Sustainability analysis in agriculture using Linguistic Pythagorean Neutrosophic number through Einstein aggregation operators

Uni e si y o New Mexico
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
Sus ainabili y analysis in ag icul u e using Linguis ic Py hago ean
Neu osophic numbe h ough Eins ein agg ega ion ope a o s
S. Annadu ai1, R. Sunda eswa an2, M. Shanmugap iya3, M. Mohanalakshmi4
1Depa men o Ma hema ics, S . Joseph's College o Enginee ing, India.
2,3Depa men o Ma hema ics, S i Si asub amaniya Nada College o Enginee ing, India.
4Depa men o Chemical Enginee ing, S i Si asub amaniya Nada College o Enginee ing, India.
Abs ac :
The Linguis ic Py hago ean Neu osophic (LPN) se is a powe ul amewo k o handling
unce ain y in assessmen s by in eg a ing linguis ic a iables wi h Py hago ean Neu osophic
numbe s (PNNs). In his s udy, we de ine new undamen al ope a ions on Linguis ic Py hago ean
Neu osophic Numbe s (LPNNs) based on Eins ein ope a ions and examine hei in e ela ionships.
To add ess he challenges o LPNN usion, we p opose se e al LPN agg ega ion ope a o s, namely
he LPN Eins ein Weigh ed A e aging (LPNEWA), and LPN Eins ein O de Weigh ed A e aging
(LPNEOWG) ope a o s, and in es iga e hei key cha ac e is ics. To demons a e he p oposed
me hodology’s use ulness, we p esen an illus a i e case s udy in sus ainabili y ag icul u e. This
case s udy highligh s he p ac icali y and e ec i eness o he p oposed decision-making model.
Keywo ds: Linguis ic Py hago ean Neu osophic se ; LPN Eins ein Weigh ed A e age Ope a o , LPN Eins ein
O de Weigh ed A e age Ope a o , Mul i-C i e ia Decision Making
i. In oduc ion
In 1998, Sma andache [1] in oduced he concep o Neu osophic se s (𝑁𝑠𝑒𝑡), as an ex ension o
in ui ionis ic uzzy se s (𝐼𝐹𝑠𝑒𝑡), which p o ides a mo e comp ehensi e amewo k o handling
unce ain y. Unlike 𝐼𝐹𝑠𝑒𝑡𝑠, hose ha a e cha ac e ized by deg ees o u h and alsi y, 𝑁𝑠𝑒𝑡
inco po a es an addi ional dimension o unce ain y, enabling decision-make s o e alua e p oblems
in e ms o independen u h (T), inde e minacy (I), and alsi y (F) alues. This independence makes
𝑁𝑠𝑒𝑡 a mo e powe ul and gene alized ma hema ical amewo k o ep esen ing and p ocessing
ague o imp ecise in o ma ion. Since i s incep ion, esea che s ha e ex ensi ely s udied [2-5] bo h
he heo e ical ounda ions and applica ions o 𝑁𝑠𝑒𝑡𝑠. Linguis ic a iables (𝐿𝑉𝑠) a e used o exp ess
quali a i e e alua ions in complex decision-making. Zadeh [6] concep o 𝐿𝑉𝑠 o p e e ence
in o ma ion in uzzy easoning gained b oad esea ch in e es and led o u he ad ancemen s in
decision-making (DM) science. Fang and Ye [7] i s in oduced linguis ic neu osophic numbe s
(𝐿𝑁𝑁𝑠), inco po a ing linguis ic alues o u h, inde e minacy, and alsi y, and enabling he use o
Neu osophic Se s and Sys ems, Vol. 94, 2025
13
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
all h ee kinds o linguis ic in o ma ion simul aneously. They u he de eloped sco e and accu acy
unc ions, along wi h agg ega ion ope a o s, o e ec i e decision-making. Recen ly, many
esea che s [8-12] ha e been explo ing he applica ions, enhancemen s, and in eg a ion o 𝐿𝑁𝑁𝑠 in o
a ious decision-making amewo ks and uzzy logic sys ems.
Zhao [13] in oduced gene alized agg ega ion ope a o s based on 𝐼𝐹𝑠𝑒𝑡𝑠 and showed ha he
a i hme ic agg ega ion (AA) and geome y agg ega ion (GA) a e special cases o hese ope a o s.
These ope a o s a e de i ed using he algeb aic sum and p oduc o numbe se s, co esponding o
he A chimedes -cono m and -no m o de ining union and in e sec ion ope a ions. Wang and Liu
[14] de eloped se e al 𝐼𝐹𝐸𝐴 ope a o s and demons a ed ha he Eins ein agg ega ion ope a o
o e s be e esul s compa ed o he AA ope a o . Zhao and Wei [15] in oduced he 𝐼𝐹𝐸𝐻𝐴 and
𝐼𝐹𝐸𝐻𝐺 ope a o s. Guo e al. [16] applied he Eins ein ope a ions o hesi an uzzy se s. La e Li e al.
[17] in oduced he gene alized Neu osophic numbe o he Eins ein agg ega ion ope a o .
Recen ly, nume ous esea che s [18-20] ha e been explo ing he Neu osophic Eins ein ope a o and
i s applica ion in a ious decision-making p ocesses. When combined wi h 𝐿𝑁𝑁𝑠 , he Eins ein
ope a o s enable e ec i e agg ega ion o linguis ic alues in ol ing u h, inde e minacy, and alsi y
p obabili ies. This in eg a ion enhances decision-making by managing unce ain y and o e ing
smoo h agg ega ion me hods, such as weigh ed o geome ic a e ages. Recen ly, many esea che s
[21-23] ha e ocused on he use o he Eins ein ope a o wi h 𝐿𝑁𝑁𝑠 o manage unce ain o ague
da a in eal-wo ld decision-making se ings.
1.2 Mo i a ion
Agg ega ion ope a o s a e i al in decision suppo sys ems o consolida ing in o ma ion and
anking al e na i es. While adi ional algeb aic T-no m and S-no m ope a o s lack lexibili y and
obus ness, Eins ein T-no m and S-no m p o ide a supe io al e na i e wi h smoo h app oxima ion
p ope ies. To enhance decision suppo sys ems, we de elop Linguis ic Py hago ean Neu osophic
Eins ein Ope a o s (LPNEO), enabling mo e e ec i e agg ega ion o unce ain in o ma ion. In
sus ainable ag icul u e, decision-making is o en challenged by imp ecise da a, con lic ing expe
opinions, and dynamic en i onmen al condi ions. Tasks such as selec ing app op ia e c op a ie ies,
op imizing esou ce alloca ion and in as uc u e, o assessing he en i onmen al impac o a ming
p ac ices ypically in ol e unce ain, incomple e, o ambiguous in o ma ion. By in eg a ing LPNEO,
hese challenges a e e ec i ely add essed, enabling mo e accu a e handling o unce ain y and
agueness in ag icul u al decision p ocesses.
1.3 No el y
➢ This s udy ex ends he Eins ein T-no m and T-cono m o LPNEO, imp o ing hei capabili y
o manage unce ain y and imp ecision mo e e ec i ely.
➢ Es ablish a Mul i-A ibu e G oup Decision-Making (MAGDM) amewo k based on he
newly in oduced Eins ein ope a o s, p o iding a mo e e icien and accu a e app oach o
decision-making in unce ain en i onmen s.
1.4 Objec i e
The key esea ch objec i es and con ibu ions o his s udy a e:
Neu osophic Se s and Sys ems, Vol. 94, 2025
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➢ Ex ending he Eins ein T-no m and T-cono m o LPNEO o enhance lexibili y and
obus ness.
➢ In oducing a ious LPNEOs, including LPN Eins ein a e aging ope a o s, LPN Eins ein
geome ic ope a o s, and LPN Eins ein hyb id ope a o s, while explo ing hei undamen al
p ope ies.
➢ De eloping a no el decision-making (DM) me hod based on he p oposed ope a o s o
e ec i ely add ess MAGDM p oblems in eal-wo ld scena ios.
2 P elimina ies
In his sec ion, some undamen al concep s ela ed o LPNS ha e been p esen ed.
De ini ion: 1 Neu osophic se (𝑁𝑠𝑒𝑡): [1] Le Θ be a uni e se se . A 𝑁𝑠𝑒𝑡, 𝐴󰆻 on Θ is de ined as 𝐴󰆻=
{〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, whe e 𝑇𝐴(𝑥):Θ→−]0,1[+ is said o be he TMF, which ep esen s he
deg ee o con idence, 𝐼𝐴(𝑥):Θ→−]0,1[+is said o be he IMF, which ep esen s he deg ee o
unce ain y, and 𝐹𝐴(𝑥):Θ→−]0,1[+ is said o be he FMF, which ep esen s he deg ee o skep icism,
espec i ely o he elemen 𝑥∈Θ in 𝐴𝑁
 , such ha 0≤𝑇𝐴(𝑥)+𝐼𝐴(𝑥)+𝐹𝐴(𝑥)≤3.
De ini ion: 2 Py hago ean Neu osophic se s (𝑃𝑁𝑠𝑒𝑡): [2] Le Θ be a uni e se se . A 𝑃𝑁𝑠𝑒𝑡 𝐴󰆻 on Θ is
de ined as 𝐴󰆻={〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, such ha ( 𝑇𝐴(𝑥))2+( 𝐼𝐴(𝑥))2+( 𝐹𝐴(𝑥))2≤2, whe e
𝑇𝐴(𝑥):Θ→−]0,1[+ is he TMF, 𝐼𝐴(𝑥):Θ→−]0,1[+is he IMF, and 𝐹𝐴(𝑥):Θ→−]0,1[+ is he FMF.
De ini ion: 3 Linguis ic Neu osophic Se (𝐿𝑁𝑠𝑒𝑡): [3] Le Θ be a uni e se se . A 𝐿𝑁𝑠𝑒𝑡 in Θ is de ined
as 𝐴󰆻={〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, whe e 𝑇𝐴(𝑥):Θ→−]0,1[+ is he LTMF, 𝐼𝐴(𝑥):Θ→−]0,1[+is he
LIMF, and 𝐹𝐴(𝑥):Θ→−]0,1[+ is he LFMF. Each membe ship unc ions 𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝑎𝑛𝑑 𝐹𝐴(𝑥) akes
linguis ic alues om a p ede ined linguis ic e m se 𝑆.
De ini ion: 4 Linguis ic Py hago ean Neu osophic Se (𝐿𝑃𝑁𝑠𝑒𝑡): [24] Le Θ be a uni e se se . A 𝐿𝑃𝑁𝑠𝑒𝑡
in Θ is de ined as 𝐴󰆻={〈𝑥,𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝐹𝐴(𝑥)〉:𝑥∈Θ}, such ha ( 𝑇𝐴(𝑥))2+( 𝐼𝐴(𝑥))2+( 𝐹𝐴(𝑥))2≤2,
whe e 𝑇𝐴(𝑥),𝐼𝐴(𝑥),𝑎𝑛𝑑 𝐹𝐴(𝑥) a e ep esen ed using linguis ic e ms.
De ini ion: 5 Eins ein T-No m and S-No m [5]: Fo a bi a y wo eal numbe s (𝑎,𝑏)∈[0.1], he
Eins ein sums and p oduc a e de ined as ollows:
𝑆󰆻𝐸(𝑎 ,𝑏
)=𝑎 ⊕𝜖𝑏=𝑎+𝑏
1+𝑎∙
𝑏 , 𝑇𝐸(𝑎 ,𝑏
)=𝑎 ⊗𝜖𝑏=𝑎∙
𝑏
1+(1−𝑎)∙(1−𝑏), ∀(𝑎 ,𝑏
)∈[0,1]2. ,
Ga g [24] in oduced new di e en unc ions o o de ing he al e na i es using he sco e unc ion
wi h an accu acy unc ion o build he compa ison app oach o LPNNs.
De ini ion: 6 Le 𝑣 = (𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) be a LPNN. Then he sco e unc ion 𝒯 and accu acy unc ion ℋ
o 𝒜 a e de ined as: 𝔗(𝒜) = 𝜁√𝑘2+𝛼12−𝛽12−𝛾12
3
ℌ(𝒜) = 𝜁√𝛼12+𝛽12−𝛾12
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
15
Fo compa ing wo LPNNs A and B, he compa ison me hod is gi en as:
i. i 𝒯(𝒜)>𝒯(ℬ), hen 𝒜≻ℬ;
ii. i 𝒯(𝒜)=𝒯(ℬ), hen
➢ i ℋ(𝒜)<ℋ(ℬ), hen 𝒜≺ ℬ;
➢ i ℋ(𝒜)=ℋ(ℬ), hen 𝒜∼ ℬ.
3 Eins ein Ope a ion o Linguis ic Py hago ean Neu osophic Numbe s (LPNNs)
Linguis ic Py hago ean Neu osophic Numbe s (LPNNs) o e a no el and powe ul amewo k o
handling unce ain y, o e coming he limi a ions o adi ional linea app oaches. Unlike p e ious
wo ks ha ocus on uzzy o s anda d neu osophic numbe s, LPNNs combine he enhanced
lexibili y o Py hago ean logic wi h he in e p e abili y o linguis ic e ms. The applica ion o
Eins ein ope a ions, known o hei nonlinea , bounded, and smoo h agg ega ion beha io , u he
s eng hens he obus ness o his app oach in complex decision-making scena ios. In his sec ion, we
in oduce he Eins ein sum (⊕𝜖) and Eins ein p oduc (⊗𝜖) ope a ions wi hin he LPNN amewo k,
along wi h wo agg ega ion ope a o s such as LPN Eins ein Weigh ed A e age (LPNEWA) ope a o ,
and LPN Eins ein O de ed Weigh ed A e age (LPNEOWA) ope a o .
De ini ion: 7 Le 𝒫=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) and 𝒬=(𝜁𝛼2,𝜁𝛽2,𝜁𝛾2) be wo LPNNs and 𝜆≥0, hen he Eins ein
ope a ion o ⊕𝜖 and ⊗𝜖 unde he LPNN a e de ined as ollows:
i. 𝒫⊕𝜖𝒬=(𝜁𝑡√𝑡2(𝛼12+𝛼22)
𝑡4+𝛼12𝛼22,𝜁𝑡𝛽1𝛽2
√𝑡4+(𝑡2−𝛽12)(𝑡2−𝛽22),𝜁𝑡𝛾1𝛾2
√𝑡4+(𝑡2−𝛾12)(𝑡2−𝛾22));
ii. 𝒫⊗𝜖𝒬=(𝜁𝑡𝛼1𝛼2
√𝑡4+(𝑡2−𝛼12)(𝑡2−𝛼22),𝜁𝑡√𝑡2(𝛽12+𝛽22)
𝑡4+𝛽12𝛽22,𝜁𝑡√𝑡2(𝛾12+𝛾22)
𝑡4+𝛾12𝛾22);
iii. 𝜆𝒫=
(
𝜁𝑡√(𝑡2+𝛼12)𝜆−(𝑡2−𝛼12)𝜆
(𝑡2+𝛼12)𝜆+(𝑡2−𝛼12)𝜆,𝜁𝑡√2 𝛽1𝜆
√(2𝑡2−𝛽12)𝜆+(𝛽12)𝜆,𝜁𝑡√2 𝛾1𝜆
√(2𝑡2−𝛾12)𝜆+(𝛾12)𝜆
)
;
i . 𝒫𝜆=
(
𝜁𝑡√2 𝛼1𝜆
√(2𝑡2−𝛼12)𝜆+(𝛼12)𝜆,𝜁𝑡√(𝑡2+𝛽12)𝜆−(𝑡2−𝛽12)𝜆
(𝑡2+𝛽12)𝜆+(𝑡2−𝛽12)𝜆,𝜁𝑡√(𝑡2+𝛾12)𝜆−(𝑡2−𝛾12)𝜆
(𝑡2+𝛾12)𝜆+(𝑡2−𝛾12)𝜆
)
.
Theo em: 1 Le 𝒫=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) and 𝒬=(𝜁𝛼2,𝜁𝛽2,𝜁𝛾2) be wo LPNNs and 𝜆1,𝜆2,𝜆3≥0, hen he
Eins ein ope a ion o ⊕𝜖 and ⊗𝜖 ha e he ollowing pe o mance:
i. 𝒫⊕𝜖𝒬=𝒬⊕𝜖𝒫;
ii. 𝒫⊗𝜖𝒬=𝒬⊗𝜖𝒫;
iii. 𝜆(𝒫⊕𝜖𝒬)=𝜆𝒫⊕𝜖𝜆𝒬;
i . (𝒫⊗𝜖𝒬)𝜆=𝒫𝜆⊗𝜖𝒬𝜆;
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
16
. (𝜆1⊕𝜖 𝜆2)𝒫=𝜆1𝒫⊕𝜖𝜆2𝒫;
i. 𝒫𝜆1⊗𝜖𝒫𝜆2=𝒫𝜆1+𝜆2.
P oo : Pe o mance (𝑖) 𝑎𝑛𝑑 (𝑖𝑖) 𝑎𝑟𝑒 𝑒𝑎𝑠𝑦.𝑆𝑜,𝑤e p o es (𝑖𝑖𝑖),𝑎𝑛𝑑 (𝑣).
Acco ding o De ini ion 5, we can ge
𝒫⊕𝜖𝒬=(𝜁𝑡√𝑡2(𝛼12+𝛼22)
𝑡4+𝛼12𝛼22,𝜁𝑡𝛽1𝛽2
√𝑡4+(𝑡2−𝛽12)(𝑡2−𝛽22),𝜁𝑡𝛾1𝛾2
√𝑡4+(𝑡2−𝛾12)(𝑡2−𝛾22));
=(𝜁𝑡√(𝑡2+𝛼12)(𝑡2+𝛼22)−(𝑡2−𝛼12)(𝑡2−𝛼22)
(𝑡2+𝛼12)(𝑡2+𝛼22)+(𝑡2−𝛼12)(𝑡2−𝛼22),𝜁𝑡√2 𝛽12𝛽22
𝛽12𝛽22+(2𝑡2−𝛽12)(2𝑡2−𝛽22),𝜁𝑡√2𝛾12𝛾22
𝛾12𝛾22+(2𝑡2−𝛾12)(2𝑡2−𝛾22));
=(𝜁𝑡√𝑎−𝑏
𝑎+𝑏,𝜁𝑡√2𝑐
𝑐+𝑑,𝜁𝑡√2 𝑒
𝑒+𝑓󰆻)
whe e 𝑎=(𝑡2+𝛼12)(𝑡2+𝛼22),𝑏=(𝑡2−𝛼12)(𝑡2−𝛼22),𝑐=𝛽12𝛽22,𝑑󰆻=(2𝑡2−𝛽12)(2𝑡2−𝛽22),𝑒=
𝛾12𝛾22,𝑓󰆻=(2𝑡2−𝛾12)(2𝑡2−𝛾22).
𝒫⊕𝜖𝒬=𝜆(𝜁𝑡√𝑎−𝑏
𝑎+𝑏,𝜁𝑡√2𝑐
𝑐+𝑑,𝜁𝑡√2 𝑒
𝑒+𝑓󰆻)
=(𝜁𝑡√(𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆−(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆
(𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆+(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆,𝜁𝑡√2 (𝛽12)𝜆(𝛽22)𝜆
(𝛽12)𝜆(𝛽22)𝜆+(2𝑡2−𝛽12)𝜆(2𝑡2−𝛽22)𝜆,𝜁𝑡√2(𝛾12)𝜆(𝛾22)𝜆
(𝛾12)𝜆(𝛾22)𝜆+(2𝑡2−𝛾12)𝜆(2𝑡2−𝛾22)𝜆)
=(𝜁𝑡√ 𝑎
𝜆−𝑏𝜆
𝑎𝜆+𝑏𝜆,𝜁𝑡√2𝑐𝜆
𝑐𝜆+𝑑
𝜆,𝜁 𝑡√2 𝑒𝜆
𝑒𝜆+𝑓󰆻𝜆).
Now,
𝜆𝒫=
(
𝜁𝑡√(𝑡2+𝛼12)𝜆−(𝑡2−𝛼12)𝜆
(𝑡2+𝛼12)𝜆+(𝑡2−𝛼12)𝜆,𝜁𝑡√2 (𝛽12)𝜆
(𝛽12)𝜆+(2𝑡2−𝛽12)𝜆,𝜁𝑡√2(𝛾12)𝜆
(𝛾12)𝜆+(2𝑡2−𝛾12)𝜆
)
=(𝜁𝑡√𝑎1−𝑏1
𝑎1+𝑏1,𝜁𝑡√2𝑐1
𝑐1+𝑑1,𝜁𝑡√2 𝑒1
𝑒1+𝑓󰆻1)
and 𝜆𝒬=(𝜁𝑡√(𝑡2+𝛼22)𝜆−(𝑡2−𝛼12)𝜆
(𝑡2+𝛼22)𝜆+(𝑡2−𝛼12)𝜆,𝜁𝑡√2 (𝛽22)𝜆
(𝛽22)𝜆+(2𝑡2−𝛽22)𝜆,𝜁𝑡√2(𝛾22)𝜆
(𝛾22)𝜆+(2𝑡2−𝛾22)𝜆)=(𝜁𝑡√𝑎2−𝑏2
𝑎2+𝑏2,𝜁𝑡√2𝑐2
𝑐2+𝑑2,𝜁𝑡√2 𝑒2
𝑒2+𝑓2)
hen 𝜆𝒫⊕𝜖𝜆𝒬=(𝜁𝑡√𝑎1−𝑏1
𝑎1+𝑏1,𝜁𝑡√2𝑐1
𝑐1+𝑑1,𝜁𝑡√2 𝑒1
𝑒1+𝑓1)⊕𝜖(𝜁𝑡√𝑎2−𝑏2
𝑎2+𝑏2,𝜁𝑡√2𝑐2
𝑐2+𝑑2,𝜁𝑡√2 𝑒2
𝑒2+𝑓2)
=(𝜁𝑡√𝑎1𝑎2−𝑏1𝑏2
𝑎1𝑎2+𝑏1𝑏
2,𝜁𝑡√2𝑐1𝑐2
𝑐1𝑐
2+𝑑1𝑑2,𝜁𝑡√2 𝑒1𝑒2
𝑒1𝑒2+𝑓󰆻1𝑓
2)
=(𝜁𝑡√(𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆−(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆
(𝑡2+𝛼12)𝜆(𝑡2+𝛼22)𝜆+(𝑡2−𝛼12)𝜆(𝑡2−𝛼22)𝜆,𝜁𝑡√2 (𝛽12)𝜆(𝛽22)𝜆
(𝛽12)𝜆(𝛽22)𝜆+(2𝑡2−𝛽12)𝜆(2𝑡2−𝛽22)𝜆,𝜁𝑡√2(𝛾12)𝜆(𝛾22)𝜆
(𝛾12)𝜆(𝛾22)𝜆+(2𝑡2−𝛾12)𝜆(2𝑡2−𝛾22)𝜆)
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s

17
whe e 𝑎1=(𝑡2+𝛼12)𝜆,𝑎2=(𝑡2+𝛼22)𝜆,𝑏1=(𝑡2−𝛼12)𝜆,𝑏2=(𝑡2−𝛼22)𝜆,𝑐1=(𝛽12)𝜆,𝑐2=(𝛽22)𝜆,𝑑1=(2𝑡2−
𝛽12)𝜆(2𝑡2−𝛽22)𝜆,𝑑2=(2𝑡2−𝛽22)𝜆,𝑒1=(𝛾12)𝜆,𝑒2=(𝛾22)𝜆 ,𝑓1=(2𝑡2−𝛾12)𝜆,𝑓2=(2𝑡2−𝛾22)𝜆.
Hence, we can ob ain 𝜆(𝒫⊕𝜖𝒬)=𝜆𝒫⊕𝜖𝜆𝒬.
Now, we p o e he pe o mance o (𝑣):
𝜆1𝒫=
(
𝜁𝑡√(𝑡2+𝛼12)𝜆1−(𝑡2−𝛼12)𝜆1
(𝑡2+𝛼12)𝜆1+(𝑡2−𝛼12)𝜆1,𝜁𝑡√2 (𝛽12)𝜆1
(𝛽12)𝜆1+(2𝑡2−𝛽12)𝜆1,𝜁𝑡√2(𝛾12)𝜆1
(𝛾12)𝜆1+(2𝑡2−𝛾12)𝜆1
)
=(𝜁𝑡√𝑎1−𝑏1
𝑎1+𝑏1,𝜁𝑡√2𝑐1
𝑐1+𝑑1,𝜁𝑡√2 𝑒1
𝑒1+𝑓1),
𝜆2𝒫=
(
𝜁𝑡√(𝑡2+𝛼12)𝜆2−(𝑡2−𝛼12)𝜆2
(𝑡2+𝛼12)𝜆2+(𝑡2−𝛼12)𝜆2,𝜁𝑡√2 (𝛽12)𝜆2
(𝛽12)𝜆2+(2𝑡2−𝛽12)𝜆2,𝜁𝑡√2(𝛾12)𝜆2
(𝛾12)𝜆2+(2𝑡2−𝛾12)𝜆2
)
=(𝜁𝑡√𝑎1−𝑏1
𝑎1+𝑏1,𝜁𝑡√2𝑐1
𝑐1+𝑑1,𝜁𝑡√2 𝑒1
𝑒1+𝑓1),
whe e 𝑎1=(𝑡2+𝛼12)𝜆1,𝑎2=(𝑡2+𝛼22)𝜆2,𝑏1=(𝑡2−𝛼12)𝜆1,𝑏1=(𝑡2−𝛼22)𝜆2,𝑐1=(𝛽12)𝜆1,𝑐2=(𝛽22)𝜆2,𝑑1=
(2𝑡2−𝛽12)𝜆1,𝑑2=(2𝑡2−𝛽22)𝜆2,𝑒1=(𝛾12)𝜆2,𝑒2=(𝛾22)𝜆1 ,𝑓1=(2𝑡2−𝛾12)𝜆1,𝑓2=(2𝑡2−𝛾22)𝜆2.
𝜆1𝒫⊕𝜖𝜆2𝒫=(𝜁𝑡√𝑎1−𝑏1
𝑎1+𝑏1,𝜁𝑡√2𝑐1
𝑐1+𝑑1,𝜁𝑡√2 𝑒1
𝑒1+𝑓1)⊕𝜖(𝜁𝑡√𝑎1−𝑏1
𝑎1+𝑏1,𝜁𝑡√2𝑐1
𝑐1+𝑑1,𝜁𝑡√2 𝑒1
𝑒1+𝑓1)
=(𝜁𝑡√𝑎1𝑎2−𝑏1𝑏2
𝑎1𝑎2+𝑏1𝑏2,𝜁𝑡√2𝑐1𝑐1
𝑐1𝑐2+𝑑1𝑑2,𝜁𝑡√2 𝑒1𝑒2
𝑒1𝑒2+𝑓1𝑓2)
=(𝜁𝑡√(𝑡2+𝛼12)𝜆1+𝜆2−(𝑡2−𝛼12)𝜆1+𝜆2
(𝑡2+𝛼12)𝜆1+𝜆2+(𝑡2−𝛼12)𝜆1+𝜆2,𝜁𝑡√2 (𝛽12)𝜆1+𝜆2
(𝛽12)𝜆1+𝜆2+(2𝑡2−𝛽12)𝜆1+𝜆2,𝜁𝑡√2(𝛾12)𝜆1+𝜆2
(𝛾12)𝜆1+𝜆2+(2𝑡2−𝛾12)𝜆1+𝜆2)=(𝜆1⊕𝜖𝜆2)𝒫.
Hence, 𝜆1𝒫⊕𝜖𝜆2𝒫=(𝜆1⊕𝜖𝜆2)𝒫.
4. LPN Eins ein Agg ega ion Ope a o s
4.1 LPN Eins ein weigh ed a e age (LPNEWA) ope a o
De ini ion: 8 Le LPNN 𝒫𝑖=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) in 𝜁, o 𝑖=1,2,3,…𝑛. Then he LPNEWA ope a o is
de ined as: LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=𝜔1𝒫1⊕𝜖𝜔2𝒫2⊕𝜖𝜔3𝒫3⊕𝜖…⊕𝜖𝜔𝑛𝒫𝑛, wi h he weigh ec o
𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1].
Theo em: 2 Se a collec ion 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) in 𝜁, o 𝑖=1,2,3,…𝑛, hen he usion alue gene a ed
by LPNEWA ope a o is also a LPNN and
LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=(𝜁𝑡√∏(𝑡2+𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖
∏(𝑡2+𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛽𝑖2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖
𝑛
𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛾𝑖2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖
𝑛
𝑖=1 ) wi h
he weigh ec o 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1].
P oo : When 𝑛=2,LPNEWA(𝒫1,𝒫2)=𝜔1𝒫1⊕𝜖𝜔2𝒫2.
By de ini ion 5 , we ge
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
18
𝜔1𝒫1=(𝜁𝑡√(𝑡2+𝛼12)𝜔1−(𝑡2−𝛼12)𝜔1
(𝑡2+𝛼12)𝜔1+(𝑡2−𝛼12)𝜔1,𝜁𝑡√2 (𝛽12)𝜔1
(𝛽12)𝜔1+(2𝑡2−𝛽12)𝜔1,𝜁𝑡√2(𝛾12)𝜔1
(𝛾12)𝜔1+(2𝑡2−𝛾12)𝜔1),
𝜔2𝒫2=(𝜁𝑡√(𝑡2+𝛼22)𝜔2−(𝑡2−𝛼22)𝜔2
(𝑡2+𝛼22)𝜔2+(𝑡2−𝛼22)𝜔2,𝜁𝑡√2 (𝛽22)𝜔2
(𝛽22)𝜔2+(2𝑡2−𝛽22)𝜔2,𝜁𝑡√2(𝛾22)𝜔2
(𝛾22)𝜔2+(2𝑡2−𝛾22)𝜔2).
𝜔1𝒫1⊕𝜖𝜔2𝒫2
=
(
𝜁𝑡
√
(𝑡2+𝛼12)𝜔1−(𝑡2−𝛼12)𝜔1
(𝑡2+𝛼12)𝜔1+(𝑡2−𝛼12)𝜔1+(𝑡2+𝛼22)𝜔2−(𝑡2−𝛼22)𝜔2
(𝑡2+𝛼22)𝜔2+(𝑡2−𝛼22)𝜔2
1+((𝑡2+𝛼12)𝜔1−(𝑡2−𝛼12)𝜔1
(𝑡2+𝛼12)𝜔1+(𝑡2−𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2−(𝑡2−𝛼22)𝜔2
(𝑡2+𝛼22)𝜔2+(𝑡2−𝛼22)𝜔2),𝜁𝑡
√
(2 (𝛽12)𝜔1
(𝛽12)𝜔1+(2𝑡2−𝛽12)𝜔1)∙( 2 (𝛽22)𝜔2
(𝛽22)𝜔2+(2𝑡2−𝛽22)𝜔2)
1+((𝑡2−2 (𝛽12)𝜔1
(𝛽12)𝜔1+(2𝑡2−𝛽12)𝜔1)∙ 2 (𝛽22)𝜔2
(𝛽22)𝜔2+(2𝑡2−𝛽22)𝜔2),𝜁𝑡
√
(2 (𝛾12)𝜔1
(𝛾12)𝜔1+(2𝑡2−𝛾12)𝜔1)∙( 2 (𝛾22)𝜔2
(𝛾22)𝜔2+(2𝑡2−𝛾22)𝜔2)
1+((𝑡2−2 (𝛾12)𝜔1
(𝛾12)𝜔1+(2𝑡2−𝛾12)𝜔1)∙ 2 (𝛾22)𝜔2
(𝛾22)𝜔2+(2𝑡2−𝛾22)𝜔2)
)
=
(
𝜁𝑡√((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2)−((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2)
((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2)+((𝑡2+𝛼12)𝜔1)∙((𝑡2+𝛼22)𝜔2),𝜁𝑡√2 (𝛽12)𝜔1∙(𝛽22)𝜔2
(𝛽12)𝜔1(𝛽22)𝜔2+(2𝑡2−𝛽12)𝜔1∙(2𝑡2−𝛽22)𝜔2,𝜁𝑡√2 (𝛾12)𝜔1∙(𝛾22)𝜔2
(𝛾12)𝜔1(𝛽𝛾22)𝜔2+(2𝑡2−𝛾12)𝜔1∙(2𝑡2−𝛾22)𝜔2
)
Hence,LPNEWA(𝒫1,𝒫2)=𝜔1𝒫1⊕𝜖𝜔2𝒫2,𝑣𝑎𝑙𝑖𝑑 𝑓𝑜𝑟 𝑛=2.
When he consequence is alid o 𝑛=𝑘, we ha e
LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑘)=
(
𝜁𝑡√∏(𝑡2+𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖
∏(𝑡2+𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖
𝑘𝑖=1
∏ (𝛽𝑖2)𝜔𝑖
𝑘𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖
𝑘𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖
𝑘𝑖=1
∏ (𝛾𝑖2)𝜔𝑖
𝑘𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖
𝑘𝑖=1
)
.
When 𝑛=𝑘+1, we ha e
LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑘+1)=LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑘⊕𝜖𝜔𝑘+1𝒫𝑘+1)
=
(
𝜁𝑡√∏(𝑡2+𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖
∏(𝑡2+𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2)
𝑘𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖
𝑘𝑖=1
∏ (𝛽𝑖2)𝜔𝑖
𝑘𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖
𝑘𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖
𝑘𝑖=1
∏ (𝛾𝑖2)𝜔𝑖
𝑘𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖
𝑘𝑖=1
)
⊕𝜖
(
𝜁𝑡√(𝑡2+𝛼𝑘+1
2)𝜔𝑘+1−(𝑡2−𝛼𝑘+1
2)𝜔𝑘+1
(𝑡2+𝛼𝑘+1
2)𝑘+1+(𝑡2−𝛼𝑘+1
2)𝜔𝑘+1,𝜁𝑡√2 (𝛽𝑘+1
2)𝜔𝑘+1
(𝛽𝑘+1
2)𝜔𝑘+1+(2𝑡2−𝛽𝑘+1
2)𝜔𝑘+1,𝜁𝑡√2(𝛾𝑘+1
2)𝜔𝑘+1
(𝛾𝑘+1
2)𝜔𝑘+1+(2𝑡2−𝛾𝑘+1
2)𝜔𝑘+1
)
,
=
(
𝜁𝑡√∏(𝑡2+𝛼𝑖2)
𝑘+1
𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2)
𝑘+1
𝑖=1 𝜔𝑖
∏(𝑡2+𝛼𝑖2)
𝑘+1
𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2)
𝑘+1
𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖
𝑘+1
𝑖=1
∏ (𝛽𝑖2)𝜔𝑖
𝑘+1
𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖
𝑘+1
𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖
𝑘+1
𝑖=1
∏ (𝛾𝑖2)𝜔𝑖
𝑘+1
𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖
𝑘+1
𝑖=1
)
.
The e o e, LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛) holds o any 𝑛.
Hence, Theo em 2 is p o ed.
Theo em: 3 Se a collec ion 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)in 𝜁, o 𝑖=1,2,3,…𝑛, be wo LPNNs
wi h he weigh ec o 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1]. We can deduce he
ollowing p ope ies:
i. 𝐼𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑐𝑦: 𝐼𝑓 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)=(𝜁𝛼,𝜁𝛽,𝜁𝛾) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖,𝑡ℎ𝑒𝑛
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
19
LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=(𝜁𝛼,𝜁𝛽,𝜁𝛾).
ii. 𝑀𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑖𝑡𝑦:𝐼𝑓 𝒫𝑖≤ 𝒬𝑖,𝑡ℎ𝑎𝑡 𝑖𝑠,𝜁𝛼𝑖≤𝜁𝛼𝑖,𝜁𝛽𝑖≥𝜁𝛽𝑖,𝑎𝑛𝑑 𝜁𝛾𝑖≥𝜁𝛾𝑖,𝑡ℎ𝑒𝑛 𝑤𝑒 ℎ𝑎𝑣𝑒
LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤LPNEWA(𝒬1,𝒬2,𝒬3,…𝒬𝑛).
iii. Boundedness: Suppose 𝒫−=𝑚𝑖𝑛(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝒫+=𝑚𝑎𝑥(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝑡ℎ𝑒𝑛
𝒫−≤LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤ 𝒫+.
P oo : Le 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)in 𝜁, o 𝑖=1,2,3,…𝑛, be wo collec ions o LPNNs.
Then
i. 𝑤ℎ𝑒𝑛 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)=(𝜁𝛼,𝜁𝛽,𝜁𝛾) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖,𝑜𝑛𝑒 ℎ𝑎𝑠
LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)
=
(
𝜁𝑡√∏(𝑡2+𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖
∏(𝑡2+𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝑖2)
𝑛
𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝑖2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛽𝑖2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛽𝑖2)𝜔𝑖
𝑛
𝑖=1 ,𝜁𝑡√2∏ (𝛾𝑖2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛾𝑖2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛾𝑖2)𝜔𝑖
𝑛
𝑖=1
)
,
=
(
𝜁𝑡√(𝑡2+𝛼𝑖2)∏𝜔𝑖
𝑛
𝑖=1 −(𝑡2+𝛼𝑖2)∏𝜔𝑖
𝑛
𝑖=1
(𝑡2+𝛼𝑖2)∏𝜔𝑖
𝑛
𝑖=1 +(𝑡2+𝛼𝑖2)∏𝜔𝑖
𝑛
𝑖=1 ,𝜁𝑡√2(𝛽𝑖2)∏ 𝜔𝑖
𝑘𝑖=1
(𝛽𝑖2)∏ 𝜔𝑖
𝑘𝑖=1 +(2𝑡2−𝛽𝑖2)∏ 𝜔𝑖
𝑘𝑖=1 ,𝜁𝑡√2(𝛾𝑖2)∏ 𝜔𝑖
𝑘𝑖=1
(𝛾𝑖2)∏ 𝜔𝑖
𝑘𝑖=1 +(2𝑡2−𝛾𝑖2)∏ 𝜔𝑖
𝑘𝑖=1
)
,
=(𝜁𝑡(𝛼𝑖
𝑡),𝜁𝑡(𝛽𝑖
𝑡),𝜁𝑡(𝛾𝑖
𝑡))=𝒫𝑖.
ii. Fo 𝒫𝑖≤ 𝒬𝑖,𝑡ℎ𝑒𝑛 𝜔𝑖𝒫𝑖≤ 𝜔𝑖𝒬𝑖.
So, we can ob ain ⊕𝜖𝑖=1
𝑛𝜔𝑖𝒫𝑖≤ ⊕𝜖𝑖=1
𝑛𝜔𝑖𝒬𝑖. Fo LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=⊕𝜖𝑖=1
𝑛𝜔𝑖𝒫𝑖,
𝑎𝑛𝑑 LPNEWA(𝒬1,𝒬2,𝒬3,…𝒬𝑛)=⊕𝜖𝑖=1
𝑛𝜔𝑖𝒬𝑖, hen we can ge LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤
LPNEWA(𝒬1,𝒬2,𝒬3,…𝒬𝑛).
iii. Since 𝒫−=𝑚𝑖𝑛(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝒫+=𝑚𝑎𝑥(𝒫1,𝒫2,𝒫3,…𝒫𝑛). By he p e ious p oo (𝑖𝑖), we ha e
LPNEWA(𝒫−,𝒫−,𝒫−,…𝒫−)≤LPNEWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤LPNEWA(𝒫+,𝒫+,𝒫+,…𝒫+).
In addi ion, by he p e ious p oo (𝑖), we ha e
LPNEWA(𝒫+,𝒫+,𝒫+,…𝒫+)=𝒫+,𝑎𝑛𝑑 LPNEWA(𝒫−,𝒫−,𝒫−,…𝒫−)=𝒫−.
F om all he abo e, we can ge 𝒫−≤LPNEWA(𝒫+,𝒫+,𝒫+,…𝒫+)=𝒫+.
4.2 LPN Eins ein o de weigh ed a e age (LPNEOWA) ope a o
De ini ion: 9 Se a LPNNs 𝒫𝑖=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) in 𝜁, o 𝑖=1,2,3,…𝑛, hen he LPNEOWA ope a o is
de ined as: LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=𝜔1𝒫𝜌(1)⊕𝜖𝜔2𝒫𝜌(2)⊕𝜖𝜔3𝒫𝜌(3)⊕𝜖…⊕𝜖𝜔𝑛𝒫𝜌(𝑛),
whe e (𝜌(1),𝜌(2),𝜌(3),…𝜌(𝑛)) is a pe mu a ion o (𝑖=1,2,3,…𝑛),𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝒫𝜌(𝑖−1)≥𝒫𝜌(𝑖) o each
𝑖, wi h he weigh ec o 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1].
Theo em: 4 Se a collec ion 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) in 𝜁, o 𝑖=1,2,3,…𝑛, hen he usion esul by
LPNEOWA ope a o is ob ained as:
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
20
LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=
(
𝜁𝑡√∏(𝑡2+𝛼𝜌(𝑖)
2)
𝑛
𝑖=1 𝜔𝑖−∏(𝑡2−𝛼𝜌(𝑖)
2)
𝑛
𝑖=1 𝜔𝑖
∏(𝑡2+𝛼𝜌(𝑖)
2)
𝑛
𝑖=1 𝜔𝑖+∏(𝑡2−𝛼𝜌(𝑖)
2)
𝑛
𝑖=1 𝜔𝑖,𝜁𝑡√2∏ (𝛽𝜌(𝑖)
2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛽𝜌(𝑖)
2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛽𝜌(𝑖)
2)𝜔𝑖
𝑛
𝑖=1 ,𝜁𝑡√2∏ (𝛾𝜌(𝑖)
2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛾𝜌(𝑖)
2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛾𝜌(𝑖)
2)𝜔𝑖
𝑛
𝑖=1
)
,
whe e (𝜌(1),𝜌(2),𝜌(3),…𝜌(𝑛)) is a pe mu a ion o (𝑖=1,2,3,…𝑛),𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝒫𝜌(𝑖−1)≥𝒫𝜌(𝑖) o each
𝑖, wi h he weigh ec o 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1]. E iden ly, i 𝜔=
(1𝑛,1𝑛,1𝑛,…,1𝑛 ),𝑡ℎ𝑒 LPNEOWA ope a o will educe o LPNWA ope a o .
Theo em: 5 Se a collec ion 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖), 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)in 𝜁, o 𝑖=1,2,3,…𝑛, be wo LPNNs
wi h he weigh ec o 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1]. We can deduce he
ollowing p ope ies:
i. 𝐼𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑐𝑦: 𝐼𝑓 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖)=(𝜁𝛼,𝜁𝛽,𝜁𝛾) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖,𝑡ℎ𝑒𝑛
LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=(𝜁𝛼,𝜁𝛽,𝜁𝛾).
ii. 𝑀𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑖𝑡𝑦:𝐼𝑓 𝒫𝑖≤ 𝒬𝑖,𝑡ℎ𝑎𝑡 𝑖𝑠,𝜁𝛼𝑖≤𝜁𝛼𝑖,𝜁𝛽𝑖≥𝜁𝛽𝑖,𝑎𝑛𝑑 𝜁𝛾𝑖≥𝜁𝛾𝑖,𝑡ℎ𝑒𝑛
LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤LPNEWOA(𝒬1,𝒬2,𝒬3,…𝒬𝑛).
iii. Boundedness: Suppose 𝒫−=𝑚𝑖𝑛(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝒫+=𝑚𝑎𝑥(𝒫1,𝒫2,𝒫3,…𝒫𝑛),𝑡ℎ𝑒𝑛
𝒫−≤LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)≤ 𝒫+.
i . Commu a i i y: 𝒬𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) (𝑖=1,2,3,…𝑛) is any pe mu a ion o 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖),
hen LPNEOWA(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=LPNEWOA(𝒬1,𝒬2,𝒬3,…𝒬𝑛). The p oo is simila o ha o
Theo em 3; he e o e, we omi i he e.
4.3 LPN Eins ein weigh ed geome y (LPNEWG) ope a o
De ini ion: 10 Le LPNNs 𝒫𝑖=(𝜁𝛼1,𝜁𝛽1,𝜁𝛾1) in 𝜁, o 𝑖=1,2,3,…𝑛. Then he LPNEWG ope a o is
de ined as: LPNEWG (𝒫1,𝒫2,𝒫3,…𝒫𝑛)=𝒫1𝜔1⊗𝜖𝒫2𝜔2⊗𝜖…𝒫𝑛𝜔𝑛, wi h he weigh ec o 𝜔=
(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1].
Theo em: 6 Se a collec ion 𝒫𝑖=(𝜁𝛼𝑖,𝜁𝛽𝑖,𝜁𝛾𝑖) in 𝜁, o 𝑖=1,2,3,…𝑛, hen he usion alue gene a ed
by LPNEWG ope a o is also a LPNN and
LPNEWG(𝒫1,𝒫2,𝒫3,…𝒫𝑛)=( 𝜁𝑡√2∏ (𝛼𝑖2)𝜔𝑖
𝑛
𝑖=1
∏ (𝛼𝑖2)𝜔𝑖
𝑛
𝑖=1 +∏ (2𝑡2−𝛼𝑖2)𝜔𝑖
𝑛
𝑖=1 ,𝜁𝑡√∏(𝑡2+𝛽𝑖2)
𝑛
𝑖=1 𝜔𝑖−∏(𝑡2−𝛽𝑖2)
𝑛
𝑖=1 𝜔𝑖
∏(𝑡2+𝛽𝑖2)
𝑛
𝑖=1 𝜔𝑖+∏(𝑡2−𝛽𝑖2)
𝑛
𝑖=1 𝜔𝑖,𝜁𝑡√∏(𝑡2+𝛾𝑖2)
𝑛
𝑖=1 𝜔𝑖−∏(𝑡2−𝛾𝑖2)
𝑛
𝑖=1 𝜔𝑖
∏(𝑡2+𝛾𝑖2)
𝑛
𝑖=1 𝜔𝑖+∏(𝑡2−𝛾𝑖2)
𝑛
𝑖=1 𝜔𝑖) wi h
he weigh ec o 𝜔=(𝜔1,𝜔2,𝜔3,…𝜔𝑛)𝑇,∑𝜔𝑖=1
𝑛𝑖=1 and 𝜔𝑖∈[0,1].
P oo : When 𝑛=2,LPNEWA(𝒫1,𝒫2)=𝒫1𝜔1⊗𝜖𝒫2𝜔2.
By de ini ion 5, we ge
𝒫1𝜔1=(𝜁𝑡√2(𝛼12)𝜔1
(𝛼12)𝜔1(2𝑡2−𝛼12)𝜔1,𝜁𝑡√(𝑡2+𝛽12)𝜔1−(𝑡2−𝛽12)𝜔1
(𝑡2+𝛽12)𝜔1+(𝑡2−𝛽12)𝜔1,𝜁𝑡√(𝑡2+𝛾12)𝜔1−(𝑡2−𝛾12)𝜔1
(𝑡2+𝛾12)𝜔1+(𝑡2−𝛾12)𝜔1),
𝒫2𝜔2=(𝜁𝑡√2(𝛼22)𝜔2
(𝛼22)𝜔2(2𝑡2−𝛼22)𝜔2,𝜁𝑡√(𝑡2+𝛽22)𝜔2−(𝑡2−𝛽22)𝜔2
(𝑡2+𝛽22)𝜔2+(𝑡2−𝛽22)𝜔2,𝜁𝑡√(𝑡2+𝛾22)𝜔2−(𝑡2−𝛾22)𝜔2
(𝑡2+𝛾22)𝜔2+(𝑡2−𝛾22)𝜔2)
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S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
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SDG2 ocuses on ending hunge , achie ing ood secu i y and imp o ed nu i ion and p omo ing
sus ainable ag icul u e. In he Indian ag icul u e sec o , companies like God ej Ag o e , AgNex
Technologies, and Co omandel In e na ional a e known o hei sus ainabili y ini ia i es and ocus
on sus ainable a ming p ac ices.
In his wo k, we conside ou o hese companies ℭ=(ℭ1,ℭ2,ℭ3,ℭ4) in India depends on hei
p oduc selling s a egies o achie ing sus ainabili y in ag icul u e. based on hese ac o s how he
companies can main ain hei sus ainabili y in he ag icul u e ield. The e a e decision make s/
expe s 𝔇=(𝔇1,𝔇2,𝔇3) a e in i ed o e alua e acco ding o six ac o s based on he company’s
pe o mance wi h weigh ec o is 𝓌 = (0.37,0.33,0.3) . The e alua ions based on he expe s make
e alua ions on he ou al e na i e ac o s 𝒮ℱ=(𝒮ℱ1, 𝒮ℱ2, 𝒮ℱ3, 𝒮ℱ4) wi h he weigh ec o 𝒲𝑓=
(0.26,024,0.21,0.29). Now, he expe s use LPNNs o make he e alua ion alues wi h a linguis ic
se 𝜁 = {𝜁0 = 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑝𝑜𝑜𝑟,𝜁1 = 𝑣𝑒𝑟𝑦 𝑝𝑜𝑜𝑟,𝜁2= 𝑝𝑜𝑜𝑟,𝜁3= 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑝𝑜𝑜𝑟,𝜁4= 𝑚𝑒𝑑𝑖𝑢𝑚 ,𝜁5=
𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑔𝑜𝑜𝑑,𝜁6= 𝑔𝑜𝑜𝑑,𝜁7= 𝑣𝑒𝑟𝑦 𝑔𝑜𝑜𝑑,𝜁8 = 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑔𝑜𝑜𝑑}. The decision e alua ion ma ix
a e gi en below ( ables 1– 4).
Table 1: The i s decision make 𝔇1 gi es he ollowing alues in he ma ix o m
𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒
ℭ1 (𝜁6,𝜁1,𝜁3) (𝜁7,𝜁1,𝜁3) (𝜁8,𝜁1,𝜁3) (𝜁5,𝜁2,𝜁3)
ℭ2 (𝜁6,𝜁2,𝜁3) (𝜁6,𝜁7,𝜁5) (𝜁6,𝜁6,𝜁3) (𝜁5,𝜁3,𝜁3)
ℭ3 (𝜁6,𝜁3,𝜁3) (𝜁6,𝜁4,𝜁3) (𝜁6,𝜁1,𝜁6) (𝜁5,𝜁3,𝜁3)
ℭ4 (𝜁6,𝜁2,𝜁2) (𝜁6,𝜁1,𝜁5) (𝜁8,𝜁1,𝜁3) (𝜁6,𝜁3,𝜁3)
Table 2: The second decision make 𝔇2 gi es he ollowing alues in he ma ix o m
𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒
ℭ1 (𝜁6,𝜁2,𝜁3) (𝜁7,𝜁4,𝜁3) (𝜁6,𝜁1,𝜁3) (𝜁6,𝜁3,𝜁3)
ℭ2 (𝜁6,𝜁2,𝜁3) (𝜁7,𝜁1,𝜁4) (𝜁6,𝜁1,𝜁3) (𝜁7,𝜁2,𝜁3)
ℭ3 (𝜁5,𝜁3,𝜁3) (𝜁5,𝜁2,𝜁4) (𝜁6,𝜁1,𝜁3) (𝜁8,𝜁3,𝜁3)
ℭ4 (𝜁6,𝜁4,𝜁3) (𝜁5,𝜁4,𝜁3) (𝜁6,𝜁1,𝜁3) (𝜁5,𝜁2,𝜁3)
Table 3: The hi d decision make 𝔇3 gi es he ollowing alues in he ma ix o m
𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒
ℭ1 (𝜁7,𝜁1,𝜁3) (𝜁6,𝜁2,𝜁5) (𝜁3,𝜁3,𝜁3) (𝜁8,𝜁1,𝜁3)
ℭ2 (𝜁6,𝜁4,𝜁3) (𝜁6,𝜁2,𝜁5) (𝜁5,𝜁5,𝜁5) (𝜁6,𝜁2,𝜁2)
ℭ3 (𝜁6,𝜁1,𝜁4) (𝜁6,𝜁5,𝜁3) (𝜁6,𝜁5,𝜁3) (𝜁6,𝜁2,𝜁3)
ℭ4 (𝜁6,𝜁1,𝜁5) (𝜁6,𝜁5,𝜁3) (𝜁4,𝜁4,𝜁3) (𝜁6,𝜁3,𝜁3)
Based on he 𝐋𝐏𝐍𝐄𝐖𝐀 and 𝐋𝐏𝐍𝐄𝐖𝐆 ope a o s, we sol e he abo e decision-making p oblem in he
ollowing manne and he ob ained alues a e in Table 5 and Table 6.
Table 5. The o e all decision ma ix
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s

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𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒
ℭ1 (𝜁6.0397,𝜁1.1230,𝜁3) (𝜁6.6098,𝜁1.4128,𝜁3.5219) (𝜁8,𝜁1.1862,𝜁3) (𝜁8,𝜁1.2946,𝜁3)
ℭ2 (𝜁5.7547,𝜁1.17956,𝜁3) (𝜁6.2061,𝜁2.6956,𝜁4.6548) (𝜁5.5650,𝜁2.5979,𝜁3.5219) (𝜁5.3915,𝜁1.6758,𝜁2.6612)
ℭ3 (𝜁5.4334,𝜁1.81.42,𝜁3.2762) (𝜁5.4334,𝜁2.4168,𝜁3.3049) (𝜁5.7547,𝜁1.3046,𝜁3.9515) (𝜁5.3915,𝜁1.6758,𝜁3)
ℭ4 (𝜁5.7547,𝜁1.6443,𝜁3.0485) (𝜁5.4334,𝜁1.6460,𝜁3.6536) (𝜁8,𝜁1.2478,𝜁3) (𝜁5.4334,𝜁1.9888,𝜁3)
S ep 2: The o al collec i e LPNN ℭ𝑖 (𝑖 = 1,2, …, 𝑚) can be ob ained by he LPNEWA ope a o :
ℭ1=(𝜁8,𝜁2.3655,𝜁3.1190);ℭ2=(𝜁5.0618,𝜁3.1022,𝜁3.3464);
ℭ3=(𝜁4.7291,𝜁3.2301,𝜁3.3319); ℭ4= (𝜁7.6441,𝜁3.0416,𝜁3.1604)
S ep 3: By using de ini ion 5, we calcula e he expec ed alues o 𝔗(ℭ𝑖) o ℭ𝑖 (𝑖 = 1,2,3,4)
𝔗(ℭ1)=6.1285; 𝔗(ℭ2)= 4.7889 ;𝔗(ℭ3)=4.6486 ; 𝔗(ℭ4)=5.8649.
Based on he expec ed alues, ou al e na i es can be anked ℭ1 ≻ ℭ4≻ ℭ2≻ ℭ3,. Thus, company
ℭ3 is he op imal choice.
Now, we ind he op imal choice using he LPNEWG ope a o .
Table 6. The o e all decision ma ix
𝓢𝓕𝟏 𝓢𝓕𝟐 𝓢𝓕𝟑 𝓢𝓕𝟒
ℭ1 (𝜁3.5005,𝜁1.3752,𝜁2.848) (𝜁3.7147,𝜁2.558,𝜁3.391) (𝜁3.589,𝜁1.585,𝜁3.391) (𝜁3.392,𝜁1.418,𝜁2.848)
ℭ2 (𝜁3.4122,𝜁2.4606,𝜁2.848) (𝜁3.5091,𝜁4.968,𝜁4.447) (𝜁3.327,𝜁4.471,𝜁3.391) (𝜁3.197,𝜁2.116,𝜁2.667)
ℭ3 (𝜁3.3187,𝜁2.5533,𝜁3.089) (𝜁3.3187,𝜁3.534,𝜁4.447) (𝜁3.412,𝜁2.442,𝜁4.388) (𝜁3.197,𝜁2.116,𝜁2.848)
ℭ4 (𝜁3.4122,𝜁2.6547,𝜁3.111) (𝜁3.3187,𝜁3.307,𝜁3.785) (𝜁3.685,𝜁1.991,𝜁2.848) (𝜁3.319,𝜁2.542,𝜁2.848)
S ep 2: The o al collec i e LPNN ℭ𝑖 (𝑖 = 1,2, …, 𝑚) can be ob ained by he LPNEWA ope a o :
ℭ1=(𝜁4.8605,𝜁1.5982,𝜁2.593);ℭ2=(𝜁4.6882,𝜁3.4566,𝜁3.093);
ℭ3=(𝜁4.6193,𝜁2.4321,𝜁3.054); ℭ4= (𝜁4.7284,𝜁2.2984,𝜁2.803)
S ep 3: By using de ini ion 5, we calcula e he expec ed alues o 𝔗(ℭ𝑖) o ℭ𝑖 (𝑖 = 1,2,3,4)
𝔗(ℭ1)=5.1103; 𝔗(ℭ2)= 4.6356 ;𝔗(ℭ3)=4.8338 ; 𝔗(ℭ4)=4.9402.
Based on he expec ed alues, ou al e na i es can be anked ℭ1 ≻ ℭ4≻ ℭ3≻ ℭ2. Thus, company
ℭ2 is he op imal choice.
6.2 Compa a i e Analysis
We compa e he p oposed LPNEWA and LPNEWG me hods wi h o he LIFEWA and LPFEWA
app oaches. The esul s o his compa ison a e p esen ed in Figu e 4.
F om Figu e 4, i is e iden ha al e na i es ℭ3 and ℭ2 eme ge as he mos op imal choices when
e alua ed using he LPNEWA and LPNEWG me hods. The anking o de s p oduced by hese wo
me hods a e: ℭ1 ≻ ℭ4≻ ℭ2≻ ℭ3 o LPNEWA, and ℭ1 ≻ ℭ4≻ ℭ3≻ ℭ2. o LPNEWG. To
alida e he e ec i eness o he p oposed me hod, a compa ison is made wi h exis ing app oaches,
including he linguis ic in ui ionis ic uzzy weigh ed a e age (LIFWA) ope a o in oduced by Chen
e al. [25], he LPF weigh ed a e age (LPFWA) ope a o de eloped by Ga g [26], Sine Single-Valued
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S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
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Neu osophic Eins ein Weigh ed A e aging (S-S NVEWA) and Sine Single-Valued Neu osophic
Eins ein Weigh ed Geome ic (S-S NVEWG) agg ega ion ope a o s de eloped by Zhang e al. [27].
Unlike hese ea lie me hods [25–27], he p oposed LPNN-based app oach can e ec i ely ep esen
and handle pu ely linguis ic e alua ion alues—some hing ha adi ional MCDM me hods canno
achie e. By in eg a ing LPNS wi h Eins ein ope a ions, he p oposed me hod clea ly demons a es
i s lexibili y and e ec i eness.
Fig. 4. Compa a i e analysis o di e en MCDM me hods
7. Conclusion
This pape p oposed a no el app oach o sol ing MCDM p oblems. Ini ially, he Eins ein ope a ion
was applied o Linguis ic Py hago ean Neu osophic Numbe s (LPNNs), and new ope a ional ules
we e es ablished based on his ope a o . Subsequen ly, se e al agg ega ion ope a o s we e in eg a ed
wi h he LPNNs o de ine he Linguis ic Py hago ean Neu osophic Eins ein Weigh ed A e age
(LPNEWA) ope a o and he Linguis ic Py hago ean Neu osophic Eins ein Weigh ed Geome ic
(LPNEWG) ope a o , in acco dance wi h he newly de eloped ules. Using he LPNEWA and
LPNEWG ope a o s, wo me hods we e in oduced o e ec i ely add ess MCDM p oblems. To
demons a e he p ac icali y and bene i s o he p oposed me hods, hey we e applied o a eal-wo ld
example.
Acknowledgmen s: The au ho s wish o exp ess g a i ude o he Managemen , P incipal, S i
Si asub amaniya Nada College o Enginee ing, Chennai, India.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Neu osophic Se s and Sys ems, Vol. 94, 2025
S. Annadu ai, R. Sunda eswa an, M. Shanmugap iya, M. Mohanalakshmi , Sus ainabili y analysis in ag icul u e using
Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s
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Funding: This esea ch ecei ed no ex e nal unding.
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Linguis ic Py hago ean Neu osophic numbe h ough Eins ein agg ega ion ope a o s