A Machine Learning Approach Using Principal Component Analysis and Cubic Spherical Neutrosophic Sets for MCDM
Full text
Neu osophic Se s and Sys ems, Vol. 97, 2026
Uni e si y o New Mexico
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
A Machine Lea ning App oach Using P incipal Componen
Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
S. Goma hi1*, A. Ka i ha2, R. Ramesh3, E. Ka uppusamy4 and L. Meenachi5
1 Depa men o Ma hema ics, Ra hinam Technical Campus, Coimba o e, India; [email p o ec ed]
2 Depa men o Ma hema ics, Ka pagam College o Enginee ing, Coimba o e, India; [email p o ec ed]
3 Depa men o Ma hema ics, D . Mahalingam College o Enginee ing and Technology, Pollachi, India;
[email p o ec ed]
4 Depa men o Ma hema ics, S i K ishna College o Enginee ing and Technology, Coimba o e, India;
[email p o ec ed]
5 Depa men o In o ma ion Technology, D . Mahalingam College o Enginee ing and Technology, Pollachi, India;
[email p o ec ed]
* Co espondence: [email p o ec ed]
Abs ac : This s udy in oduces wo no el agg ega ion ope a o s, namely he cubic sphe ical
neu osophic weigh ed a i hme ic ope a o and he cubic sphe ical neu osophic weigh ed geome ic
ope a o , o add ess mul i-c i e ia decision-making p oblems unde unce ain y. The p oposed
amewo k is buil upon he concep o he cubic sphe ical neu osophic se , whe e he e alua ions o
decision make s a e ans o med in o a sphe ical ep esen a ion by compu ing he cen e and adius
o he sphe e a he han simply a e aging decision alues. This ans o ma ion enables a mo e
comp ehensi e modelling o u h, inde e minacy, and alsi y deg ees, while p ese ing he geome ic
s uc u e o unce ain y. The cubic sphe ical neu osophic weigh ed a i hme ic ope a o and he cubic
sphe ical neu osophic weigh ed geome ic ope a o s sa is y essen ial ma hema ical p ope ies such
as idempo ency, mono onici y, and boundedness, ensu ing heo e ical soundness. To de e mine he
ela i e signi icance o decision c i e ia, p incipal componen analysis is employed o dimensionali y
educ ion and objec i e weigh es ima ion based on a iance con ibu ion. A nume ical case s udy on
p ima y school selec ion in a pa icula egion is p o ided, whe e he p oposed ope a o s combined
wi h p incipal componen analysis a e used o ank he al e na i es. Finally, a compa a i e analysis
wi h exis ing MCDM app oaches demons a es s ong co ela ion wi h benchma k esul s, while
highligh ing he enhanced disc imina ion powe and obus ness o he p oposed me hodology.
Keywo ds: Machine lea ning; P incipal componen analysis; Cubic sphe ical neu osophic se s.
1. In oduc ion
Decision-making in p ima y educa ion policy and planning in ol es assessing mul iple al e na i es
ac oss a wide ange o c i e ia, such as lea ning ou come imp o emen , cos e iciency, communi y
accep ance, and sus ainabili y. In u al and semi-u ban egions, whe e esou ces a e limi ed, he
p io i iza ion o in e en ions mus be bo h accu a e and equi able. Con en ional Mul i-C i e ia
Decision-Making (MCDM) me hods o en ail o handle he high deg ee o unce ain y and
inconsis ency in expe opinions, especially when quali a i e judgmen s a e con e ed in o quan i a i e
scales.
Neu osophic se heo y, in oduced by Sma andache [8], ex ends uzzy [10] and in ui ionis ic uzzy
[2] se s by inco po a ing u h, inde e minacy, and alsi y membe ship unc ions, he eby p o iding a
Neu osophic Se s and Sys ems, Vol. 97, 2026 295
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
iche ma hema ical ep esen a ion o unce ain y. Wi hin his amewo k, he Cubic Sphe ical
Neu osophic Se [4] (CSNS) is a geome ic ep esen a ion ha models a collec ion o neu osophic se s
wi hin a uni cube. Unlike adi ional sphe ical neu osophic numbe s ha assign membe ship deg ees
indi idually, CSNS cap u es he en i e se o neu osophic alues by calcula ing a cen e ( ep esen a i e
neu osophic se ) o he collec ion. The adius o he CSNS is hen de ined as he maximum dis ance
be ween his cen e and any neu osophic se wi hin he collec ion, he eby enclosing all elemen s inside
a sphe e in he neu osophic space. This sphe ical geome ic ep esen a ion enables simul aneous
handling o mul iple neu osophic alues and hei unce ain y by conside ing bo h he collec i e
a e age and hei sp ead. The CSNS hus o e s a mo e holis ic and in ui i e way o agg ega e and
analyze mul iple neu osophic se s in decision-making p oblems.
The concep o Cubic Sphe ical Neu osophic Se s (CSNSs) has eme ged as an ad anced amewo k o
ep esen ing and p ocessing unce ain y in complex decision-making en i onmen s [4]. A CSNS
cha ac e izes each elemen h ough a sphe ical ep esen a ion, de ined by i s cen e —cap u ing u h,
inde e minacy, and alsi y deg ees—and a adius ha e lec s he neu ali y ma gin, enabling a mo e
lexible and geome ic in e p e a ion o imp ecise in o ma ion. The ini ial o mula ion o CSNSs
in oduced hei s uc u e, basic ope a ions, and applica ions in mul i-c i e ia decision-making
(MCDM), suppo ed by cosine dis ance-based anking me hods [4]. Building on his ounda ion,
u he s udies ex ended CSNSs in o cubic sphe ical neu osophic opological spaces, in eg a ing
opological p inciples o analyze con inui y, con e gence, and ela ed spa ial p ope ies in unce ain
sys ems [3]. Addi ionally, no el weigh ed addi i e and weigh ed geome ic agg ega ion ope a o s ha e
been de eloped o combine in o ma ion om mul iple expe s o c i e ia, he eby enhancing g oup
decision-making accu acy [5]. The in eg a ion o A chimedean -no ms and -cono ms wi hin he CSNS
amewo k has p o ided obus algeb aic ools o using unce ain da a, wi h applica ions
demons a ed in domains such as ag icul u al planning and elec ic uck selec ion [6]. Collec i ely,
hese ad ancemen s posi ion CSNSs as a e sa ile and ma hema ically igo ous app oach, capable o
add essing eal-wo ld p oblems in ol ing ambigui y, con lic ing opinions, and mul i-dimensional
unce ain y.
In 1987, Wold, Esbensen, and Geladi [9] published a ounda ional u o ial on P incipal Componen
Analysis (PCA) in Chemome ics and In elligen Labo a o y Sys ems, es ablishing PCA as a pi o al
me hod o mul i a ia e da a analysis. Thei wo k explained how PCA ans o ms high-dimensional,
possibly co ela ed a iables in o a educed se o unco ela ed p incipal componen s ha cap u e he
mos signi ican a iance— acili a ing pa e n ecogni ion, da a educ ion, and in e p e abili y in
complex da ase s. Fas o wa d o 2019, Aslam and Albassam [1] applied neu osophic logic in
epidemiology, speci ically o examine he unce ain ela ionship be ween die a y a consump ion and
p os a e cance mo ali y ac oss 30 coun ies. Thei s udy demons a ed ha neu osophic eg ession
and co ela ion-unlike classical s a is ical models-e icien ly handle inde e mina e, in e al- alued
da a, e ealing a posi i e associa ion be ween highe die a y a le els and inc eased p os a e cance
dea h a es. Mos ecen ly, in 2025, Rod íguez, Solis, and Jiménez [7] ad anced he in eg a ion o PCA
wi h neu osophic logic by conduc ing a compa a i e sociocul u al analysis o eu hanasia legisla ion in
Ecuado and o he La in Ame ican coun ies. By combining PCA’s abili y o unco e la en s uc u es
wi h neu osophic me hods’ handling o e hical ambigui y, hey mapped how ac o s like Human
De elopmen Index (HDI), mo al a i udes, and eligiosi y clus e ac oss con ex s-demons a ing ha
ackling mul idimensional, alue-laden issues equi es bo h la en pa e n de ec ion and unce ain y
modelling.
In he p oposed wo k, decision-make judgmen s a e di ec ly con e ed in o cubic sphe ical
neu osophic alues ins ead o using a e age-based agg ega ion, hus a oiding he po en ial loss o
indi idual decision pa e ns. To manage he la ge numbe o e alua ion c i e ia, P incipal Componen
Analysis (PCA) is applied o educe dimensionali y and de e mine objec i e weigh s o he educed
Neu osophic Se s and Sys ems, Vol. 97, 2026 296
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
se o c i e ia. Finally, we u ilize cubic sphe ical neu osophic a i hme ic and geome ic agg ega ion
ope a o s o syn hesize he e alua ions in o o e all sco es, ollowed by a compa a i e analysis wi h
exis ing me hods o alida e he e ec i eness o he app oach.
1.1 Resea ch Gap
While se e al MCDM me hods inco po a ing uzzy and neu osophic se s ha e been applied in
educa ion and o he domains, he ollowing limi a ions emain:
1. Loss o expe in o ma ion – Mos me hods agg ega e expe judgmen s using a e ages,
leading o in o ma ion dis o ion and loss o decision-make -speci ic nuances.
2. Limi ed applica ion o cubic sphe ical neu osophic se s – The in eg a ion o cubic sphe ical
neu osophic concep s o iche unce ain y modelling in educa ional decision-making is
unde explo ed.
3. Unde u iliza ion o PCA o c i e ia weigh ing – PCA is p ima ily used o da a educ ion in
MCDM bu a ely o bo h educe dimensionali y and de i e objec i e c i e ia weigh s in a
neu osophic con ex .
This esea ch add esses hese gaps by combining cubic sphe ical neu osophic ep esen a ion, PCA-
based c i e ia weigh ing, and agg ega ion ope a o s, ollowed by a compa a i e e alua ion agains
baseline me hods.
1.2 Mo i a ion
In complex eal-wo ld decision-making scena ios, such as supply chain managemen , heal hca e
planning, and enginee ing design, unce ain y and imp ecision o en hinde he eliabili y o esul s.
T adi ional MCDM app oaches a e limi ed in handling he agueness o human judgmen s and he
p esence o mul iple con lic ing c i e ia. Cubic Sphe ical Neu osophic Se s (CSNS) p o ide a powe ul
amewo k o model u h, inde e minacy, and alsi y simul aneously, he eby cap u ing mo e ealis ic
in o ma ion om decision make s. Howe e , when decision p oblems in ol e a la ge numbe o
c i e ia, exis ing CSNS-based app oaches ace challenges o compu a ional complexi y and educed
accu acy. To o e come hese issues, he e is a p essing need o an in eg a ed me hodology ha no
only le e ages he exp essi e powe o CSNS bu also educes dimensionali y in a meaning ul way.
1.3 Objec i e
This s udy aims o de elop a no el hyb id MCDM amewo k ha in eg a es CSNS ep esen a ion,
P incipal Componen Analysis (PCA), and weigh ed agg ega ion ope a o s o imp o e decision-
making unde unce ain y. The speci ic objec i es a e o:
1. Con e linguis ic e alua ions om mul iple decision make s in o cubic sphe ical
neu osophic ep esen a ions.
2. Apply sco e unc ions o ans o m neu osophic in o ma ion in o c isp alues o u he
p ocessing.
3. Employ PCA o s anda dize and educe he dimensionali y o he c i e ia while p ese ing
signi ican a iance.
4. Use CSNS weigh ed agg ega ion ope a o s (CSNWA and CSNWG) o e ec i ely combine
in o ma ion ac oss educed c i e ia.
5. In oduce Hamming dis ance–based simila i y wi h he ideal al e na i e o anking.
6. Demons a e he applicabili y o he p oposed amewo k h ough a eal-wo ld case s udy ,
ensu ing bo h heo e ical igo and p ac ical ele ance.
1.4 Ou come
The p oposed me hodology o e s a compu a ionally e icien and accu a e decision-making p ocess
ha enhances he applicabili y o CSNS in high-dimensional MCDM p oblems. By in oducing PCA
Neu osophic Se s and Sys ems, Vol. 97, 2026 297
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
in o he neu osophic decision amewo k, he model educes edundancy, imp o es s abili y o
weigh s, and enhances he eliabili y o agg ega ed esul s. The no el y o his wo k lies in he i s
sys ema ic in eg a ion o PCA wi h CSNS-based MCDM, which no only imp o es decision accu acy
bu also signi ican ly educes compu a ional o e head. The amewo k enables decision make s o
ank and selec he bes al e na i e wi h g ea e con idence, while sensi i i y analysis alida es he
obus ness o he esul s. O e all, his s udy con ibu es a p ac ical and gene alizable me hodology
sui able o di e se applica ions in enginee ing, managemen , and heal hca e decision-making.
1.2 P elimina ies
De ini ion 1.3.2: [4] Le 𝑋
be a ixed uni e se and 𝑥⊆
𝑋. A Cubic Sphe ical Neu osophic Se (CSNS),
deno ed by
𝑥
is de ined as:
𝑥
={<
𝑥
,
T
(
𝑥
),
I
(
𝑥
),
F
(
𝑥
);
𝑅
>:
𝑥
∈
𝑋
}
whe e 𝐓(𝒙)
,
𝐈(𝒙)
,
𝐅(𝒙)
∶
𝑿
→ [0,1] a e he u h, inde e minacy, and alsi y-membe ship unc ions
sa is ying 0 ≤𝐓(𝒙)+𝐈(𝒙)
+
𝐅(𝒙)
≤ 3 and 𝐑∈[0,1] is he adius o he sphe e cen ed a he poin
(𝐓(𝐱),𝐈(𝐱)
,
𝐅(𝐱)
)
in he neu osophic space.
Gi en e alua ions
{<
𝑥
i,
∶T
i,j
,
I
i,j
,
F
i,j
,>𝑖=1,2,…𝑗=1,2,,,,,k
i
}
he sphe e cen e is:
<
T
(
𝑥
i
),
I
(
𝑥
i
),
F
(
𝑥
i
)>=<1
k
i
j=1
k
i
T
i,j
,1
k
i
j=1
k
i
I
i,j
,1
k
i
j=1
k
i
F
i,j
>
and he adius
Ri
is
Ri
=minimum {maximum
1≤j≤k
i
√(T(
𝑥
i
)
−
T
i,j
)
2
+(
I(
𝑥
i
)
−
I
i,j
)
2
+(
F(
𝑥
i
)
−
F
i,j
)
2
,1}
De ini ion 1.3.2 [4] Le 𝛼1=〈𝑇1,𝐼1,𝐹1;𝑅1〉, 𝛼2=〈𝑇1, 𝐼1,𝐹1;𝑅1〉 a e he wo CSNSs o e he uni e sal se
𝑋 and γ(minimum,maximum ),a>0. The ope a o s de ined as ollows:
1. 𝑎
𝛼1=〈1−(1−𝑇1)𝑎,𝐼1𝑎,𝐹1𝑎;𝑅1𝑎〉.
2. 𝛼1𝑎=〈𝑇1𝑎,1−(1−𝐼1)𝑎,1−(1−𝐹1)𝑎; 𝑅1𝑎〉.
2. Cubic Sphe ical Neu osophic Agg ega ion Ope a o s
De ini ion 2.1: Le 𝛼1=〈𝑇1,𝐼1,𝐹1;𝑅1〉, 𝛼2=〈𝑇1, 𝐼1,𝐹1;𝑅1〉 a e he wo CSNSs o e he uni e sal se 𝑋
and γ(minimum,maximum). The ope a o s de ined as ollows:
1. 𝛼1⨁𝛼2=〈 𝑇1+𝑇2−𝑇1𝑇2, 𝐼1𝐼2,𝐹1𝐹2; γ(𝑅1,𝑅2)〉.
2. ⨁𝑖=1
𝑛𝛼𝑖 =〈1−∏(1−𝑇1),
𝑛𝑖=1 ∏𝐼1,
𝑛𝑖=1 ∏𝐹1;
𝑛𝑖=1 γ(𝑅𝑖)〉.
3. 𝛼1⨂𝛼2=〈𝑇1𝑇2,𝐼1+𝐼2−𝐼1𝐼2,𝐹1+𝐹2−𝐹1𝐹2; γ(𝑅1,𝑅2)〉.
4. ⨂𝑖=1
𝑛 𝛼𝑖 =〈∏ 𝑇𝑖,
𝑛𝑖=1 1−∏(1−𝐼𝑖),1−∏(1−𝐹𝑖); γ (𝑅𝑖)
𝑛𝑖=1
𝑛𝑖=1 〉.
5. 𝑎
𝛼1=〈1−(1−𝑇1)𝑎,𝐼1𝑎,𝐹1𝑎;𝑅1𝑎〉.
6. 𝛼1𝑎=〈𝑇1𝑎,1−(1−𝐼1)𝑎,1−(1−𝐹1)𝑎; 𝑅1𝑎〉.
Theo em 2.2: Le 𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 be he CSNSs and 𝑤={𝑤1,𝑤2,…,𝑤𝑛}𝑇 be he weigh s o he se
𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 (i = 1, 2, …, n) whe e 𝑤𝑖 is lying be ween 0 o 1, 𝑤1+𝑤2+⋯+𝑤𝑛=1 and
γ(minimum,maximum). Then, CSN- weigh ed a i hme ic (CSNWA) ope a o is
Neu osophic Se s and Sys ems, Vol. 97, 2026 298
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)=⨁𝑖=1
𝑛𝑤𝑖𝛼𝑖=
{
1−∏ (1−𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
∏(𝐼𝑖)𝑤
𝑖
𝑛𝑖=1
∏(𝐹𝑖)𝑤
𝑖
𝑛𝑖=1
γ (𝑅𝑖)𝑤
𝑖
P oo :
Le 𝛼1=〈𝑇1,𝐼1,𝐹1;𝑅1〉 and 𝛼2=〈𝑇2,𝐼2,𝐹2;𝑅2〉
By he basic ope a ions,
𝑤1 𝛼1=〈1−(1−𝑇1)𝑤
1,𝐼1𝑤
1,𝐹1𝑤
1;𝑅1𝑤
1〉.
𝑤2 𝛼2=〈1−(1−𝑇2)𝑤
2,𝐼2𝑤
2,𝐹2𝑤
2;𝑅2𝑤
2〉.
𝑤1𝛼1⨁𝑤2𝛼2=〈𝑇1𝑤
1+𝑇2𝑤
2−𝑇1𝑤
1𝑇2𝑤
2,𝐼1𝑤
1 𝐼2𝑤
2,𝐹1𝑤
1𝐹2𝑤
2;γ(𝑅1𝑤
1,𝑅2𝑤
2)〉
Le us p o e,
⨁𝑖=1
𝑛𝑤𝑖𝛼𝑖=
{
1−∏ (1−𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
∏(𝐼𝑖)𝑤
𝑖
𝑛𝑖=1
∏(𝐹𝑖)𝑤
𝑖
𝑛𝑖=1
γ (𝑅𝑖)𝑤
𝑖
Hence p o ed.
Theo em 2.3 (Idempo ency)
Le 𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 (i =1, 2, …, n) be he CSNS and hen 𝛼𝑖=𝛼=〈𝑇,𝐼,𝐹;𝑅〉 o all i, hen
𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)=𝐶𝑆𝑁𝑊𝐴(𝛼,𝛼,…,𝛼).
P oo :
Since 𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 (i =1, 2, …, n)
𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)=𝐶𝑆𝑁𝑊𝐴(𝛼,𝛼,…,𝛼)=
{
1−∏ (1−𝑇)𝑤
𝑛𝑖=1
∏(𝐼)𝑤
𝑛𝑖=1
∏(𝐹)𝑤
𝑛𝑖=1
γ (𝑅)𝑤
=〈𝑇,𝐼,𝐹;𝑅〉
Hence p o ed.
Theo em 2.4 (Mono onici y)
Le 𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 and 𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖; 𝑅𝑖〉 (i = 1, 2, …, n ) be he wo collec ion o CSNSs. I 𝛼𝑖≥𝛼𝑖
o all i, suppose 𝑇𝑖≥𝑇𝑖, 𝐼𝑖≤𝐼𝑖, 𝐹𝑖≤𝐹𝑖 and 𝑅𝑖≥𝑅𝑖 hen 𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)=𝐶𝑆𝑁𝑊𝐴(𝛼,𝛼,…,𝛼).
P oo :
(I). Since 𝑇𝑖≥𝑇𝑖 o all i, 𝑇𝑖≥𝑇𝑖
1−𝑇𝑖≤1−𝑇𝑖
Neu osophic Se s and Sys ems, Vol. 97, 2026 299
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
(1−𝑇𝑖)𝑤
𝑖≤(1−𝑇𝑖)𝑤
𝑖
∏(1−𝑇𝑖)𝑤
𝑖≤
𝑛𝑖=1 ∏ (1−𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
1−∏ (1−𝑇𝑖)𝑤
𝑖≥1−
𝑛𝑖=1 ∏ (1−𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
Simila ly, we can p o e adius.
(II). Since 𝐼𝑖≤𝐼𝑖 o all i, 𝐼𝑖≤𝐼𝑖
(𝐼𝑖)𝑤
𝑖≤(𝐼𝑖)𝑤
𝑖
∏(𝐼𝑖)𝑤
𝑖≤
𝑛𝑖=1 ∏(𝐼𝑖)𝑤
𝑖
𝑛𝑖=1
Simila ly, p o e he adius.
Theo em 2.5 (Boundedness)
𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖; 𝑅𝑖〉 be he collec ion o CSNS and 𝛼𝑖−=(𝑚𝑖𝑛(𝑇𝑖),𝑚𝑎𝑥(𝐼𝑖),𝑚𝑎𝑥(𝐹𝑖);𝑚𝑖𝑛(𝑅𝑖)) and
𝛼𝑖+=(𝑚𝑎𝑥(𝑇𝑖),𝑚𝑖𝑛(𝐼𝑖),𝑚𝑖𝑛(𝐹𝑖);𝑚𝑎𝑥(𝑅𝑖)) hen 𝛼𝑖−≤𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)≤𝛼𝑖+.
P oo :
Since 𝛼𝑖≥𝛼𝑖, hen based on he abo e heo ems,
𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)≥𝐶𝑆𝑁𝑊𝐴(𝛼1−,𝛼2−,…,𝛼𝑛−)=𝛼𝑖−
𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)≤𝐶𝑆𝑁𝑊𝐴(𝛼1+,𝛼2+,…,𝛼𝑛+)=𝛼𝑖+. Then 𝛼𝑖−≤𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)≤𝛼𝑖+.
Hence p o ed.
Theo em 2.6 : Le 𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 be he CSNSs and 𝑤={𝑤1,𝑤2,…,𝑤𝑛}𝑇 be he weigh s o he se
𝛼𝑖=〈𝑇𝑖,𝐼𝑖,𝐹𝑖;𝑅𝑖〉 (i = 1, 2, …, n) whe e 𝑤𝑖 is lying be ween 0 o 1, 𝑤1+𝑤2+⋯+𝑤𝑛=1 and
γ(minimum,maximum). Then, CSN- weigh ed geome ic (CSNWG) ope a o is
𝐶𝑆𝑁𝑊𝐺(𝛼1,𝛼2,…,𝛼𝑛)=⨂𝑖=1
𝑛(𝑤𝑖𝛼𝑖)=
{
∏(𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
1−∏ (1−𝐼𝑖)𝑤
𝑖
𝑛𝑖=1
1−∏ (1−𝐹𝑖)𝑤
𝑖
𝑛𝑖=1
γ (𝑅𝑖)𝑤
𝑖
P oo :
This p oo same as he heo em 2.2 and e i ied he idempo ency, mono onici y and boundedness
p ope ies.
3. Mul i-C i e ia Decision-Making
In mul i-c i e ia decision-making (MCDM) p oblems in ol ing unce ain y and imp ecision, cubic
sphe ical neu osophic se s (CSNS) p o ide a obus ma hema ical amewo k o ep esen ing and
p ocessing complex e alua ion da a. By in eg a ing linguis ic assessmen s om mul iple decision
make s wi h hei co esponding neu osophic ep esen a ions, i becomes possible o cap u e u h,
inde e minacy, and alsi y in o ma ion mo e comp ehensi ely. Howe e , he p esence o many c i e ia
o en inc eases compu a ional complexi y and may educe decision accu acy. To add ess his, P incipal
Neu osophic Se s and Sys ems, Vol. 97, 2026 300
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
Componen Analysis (PCA) can be applied o dimensionali y educ ion while p ese ing he mos
signi ican a iance in he da a. Fu he mo e, CSNS weigh ed agg ega ion ope a o s enable e ec i e
usion o in o ma ion, and cosine dis ance om an ideal sphe e aid in iden i ying he bes al e na i e.
The p oposed me hodology ollows he s eps below.
1. Selec MCDM p oblem: de ine objec i e, al e na i es, c i e ia, decision make s, linguis ic scale
and neu osophic mapping. Collec decision make s’ e alua ions o each al e na i e unde
each c i e ion (linguis ic e ms). Map each linguis ic e m o i s neu osophic in e al/ iple.
2.
Con e decision make s decision in o cubic sphe ical neu osophic ep esen a ion by inding
cen e
<
T
(
𝑥
i
),
I
(
𝑥
i
),
F
(
𝑥
i
)>=<1
k
i
j=1
k
i
T
i,j
,1
k
i
j=1
k
i
I
i,j
,1
k
i
j=1
k
i
F
i,j
>
and adius
Ri
=minimum {maximum
1≤j≤k
i
√(T(
𝑥
i
)
−
T
i,j
)
2
+(
I(
𝑥
i
)
−
I
i,j
)
2
+(
F(
𝑥
i
)
−
F
i,j
)
2
,1}
.
3. Apply a sco e unc ion 2+𝑇1−𝐼𝑖−𝐹1−𝑅1
3 o ans o m each cubic-sphe ical neu osophic alue in o
a c isp scala pe al e na i e - c i e ion.
4. S anda dize he c isp c i e ion ma ix and pe o m PCA; choose componen s co e ing he
a iance h eshold and ob ain educed-dimension ep esen a ion and/o PCA-based c i e ion
weigh s.
5. Compu e CSNS weigh ed agg ega ion (a i hme ic and geome ic) ac oss c i e ia using he
de i ed weigh s o ge an agg ega ed sphe e o each al e na i e.
𝐶𝑆𝑁𝑊𝐴(𝛼1,𝛼2,…,𝛼𝑛)=⨁𝑖=1
𝑛(𝑤𝑖𝛼𝑖)=
{
1−∏ (1−𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
∏(𝐼𝑖)𝑤
𝑖
𝑛𝑖=1
∏(𝐹𝑖)𝑤
𝑖
𝑛𝑖=1
γ (𝑅𝑖)𝑤
𝑖
𝐶𝑆𝑁𝑊𝐺(𝛼1,𝛼2,…,𝛼𝑛)=⨂𝑖=1
𝑛(𝑤𝑖𝛼𝑖)=
{
∏(𝑇𝑖)𝑤
𝑖
𝑛𝑖=1
1−∏ (1−𝐼𝑖)𝑤
𝑖
𝑛𝑖=1
1−∏ (1−𝐹𝑖)𝑤
𝑖
𝑛𝑖=1
γ (𝑅𝑖)𝑤
𝑖
6. De ine he ideal sphe e 𝛼1= (1, 0, 0; 1) and compu e Hamming dis ance
H(𝛼1, 𝛼2) =12(|𝑅1−𝑅2|
√3+1𝑛∑|𝑇1−𝑇2|+| 𝐼1−𝐼2|+|𝐹1−𝐹2|
𝑛)
each agg ega ed sphe e o he ideal.
7. Rank al e na i es by dis ance (smalle = be e ).
8. Selec he op al e na i e(s) and epo esul s wi h sensi i i y analysis.
Neu osophic Se s and Sys ems, Vol. 97, 2026 301
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
3.1 PCA’s Role in Decision P ocess wi h Machine Lea ning Techniques:
S ep 1: C ea e he Decision Ma ix.
S ep 2: S anda dize he Da a
To pu all c i e ia on he same scale: zij= 𝐶𝑥𝑖𝑗−𝐶𝑗
σj ,
whe e 𝐶𝑖𝑗 = alue o c i e ion j o al e na i e i, 𝐶𝑗= mean o c i e ion j, σj = s anda d de ia ion o
c i e ion j
S ep 3: Compu e he Co a iance Ma ix
Measu es ela ionships be ween c i e ia: S= 1
𝑛−1 𝑍𝑇𝑍,
whe e: Z = s anda dized da a ma ix, n = numbe o al e na i es
S ep 4: Find Eigen alues o Ma ix S.
S ep 5: Selec P incipal Componen s.
S ep 6: Explained Va iance Ra io.
The lowcha gi en below explains he Mul i-C i e ia Decision-Making (MCDM) p ocess using Cubic
Sphe ical Neu osophic Se s (CSNS). I ou lines he sys ema ic s eps s a ing om he selec ion o
al e na i es, c i e ia, and decision make s, ollowed by linguis ic e alua ion, da a ans o ma ion,
dimensionali y educ ion using PCA, and agg ega ion o alues. Finally, he al e na i es a e anked,
and he bes op ion is selec ed based on he compu ed esul s.
Figu e 0: Flow Cha o MCDM using CSNS.
S a Selec al e na i es,
c i e ia, and decision
make s
Pe o m linguis ic
e alua ion Compu e cen e and
adius
Rep esen using
Cubic Sphe ical
Neu osophic Se
(CSNS)
Apply sco e unc ion
o ob ain c isp alues S anda dize he c isp
c i e ion ma ix
Pe o m P incipal
Componen Analysis
(PCA)
Compu e weigh s o
each c i e ion
Reduce he
dimensionali y o
c i e ia using PCA
Agg ega e alues
using CSNWA and
CSNWG
Compu e Hamming
dis ance
Rank al e na i es Selec he bes
al e na i e
Neu osophic Se s and Sys ems, Vol. 97, 2026 302
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
3.2 Nume ical Example
Selec ing he mos sui able p ima y school in a pa icula a ea is a c i ical decision o pa en s,
local au ho i ies, and educa ional planne s. In oday’s compe i i e and esou ce-cons ained
en i onmen , p ima y educa ion ins i u ions a y signi ican ly in hei in as uc u e, eaching quali y,
ex acu icula oppo uni ies, sa e y, a o dabili y, and accessibili y. The choice o he bes school
canno be made by elying on a single c i e ion; a he , i in ol es simul aneously e alua ing mul iple
ac o s ha e lec bo h angible and in angible quali ies. This inhe en ly makes i a Mul i-C i e ia
Decision-Making (MCDM) p oblem.
In his s udy, we conside six al e na i e schools:
𝐴₁: A-Public School, 𝐴₂: B -School, 𝐴₃: C-P ima y Academy, 𝐴₄: D-P ima y School, 𝐴₅: E-In e na ional
School, 𝐴₆: F-P ima y School
The decision is based on se en e alua ion c i e ia, iden i ied h ough consul a ion wi h educa ional
expe s and pa en s:
𝐶1: Academic Pe o mance: A e age s uden pe o mance in s anda d assessmen s.
𝐶2: Teache Quali y: Expe ience, quali ica ions, and eaching e ec i eness o acul y.
𝐶3: – In as uc u e: Class oom condi ions, labo a o ies, lib a ies, and digi al acili ies.
𝐶4: – Ex acu icula Ac i i ies: Range and quali y o spo s, a s, and clubs.
𝐶5: – Sa e y and Secu i y: Measu es o s uden sa e y, including su eillance and ained s a .
𝐶6: – Accessibili y: Ease o commu e, anspo a ion acili ies, and dis ance om home.
𝐶7: – A o dabili y: Tui ion ees and addi ional cos s ela i e o pe cei ed alue.
The aim is o de elop a obus decision-making amewo k ha in eg a es Cubic Sphe ical
Neu osophic Se s (CSNS) and P incipal Componen Analysis (PCA). The CSNS amewo k will
cap u e he inhe en unce ain y and imp ecision in decision-make s’ e alua ions, while PCA will
educe he dimensionali y o he p oblem and de i e da a-d i en weigh s o he educed c i e ia se .
The agg ega ed e alua ions will hen be p ocessed using cubic sphe ical neu osophic a i hme ic and
geome ic agg ega ion ope a o s o de e mine he mos sui able p ima y school.
To ope a ionalize he p oposed me hodology, he i s s ep is o de ine he linguis ic scale and i s
neu osophic mapping, ollowed by he collec ion o e alua ions om mul iple decision make s o
each al e na i e unde he speci ied c i e ia. Each linguis ic e m is associa ed wi h a cubic sphe ical
neu osophic iple (T,I,F) which cap u es he u h, inde e minacy, and alsi y deg ees o he
assessmen . These mappings allow o sys ema ic con e sion o quali a i e judgmen s in o quan i a i e
o m, enabling u he p ocessing using PCA and agg ega ion ope a o s. The ollowing able p esen s
he linguis ic scale used in his s udy, along wi h he co esponding e alua ions o he six al e na i e
schools p o ided by h ee decision make s.
Linguis ic Te m
No a ion
(T, I, F)
Ve y Low Impac
𝑉𝐿𝐼
(0.099, 0.977, 0.880)
Low Impac
𝐿𝐼
(0.345, 0.655, 0.633)
Mode a e Impac
𝑀𝐼
(0.545, 0.455, 0.433)
High Impac
𝐻𝐼
(0.745, 0.255, 0.233)
Ve y High Impac
𝑉𝐻𝐼
(1.000, 0.000, 0.000)
Table 1. Linguis ic e ms, hei neu osophic mappings, and decision make s’ e alua ions
o al e na i es ac oss c i e ia
Neu osophic Se s and Sys ems, Vol. 97, 2026 309
S. Goma hi, A. Ka i ha, R. Ramesh, E. Ka uppusamy and L. Meenachi, A Machine Lea ning App oach Using P incipal
Componen Analysis and Cubic Sphe ical Neu osophic Se s o MCDM
9. Wold, S., Esbensen, K., & Geladi, P. (1987). P incipal componen analysis. Chemome ics and
in elligen labo a o y sys ems, 2(1-3), 37-52.
10. Zadeh, L. A. (1965). Fuzzy se s. In o ma ion and con ol, 8(3), 338-353.
Recei ed: Ap il 30, 2025. Accep ed: Sep 29, 2025