Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets

Uni e si y o New Mexico
So se models applied o pa e n ecogni ion ia 3- alued
ex ension o neu osophic so se s
Ahmad A. Abubake ,1, M. Palanikuma (2,∗), Abdallah Al-Husban3,4
1Facul y o Compu e S udies, A ab Open Uni e si y, Saudi A abia; a.abubake @a abou.edu.sa
2Depa men o Ma hema ics, Sa ee ha School o Enginee ing, Sa ee ha Ins i u e o Medical and Technical
Sciences, Chennai-602105, India; [email p o ec ed]
3Depa men o Ma hema ics, Facul y o Science and Technology, I bid Na ional Uni e si y, P.O. Box: 2600
I bid, Jo dan; d [email protected]
4Jada a Resea ch Cen e , Jada a Uni e si y, I bid 21110, Jo dan.
∗Co espondence: [email p o ec ed];
In his communica ion, we gi e he heo y o he 3- alued ex ension o neu osophic so
se (3- alued ENSS) and de ine ce ain ope a ions. No ably, we demons a ed an algo i hm o
add ess he decision-making p oblem using a so se model. We p esen a simila i y measu e
o wo 3- alued ENSSs and desc ibe i s use in a pa e n ecogni ion challenge. Illus a i e
examples a e p o ided o demons a e hei e ec i eness in sol ing p oblems wi h unce ain-
ies.
Keywo ds: in e al alued uzzy so se , uzzy so se , decision making p oblem, agg ega-
ion ope a o .
————————————————————————————————
1. In oduc ion
E e y day, machine lea ning, in elligence collec ion, knowledge compila ion, and o he p o-
cesses a e used in he con lic esolu ion p ocess. One o he bigges challenges is demons a ing
he e ec i eness o s a egic planning. Ma hema ical heo y enables he adop ion o app o-
p ia e decision-making echniques. The DM concep migh be ad an ageous o businesses
Neu osophic Se s and Sys ems, Vol. 94, 2025
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
since i assesses and anks di e en poin s o iew based on hei quali ies. We can hen e -
ec i ely selec , classi y, gene a e, and e alua e ou op ions. MADM conside s e e y ea u e
and componen o p o ide he op imal esponse. I used o be widely accep ed ha weigh s
and a ibu es equi ed o be ep esen ed as dis inc nume ical alues. Nume ous assessmen s
and DM p oblems equi e he analysis o nume ous a iables and indica ions. Assessing and
collec ing da a o e alua ion indica o s can help o e come assessmen and DM p oblems. The
complex s uc u e o eal-wo ld sys ems equen ly leads o MADM issues, which obscu e e al-
ua ion speci ics. To explain unce ain y, many heo ies ha e been p oposed, such as uzzy se s
(FSs) [1], which ha e membe ship g ades (MG) om 0 o 1. Each elemen in he in ui ionis ic
FS (IFS) de eloped by A anasso [2] he condi ion ha 0 ℘+[1 , o ℘, [ ∈[0,1] and
posi i e ℘and nega i e [. Yage [3] in en ed he Py hago ean FSs (PFS) concep , which is
cha ac e ized by i s MG and non-membe ship g ade (NMG), wi h he es ic ion ha ℘+[1
o ℘2+[21. Many esea ch ha e been conduc ed on he use o IFSs and PFSs in many
ields. The cons ain ha he squa e sum o i s MG and NMG deg ees no exceed uni y
de ines he ex ended IFSs. The concep o pic u e FSs is ex ended by FSs and IFSs [4]. Thei
abili y o communica e in o ma ion is s ill es ic ed. Because o his, he expe s we e s ill
ha ing ouble explaining he da a in hese se s and he associa ed da a. Wang e al. [5]
in es iga ed he concep o complex IFS wi h DOMBI p io i ized AOs and i s applica ion o
us wo hy g een supplie selec ion. Using he in e al- alued IFS dis ance-based MAIRCA
me hodology, Mish a e al. [6] in es iga e a way o assessing sus ainable was ewa e ea men
sys ems. Posi i e MG (℘), neu al MG (∝), and nega i e MG ([) a e he h ee basic concep s
o he pic u e FS, acco ding o Cuong e al. [7].
Addi ionally, i o e s mo e bene i s han PFS and IFS. Since ℘, ∝, [ ∈[0,1], i has been
no ed ha he pic u e FS is an enhancemen o he IFS ha may handle mo e inconsis ency
and 0 ℘+∝+[1. Acco ding o he pic u e FS desc ip ion, expe opinions like ”yes,”
”abs ain,” ”no,” and ” e usal” will be con eyed. I will also encou age uni o mi y be ween
he assessmen da a and he ac ual decision en i onmen and s op e alua ion in o ma ion
om being h own away. Molod so in oduced he concep o so se s (SOSs) [8]. SOSs
mo e accu a ely he complexi y and objec i i y o DM in eal wo ld scena ios han o he
unce ain heo ies. Fu he mo e, a c ucial a ea o s udy is he in eg a ion o SOSs wi h o he
ma hema ical models. Maji sugges ed FSOSs [9] and in ui ionis ic FSOSs [10]. Recen ly, Al-
Husban e al. [11–14] discussed he a ious ex ension algeb aic s uc u es ia FS, IVFS and
AO. These wo heo ies a e used o add ess a ange o DM p oblems. Fo he es o he wo k,
I shall adhe e o he s uc u e men ioned below. Sec ion 1 deals ha he in oduc ion. Sec ion
2 discussed PFS and i s basic concep s. The new idea o 3- alued ENSSs and i s undamen al
unc ions a e explained in Sec ion 3. The simila i y measu es be ween 3- alued ENSSss a e
Neu osophic Se s and Sys ems, Vol. 94, 2025 33
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
discussed in Sec ion 4. The selec ion ools o pa e n ecogni ion and i s p ac ical applica ions
we e co e ed in Sec ion 5. The conclusion is co e ed in sec ion 6.
2. P elimina ies
Le Xbe a uni e sal se o he en i e y o his sec ion. The undamen al concep s o he
neu osophic se (NSS), which a e well-known in he li e a u e, a e e iewed and in oduced
in his sec ion.
De ini ion 2.1. The neu osophic in e al
alued FS (NIVFS) Z={x, >Z(∂),=Z(∂),`Z(∂)|x∈X}, whe e >Z(∂)=[>l
Z(∂),>u
Z(∂)] and
=Z(∂) = [=l
Z(∂),=u
Z(∂)] and `Z(∂) = [`l
Z(∂),`u
Z(∂)] called he alue o u h, in e minacy
and alse membe ship o Z, espec i ely. The unc ion >Z:X→D[0,1], =Z:X→D[0,1],
`Z:X→D[0,1] and 0 (>Z(∂)) + (=Z(∂)) + (`Z(∂)) 3.
He e Z=D[>l
Z,>u
Z],[=l
Z,=u
Z],[`l
Z,`u
Z]Eis men ioned a NIVF numbe (NIVFN).
De ini ion 2.2. Suppose ha Z=h>Z,=Z,`Ziand 0=h>0,=0,`0ia e any wo NIVFNs
o e (X, E). Then
(1) Zc=h`Z,=Z,>Zi
(2) Z 0=Dmax(>Z,>0),min(=Z,=0),min(`Z,`0)E
(3) Zu0=Dmin(>Z,>0),min(=Z,=0),max(`Z,`0)E
(4) Z0i >Z >0and =Z =0and `Z`0
(5) Z=0i >Z=>0and =Z==0and `Z=`0.
De ini ion 2.3. Le Ebe a se o pa ame e . Gi en Z Eand F:Z→PF(X),
whe e PF(X) is he collec ion o all uzzy subse s o X, he pai (F, Z) is e e ed o as
a Py hago ean FSOS (PFSOS) on X.
De ini ion 2.4. A neu osophic e ined se (NRS) Z={hx, (>1
Z(∂),>2
Z(∂), ..., , >P
Z(∂)),
(=1
Z(∂),=2
Z(∂), ..., , =P
Z(∂)),(`1
Z(∂),`2
Z(∂), ..., , `P
Z(∂))i:∂∈E},
whe e, >1
Z(∂),>2
Z(∂), ..., , >P
Z(∂) : E→[0,1],=1
Z(∂),=2
Z(∂), ..., , =P
Z(∂) : E→
[0,1],`1
Z(∂),`2
Z(∂), ..., , `P
Z(∂) : E→[0,1] such ha 0  >i
Z(∂) + =i
Z(∂) + `i
Z(∂)3(i=
1,2, ..., P) and >1
Z(∂) >2
Z(∂)...  >i
Z(∂), o any ∂∈E. He e >1
Z(∂),>2
Z(∂),>P
Z(∂),
=1
Z(∂),=2
Z(∂),=P
Z(∂) and `1
Z(∂),`2
Z(∂),`P
Z(∂) ep esen s he TMG, IMG, and FMG sequences
o he elemen x. Addi ionally, Pis e e ed o as he dimension o NRS Z. TMG sequences
g ow, bu o he sequences (IMG, FMG) do no inc ease o dec ease. Howe e , he e is no ise
o dec ease in TMG, IMG, o FMG sequences h oughou his pape . NRS(E) deno es he
collec ion o all neu osophic e ined se s in E.
Neu osophic Se s and Sys ems, Vol. 94, 2025 34
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
3. No el app oach owa ds 3- alued ENS
The no ion o 3- alued ex ended neu osophic so se s will be p esen ed.
De ini ion 3.1. Le E={κ1, κ2, ..., κm}be a se o pa ame e s and Xbe he uni e sal se .
A so uni e se is he pai (X, E). Le Tand Fbe unchanged, and le he inde e minacy Ibe
e ined as ei he he same o (Unknown, con adic ion), which a e subse s o [0,1]. Nex , we
ob ain he 3- alued neu osophic so se scena io as ollows:
We p o ide he ollowing nume ical example o demons a e he De ini ion 3.1:
Example 3.2. A collec ion o pa ame e s is E={κ1, κ2, κ3}. Assume M:E→M(X) is
p o ided by
M1(∂) = 



κ1
(h0.35,0.45i,h0.05,0.10i,h0.45,0.60i)
κ2
(h0.50,0.55i,h0.05,0.15i,h0.40,0.45i)
κ3
(h0.30,0.35i,h0.20,0.25i,h0.40,0.50i)




;M2(∂) = 



κ1
(h0.30,0.35i,h0.10,0.15i,h0.50,0.60i)
κ2
(h0.35,0.45i,h0.20,0.30i,h0.40,0.50i)
κ3
(h0.30,0.40i,h0.10,0.20i,h0.45,0.60i)




;
M3(∂) = 



κ1
(h0.05,0.10i,h0.30,0.35i,h0.50,0.60i)
κ2
(h0.15,0.20i,h0.40,0.45i,h0.35,0.40i)
κ3
(h0.05,0.10i,h0.15,0.25i,h0.45,0.50i)




;
De ini ion 3.3. Assume ha wo 3- alued ENSs on (X, E) a e Mand N. As indica ed by
M Ni and only i Mi(∂) Ni(∂) i >i(∂) >i(∂),=i(∂) =i(∂),`i(∂)`i(∂),
∀∂∈X.
We p opose he ollowing nume ical example o demons a e he p e iously men ioned de -
ini ion:
Example 3.4. Conside he 3- alued ENS Min Example 3.2. Le Nbe ano he 3- alued
ENS is de ined as ollows:
N1(∂) = 



κ1
(h0.53,0.63i,h0.18,0.38i,h0.27,0.32i)
κ2
(h0.63,0.78i,h0.18,0.38i,h0.07,0.12i)
κ3
(h0.48,0.68i,h0.38,0.58i,h0.17,0.22i)




;N2(∂) = 



κ1
(h0.48,0.58i,h0.28,0.48i,h0.12,0.17i)
κ2
(h0.53,0.68i,h0.43,0.68i,h0.02,0.07i)
κ3
(h0.48,0.78i,h0.38,0.48i,h0.17,0.22i)




;
N3(∂) = 



κ1
(h0.33,0.48i,h0.48,0.63i,h0.07,0.12i)
κ2
(h0.43,0.58i,h0.68,0.68i,h0.07,0.12i)
κ3
(h0.33,0.58i,h0.43,0.53i,h0.12,0.17i)




;
De ini ion 3.5. Suppose Mand Na e wo 3- alued ENSs on (X, E). These wo 3- alued
ENSs a e iden ical (deno ed by M=N) i and only i Mi Niand MiwNi.
Neu osophic Se s and Sys ems, Vol. 94, 2025 35
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
De ini ion 3.6. Le Mand Ndeno e wo 3- alued ENSs on (X, E). The union and in e sec-
ion o Mand No e (X, E) a e indica ed by M Nand MuN, espec i ely, J:E→M(X),
I:E→M(X) such ha J(∂) = M(∂) N(∂), I(∂) = M(∂)uN(∂), o all ∂∈X.
Example 3.7. Le Mand Nbe he wo 3- alued ENSs on (X, E) ha a e de ined as
M1(∂) = 



κ1
(h0.30,0.50i,h0.20,0.30i,h0.30,0.50i)
κ2
(h0.20,0.40i,h0.50,0.60i,h0.30,0.40i)
κ3
(h0.30,0.60i,h0.20,0.40i,h0.20,0.30i)




;M2(∂) = 



κ1
(h0.20,0.50i,h0,0.10i,h0.50,0.60i)
κ2
(h0.30,0.40i,h0.10,0.30i,h0.50,0.60i)
κ3
(h0.40,0.60i,h0.10,0.20i,h0.30,0.50i)




;
and
N1(∂) = 



κ1
(h0.30,0.40i,h0.20,0.40i,h0.40,0.60i)
κ2
(h0.20,0.30i,h0,0.20i,h0.50,0.70i)
κ3
(h0.40,0.50i,h0.30,0.40i,h0.20,0.30i)




;N2(∂) = 



κ1
(h0.10,0.40i,h0.20,0.30i,h0.50,0.60i)
κ2
(h0.20,0.30i,h0.30,0.40i,h0.40,0.50i)
κ3
(h0.20,0.40i,h0.30,0.50i,h0.40,0.50i)




;
M1(∂) N1(∂) = 



κ1
(h0.30,0.50i,h0.20,0.30i,h0.30,0.50i)
κ2
(h0.20,0.40i,h0,0.20i,h0.30,0.40i)
κ3
(h0.40,0.60i,h0.20,0.40i,h0.20,0.30i)




;M2(∂) N2(∂) = 



κ1
(h0.20,0.50i,h0,0.10i,h0.50,0.60i)
κ2
(h0.30,0.40i,h0.10,0.30i,h0.40,0.50i)
κ3
(h0.40,0.60i,h0.10,0.20i,h0.30,0.50i)




;
M1(∂)uN1(∂) = 



κ1
(h0.30,0.40i,h0.20,0.30i,h0.40,0.60i)
κ2
(h0.20,0.30i,h0,0.20i,h0.50,0.70i)
κ3
(h0.30,0.50i,h0.20,0.40i,h0.20,0.30i)




;M2(∂)uN2(∂) = 



κ1
(h0.10,0.40i,h0,0.10i,h0.50,0.60i)
κ2
(h0.20,0.30i,h0.10,0.30i,h0.50,0.60i)
κ3
(h0.20,0.40i,h0.10,0.20i,h0.40,0.50i)




;
4. De e mine simila i y measu e
Me hods: Le Zand Ybe he wo 3- alued ENSs. The simila i y measu e be ween Zand
Yis de ined as Sim(Z, Y ) = Φ(Z, Y ). Since
Φ(Z, Y ) = 1
n0n
n0
M
k=p
n
M
i=p






min (Tl
1(Z(∂k), Y (∂k)) , Tl
2(Z(∂k), Y (∂k)) , Sl(Z(∂k), Y (∂k)) ),
max (Tu
1(Z(∂k), Y (∂k)) , Tu
2(Z(∂k), Y (∂k)) , Su(Z(∂k), Y (∂k)) )





(1)
and T1(Z(∂k), Y (∂k)) =
Ln0
k=pLn
i=p>il
Z(∂k)· >il
Y(∂k)
Ln0
k=pLn
i=pp−qp−>i2l
Z(∂k)·p−>i2l
Y(∂k),Ln0
k=pLn
i=p>iu
Z(∂k)· >iu
Y(∂k)
Ln0
k=pLn
i=pp−qp−>i2u
Z(∂k)·p−>i2u
Y(∂k)!
(2)
T2(Z(∂k), Y (∂k)) =
Ln0
k=pLn
i=p=i2l
Z(∂k)· =i2l
Y(∂k)
Ln0
k=pLn
i=pp−qp−=i4l
Z(∂k)·p−=i4l
Y(∂k),Ln0
k=pLn
i=p=i2u
Z(∂k)· =i2u
Y(∂k)
Ln0
k=pLn
i=pp−qp−=i4u
Z(∂k)·p−=i4u
Y(∂k)!
(3)
Neu osophic Se s and Sys ems, Vol. 94, 2025 36
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s

S(Z(∂k), Y (∂k)) =
p−
u
u
Ln0
k=pLn
i=p(`i2l
Z(∂k)−`i2u
Y(∂k))
Ln0
k=pLn
i=pp+(`i2u
Z(∂k)·`i2u
Y(∂k)),Ln0
k=pLn
i=p(`i2u
Z(∂k)−`i2l
Y(∂k))
Ln0
k=pLn
i=pp+(`i2l
Z(∂k)·`i2l
Y(∂k))!(4)
Theo em 4.1. Le Z, Y and Wdeno e any h ee 3- alued ENSs o e (X, E). Show ha
Z⊆Y W=⇒Sim(Z, W )Sim(Y, W ).
P oo . Fo i= 1,2, ..., n and k= 1,2, ..., n0















































Z Y=⇒








>il
Z(∂k),>iu
Z(∂k)>il
Y(∂k),>iu
Y(∂k)
=il
Z(∂k),=iu
Z(∂k)=il
Y(∂k),=iu
Y(∂k)
`il
Z(∂k),`iu
Z(∂k)`il
Y(∂k),`iu
Y(∂k)









Z W=⇒








>il
Z(∂k),>iu
Z(∂k)>il
W(∂k),>iu
W(∂k)
=il
Z(∂k),=iu
Z(∂k)=il
W(∂k),=iu
W(∂k)
`il
Z(∂k),`iu
Z(∂k)`il
W(∂k),`iu
W(∂k)









Y W=⇒








>il
Y(∂k),>iu
Y(∂k)>il
W(∂k),>iu
W(∂k)
=il
Y(∂k),=iu
Y(∂k)=il
W(∂k),=iu
W(∂k)
`il
Y(∂k),`iu
Y(∂k)`il
W(∂k),`iu
W(∂k)
























































(5)
Clea ly,
>il
Z(∂k)· >il
W(∂k),>iu
Z(∂k)· >iu
W(∂k)>il
Y(∂k)· >il
W(∂k),>iu
Y(∂k)· >iu
W(∂k)
n0
M
k=p
n
M
i=p>il
Z(∂k)· >il
W(∂k),
n0
M
k=p
n
M
i=p>iu
Z(∂k)· >iu
W(∂k)n0
M
k=p
n
M
i=p>il
Y(∂k)· >il
W(∂k),
n0
M
k=p
n
M
i=p>iu
Y(∂k)· >iu
W(∂k)
(6)
Clea ly,
>i2l
Z(∂k),>i2u
Z(∂k)>i2l
Y(∂k),>i2u
Y(∂k)>i2l
W(∂k),>i2u
W(∂k)
and
− >i2u
Z(∂k),−>i2l
Z(∂k)− >i2u
Y(∂k),−>i2l
Y(∂k)− >i2u
W(∂k),−>i2l
W(∂k)
and
p−>i2l
Z(∂k),p−>i2u
Z(∂k)p−>i2l
Y(∂k),p−>i2u
Y(∂k)p−>i2l
W(∂k),1− >i2u
W(∂k)
and
p−>i2l
Z(∂k)·p−>i2l
W(∂k),p−>i2u
Z(∂k)·p−>i2u
W(∂k)
p−>i2l
Y(∂k)·p−>i2l
W(∂k),p−>i2u
Y(∂k)·p−>i2u
W(∂k)
Neu osophic Se s and Sys ems, Vol. 94, 2025 37
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
and
qp−>i2l
Z(∂k)·p−>i2l
W(∂k),qp−>i2u
Z(∂k)·p−>i2u
W(∂k)
qp−>i2l
Y(∂k)·p−>i2l
W(∂k),qp−>i2u
Y(∂k)·p−>i2u
W(∂k)
and
p−qp−>i2l
Z(∂k)·p−>i2l
W(∂k)qp−>i2u
Z(∂k)·p−>i2u
W(∂k)
p−qp−>i2l
Y(∂k)·p−>i2l
W(∂k)qp−>i2u
Y(∂k)·p−>i2u
W(∂k)
p−qp−>i2u
Z(∂k)·p−>i2u
W(∂k)p−qp−>i2l
Z(∂k)·p−>i2l
W(∂k)
p−qp−>i2u
Y(∂k)·p−>i2u
W(∂k)p−qp−>i2l
Y(∂k)·p−>i2l
W(∂k)
and











Ln0
k=pLn
i=pp−qp−>i2u
Z(∂k)·p−>i2u
W(∂k) ,Ln0
k=pLn
i=pp−qp−>i2l
Z(∂k)·p−>i2l
W(∂k) !
Ln0
k=pLn
i=pp−qp−>i2u
Y(∂k)·p−>i2u
W(∂k) ,Ln0
k=pLn
i=pp−qp−>i2l
Y(∂k)·p−>i2l
W(∂k) !











(7)
Equa ions (6) is di ided by (7),
Ln0
k=pLn
i=p>il
Z(∂k)· >il
W(∂k)
Ln0
k=pLn
i=pp− p−>i2u
Z(∂k)·p−>i2u
W(∂k) ,Ln0
k=pLn
i=p>iu
Z(∂k)· >iu
W(∂k)
Ln0
k=pLn
i=pp− p−>i2l
Z(∂k)·p−>i2l
W(∂k) 
Ln0
k=pLn
i=p>il
Y(∂k)· >il
W(∂k)
Ln0
k=pLn
i=pp− p−>i2u
Y(∂k)·p−>i2u
W(∂k) ,Ln0
k=pLn
i=p>iu
Y(∂k)· >iu
W(∂k)
Ln0
k=pLn
i=pp− p−>i2l
Y(∂k)·p−>i2l
W(∂k) 
Hence
Neu osophic Se s and Sys ems, Vol. 94, 2025 38
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s







































Ln0
k=pLn
i=p(>il
Z(∂k)·>il
W(∂k))!
Ln0
k=pLn
i=pp−q((p−>i2l
Z(∂k))·(p−>i2l
W(∂k))) !, Ln0
k=pLn
i=p(>iu
Z(∂k)·>iu
W(∂k))!
Ln0
k=pLn
i=pp−q((p−>i2u
Z(∂k))·(p−>i2u
W(∂k))) !













Ln0
k=pLn
i=p(>il
Y(∂k)·>il
W(∂k))!
Ln0
k=pLn
i=pp−q((p−>i2l
Y(∂k))·(p−>i2l
W(∂k))) !, Ln0
k=pLn
i=p(>iu
Y(∂k)·>iu
W(∂k))!
Ln0
k=pLn
i=pp−q((p−>i2u
Y(∂k))·(p−>i2u
W(∂k))) !







































;
(8)
The e o e
T1(Z(∂k), W(∂k)) T1(Y(∂k), W (∂k))
Clea ly,
=i2l
Z(∂k)· =i2l
W(∂k),=i2u
Z(∂k)· =i2u
W(∂k)=i2l
Y(∂k)· =i2l
W(∂k),=i2u
Y(∂k)· =i2u
W(∂k).
Hence
n0
M
k=p
n
M
i=p=i2l
Z(∂k)· =i2l
W(∂k),
n0
M
k=p
n
M
i=p=i2u
Z(∂k)· =i2u
W(∂k)
n0
M
k=p
n
M
i=p=i2l
Y(∂k)· =i2l
W(∂k),
n0
M
k=p
n
M
i=p=i2u
Y(∂k)· =i2u
W(∂k)(9)
Clea ly,
=i4l
Z(∂k),=i4u
Z(∂k)=i4l
Y(∂k),=i4u
Y(∂k)=i4l
W(∂k),=i4u
W(∂k)
implies ha
− =i4u
Z(∂k),−=i4l
Z(∂k)− =i4u
Y(∂k),−=i4l
Y(∂k)− =i4u
W(∂k),−=i4l
W(∂k)
p−=i4l
Z(∂k),p−=i4u
Z(∂k)p−=i4l
Y(∂k),p−=i4u
Y(∂k)p−=i4l
W(∂k),1− =i4u
W(∂k)
and
p−=i4l
Z(∂k)·p−=i4l
W(∂k),p−=i4u
Z(∂k)·p−=i4u
W(∂k)
p−=i4l
Y(∂k)·p−=i4l
W(∂k),p−=i4u
Y(∂k)·p−=i4u
W(∂k)
Neu osophic Se s and Sys ems, Vol. 94, 2025 39
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
qp−=i4l
Z(∂k)·p−=i4l
W(∂k),qp−=i4u
Z(∂k)·p−=i4u
W(∂k)
qp−=i4l
Y(∂k)·p−=i4l
W(∂k),qp−=i4u
Y(∂k)·p−=i4u
W(∂k)
p−qp−=i4l
Z(∂k)·p−=i4l
W(∂k),qp−=i4u
Z(∂k)·p−=i4u
W(∂k)
p−qp−=i4l
Y(∂k)·p−=i4l
W(∂k),qp−=i4u
Y(∂k)·p−=i4u
W(∂k)
p−qp−=i4u
Z(∂k)·p−=i4u
W(∂k),p−qp−=i4l
Z(∂k)·p−=i4l
W(∂k)
p−qp−=i4u
Y(∂k)·p−=i4u
W(∂k),p−qp−=i4l
Y(∂k)·p−=i4l
W(∂k)











Ln0
k=pLn
i=pp− p−=i4u
Z(∂k)·p−=i4u
W(∂k) ,Ln0
k=pLn
i=pp− p−=i4l
Z(∂k)·p−=i4l
W(∂k) !
Ln0
k=pLn
i=pp− p−=i4u
Y(∂k)·p−=i4u
W(∂k) ,Ln0
k=pLn
i=pp− p−=i4l
Y(∂k)·p−=i4l
W(∂k) !











;
(10)
Equa ion (9) is di ided by (10),
Ln0
k=pLn
i=p=i2l
Z(∂k)· =i2l
W(∂k),Ln0
k=pLn
i=p=i2u
Z(∂k)· =i2u
W(∂k)
Ln0
k=pLn
i=pp− p−=i4u
Z(∂k)·p−=i4u
W(∂k) ,Ln0
k=pLn
i=pp− p−=i4l
Z(∂k)·p−=i4l
W(∂k) 

Ln0
k=pLn
i=p=i2l
Y(∂k)· =i2l
W(∂k),Ln0
k=pLn
i=p=i2u
Y(∂k)· =i2u
W(∂k)
Ln0
k=pLn
i=pp− p−=i4u
Y(∂k)·p−=i4u
W(∂k) ,Ln0
k=pLn
i=pp− p−=i4l
Y(∂k)·p−=i4l
W(∂k) 
Hence







































Ln0
k=pLn
i=p(=i2l
Z(∂k)·=i2l
W(∂k))!
Ln0
k=pLn
i=pp−q((p−=i4l
Z(∂k))·(p−=i4l
W(∂k))) !, Ln0
k=pLn
i=p(=i2u
Z(∂k)·=i2u
W(∂k))!
Ln0
k=pLn
i=pp−q((p−=i4u
Z(∂k))·(p−=i4u
W(∂k))) !













Ln0
k=pLn
i=p(=i2l
Y(∂k)·=i2l
W(∂k))!
Ln0
k=pLn
i=pp−q((p−=i4l
Y(∂k))·(p−=i4l
W(∂k))) !, Ln0
k=pLn
i=p(=i2u
Y(∂k)·=i2u
W(∂k))!
Ln0
k=pLn
i=pp−q((p−=i4u
Y(∂k))·(p−=i4u
W(∂k))) !







































;
The e o e
T2(Z(∂k), W(∂k)) T2(Y(∂k), W (∂k)) (11)
Clea ly,
`i2l
Z(∂k),`i2u
Z(∂k)`i2l
Y(∂k),`i2u
Y(∂k)`i2l
W(∂k),`i2u
W(∂k)
and
Neu osophic Se s and Sys ems, Vol. 94, 2025 40
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s
Table 12. 3- alued ENS o he inancial sec o Ti
T(∂)κ4κ5
T1(∂)h0.3,0.4i,h0.65,0.7i,h0.2,0.4i h0.25,0.35i,h0.5,0.55i,h0.2,0.3i
T2(∂)h0.5,0.55i,h0.45,0.5i,h0.25,0.45i h0.4,0.45i,h0.3,0.7i,h0.3,0.5i
T3(∂)h0.5,0.55i,h0.15,0.4i,h0.2,0.3i h0.4,0.5i,h0.55,0.6i,h0.3,0.35i
The expe s p esen he 3- alued ENS and i s alues in Tables 3 o 12, based on hei assessmen
o he al e na i es agains he c i e ia unde discussion. In his example, we should compu e he
simila i y measu e o he 3- alued ENSs in Table 3 o Table 12 wi h he one in Table 1 using me hod.
The simila i y measu e o pa e n ecogni ion Pi o Tiis calcula ed as shown in he able below.
Tables 13 and 14 show di e en alues.
Table 13. Di e en alues
T1
1(x1)T1
2(x1)S1(x1)T2
1(x2)T2
2(x2)
(L, P)h0.800316,0.84237i h0.758793,0.762018i h0.509328,0.765444i h0.671653,0.740835i h0.70213,0.791527i
(L, Q)h0.719578,0.794162i h0.792615,0.873883i h0.84641,0.579972i h0.916231,0.932051i h0.844602,0.866913i
(L, R)h0.895836,0.91482i h0.867642,0.895062i h0.648341,0.923459i h0.888981,0.935176i h0.557979,0.741576i
(L, S)h0.781427,0.819973i h0.679001,0.781781i h0.477713,0.693705i h0.854992,0.884033i h0.550141,0.763987i
(L, T)h0.939365,0.948507i h0.728212,0.806633i h0.512322,0.737975i h0.869847,0.895939i h0.785653,0.815063i
Table 14. Di e en alues
S2(x2)T3
1(x3)T3
2(x3)S3(x3)Simila i y
(L, P)h0.535115,0.893521i h0.652244,0.706035i h0.470374,0.533671i h0.481875,0.652386i0.142049
(L, Q)h0.457751,0.574692i h0.782963,0.865623i h0.700021,0.790201i h0.463118,0.589981i0.143720
(L, R)h0.600575,0.865618i h0.856691,0.877117i h0.665936,0.684441i h0.595246,0.773675i0.132584
(L, S)h0.46486,0.572492i h0.802296,0.841273i h0.709213,0.794079i h0.676507,0.950697i0.14246
(L, T)h0.478567,0.666398i h0.828071,0.890342i h0.73,0.82751i h0.640519,0.901223i0.145903
The pa e n ecogni ion simila i y measu e ollows he o de T > Q > S > P > R, as shown in he
p eceding esul s. As a esul , we conclude ha he pa ien T ep esen s he bes al e na i e.
6. Conclusion
The p ima y pu pose o his s udy is o p o ide a h ee- alued ENS and in es iga e some o i s
ea u es. The simila i y measu e o wo 3- alued ENS is add essed, and an example o el-li e is shown.
In he u u e, we shall use he gene alized hesi an cubic uzzy so se s and gene alized hesi an
in e alued uzzy so se s heo ies.
Acknowledgmen s: The au ho s ex end hei app ecia ion o he A ab Open Uni e si y o suppo -
ing his wo k.
Neu osophic Se s and Sys ems, Vol. 94, 2025 47
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
pa e n ecogni ion ia 3- alued ex ension o neu osophic so se s

Con lic s o In e es : The au ho decla es no con lic o in e es .
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Recei ed: May 7, 2025. Accep ed: Aug 25, 2025
Ahmad A. Abubake , M. Palanikuma , Abdallah Al-Husban, So se models applied o
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