Neu osophic Se s and Sys ems, Vol. 93, 2025
Uni e si y o New Mexico
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
A Decision-Making Model o he T a elling Salesman P oblem
Based on Neu osophic Edge Connec i i y
Apa na T ipa hy1, Ama esh Chand a Panda1*, Si a P asad Behe a1, P asan a Kuma
Rau 2
1Depa men o Ma hema ics, C.V. Raman Global Uni e si y, Bhubaneswa , Odisha, India.
1Email: [email p o ec ed], [email p o ec ed],
[email p o ec ed]
2Depa men o Ma hema ics, T iden Academy o Technology, Bhubaneswa , Odisha,
India
2Email: [email p o ec ed]
Co espondence: [email p o ec ed]
Abs ac
Neu osophic edge connec i i y is a new idea in g aph heo y ha adds inde e minacy and
unce ain y o adi ional edge connec i i y. I is especially use ul o challenging
op imiza ion p oblems ha happen in he eal wo ld. The use o neu osophic edge
connec i i y in sol ing he well-known NP-ha d issue in combina o ial op imiza ion, he
T a elling Salesman issue (TSP), is examined in his wo k. We make a new amewo k by
combining neu osophic logic, which makes TSP solu ions mo e lexible and ealis ic,
especially in en i onmen s ha a e unce ain and changing quickly, whe e adi ional
de e minis ic models don' wo k. By aking in o accoun he le els o u h, inde e minacy,
and alsi y in edge connec i i y, he sugges ed me hod helps people make be e ou e
Neu osophic Se s and Sys ems, Vol. 93, 2025 427
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
op imiza ion decisions. He e in his esea ch pape , we p o ide a ho ough heo e ical s udy
along wi h ma hema ical examples o TSP (T a eling Salesman P oblem) si ua ions o e i y
he me hod's e iciency. Fu he mo e, we p oposed a p oblem s a emen . A logis ics business
mus choose he bes deli e y pa h among di e en ci ies. I u ns ou ha ou neu osophic-
based app oach makes solu ions mo e s able and opens up a bene icial pa h o mo e esea ch
in eal-wo ld logis ics and unce ain y-based op imiza ion.
Keywo ds:
Edge Connec i i y; Neu osophic G aph Theo y; Op imiza ion, Inde e minacy; T a elling
Salesman P oblem.
1. In oduc ion
G aph heo y has g ea use in many ields, including sys em analysis, compu e science,
ne wo king, and anspo a ion sys ems. Ac ing as a basic ela ional model, i shows links
among ac ual objec s. In a g aph, e ices ep esen en i ies, and edges show hei
ela ionships. Howe e , challenges such as missing da a, lack o suppo ing e idence, and
insu icien knowledge o en plague eal-wo ld op imiza ion issues, leading o e o s in
decision-making. Zadeh [1] de eloped he idea o uzzy se s, whe e e e y elemen has a
membe ship deg ee be ween 0 and 1 and i also o help ma hema ical modeling o e come
ambigui y. Examining uzzy g aphs in excellen de ail, Si a a [2] highligh ed hei main
ea u es and hei capaci y o mo e success ully con ol ambigui y in complica ed ne wo ks.
Al hough uzzy se s help o handle imp ecision, A anasso [3] no ed a d awback in hei
me hod as hey only con ol unce ain y in one di ec ion, he e o e neglec ing he whole
complexi y o human hinking. A anasso de eloped in ui ionis ic uzzy se s, an ex ension o
uzzy se s including bo h membe ship and non-membe ship unc ions, o ge o e his
disad an age. La e , Sma andache [4] de eloped his idea by sugges ing neu osophic se s, a
mo e gene ic amewo k able o manage inconsis en and unce ain in o ma ion, which is
usually ound in p ac ical si ua ions. Wang [5] imp o ed his idea e en u he by adding
single- alued neu osophic se s (S-VNS) o inc ease i s use ul ele ance. B oumi [6] and
associa es hen expanded on his concep , assis ing in he de elopmen o single- alued
neu osophic g aphs o use in decision-making and op imiza ion. Mo e ecen ly, Rau , P. K.,
in es iga ed he use o neu osophic se s in many applica ions, including sho es -pa h
p oblems [8–17].
Unde s anding he dependabili y and esilience o ne wo ks depends in la ge pa on edge
connec ions. Edge connec i i y in classical g aph heo y is he leas numbe o edges equi ed
o disconnec a g aph. In eal-wo ld applica ions, howe e , ne wo ks o en consis o
ambiguous and unclea cha ac e is ics ha make con en ional models insu icien o co ec ly
Neu osophic Se s and Sys ems, Vol. 93, 2025 428
depic sys em beha iou . In oduced by Sma andache, neu osophic logic s e ches classical
and uzzy logic by adding h ee basic componen s: u h, alsehood, and inde e minacy,
he eby add essing his es ic ion. Including neu osophic logic in edge connec i i y analysis
helps p o ide a mo e lexible and ealis ic amewo k o handle ac ual unce ain y
Among such use ul applica ions is he well-known combina o ial op imiza ion p oblem in
compu e science and ma hema ics, he T a elling Salesman P oblem (TSP). O iginally
b ough up in he 1930s, he issue became well-known du ing he 1950s [18–20]. The TSP
calls o a salespe son o isi a se ies o ci ies and e u n o he beginning poin , he e o e
op imizing he o e all dis ance a elled. Wi h i s NP-ha d cha ac e [21, 22], he TSP emains
a benchma k issue o assessing me hods o op imiza ion. Real-wo ld ci cums ances al e
ou e a e sal by dynamic elemen s like a ic conges ion, oad condi ions, anspo a ion
p ices, and wea he a ia ions, he eby making i challenging o es ima e p ecise ip
expendi u es and ime du a ions. Decision-make s ha e o conside hese unce ain ies i hey
a e o c ea e sensible ou ing plans.
The compu a ional complexi y inc eases exponen ially as he numbe o ci ies in a TSP
ins ance ises; hence, accu a e solu ions a e useless o la ge-scale issues. O e he yea s,
many heu is ic and me aheu is ic app oaches ha e de eloped o sol e he TSP [23–29].
Because p ecise algo i hms mus e alua e all po en ial combina ions, hei e iciency declines
o big ne wo ks e en as hey can apidly iden i y e ec i e solu ions o small examples.
Wi hin an accep able compu ing pe iod, heu is ic and me aheu is ic algo i hms, such as
ha mony sea ch, a i icial bee colony algo i hms, and gene ic algo i hms, p o ide nea ly ideal
answe s. Fo example, gene ic algo i hms use selec ion, mu a ion, and c osso e p ocesses o
c ea e be e answe s while ch omosomes e lec possible solu ions.
This wo k aims o enhance TSP op imiza ion h ough he use o neu osophic edge
connec i i y, which exp esses a c leng hs as neu osophic numbe s o accoun o unce ain y.
We use neu osophic easoning in he op imiza ion p ocess since con en ional e olu iona y
app oaches a e no su icien o unknown su oundings. Ou sugges ed s uc u e allows mo e
lexible and easonable decision-making by ep esen ing a c leng hs in he TSP using
neu osophic in ege s. A ma hema ical model ha was made o he TSP wi h neu osophic
numbe a c leng hs shows how use ul neu osophic se s can be in unce ain si ua ions. We
also show he success o ou me hod wi h a nume ical example.
This pape is s uc u ed as ollows gene ally: In Sec ion-2 in oduces he undamen al
concep s o neu osophic se s, neu osophic edge connec i i y, and hei ma hema ical
cha ac e is ics. In Sec ion-3 o e s he sugges ed me hod o neu osophic edge connec ion
along wi h i s TSP sol ing applica ion. In Sec ion -4 shows a nume ical case o suppo he
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
Neu osophic Se s and Sys ems, Vol. 93, 2025 429
sugges ed app oach. In Sec ion-5 o e s a compa a i e s udy o he sugges ed me hod using
models o classical and uzzy edge connec i i y. In Sec ion6 inishes he wo k and add esses
u u e a enues o in es iga ion. This wo k sugges s a no el me hod o sol e he TSP unde
unce ain y, he eby ad ancing neu osophic g aph heo y. Ou app oach p o ides a mo e
lexible and use ul answe o eal-wo ld ou ing and anspo a ion issues by including
neu osophic edge connec ion.
2. P elimina ies:
2.1 Neu osophic Se s:
Flo en in Sma andache p esen ed a ma hema ical amewo k called a neu osophic se o
add ess unce ain y, imp ecision, ambigui y, and pa ial knowledge. Including h ee
independen membe ship unc ions, i expands classical, uzzy, and in ui ionis ic uzzy se
heo ies. and h ee membe ship alues, i.e., u h, inde e minacy, and alsi y, lie in [0,1].
E e y componen o a neu osophic se is exp essed as
𝐴 = {(𝑥,𝑇(𝑥),𝐼(𝑥),𝐹(𝑥)) ∣ 𝑥 ∈ 𝑈}
And he condi ion sa is ies h ee membe ship alues:
0
≤ 𝑇(𝑥)+ 𝐼(𝑥)+ 𝐹(𝑥)≤ 3
1
2.2 Neu osophic Edge Connec i i y: De ini ion and P ope ies
Neu osophic edge connec ion expands he con en ional idea o edge connec i i y by including
unce ain y, doub , and ambigui y in ne wo k a chi ec u es. Each edge o a neu osophic g aph
has h ee componen s: u h (T), inde e minacy (I), and alsi y (F), which, aken oge he ,
desc ibe he dependabili y and s abili y o an edge. Taking hese neu osophic ai s in o
accoun , he neu osophic edge connec i i y o a g aph is he smalles numbe o edges ha ,
when aken away, spli a g aph in o wo o mo e disconnec ed pa s.
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
Neu osophic Se s and Sys ems, Vol. 93, 2025 430
2.3 Neu osophic G aph Ma hema ical Rep esen a ion
Conside a G aph 𝐺 = (𝑉,𝐸,𝑁)
• A se o e ices V.
• A se o edges E connec ing hese e ices.
• A neu osophic unc ion 𝑁: 𝐸 → [𝑇,𝐼,𝐹], whe e each edge has an associa ed
u h, inde e minacy, and alsi y alue wi hin he ange [0,1]. The neu osophic
edge connec i i y 𝜆𝑁(𝐺) is de ined as he minimum numbe o edges whose
emo al inc eases he numbe o connec ed componen s in G, aking in o accoun
he inde e minacy componen .
2.4 P ope ies and Cha ac e is ics
• Unlike adi ional edge connec i i y, he neu osophic model allows unce ain y o
be accommoda ed, he e o e enabling a mo e ealis ic e alua ion o ne wo k
esilience.
• Dynamic Beha iou : The connec ion s eng h changes dynamically o i he shi ing
edge pa ame e cha ac e .
• Flexibili y in Op imiza ion: This me hod is e y help ul in decisions when edges
show a ying deg ees o dependabili y.
• Neu osophic edge connec ion o e s a wide iew o ne wo k esiliency han
con en ional and uzzy models.
3. P oposed Algo i hm o Neu osophic Edge Connec i i y and I s
Applica ion in he T a elling Salesman P oblem
S ep 1: Rep esen a ion o he G aph in Neu osophic En i onmen
1. De ine a weigh ed g aph whe e is he se o e ices (ci ies) and is he se o edges
( ou es be ween ci ies).
2. Assign a neu osophic weigh o each edge, ep esen ed as, whe e:
• (T u h) ep esen s he deg ee o ce ain y ha he edge exis s.
• (Inde e minacy) ep esen s he unce ain y in edge exis ence.
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
Neu osophic Se s and Sys ems, Vol. 93, 2025 431
• (Falsi y) ep esen s he deg ee o nonexis ence o he edge.
S ep 2: Compu a ion o Neu osophic Edge Connec i i y
1. Compu e he edge connec i i y o he g aph unde he neu osophic se ing by:
• Finding he minimum cu -se using neu osophic weigh s.
• Iden i ying c i ical edges based on hei neu osophic s eng h.
• Using a modi ied max- low algo i hm inco po a ing neu osophic alues.
2. No malize he edge connec i i y alues o ob ain a c isp equi alen measu e o
compa ison.
S ep 3: Fo mula ing he T a elling Salesman P oblem (TSP) unde Neu osophic
Cons ain s
1. Cons uc he dis ance ma ix using neu osophic edge weigh s.
2. De ine he objec i e unc ion:
• Minimize he o al cos conside ing neu osophic unce ain y ac o s.
• Ensu e e e y e ex is isi ed exac ly once and e u ns o he s a ing poin .
S ep 4: Sol ing Neu osophic TSP using an Op imiza ion Algo i hm
1. Apply a heu is ic o me aheu is ic app oach such as Gene ic Algo i hm (GA), An
Colony Op imiza ion (ACO), o Pa icle Swa m Op imiza ion (PSO) adap ed o
handle neu osophic numbe s.
2. Modi y he cos unc ion o inco po a e neu osophic weigh s.
3. Use a selec ion mechanism based on he ce ain y le el while conside ing and
Pe o m op imiza ion i e a ions o ind he op imal ou e unde neu osophic
unce ain y.
S ep 5: De uzzi ica ion and Decision Making
1. Con e he inal neu osophic solu ion in o a c isp alue by applying a
de uzzi ica ion echnique (e.g., sco e unc ion, cen oid me hod).
2. Compa e he esul s wi h classical TSP solu ions o analyze imp o emen s in ou e
e iciency.
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
Neu osophic Se s and Sys ems, Vol. 93, 2025 432
3. E alua e he obus ness o he p oposed model by es ing di e en le els o
unce ain y.
S ep 6: Applica ion and Case S udy Analysis
1. Implemen he p oposed model on eal-wo ld anspo a ion da a.
2. Compa e esul s wi h classical edge connec i i y and TSP solu ions.
3. Valida e e iciency imp o emen s in decision-making unde unce ain y
condi ions.
4. Ma hema ical Example Using he P oposed Algo i hm
4.1 P oblem S a emen :
A logis ics company needs o ind he op imal deli e y ou e among ou ci ies A,B,C,D. The
oads be ween hese ci ies a e ep esen ed as a neu osophic weigh ed g aph G = (V,E). The
objec i e is o de e mine he neu osophic edge connec i i y and sol e he T a elling
Salesman P oblem (TSP) conside ing unce ain y.
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
Neu osophic Se s and Sys ems, Vol. 93, 2025 433
S ep 1: Rep esen a ion o he G aph in Neu osophic En i onmen
We de ine he g aph wi h ou ci ies:
𝑉 = {𝐴,𝐵,𝐶,𝐷}
The oads (edges) be ween he ci ies ha e neu osophic weigh s in he o m:
𝑤𝑖𝑗 = (𝑇𝑖𝑗,𝐼𝑖𝑗,𝐹𝑖𝑗)
whe e:
• 𝑇𝑖𝑗 (T u h): Ce ain y o he edge's exis ence.
• 𝐼𝑖𝑗 (Inde e minacy): Unce ain y due o wea he , a ic, e c.
• 𝐹𝑖𝑗 (Falsi y): P obabili y ha he edge does no exis .
The neu osophic adjacency ma ix o he weigh ed g aph is:
𝐖=[ −−−−− (0.9,0.1,0.0) (0.7,0.2,0.1) (0.8,0.1,0.1)
(0.9,0.1,0.0) −−−−− (0.85,0.1,0.05) (0.75,0.15,0.1)
(0.85,0.1,0.05) (0.7,0.2,0.1) − − − − − (0.9,0.05,0.05)
(0.8,0.1,0.1) (0.75,0.15,0.1) (0.9,0.05,0.05) −−−−− ]
S ep 2: Compu a ion o Neu osophic Edge Connec i i y
S ep 2.1: De uzzi ica ion o Neu osophic Weigh s
To con e he neu osophic weigh s in o c isp alues, we use he de uzzi ica ion o mula:
𝑊′𝑖𝑗 = (𝑇𝑖𝑗 − 𝐼𝑖𝑗 − 𝐹𝑖𝑗)
Applying his o mula:
𝐖′=[− 0.8 0.4 0.6
0.8 − 0.7 0.5
0.4 0.7 − 0.8
0.6 0.5 0.8 −]
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
Neu osophic Se s and Sys ems, Vol. 93, 2025 434
S ep 2.2: Compu e Edge Connec i i y
The edge connec i i y 𝜆𝑁(𝐺) is he minimum numbe o edges ha need o be emo ed o
disconnec he g aph:
1. The smalles weigh is 0.4 (be ween A and C).
2. Remo ing his edge does no disconnec he g aph.
3. Remo ing wo edges: (A, C) and (B, D) (0.4 and 0.5) disconnec s he g aph.
4. Thus, he neu osophic edge connec i i y is 𝜆𝑁(𝐺)= 2.
S ep 3: Fo mula ing he T a elling Salesman P oblem (TSP) unde
Neu osophic Cons ain s
The goal is o minimize he o al cos while conside ing unce ain y. The dis ance ma ix is:
𝐖′=[− 0.8 0.4 0.6
0.8 − 0.7 0.5
0.4 0.7 − 0.8
0.6 0.5 0.8 −]
We need o ind he sho es ou e ha isi s all ci ies exac ly once and e u ns o he s a ing
ci y.
S ep 4: Sol ing Neu osophic TSP using Op imiza ion
Using he Nea es Neighbou Algo i hm (NNA):
1. S a a A.
2. Visi he nea es un isi ed ci y:
• A → C (cos : 0.4)
• C→B (cos : 0.7)
Apa na T ipa hy, Ama esh Chand a Panda, Si a P asad Behe a, P asan a Kuma Rau ,
A Decision -Making Model o he T a elling Salesman P oblem Based on Neu osophic
Edge Connec i i y
