Neu osophic Se s and Sys ems, Vol. 93, 2025
Uni e si y o New Mexico
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o
T a elling Salesman P oblem
Souhail Dhouib1, Saima Dhouib2, K. Sa i ha3, R. Rajalakshmi4, Mana Donganon 5,
P asan a Kuma Rau 7
1Highe Ins i u e o Indus ial Managemen , Uni e si y o S ax, Tunisia; [email p o ec ed]
2EPI-Poly echnique, Uni e si y o Sousse, Sousse, Tunisia
3Depa men o Ma hema ics, Panimala Enginee ing College, Poonamallee, Chennai-600123,
Tamil Nadu, India; [email p o ec ed]
4Depa men o Ma hema ics, Panimala Enginee ing College, poonamallee, Chennai-600123,
Tamil Nadu, India; [email p o ec ed]
5Depa men o Ma hema ics, School o Science, Uni e si y o Phayao, Thailand;
[email p o ec ed]
7Depa men o Ma hema ics, T iden Academy o Technology, Bhubaneswa , Odisha, India.
[email p o ec ed]
Co espondence: [email p o ec ed]
Abs ac
The main idea in his pape is o s udy he applica ion o he no el g eedy Dhouib-
Ma ix-TSP1 (DM-TSP1) me hod o sol e he T a el Salesman P oblem wi h simple
neu osophic numbe s. He e, he Euclidian dis ance is used o con e he neu osophic
numbe o c isp alues, i is conside ed ha he neu osophic numbe as a h ee-
dimensional coo dina e. DM-TSP1 is a cons uc i e me hod and equi es jus (n-1)
i e a ions o gene a e o c ea e a solu ion (whe e n is he numbe o nodes).
Compu a ional esul s on a case s udy de eloped in he li e a u e p o e ha he
Neu osophic Se s and Sys ems, Vol. 93, 2025 228
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
p oposed DM-TSP1 heu is ic can c ea e be e solu ion han he Gene ic Algo i hm
wi h a i ness imp o emen o 92.68%.
Keywo ds: Neu osophic Numbe , A i icial In elligence, Ope a ions Resea ch,
Heu is ic, Dhouib-Ma ix-TSP1, Euclidian Dis ance.
1. In oduc ion
In daily li e, we equen ly encoun e si ua ions ha a e incomple e, unclea , and
ambiguous. To add ess such challenges, ma hema ician P o . Lo i A. Zadeh
in oduced he concep o uzzy se s in 1965 [1]. Fuzzy se s e ec i ely manage
si ua ions in ol ing unce ain y by allowing pa ial membe ship alues; howe e , hey
do no accoun o non-membe ship explici ly. To add ess his limi a ion, A anasso
ex ended uzzy se heo y in 1983 by in oducing in ui ionis ic uzzy se s [2], which
inco po a e bo h membe ship and non-membe ship alues. La e , in 1995,
Sma andache u he gene alized his concep wi h neu osophic se s, which added a
hi d componen o cap u e inde e minacy, he eby allowing o he ep esen a ion o
h ee membe ship alues: u h (T), inde e minacy (I), and alsi y (F) [3]. Neu osophic
se s hus p o ide a mo e comp ehensi e amewo k o modelling unce ain y and
ambigui y in a ious complex eal-wo ld applica ions.
The T a elling Salesman P oblem (TSP) is among he mos widely esea ched
op imiza ion p oblems in ope a ions esea ch and heo e ical compu e science.
Classi ied as NP-ha d, TSP poses signi ican compu a ional challenges, as inding an
op imal solu ion in polynomial ime is in easible o la ge ins ances. Ini ially in oduced
in he 1930s, TSP gained subs an ial a en ion a e he 1950s due o i s impo ance in
bo h heo e ical and applied esea ch [1, 2]. The p oblem is con en ionally amed as
ollows: gi en a lis o ci ies and he dis ances be ween each ci y pai , he objec i e is
o de e mine he sho es possible ou e ha isi s each ci y exac ly once and e u ns o
he s a ing poin . TSP has ex ensi e applica ions in logis ics, anspo a ion, ci cui
boa d design, and DNA sequencing.
The s udy o op imiza ion p oblems is c ucial in enginee ing and compu a ional
sciences, as inding op imal solu ions unde a ious cons ain s is undamen al o
imp o ing sys em pe o mance, e iciency, and esou ce u iliza ion. Consequen ly,
nume ous esea ch s udies ha e ocused on op imiza ion, esul ing in a weal h o
published li e a u e ha explo es a ious me hods and algo i hms o achie ing op imal
solu ions in di e se applica ions [3,4,5,6,7,8,9,10,11,12,13]. These s udies con ibu e
o sol ing complex, eal-wo ld p oblems ac oss ields such as logis ics, ne wo k design,
ene gy managemen , and machine lea ning, whe e op imiza ion plays a pi o al ole.
O e he yea s, nume ous esea che s ha e explo ed TSP and p oposed a ious
solu ions. Fo ins ance, Rau e al. [14] used a andom-key gene ic algo i hm o e alua e
Neu osophic Se s and Sys ems, Vol. 93, 2025 229
he sho es pa h, while Chiung Moon e al. [15] applied an e icien gene ic algo i hm.
In-Chan Choi e al. [16] ackled he asymme ic TSP wi h a gene ic algo i hm
employing a mixed- egion sea ch, and Zaki H. Ahmed e al. [17] de eloped a gene ic
algo i hm wi h a sequen ial cons uc i e c osso e ope a o . Jayan a Majumda e al.
[18] ex ended gene ic algo i hms o sol e he asymme ic TSP wi h imp ecise a el
imes, and Sangi Cha e jee e al. [19] u ilized gene ic algo i hms o TSP solu ions.
Addi ionally, Ch-Ch Chou e al. [20] used iangula uzzy numbe s and canonical
ep esen a ions o model mul iplica ion ope a ions in TSP, u he ing he ield’s
unde s anding o how o handle unce ain y in ou ing p oblems.
The concep o h ee-dimensional Euclidean dis ance (3D-ED) p o ides a spa ially
en iched app oach o measu ing dis ances, essen ial in applica ions ha span h ee
dimensions. Unlike wo-dimensional Euclidean dis ance, 3D-ED accoun s o
a ia ions along all h ee axes, o e ing a mo e accu a e ep esen a ion o eal-wo ld
dis ances, especially whe e he hi d dimension is c i ical, such as in ai o ma i ime
na iga ion, sa elli e posi ioning, and high- ise u ban ou ing. In eg a ing 3D-ED in o
TSP allows o a mo e ealis ic ep esen a ion o dis ances in p ac ical applica ions.
While 3D-ED has been applied in a ious ields, i s in eg a ion in o TSP, pa icula ly
unde condi ions o unce ain y, is s ill a de eloping a ea o esea ch.
This pape in oduces an inno a i e app oach ha combines 3D-ED wi h neu osophic
numbe s o ackle he TSP unde unce ain condi ions. The p oposed model enhances
he classical TSP by applying neu osophic numbe s o he 3D-ED calcula ions,
allowing o unce ain dis ance alues while p ese ing he spa ial accu acy a o ded
by h ee-dimensional modelling. This hyb id app oach b ings a no el pe spec i e o
TSP esea ch, ocusing on no only op imizing ou es bu also managing he ambigui y
inhe en in eal-wo ld measu emen s.
The mo i a ion behind his s udy is wo old. Fi s , he e is a need o enhance TSP
o mula ions by inco po a ing h ee-dimensional spa ial ela ionships and
inde e mina e ac o s a ec ing dis ances. T adi ional TSP app oaches ypically ely on
exac dis ance alues, which a e o en app oxima ions o a e ages wi h a ying deg ees
o con idence in eal-wo ld se ings. By in oducing neu osophic numbe s, his model
allows dis ances o ep esen in e als o possible alues wi h associa ed deg ees o
ce ain y, be e cap u ing eal-wo ld complexi ies. Second, exis ing app oaches ha
accoun o unce ain y in TSP commonly use p obabilis ic o uzzy models, which
may no ully ep esen he spec um o inde e minacy ound in scena ios whe e
in o ma ion is incomple e o only pa ially a ailable. Neu osophic numbe s help
b idge his gap, p o iding a b oade amewo k o ep esen unce ain y h ough h ee
componen s ( u h, inde e minacy and alsi y).
The emaining o he pape is o ganized as ollows: Sec ion 2 p esen s p elimina y,
co e ing a ious de ini ions ela ed o neu osophic se s. Sec ion 3 p esen s he g eedy
DM-TSP1 me hod and Sec ion 4 p esen s a nume ical example wi h empi ical analysis
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Neu osophic Se s and Sys ems, Vol. 93, 2025 230
and esul s. Sec ion 5 concludes he s udy by explo ing i s implica ions, limi a ions,
and u u e esea ch di ec ions.
2. P elimina ies
2.1 Neu osophic Se s
A neu osophic se 𝐴
in a uni e sal se 𝑈
is de ined by h ee membe ship unc ions:
he u h-membe ship unc ion 𝑇
𝐴
(𝑥), he inde e minacy-membe ship unc ion 𝐼𝐴
(𝑥)
and he alsi y-membe ship unc ion 𝐹
𝐴
(𝑥). The neu osophic se 𝐴
is ep esen ed as:
𝐴
= (𝑥,( 𝑇
𝐴
(𝑥),𝐼𝐴
(𝑥),𝐹
𝐴
(𝑥)):𝑥 ∈ 𝑋 . whe e 𝑇
𝐴
(𝑥),𝐼𝐴
(𝑥),𝐹
𝐴
(𝑥) is u h,
Inde e minacy and alsi y membe ship deg ee
whe e:
• 𝑇
𝐴
(𝑥) ep esen s he deg ee o u h,
• 𝐼𝐴
(𝑥) ep esen s he deg ee o inde e minacy,
• 𝐹
𝐴
(𝑥) ep esen s he deg ee o alsi y.
2.2 Neu osophic Numbe s
A neu osophic numbe 𝑁
is a iple ep esen ing deg ees o u h, inde e minacy, and
alsi y in unce ain da a. I is de ined as:
𝑁
= 𝑇
𝑁
(𝑥),𝐼𝑁
(𝑥),𝐹
𝑁
(𝑥)
whe e:
• 𝑇
𝑁
(𝑥) is he u h-membe ship alue (0 ≤ 𝑇
𝑁
(𝑥)≤ 1)
• 𝐼𝑁
(𝑥) is he inde e minacy-membe ship alue (0 ≤ 𝐼𝑁
(𝑥)≤ 1)
• 𝐹
𝑁
(𝑥) is he alsi y-membe ship alue (0 ≤ 𝐹
𝑁
(𝑥)≤ 1)
The neu osophic numbe p o ides a lexible way o model unce ain dis ances by
accoun ing o a ious deg ees o ce ain y.
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Neu osophic Se s and Sys ems, Vol. 93, 2025 231
2.3 Euclidean Dis ance in Th ee Dimensions
The Euclidean dis ance be ween wo poin s 𝐴 = (𝑎1,𝑏1,𝑐1) , 𝐵 = (𝑎2,𝑏2,𝑐2) and
in h ee-dimensional space is gi en by: AB=√(𝑎2−𝑎1)2+(𝑏2−𝑏1)2+(𝑐2−𝑐1)2
This dis ance o mula cap u es he s aigh -line dis ance be ween wo poin s in a 3D
space, which is essen ial o accu a ely ep esen ing spa ial ela ionships in p oblems
whe e he hi d dimension plays a signi ican ole, such as in ce ain logis ics and
ou ing asks.
2. 4. Neu osophic Euclidean Dis ance o TSP
Fo he T a elling Salesman P oblem (TSP) in ol ing unce ain dis ances, he
Euclidean dis ance d (A,B) be ween any wo poin s A and B is ex ended o a
neu osophic dis ance 𝐷𝑁(𝐴,𝐵) de ined as a iple :
𝐷𝑁(𝐴,𝐵) = (𝑑𝑇,𝑑𝐼,𝑑𝐹)
whe e:
• 𝑑𝑇 ep esen s he u h alue dis ance.
• 𝑑𝐼 ep esen s he inde e minacy alue dis ance,
• 𝑑𝐹 ep esen s he alsi y alue dis ance.
Each componen o 𝐷𝑁(𝐴,𝐵) is calcula ed based on he applica ion con ex , and his
neu osophic dis ance allows o a mo e lexible ep esen a ion o he TSP by
inco po a ing unce ain y di ec ly in o he dis ance measu emen s.
3. The g eedy DM-TSP1 me hod
The no el g eedy Dhouib-Ma ix-TSP1 (DM-TSP1) is used o gene a e he
Hamil onian cycle and/o he open ou e (see Figu e 2). Basically, DM-TSP1 is
composed o ou s eps and epea ed wi h (n-1) i e a ions (whe e n is he numbe o
nodes) o gene a e he Hamil onian cycle. Unless, o design an open ou e, DM-TSP1
equi es only he i s h ee s eps (see Figu e 2): The i s s ep is composed o ou asks
o ini ia e he DM-TSP1 wi h he assignmen o he i s connec ion (be ween wo
ci ies); he second s ep is used o ind he z ci y cha ac e ized by i s sho es dis ance;
and he hi d s ep is equi ed o disca d he column o he z ci y and o upda e he lis
o Lis -ci ies ( o mo e cla i ica ion see[22, 23]).
DM-TSP1 is applied o sol e unce ain T a elling Salesmen P oblem in [21, 22]. In
addi ion, i is hyb idized wi h he A i icial Bee Colony me aheu is ic in [23], wi h he
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Neu osophic Se s and Sys ems, Vol. 93, 2025 232
Dhouib-Ma ix-3 me aheu is ic in [24, 25, 26] and he mul i-s a Dhouib-Ma ix-4 in
[27, 28, 29, 30, 31].
Figu e 2. The gene al s uc u e o he DM-TSP1 heu is ic
DM-TSP1 is a componen o he gene al concep o Dhouib-Ma ix whe e o he
me hods a e designed such as he Dhouib ma ix-TP1 in [32, 33], he Dhouib-Ma ix-
AP1 in [34, 35, 36, 37], he Dhouib-Ma ix-MSTP in [38, 39] and he Dhouib-Ma ix-
SPP in [40, 41, 42, 43, 44, 45].
4. Nume ical examples
In his sec ion
Le us conside he example o se en nodes (n=7) wi h neu osophic numbe in oduced
in [46]. The i ness unc ion Eq. (1) is compu ed based on he dis ance, he highes -
membe ship alue and he minimal inde e minacy-membe ship alue o he nodes
neu osophic numbe s.
( )( )
max
min
*
1
N
TD
Fi ness I
=+
(1)
Whe e:
max
T
is he highes u h-membe ship alue o he node neu osophic numbe ( o he
example p esen ed in Table 2, he highes u h-membe ship numbe is 0.7, hus,
max 0.7T=
)
min
I
is he minimal inde e minacy-membe ship alue o he node neu osophic numbe
( o he example p esen ed in Table 2, he minimal inde e minacy-membe ship numbe
is 0.2, hus,
min 0.2I=
)
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Neu osophic Se s and Sys ems, Vol. 93, 2025 233
𝐷𝑁 is he neu osophic dis ance o he gene a ed ou e ( o he solu ion gene a ed by
DM-TSP1 he D)
In addi ion, he dis ance be ween any wo ci ies is compu ed no by using i s coo dina es
(x and y) bu by using hei co esponding neu osophic numbe s (see Eq. (2)):
( )
() ( ) ( )
2 2 2
,
N i j i j i j
D i j T T I I F F
= − + − + −
(2)
Mo eo e , a i ness imp o emen unc ion Eq. 3 is used o compu e he pe o mance o
DM-TSP1 e sus GA algo i hm
()
()
11
Fi ness imp o emen = / *100
GA DM TSP DM TSP
Fi ness Fi ness Fi ness
−−
−
(3)
Whe e:
GA
Fi ness
is he i ness esul c ea ed by he GA algo i hm
1
DM TSP
Fi ness −
is he i ness esul gene a ed by DM-TSP1
Table 1 summa ize he geog aphical posi ions o he se en ci ies ( hese coo dina es a e
used only o he g aphical ep esen a ion).
Table 1. he geog aphical posi ions o he se en ci ies
Ci y X coo dina e Y coo dina e
1 0 0
2 1 1
3 2 0
4 1 2
5 3 1
6 2 3
7 4 2
Figu e 3 depic s he g aphical ep esen a ion o he case s udy o se en ci ies.
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Neu osophic Se s and Sys ems, Vol. 93, 2025 234
Figu e 3. The g aphical ep esen a ion o he se en ci ies
Ac ually, o each ci y a simple neu osophic numbe ( he alue o each unc ion is
be ween 0 and 1) is speci ied in Table 2.
Table 2. The neu osophic numbe o each ci y
Ci y Neu osophic alue
1 (0.7, 0.2, 0.1)
2 (0.6, 0.3, 0.1)
3 (0.5, 0.3, 0.2)
4 (0.2, 0.5, 0.3)
5 (0.4, 0.4, 0.2)
6 (0.3, 0.4, 0.3)
7 (0.6, 0.3, 0.1)
Be o e s a ing he DM-TSP1 me hod he con ingency ma ix (see Figu e 4) is
gene a ed by compu ing he Euclidian dis ance (using he neu osophic numbe s)
be ween all ci ies using Eq. (2). Fo example, o compu e he dis ance be ween ci ies 1
and 2 he dis ance is compu ed by
( ) ( ) ( ) ( )
2 2 2
1, 2 0.7 0.6 0.2 0.3 0.1 0.1 0.14
N
D= − + − + − =
.
0.00 0.14 0.24 0.62 0.37 0.49 0.14
0.14 0.00 0.14 0.49 0.24 0.37 0.00
0.24 0.14 0.00 0.37 0.14 0.24 0.14
0.62 0.49 0.37 0.00 0.24 0.14 0.49
0.37 0.24 0.14 0.24 0.00 0.14 0.24
0.49 0.37 0.24 0.14 0.14 0.00 0.37
0.14 0.00 0.14 0.49 0.24 0.37 0.00
Figu e 4. The con ingency ma ix
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
Neu osophic Se s and Sys ems, Vol. 93, 2025 235
The nex s ep consis s on calcula ing he maximum o each ow and inse i on he
igh -hand side o he ma ix. Besides he minimal alue (0.37) is selec ed (see Figu e
5).
Figu e 5. The maximum alue o each ow is inse ed a igh
Besides, he DM-TSP1 equi es only six (n-1) i e a ions o c ea e a solu ion. Figu e 6
illus a es a s ep-by-s ep applica ion o DM-TSP1. A i s , DM-SPP selec s he elemen
a he posi ion (d21) and disca d column 1 and 2 (see Figu e 6.1). A second, DM-SPP
selec s he elemen a he posi ion (d16) and disca d column 6 (see Figu e 6.2). A hi d,
he elemen a he posi ion (d60) is selec ed and he column 0 is disca ded (see Figu e
6.3). Besides, he elemen a he posi ion (d24) is selec ed and he column 4 is disca ded
(see Figu e 6.4). Nex , he elemen a he posi ion (d45) is selec ed and he column 5 is
disca ded (see Figu e 6.5). Finally, he elemen a he posi ion (d53) is selec ed and he
column 3 is disca ded (see Figu e 6.6).
Figu e 6. The s ep-by-s ep applica ion o DM-TSSP1 on se en nodes p oblem
Souhail Dhouib, , K. Sa i ha, R. Rajalakshmi, Saima Dhouib, P asan a Kuma Rau , Mana Donganon
, Th ee-dimensional Euclidian Dis ance o Neu osophic Numbe o T a elling Salesman P oblem
