A A A A
l m n
sL q s
− + − + − +
= = =
= + + − − + − −
(1)
is called he sco e unc ion o
L
whe e
,,l m n
a e he numbe s o mul i- alued in e al alues in
( ), ( ), ( )
q s
y y y
.
Neu osophic Se s and Sys ems, Vol. 97, 2026 666
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
2.3. So Se
De ini ion 2.5 [6]
Le
U
be an ini ial uni e se se and
E
be a se o pa ame e s. Conside
.AE
Le
()PU
deno es
he powe SS o
.U
A pai
( , )LA
is called an SS o e
U
and he unc ion
L
is a mapping de ined
by
: ( )L A P U→
such ha
( )( )Ly
=
i
.yU
He e,
()L
is called he app oxima e unc ion o he so se
( , ),LA
and he alue
( )( )Ly
is a se
called x-elemen o he SS o all
.yU
The se s can be andom, emp y, o ha e non-emp y
in e sec ions.
2.4. Mul i-Valued In e al Neu osophic So Se
De ini ion 2.6 [73]
The pai
( , )LA
is called an MVIN-SS o e
( ),PU
whe e
P
is a mapping gi en by
: ( ).L A P U→
()PU
deno es he se o all MVIN-SS o
U
wi h pa ame e s om
A
and he unc ion
()L
is a
mapping de ined by
: ( )L A P U→
such ha
( )( )Ly
=
i
.yU
( , )LA
is cha ac e ized by
( ) ( )
( ), ( )
LL
yy
and
()
()
Ly
in he o m o a subse o
]1,0[
and can
be de ined as ollows:
( ) ( ) ( )
( , ) ( ), ( ), ( ) / ; ,
q s
L L L
L A y y y y A y U
=
whe e
1 1 2 2 1 1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) [ ( ), ( )], [ ( ), ( )], , [ ( ), ( )], ( ) [ ( ), ( )],
q q q
L L L L L L L L L L
y y y y y y y y y y
− + − + − + − +
==
22
( ) ( ) ( ) ( )
[ ( ), ( )], , [ ( ), ( )]
L L L L
y y y y
− + − +
and
1 1 2 2
( ) ( ) ( ) ( ) ( )
( ) [ ( ), ( )],[ ( ), ( )],
s
L L L L L
y y y y y
− + − +
=
( ) ( )
, [ ( ), ( )]
ss
LL
yy
−+
a e he in e al u h-membe ship sequence, in e al inde e minacy-
membe ship sequence and in e al alsi y-membe ship sequence espec i ely ha objec
y
holds
on pa ame e
.
3. Agg ega ion Based on Mul i-Valued In e al Neu osophic So Se
In his pa , we in oduce he agg ega ion based on MVIN-SS which a e he mul i- alued in e al
neu osophic so -weigh ed geome ic a e age (MVINSWGA) and mul i- alued in e al
Neu osophic Se s and Sys ems, Vol. 97, 2026 667
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
neu osophic so -weigh ed a i hme ic a e age (MVINSWAA) ope a o s o agg ega e he a ibu es
and al e na i es espec i ely.
We de ine he MVINSWGA and gi e p oo o i s p ope ies.
De ini ion 3.1
Le
( ) ( ) ( )
( , ) ( ), ( ), ( ) / y; ,
q s
L L L
L A y y y A y U
=
be an MVIN-SS. A mapping
:n
MVINSWGA L L→
is called a mul i- alued in e al neu osophic so weigh ed geome ic
a e aging (MVINSWGA) ope a o i i sa is ies
12
( , , , )
n
MVINSWGA A A A =
22
( ) ( ) ( ) ( )
1 1 1 1
()
11
( ) ( )
11
, ( ( )) , ( ( )) , , ( ( )) , ( ( )) ,
1 (1
( ( )) , ( ( )) l l l lll
q q q q
qq
L L L L
l l l l
L
qq
LL
ll
y y y yyy
− + − +
= = = =
−+
==
−−
1 1 2 2
( ) ( ) ( ) ( ) ( )
1 1 1 1 1 1
( )) , 1 (1 ( )) , 1 (1 ( )) , 1 (1 ( )) , , 1 (1 ( )) , 1 (1 ( )) ,
1 (1
m m m m m m
L L L L L
m m m m m m
y y y y y y
− + − + − +
= = = = = =
− − − − − − − − − −
−−
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
1 1 1 1 1 1
( )) , 1 (1 ( )) , 1 (1 ( )) , 1 (1 ( )) , , 1 (1 ( )) , 1 (1 ( ))
n n n n n n
s s s s s s
ss
L L L L L L
n n n n n n
y y y y y y
− + − + − +
= = = = = =
− − − − − − − − − −
(2)
o all
,.A y U
Theo em 1
Le
( ) ( ) ( )
( , ) ( ), ( ), ( ) / ; ,
q s
L L L
L A y y y y A y U
=
be an MVIN-SS. Then,
(1) Idempo ency
I
j
LL=
o all
1, 2, , ,j =
hen
12
( , , , ) .
MVINSWGA L L L L=
(2) Mono onici y
I
*
jj
LL
o all
1, 2, , ,j =
hen
* * * *
1 2 1 2
( , , , ) ( , , , ).
MVINSWGA L L L MVINSWGA L L L
(3) Boundedness
12
1,2, , 1,2, ,
min { } ( , , , ) max { }
j j j
jjq
L MVINSWGA L L L L
==
.
P oo (1) Idempo ency:
Since
)
(
(
1 1 2 2 1 1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ( ), ( )], [ ( ), ( )], , [ ( ), ( )] , [ ( ), ( )],
qq
jL L L L L L L L
L L y y y y y y y y
− + − + − + − +
==
) (
2 2 1 1 2 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ( ), ( )], , [ ( ), ( )] , [ ( ), ( )], [ ( ), ( )],
L L L L L L L L
y y y y y y y y
− + − + − + − +
)
( ) ( )
, [ ( ), ( )]
ss
LL
yy
−+
o all
,j
Neu osophic Se s and Sys ems, Vol. 97, 2026 668
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
we ha e
1
( ) ( ) j
jj
j
MVINSWGA L L
=
=
( ) ( )
11
()
1 1 2 2
( ) ( ) ( ) ( )
1 1 1 1
, , ( ( )) , ( ( )) ,
1 (1
( ( )) , ( ( )) , ( ( )) , ( ( )) lll l l l
qq
qq
LL
ll
L
q q q q
L L L L
l l l l
yyy y y y
−+
==
− + − +
= = = =
−−
1 1 2 2
( ) ( ) ( ) ( ) ( )
1 1 1 1 1 1
( )) , 1 (1 ( )) , 1 (1 ( )) , 1 (1 ( )) , , 1 (1 ( )) , 1 (1 ( )) ,
1 (1
m m m m m m
L L L L L
m m m m m m
y y y y y y
− + − + − +
= = = = = =
− − − − − − − − − −
−−
22
( ) ( )
11
11
( ) ( ) ( ) ( )
1 1 1 1
, 1 (1 ( )) , 1 (1 ( )) , ,( )) , 1 (1 ( )) 1 (1 ( )) , 1 (1 ( ))
nnn n n n
ss
LL
nn
s s s s
ss
L L L L
n n n n
yyy y y y
−+
==
− + − +
= = = =
− − − −
− − − − − −
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( ( )) , ( ( )) , ( ( )) , ( ( )) , , ( ( )) , ( ( )) ,
1 (1
q q q q q q
l l l l l l
l l l l l l
qq
L L L L L L
L
y y y y y y
− + − + − +
−−
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( )) , 1 (1 ( )) , 1 (1 ( )) , 1 (1 ( )) , , 1 (1 ( )) , 1 (1 ( )) ,
1 (1
m m m m
m m m m
mm
mm
L L L L L
y y y y y y
− + − + − +
− − − − − − − − − −
−−
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( )) , 1 (1 ( )) , 1 (1 ( )) , 1 (1 ( )) , , 1 (1 ( )) , 1 (1 ( ))
s s s s s s
n n n n n n
n n n n n n
ss
L L L L L L
y y y y y y
− + − + − +
− − − − − − − − − −
Since
1, 1, 1,
q s
l m n
l m n
= = =
we ha e
( )
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
1 1 2
( ) ( ) ( )
( ( )), ( ( )) , ( ( )), ( ( )) , , ( ( )), ( ( )) ,
1 (1 ( )), 1 (1 ( )) , 1 (1 ( )
qq
L L L L L L
L L L
y y y y y y
y y y
− + − + − +
− + −
− − − − − −
( )
(
2
( ) ( ) ( )
1 1 2 2
( ) ( ) ( ) ( ) ( )
), 1 (1 ( )) , , 1 (1 ( )), 1 (1 ( )) ,
1 (1 ( )), 1 (1 ( )) , 1 (1 ( )), 1 (1 ( )) , , 1 (1
L L L
s
L L L L L
y y y
y y y y
+ − +
− + − +
− − − − − −
− − − − − − − − − −
)
()
( )), 1 (1 ( ))
s
L
yy
−+
−−
)
(
(
1 1 2 2 1 1 2 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
()
[ ( ), ( )], [ ( ), ( )], , [ ( ), ( )] , [ ( ), ( )], [ ( ), ( )],
, [ (
qq
L L L L L L L L L L
L
y y y y y y y y y y
− + − + − + − + − +
−
)
)
(
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( ) ( )
), ( )] [ ( ), ( )], [ ( ), ( )], , [ ( ), ( )],
s s
L L L L L L L
y y y y y y y y
+ − + − + − +
( ) ( ) ( )
( ), ( ), ( )
l m n
L L L
y y y L
=
which p o es he Theo em 1 (1).
P oo (2) Mono onici y:
Since
*
( ) ( )
( ) ( )
ll
LL
yy
−−
o all
,j
hen we ha e
*
( ) ( )
1 ( ) 1 ( )
ll
LL
yy
−−
− −
()
1
*
()
1
(1 ( )) (1 ( ))
ll
q
l
L
l
q
l
L
l
yy
−
=
−
=
− −
Since
*
( ) ( )
( ) ( )
ll
LL
yy
++
o all
,j
hen we ha e
*
( ) ( )
1 ( ) 1 ( )
ll
LL
yy
++
− −
()
1
*
()
1
(1 ( )) (1 ( ))
ll
q
l
L
l
q
l
L
l
yy
+
=
+
=
− −
Since
*
( ) ( )
( ) ( )
mm
LL
yy
−−
o all
,j
hen we ha e
*
( ) ( )
1 ( ) 1 ( )
mm
LL
yy
−−
− −
Neu osophic Se s and Sys ems, Vol. 97, 2026 675
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
2
D
U
1con ibu ion o o ganiza ion pe o mance
=
2e o o ans o m om cu en sys em
=
1
([0.4, 0.7],[0.2, 0.5]), ([0.8, 0.9],[0.2, 0.6]), ([0.7, 0.9],[0.4, 0.7])
([0.2, 0.5],[0.2, 0.6]), ([0.4, 0.8],[0.2, 0.7]), ([0.1, 0.5],[0.8, 0.9])
2
([0.3, 0.6],[0.2, 0.5]), ([0.7, 0.9],[0.4, 0.6]), ([0.6, 0.9],[0.1, 0.3])
([0.2, 0.6],[0.6, 0.8]), ([0.1, 0.4],[0.2, 0.5]), ([0.3, 0.6],[0.7, 0.9])
3
([0.2, 0.5],[0.3, 0.5]), ([0.5, 0.7],[0.8, 0.9]), ([0.4, 0.7],[0.1, 0.4])
([0.1, 0.5],[0.2, 0.6]), ([0.2, 0.4],[0.5, 0.8]), ([0.1, 0.4],[0.7, 0.9])
4
([0.4, 0.6],[0.3, 0.7]), ([0.5, 0.9],[0.1, 0.4]), ([0.2, 0.6],[0.3, 0.4])
([0.2, 0.4],[0.1, 0.5]), ([0.5, 0.7],[0.1, 0.4]), ([0.6, 0.9],[0.1, 0.4])
5
([0.2, 0.4],[0.1, 0.5]), ([0.4, 0.8],[0.6, 0.9]), ([0.4, 0.8],[0.4, 0.6])
([0.1, 0.5],[0.2, 0.5]), ([0.6, 0.9],[0.3, 0.7]), ([0.3, 0.8],[0.6, 0.9])
2
D
U
3ha dwa e/so wa e in es men cos
=
4ou sou cing so wa e de elope eliabili y
=
1
([0.3, 0.7],[0.1, 0.4]), ([0.2, 0.6],[0.2, 0.5]), ([0.4, 0.7],[0.3, 0.6])
([0.5, 0.7],[0.3, 0.7]), ([0.4, 0.8],[0.2, 0.7]), ([0.4, 0.7],[0.5, 0.9])
2
([0.4, 0.6],[0.2, 0.5]), ([0.7, 0.9],[0.3, 0.6]), ([0.2, 0.5],[0.8, 0.9])
([0.6, 0.7],[0.2, 0.5]), ([0.5, 0.8],[0.1, 0.4]), ([0.1, 0.6],[0.5, 0.8])
3
([0.1, 0.4],[0.2, 0.6]), ([0.1, 0.5],[0.8, 0.9]), ([0.5, 0.9],[0.4, 0.9])
([0.1, 0.6],[0.5, 0.8]), ([0.1, 0.4],[0.8, 0.9]), ([0.5, 0.7],[0.7, 0.8])
4
([0.1, 0.4],[0.5, 0.7]), ([0.2, 0.4],[0.6, 0.8]), ([0.6, 0.9],[0.3, 0.6])
([0.3, 0.5],[0.2, 0.5]), ([0.2, 0.4],[0.6, 0.9]), ([0.3, 0.5],[0.1, 0.4])
5
([0.4, 0.7],[0.1, 0.5]), ([0.2, 0.7],[0.7, 0.9]), ([0.2, 0.7],[0.2, 0.7])
([0.2, 0.6],[0.2, 0.6]), ([0.4, 0.7],[0.2, 0.6]), ([0.7, 0.9],[0.1, 0.4])
3
D
U
1con ibu ion o o ganiza ion pe o mance
=
2e o o ans o m om cu en sys em
=
1
([0.1, 0.4],[0.2, 0.7]), ([0.2, 0.7],[0.5, 0.8]), ([0.2, 0.7],[0.3, 0.6])
([0.7, 0.9],[0.2, 0.5]), ([0.4, 0.7],[0.5, 0.7]), ([0.8, 0.9],[0.2, 0.5])
2
([0.2, 0.5],[0.3, 0.8]), ([0.4, 0.8],[0.2, 0.7]), ([0.1, 0.7],[0.2, 0.4])
([0.7, 0.9],[0.1, 0.4]), ([0.4, 0.7],[0.3, 0.5]), ([0.8, 0.9],[0.2, 0.4])
3
([0.3, 0.4],[0.1, 0.7]), ([0.4, 0.5],[0.3, 0.5]), ([0.7, 0.9],[0.2, 0.5])
([0.5, 0.7],[0.3, 0.5]), ([0.3, 0.6],[0.1, 0.4]), ([0.6, 0.9],[0.7, 0.9])
4
([0.2, 0.4],[0.1, 0.5]), ([0.3, 0.7],[0.3, 0.6]), ([0.3, 0.7],[0.4, 0.8])
([0.6, 0.9],[0.2, 0.6]), ([0.3, 0.8],[0.4, 0.8]), ([0.3, 0.7],[0.1, 0.3])
5
([0.3, 0.7],[0.3, 0.6]), ([0.1, 0.4],[0.2, 0.8]), ([0.1, 0.4],[0.2, 0.6])
([0.4, 0.7],[0.2, 0.5]), ([0.1, 0.7],[0.2, 0.5]), ([0.5, 0.9],[0.2, 0.7])
3
D
U
3ha dwa e/so wa e in es men cos
=
4ou sou cing so wa e de elope eliabili y
=
1
([0.1, 0.3],[0.4, 0.7]), ([0.2, 0.6],[0.2, 0.4]), ([0.1, 0.8],[0.6, 0.8])
([0.2, 0.6],[0.2, 0.5]), ([0.3, 0.5],[0.2, 0.4]), ([0.7, 0.9],[0.2, 0.6])
2
([0.5, 0.8],[0.5, 0.8]), ([0.3, 0.6],[0.4, 0.7]), ([0.3, 0.6],[0.4, 0.7])
([0.3, 0.7],[0.1, 0.4]), ([0.6, 0.8],[0.3, 0.7]), ([0.2, 0.6],[0.4, 0.8])
3
([0.3, 0.6],[0.4, 0.7]), ([0.3, 0.7],[0.3, 0.7]), ([0.8, 0.9],[0.5, 0.8])
([0.2, 0.7],[0.2, 0.6]), ([0.4, 0.7],[0.5, 0.9]), ([0.7, 0.8],[0.4, 0.8])
4
([0.3, 0.5],[0.6, 0.8]), ([0.1, 0.4],[0.2, 0.6]), ([0.3, 0.6],[0.4, 0.7])
([0.7, 0.9],[0.3, 0.6]), ([0.4, 0.7],[0.4, 0.7]), ([0.1, 0.5],[0.7, 0.9])
5
([0.6, 0.9],[0.3, 0.6]), ([0.7, 0.8],[0.2, 0.6]), ([0.4, 0.7],[0.8, 0.9])
([0.4, 0.7],[0.4, 0.5]), ([0.6, 0.7],[0.2, 0.5]), ([0.1, 0.3],[0.4, 0.5])
Neu osophic Se s and Sys ems, Vol. 97, 2026 676
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
S ep 2: Agg ega e he a ibu es and al e na i es using MVINSWGA and MVINSWAA ope a o .
By applying he equa ion in De ini ion 3.1, he agg ega ed a ibu es a e p esen ed in Table 2.
Table 2 Agg ega ed A ibu es
Re e o he equa ion in De ini ion 3.2, he agg ega ed al e na i es a e gi en in Table 3.
Table 3 Agg ega ed Al e na i es
D
U
1con ibu ion o o ganiza ion pe o mance
=
2e o o ans o m om cu en sys em
=
1
([0.18, 0.50],[0.15, 0.56]), ([0.67, 0.89], [0.40, 0.80]), ([0.69, 0.92],[0.65, 0.85])
([0.17, 0.47], [0.09, 0.39]), ([0.54, 0.87], [0.43, 0.84]), ([0.62, 0.84],[0.62, 0.88])
2
([0.11, 0.35],[0.17, 0.57]), ([0.62, 0.94], [0.51, 0.89]), ([0.43, 0.87],[0.20, 0.54])
([0.24, 0.61], [0.15, 0.51]), ([0.541, 0.87],[0.29, 0.58]), ([0.76, 0.89], [0.56, 0.83])
3
([0.15, 0.37],[0.11, 0.53]), ([0.54, 0.76],[0.71, 0.86]), ([0.73, 0.95], [0.34, 0.70])
([0.10, 0.53],[0.08, 0.35]), ([0.47, 0.73],[0.40, 0.78]), ([0.50, 0.85],[0.75, 0.94])
4
([0.09, 0.35],[0.08, 0.37]), ([0.47, 0.88], [0.29, 0.62]), ([0.37, 0.73],[0.59, 0.89])
([0.24, 0.54], [0.08, 0.49]), ([0.44, 0.87],[0.34, 0.76]), ([0.71, 0.92], [0.30, 0.59])
5
([0.08, 0.44],[0.09, 0.35]), ([0.34, 0.76], [0.49, 0.92]), ([0.34, 0.71],[0.56, 0.87])
([0.06, 0.35],[0.14, 0.42]), ([0.46, 0.88],[0.33, 0.73]), ([0.54, 0.92],[0.56, 0.92])
D
U
3ha dwa e/so wa e in es men cos
=
4ou sou cing so wa e de elope eliabili y
=
1
([0.08, 0.32],[0.15, 0.50]), ([0.24, 0.67], [0.28, 0.65]), ([0.43, 0.87], [0.59, 0.82])
([0.20, 0.50], [0.17, 0.53]), ([0.42, 0.76], [0.24, 0.67]), ([0.70, 0.92], [0.55, 0.89])
2
([0.32, 0.62],[0.14, 0.53]), ([0.68, 0.91], [0.65, 0.89]), ([0.42, 0.76],[0.71, 0.88])
([0.19, 0.38],[0.10, 0.37]), ([0.72, 0.91], [0.34, 0.73]), ([0.24, 0.75], [0.65, 0.94])
3
([0.09, 0.35],[0.13, 0.46]), ([0.44, 0.88],[0.69, 0.91]), ([0.83, 0.97], [0.70, 0.94])
([0.12, 0.61],[0.24, 0.66]), ([0.43, 0.73],[0.76, 0.93]), ([0.65, 0.80],[0.60, 0.83])
4
([0.08, 0.32],[0.24, 0.58]), ([0.29, 0.58],[0.53, 0.85]), ([0.63, 0.94],[0.65, 0.89])
([0.20, 0.52], [0.08, 0.39]), ([0.34, 0.67],[0.65, 0.92]), ([0.50, 0.73],[0.51, 0.80])
5
([0.38, 0.75],[0.08, 0.35]), ([0.54, 0.81],[0.65, 0.89]), ([0.38, 0.79], [0.67, 0.91])
([0.18, 0.46],[0.13, 0.39]), ([0.54, 0.79], [0.49, 0.80]), ([0.60, 0.92],[0.34, 0.65])
U
Agg ega ed Ma ix
1
([0.29, 0.70], [0.27, 0.75]), ([0.19, 0.62],[0.11, 0.54]), ([0.36, 0.79], [0.36, 0.74])
2
([0.39, 0.76], [0.27, 0.75]), ([0.40, 0.82],[0.18, 0.58]), ([0.18, 0.66], [0.23, 0.61])
3
([0.22, 0.73],[0.26, 0.76]), ([0.22, 0.60], [0.38, 0.75]), ([0.44, 0.78],[0.33, 0.72])
4
([0.29, 0.68], [0.23, 0.71]), ([0.14, 0.54],[0.18, 0.60]), ([0.29, 0.68],[0.24, 0.61])
5
([0.34, 0.77], [0.21, 0.61]), ([0.21, 0.65], [0.23, 0.69]), ([0.21, 0.69],[0.27, 0.69])
Neu osophic Se s and Sys ems, Vol. 97, 2026 677
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
S ep 3: The mul i- alued in e al neu osophic e e ence o posi i e-ideal solu ion (MVINRPIS,
A+
)
and nega i e-ideal solu ion (MVINRNIS,
A−
) a e iden i ied espec i ely.
The
A+
and
A−
a e de e mined as ollows:
( )
[1, 1], [1, 1],[0, 0], [0, 0], [0, 0], [0, 0]A+=
( )
[0, 0], [0, 0], [1, 1], [1, 1], [1, 1], [1, 1]A−=
The Euclidean dis ance o each al e na i e om
A+
and
A−
is calcula ed by using eqn (4) and (5)
as p esen ed in Table 4.
Table 4 The dis ance o each al e na i e om
A+
and
A−
Al e na i es
d+
d−
1
0.5279
0.5734
2
0.5098
0.5873
3
0.5650
0.5269
4
0.4969
0.5932
5
0.5273
0.5705
Then, he closeness coe icien o each al e na i e is calcula ed by using eqn (6) as shown in Table 5.
Table 5 The closeness coe icien s o each al e na i e
Al e na i es
n
CC
Ranking
1
0.5207
3
2
0.5353
2
3
0.4825
5
4
0.5442
1
5
0.5197
4
S ep 4: Acco ding o he alue o closeness coe icien in Table 5, we can ank he al e na i es in
descending o de as
4 2 1 5 3
whe e he symbol
""
e e s o ‘supe io o’. So, i can be concluded ha
4
is he bes al e na i e.
Wi h simila compu a ion, he alue o sco e unc ion ( om eqn 1) is calcula ed and p esen ed in
Table 6.
Neu osophic Se s and Sys ems, Vol. 97, 2026 678
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
Table 6 The sco e unc ion o each al e na i e
Al e na i es
Sco e
Func ion
Ranking
1
0.5251
3
2
0.5418
2
3
0.4794
5
4
0.5522
1
5
0.5237
4
I can be seen ha he highes alue o closeness coe icien is 0.5442 and he highes alue o sco e
unc ion is 0.5522 (see Table 5 and Table 6 espec i ely). This ob iously shows ha
4
is he bes
al e na i e.
Figu e 2 G aphical compa ison o al e na i es based on closeness coe icien and sco e unc ion.
The al e na i es can be anked as
4 2 1 5 3
acco ding o he sco e unc ion
alue. Figu e 2 p esen s a line cha compa ing he closeness coe icien s and sco e unc ion alues o
each al e na i e. As obse ed, Al e na i e
4
has he highes alues in bo h me ics, con i ming i
as he bes op ion. This g aphical ep esen a ion acili a es an in ui i e unde s anding o he anking
consis ency ac oss di e en e alua ion measu es. No only he bes choice, in ac , he o de o
p e e ence o he p oposed agg ega ion me hod unde MVIN-SS in o ma ion ei he using sco e
unc ion o closeness coe icien is consis en .
0.44
0.46
0.48
0.5
0.52
0.54
0.56
C1 C2 C3 C4 C5
G aphical compa ison o al e na i es based on closeness
coe icien and sco e unc ion
Closeness Coe icien sco e unc ion
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No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
6. Compa a i e Analysis
By adop ing he same case s udy o so wa e selec ion p oblems, a compa a i e analysis wi h
o he me hods is conduc ed o alida e he e icacy and easibili y o he p oposed decision-making
app oach based on MVINSWGA and MVINSWAA ope a o s. The compa isons be ween he
p oposed me hods wi h he exis ing me hods a e shown in Table 7.
Table 7 The compa ison wi h he exis ing me hods
Se
Agg ega ion
Ope a o
Weigh
Measu emen
Func ion
Ranking O de
Conside a ion
o
,,
T iangula
in ui ionis ic
uzzy se [77]
TIFOWG
✓
Sco e unc ion
4 1 5 2 3
T apezoidal
in ui ionis ic
uzzy se [76]
TIFPWA
TIFPWG
✓
✓
Sco e unc ion
Sco e unc ion
4 1 2 5 3
4 1 2 5 3
T apezoidal
neu osophic
se [83]
TNNWAA
TNNWGA
✓
✓
Sco e unc ion
Sco e unc ion
4 1 5 2 3
4 1 5 2 3
✓
✓
MVIN-SS
(P oposed se )
MVINSWAA
&
MVINSWGA
✓
Euclidean
Dis ance
4 2 1 5 3
✓
MVINSWAA
&
MVINSWGA
✓
Sco e unc ion
4 2 1 5 3
✓
I can be seen ha he e a e di e en ypes o se s and agg ega ion ope a o s used in o de
o ob ain he inal anking o de . As a esul o he compa a i e s udy desc ibed abo e, wo issues
may be conside ed. Fi s , he p oposed me hod yields a di e en ou come han he cu en
agg ega ion ope a o s, which ook in o accoun di e en ypes o se s. Al hough mul iple agg ega ion
Neu osophic Se s and Sys ems, Vol. 97, 2026 680
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
ope a o s may be employed o handle he a ious ela ionships o he agg ega ed a gumen s, he
numbe o ope a ions and he size o he esul s will g ow exponen ially as mo e MVINSNs a e
engaged in he p ocesses. Fu he mo e, a ious agg ega ing ope a o s migh p oduce dispa a e
ou comes. These migh be because he p oposed ope a o s conside u h, inde e minacy, and alsi y
membe ships o he new se MVIN-SS while p oposing geome ic and a i hme ic based agg ega ion
ope a o s. The de e io a ion b ough on by hese di icul ies may limi he pe o mance o
agg ega ion ope a o s. One o he sho comings wi h he exis ing me hod is ha he u h,
inde e minacy, and alsi y membe ships we e no conside ed. Fo example, Wang [77] used TIFOWG
ope a o and sco e unc ion using in ui ionis ic uzzy se . Howe e , he u h, inde e minacy, and
alsi y a e no p esen as he au ho used iangula in ui ionis ic uzzy se . The anking o de
gene a ed by using a iangula in ui ionis ic uzzy se is compa able o he o he se s. Second, i is
also wo h no ing ha some agg ega ion ope a o s employed sco e unc ions wi hou aking in o
conside a ion he h ee membe ships, whe eas he p oposed agg ega ion ope a o s used sco e
unc ion and Euclidean dis ance in measu emen . Mo e impo an ly, he p oposed ope a o s conside
he u h, inde e minacy, and alsi y membe ships o MVIN-SS o which he in e als o h ee
membe ships a e he dis inc ea u e o he p oposed ope a o s and can handle unce ain and
inde e minacy in o ma ion especially when he assessmen s p o ided by decision-make s a e gi en
in mul iple alues, in e al scale, and bi u ca ed.
Howe e , he p oposed app oach using MVIN-SS a ies om exis ing me hods, which always en ail
ope a ions whose in luence on he inal solu ion may be ega ded as p e iously indica ed, since he
p oposed me hod may o e come hese d awbacks. I is possible o a oid loss and dis o ion o he
gi en p e e ence in o ma ion, which imp o es he inal indings' co espondence wi h genuine
decision-making issues. Fu he mo e, he p oposed me hod is a ou ed o sol ing issues when he
numbe o c i e ia obse ably su passes he numbe o al e na i es. As a esul , he p oposed me hod
can success ully deal wi h he p e e ence in o ma ion p esen ed by MVIN-SS, which is mean o
ensu e he alidi y o he inal ankings. In o he wo ds, he p oposed me hod can deal wi h
in o ma ion ha is cha ac e ized by uzziness, inde e minacy, and unce ain y, he eby ge mane o
sol e complex MCDM p oblems.
Neu osophic Se s and Sys ems, Vol. 97, 2026 681
No Liyana Amalini Mohd Kamal1*, Lazim Abdullah2, Ilyani Abdullah3, Vakkas Uluçay4, and Khalid Naeem5, Mul i-
Valued In e al Neu osophic So Se s and Thei Agg ega ion Ope a o s
7. Conclusions
In his pape , wo no el agg ega ion ope a o s—MVINSWAA and MVINSWGA—we e
p oposed wi hin he mul i- alued in e al neu osophic so se (MVIN-SS) amewo k o acili a e
mo e obus mul i-c i e ia g oup decision-making (MCGDM). Theo e ical alida ion o hese
ope a o s was es ablished ia p oo s o idempo ency, mono onici y, and boundedness. A s uc u ed
decision-making p ocedu e and a so wa e selec ion case s udy demons a ed he p ac ical
applicabili y and consis ency o he p oposed me hods. Compa a i e analysis u he con i med he
supe io i y o ou app oach in handling unce ain y, inde e minacy, and mul i- alued in o ma ion.
Fu u e esea ch could explo e se e al di ec ions. Fi s , he in oduced agg ega ion ope a o s can be
enhanced o handle dynamic o ime-dependen decision-making en i onmen s, whe e e alua ion
c i e ia may e ol e o e ime. Second, in eg a ion wi h machine lea ning echniques could enable
au oma ed weigh ing o anking o al e na i es based on his o ical da a o use eedback. Thi d,
u u e wo k could adap he MVIN-SS amewo k o dis ibu ed decision-making sys ems,
pa icula ly in con ex s in ol ing au onomous agen s o decen alized sys ems. Finally, applying he
p oposed me hods o domain-speci ic applica ions such as heal hca e diagnos ics, en i onmen al isk
assessmen s, and sma ci y in as uc u e planning would u he alida e hei u ili y in eal-wo ld
scena ios.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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