Enginee ing Business Managemen o Decision Making - O iginal Resea ch A icle
In e na ional Jou nal o Enginee ing
Business Managemen
Volume 17: 1–15
© The Au ho (s) 2025
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DOI: 10.1177/18479790251314990
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A uzzy logic app oach o ie s eng h
assessmen in ela ionship managemen :
Design and pe o mance compa ison o wo
implemen ed uzzy-based models
Shunya Higashi
1
, Phudi Amp i i
2
, Mako o Ikeda
2
, Kei a Ma suo
2
, Leona d Ba olli
2
and Fa os Xha a
3
Abs ac
In an inc easingly digi alised and in e connec ed wo ld, assessing he s eng h o in e pe sonal ies in social ne wo ks is
c ucial o fields such as business, ma ke ing and sociology. T adi ional me hods o e alua ing Tie S eng h (TS), which
o en classi y ela ionships as ei he ”s ong”o ”weak”, ail o cap u e he unce ain y and ambigui y o human in e ac ions.
This s udy p oposes a Fuzzy-based Sys em o Assessmen o Tie S eng h (FSATS). We de elop and e alua e wo models:
FSATSM1, which u ilises h ee inpu pa ame e s In e ac ion Time (IT), Le el o In imacy (LoI) and Emo ional In ensi y (EI);
and FSATSM2, whe e we in oduce Recip oci y (Rc) as an addi ional pa ame e . Th ough simula ions, we compa e he
pe o mance o bo h models o he assessmen o TS. The simula ion esul s show ha o FSATSM1, when IT is 0.9 and EI
is 0.7 o all alues o LoI, he TS alues a e mo e han 0.5. While, o FSATSM2, when IT is 0.9, o EI 0.1 (Rc mo e han 0.8),
EI 0.5 (Rc mo e han 0.5) and EI 0.9 (Rc mo e han 0.2), all alues o TS a e mo e han 0.5, indica ing a s ong
ela ionship. The esul s sugges ha FSATSM2 p o ides a mo e accu a e eflec ion o eal-wo ld ela ionships, which can
be applied in con ex s such as Social Cus ome Rela ionship Managemen (SCRM), enabling businesses o enhance
cus ome engagemen s a egies.
Keywo ds
Social ne wo k, uzzy logic, ie s eng h, business ma ke ing s a egies, social cus ome ela ionship managemen
Da e ecei ed: 27 No embe 2024; accep ed: 17 Decembe 2024
In oduc ion
In an inc easingly in e connec ed wo ld, unde s anding he
in e pe sonal ela ionships in social ne wo ks is c ucial. This
a ea o s udy spans mul iple disciplines, including sociology,
economics, and business managemen . The concep o Tie
S eng h (TS) plays a key ole in hese analyses, as i p o ides
insigh s in o he quali y and influence o ela ionships be ween
indi iduals o en i ies wi hin a ne wo k.
1
T adi ionally, TS is
classified in o wo ca ego ies: s ong ies, cha ac e ised by
equen and in ima e in e ac ions, and weak ies, which in-
ol e less equen and mo e supe ficial connec ions.
2
1
G adua e School o Enginee ing, Fukuoka Ins i u e o Technology,
Fukuoka, Japan
2
Depa men o In o ma ion and Communica ion Enginee ing, Fukuoka
Ins i u e o Technology, Fukuoka, Japan
3
Depa men o Compu e Science, Technical Uni e si y o Ca alonia,
Ba celon, Spain
Co esponding au ho :
Leona d Ba olli, Depa men o In o ma ion and Communica ion
Enginee ing, Fukuoka Ins i u e o Technology, 3-30-1 Waji o-Higashi,
Higashi-Ku, Fukuoka 811-0295, Japan.
Email: ba olli@fi .ac.jp
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The bina y classifica ions o en o e simpli y he com-
plex and mul i ace ed na u e o human ela ionships. In
eali y, ela ionships can a y significan ly in e ms o in-
ensi y and closeness. In social ne wo ks, some ela ionships
may no fi in o he ca ego ies o ei he s ong o weak ies
bu ins ead all somewhe e in be ween, exhibi ing cha ac-
e is ics o bo h, as illus a ed in Figu e 1. These ”mode a e
ies”can play unique oles, such as acili a ing he flow o
no el in o ma ion while main aining a le el o us and
eliabili y ha weake ies migh lack.
3
The limi a ions o bina y classifica ions a e pa icula ly
appa en in mode n digi al en i onmen s, whe e he
bounda ies be ween pe sonal and p o essional ela ionships
a e inc easingly blu ed.
4–6
Social media pla o ms acili a e
a wide ange o in e ac ion ypes, om casual likes and
commen s o in-dep h, p i a e con e sa ions, making i
di ficul o flexibly assess TS using adi ional me hods.
Fu he mo e, he ise o emo e wo k and globalisa ion has
added o his complexi y, highligh ing he need o mo e
sophis ica ed ools o e alua e he s eng h o ela ionships
ac oss geog aphical and cul u al bounda ies.
In business and managemen , accu a ely assessing TS is
e y impo an .
7
Fo ins ance, in Social Cus ome Rela-
ionship Managemen (SCRM), unde s anding he a ying
deg ees o connec ion be ween cus ome s and a b and is
c ucial o ailo ing ma ke ing s a egies.
8
Cus ome s wi h
s ong ies o a b and a e mo e likely o exhibi loyal y and
ad ocacy, whe eas hose wi h weake ies may equi e
di e en engagemen s a egies o main ain hei in e es .
Simila ly, wi hin o ganisa ions, iden i ying he TS among
employees is essen ial o designing e ec i e eams, os-
e ing collabo a ion and ensu ing he smoo h flow o
in o ma ion.
The e a e some esea ch wo ks ela ed wi h TS.
Howe e , mos o hem conside some ques ionnai es, da a
se s, models and amewo ks and p esen some analy ical
esul s. Fo ins ance, Pe ikos and Michael p esen a su ey
on TS es ima ion me hods in online social ne wo ks. They
analyze TS and s udy he dimensions o TS. Then, hey
ca y ou a compa a i e s udy o me hodologies o model
TS and examine he key findings.
9
Gup e and Eliassi-Rad
classi y exis ing TS measu es acco ding o he axioms ha
hey sa is y. They show he comple eness and soundness o
he axioms, and p esen Kendall Tau Rank Co ela ion
be ween a ious TS measu es.
10
The ne wo k da a ela ed
o TS was collec ed by su ey me hods wi h ques ion-
nai es, ollowedupby i ual‘ ocus g oup’discussion o
e i y he ques ionnai e esul s. The 12-ques ion su ey
used nomina ion echnique wi h non-specificaided e-
call.
11
U eña-Ca ion e al. sys ema ically s udy how
ea u es o he con ac ime se ies a e ela ed o opological
ea u es usually associa ed wi h TS. They ocus on a la ge
mobile-phone da ase and measu e a numbe o p ope ies
o he con ac ime se ies o each ie and use hese o
p edic he so-called neighbou hood o e lap, which is a
ea u e ela ed o TS.
12
In all hese s udies, he e is no p esen ed o implemen ed
a sys em o measu e o assess he TS. In his s udy, we
conside he applica ion o Fuzzy Logic (FL) as a solu ion.
The FL wi h i s abili y o handle unce ain y and ambigui y,
can o e a flexible amewo k o assessing ela ionships,
enabling he ep esen a ion o TS on a con inuum in eal-
wo ld social ne wo ks. This app oach no only aligns mo e
closely wi h he complex eali y o human ela ionships bu
also enhances he accu acy and applicabili y o TS as-
sessmen s in p ac ical con ex s. By conside ing FL o he
e alua ion o TS, his esea ch aims o de elop a new in-
elligen sys em applicable ac oss a ious domains. In he
analysis o social ne wo ks, he design o ma ke ing
s a egies, o he op imisa ion o o ganisa ional communi-
ca ion, he p oposed sys em can se e as a ool o un-
de s anding he dynamic na u e o ela ionships.
FL has many applica ions ac oss a ious domains. Bose
and Mali emphasize e olu ion o uzzy ime se ies and hei
abili y o add ess challenges associa ed wi h unce ain y in
decision-making con ex s, anging om economics o cli-
ma ology. Also, he in eg a ion o FL wi h o he echniques,
such as in hyb id uzzy sys ems, has shown imp o emen s
in decision-making amewo ks.
13
Fayek explo es ole o
uzzy hyb id echniques in cons uc ion enginee ing and
managemen , highligh ing hei abili y o model complex
ela ionships and p o ide enhanced decision suppo in
unce ain en i onmen s.
14
Beyond hese fields, applica ions o FL ha e been ex-
ended o he heal hca e indus y. Wan and Chin p opose an
IoT-based ca e link sys em u ilising Adap i e Neu o-Fuzzy
In e ence Sys ems (ANFIS) o enhance decision suppo
Figu e 1. TS o e iew.
2In e na ional Jou nal o Enginee ing Business Managemen
unc ionali ies in elde ly ca e managemen , showcasing he
e sa ili y o FL sys ems in dynamic and eal-wo ld sce-
na ios.
15
In business and supply chain managemen , by he
combina ion o uzzy me hods wi h analy ic echniques can
be add essed complex decision-making p oblems. Okwu
and Ta ibu implemen a hyb id model o sus ainable
supplie selec ion, demons a ing he obus ness o FL in
achie ing sus ainabili y- ocused decisions.
16
These s udies collec i ely illus a e he g owing appli-
ca ion o FL in di e se fields, unde sco ing i s adap abili y
and ele ance in sol ing mul i ace ed p oblems.
In his pape , we implemen a Fuzzy-based Sys em o
Assessmen o TS (FSATS). We de eloped wo models:
FSATSM1 and FSATSM2. FSATSM1 conside s h ee inpu
pa ame e s: In e ac ion Time (IT), Le el o In imacy (LoI)
and Emo ional In ensi y (EI), and he ou pu pa ame e is
TS. In FSATSM2, we in oduce Recip oci y (Rc) as a new
inpu pa ame e . We e alua ed he p oposed sys em h ough
a se ies o simula ions and p o ided a compa a i e analysis
o bo h models.
The main con ibu ions o his esea ch wo k a e sum-
ma ized as ollows.
·We in es iga e di e en pa ame e s ha a ec TS.
·We implemen a new in elligen sys em based on FL
o decision o TS.
·We implemen wo models and compa e hei pe -
o mance in o de o show he e ec s o each pa-
ame e on TS.
·We p esen a p ac ical use case in he domain o
SCRM, showcasing he sys em’s u ili y in business
con ex s and p o iding a angible pa hway o in e-
g a ing he model in o eal-wo ld decision-making
p ocesses.
This esea ch dis inguishes i sel om exis ing wo ks by
le e aging he flexibili y o FL o make a be e ep esen-
a ion o TS and cap u ing he nuances o eal-wo ld in-
e pe sonal ela ionships, enabling mo e p ecise and
adap able assessmen s. Fu he mo e, unlike many s udies
ha ocus solely on heo e ical modeling, his esea ch
demons a es a p ac ical use case in he domain o SCRM,
showcasing he sys em’s u ili y in business con ex s and
p o iding a angible pa hway o in eg a ing he model in o
eal-wo ld decision-making p ocesses.
The s uc u e o his pape is as ollows. Nex ,
we in oduce he concep ual amewo k o TS and examines
he limi a ions o adi ional me ics. Then, we ou line he
p inciples o FL u ilised o he p oposed app oach. A e
ha is desc ibed he p oposed FSATS. In ollowing,
we p esen he simula ion esul s. Then is shown a use case
o he applica ion o FSATS in SCRM. Finally,
we conclude he pape wi h a discussion o he findings and
sugges ions o u u e esea ch.
Concep and me ics o TS
Defini ion and impo ance o TS
TS is a undamen al concep in social ne wo k analysis,
e e ing o he s eng h o a connec ion be ween wo in-
di iduals o en i ies wi hin a ne wo k. The concep was fi s
o malised by G ano e e in his seminal wo k,
17
”The
S eng h o Weak Ties”, whe e he emphasised he signifi-
cance o bo h s ong and weak ies in he dissemina ion o
in o ma ion and he main enance o social s uc u es. S ong
ies a e ypically cha ac e ised by equen , close, and
emo ionally in ense in e ac ions, o en in ol ing amily
membe s, close iends, o us ed colleagues. In con as ,
weak ies in ol ing less equen in e ac ion a e mo e casual
and equi e a lowe le el o emo ional in es men such as
wi h acquain ances o dis an colleagues.
The impo ance o unde s anding TS lies in i s abili y o
influence a ious ou comes wi hin social ne wo ks.
18
S ong ies a e essen ial o p o iding social suppo , a-
cili a ing us , and ensu ing he eliabili y o in o ma ion.
Weak ies, on he o he hand, a e c ucial o in oducing
no el in o ma ion and b idging di e en social g oups. In
p o essional con ex s, a balance o s ong and weak ies is
necessa y o os e inno a ion, collabo a ion, and e ec i e
communica ion. The e o e, accu a ely assessing TS is i al
o bo h heo e ical unde s anding and p ac ical applica ions
in fields such as sociology, business managemen and
ma ke ing.
Analysis and measu emen o TS
The accu a e measu emen o TS is a complex bu essen ial
ask in social ne wo k analysis. TS is a mul i ace ed concep
ha encompasses a ious dimensions o ela ionships, in-
cluding in e ac ion equency, emo ional in ensi y, and he
con ex ual se ing o he ela ionship. To cap u e he mul-
i ace ed na u e o TS, esea che s ha e de eloped a a ie y
o measu emen app oaches, each designed o highligh
di e en aspec s o social in e ac ions. In his subsec ion,
we will explo e se e al key app oaches o TS measu emen ,
ocusing on hei unde lying p inciples and applicabili y.
S uc u al analysis o ne wo k nodes. Es ima ing TS based on
node pai in o ma ion is a s aigh o wa d app oach. Ac-
co ding o he ”weak ie hypo hesis”,
17
he local ne wo k
s uc u e su ounding a node pai is closely co ela ed wi h
TS. Es ablished node simila i y indices, such as pa h dis-
ance, he numbe o sha ed common neighbou s and he
p oximi y be ween nodes,
19
a e commonly used o es ima e
link weigh and TS, pa icula ly in link p edic ion asks.
Liben-Nowell and Kleinbe g p o ided a e iew on TS
measu emen , ocusing on he local p oximi y o node
pai s.
20
While node-based me ics a e undamen al and easy
Higashi e al. 3
o apply, inco po a ing addi ional a ibu es in o his ap-
p oach can significan ly imp o e he accu acy o TS
es ima ion.
21
In e ac ion-based analysis. Online social ne wo ks c ea e an
IT-enabled communica ion en i onmen whe e indi iduals
engage and o m social s uc u es like communi ies, soci-
e ies and o ganisa ions, each wi h dis inc in e ac ion pa -
e ns. Communica ion h ough messages, emails, and calls
can be modelled as edge a ibu es. Thus, analysing hese
communica ion pa e ns p o ides aluable insigh s in o
es ima ing TS in such con ex s. Onnela e al. examined TS
by conside ing call du a ion, he cumula i e numbe o calls
be ween indi iduals, and he local opology wi hin mobile
communica ion ne wo ks.
22
O he s udies ha e also ound a
co ela ion be ween communica ion pa e ns and TS. Ad-
di ionally, in eg a ing con en in o ma ion om communi-
ca ion links can enhance he accu acy o his me ic.
23,24
Con en and con ex analysis. The apid expansion o online
pla o ms has significan ly inc eased he c ea ion and
sha ing o con en on social ne wo ks. Use s gene a e a wide
ange o con en , including commen s on social ne wo king
si es like Facebook and Th eads, as well as messages,
pho os, and pos s.
25
This con en , o en e e ed o as
linkage ea u es o in e ac ion con en , adds a schema ic
dimension o he node a ibu es wi hin social ne wo ks. The
in e ac ion con en o indi iduals in online social ne wo ks
has been ecognised as a key ac o in he analysis o TS
es ima ion.
26,27
Mo eo e , he applica ion o a ious ma-
chine lea ning echniques (e.g., mul i a ia e ea u es and
use ac i i y) allows o he es ima ion o ela ionship
s eng h on hese a ibu ed g aphs.
Limi a ions o adi ional app oaches
While adi ional app oaches o measu ing TS ha e been
undamen al o unde s anding social ne wo ks, hey a e no
wi hou limi a ions. One o he p ima y challenges is he
o e simplifica ion o he complexi y o human ela ionships
in hese me hods. This can lead o misin e p e a ions,
pa icula ly in dynamic and di e se social ne wo ks whe e
TS may fluc ua e o e ime o a y ac oss di e en
con ex s.
28
Mo eo e , hese app oaches end o ocus hea ily on
s uc u al and in e ac ional da a, o en neglec ing sub le ies
such as emo ional dep h, cul u al influences, and si ua ional
a ia ions, which a e c ucial in shaping ela ionships. While
con en and con ex analysis a emp s o add ess some o
hese gaps, i s uggles wi h he inhe en complexi y and
unce ain y o human in e ac ions. Machine lea ning
echniques,
29
hough powe ul, a e o en limi ed by he
quali y and quan i y o a ailable da a and may equi e
ex ensi e compu a ional esou ces o p ocess la ge da ase s.
Ano he significan limi a ion is he inabili y o adi-
ional app oaches o handle ambigui y and unce ain y in
ela ionships. Human in e ac ions a e inhe en ly uzzy, wi h
a ying deg ees o emo ional in ensi y, ecip oci y and
commi men ha canno be easily quan ified o ca ego ised
using con en ional me hods. The igid amewo ks o
adi ional app oaches hus all sho in cap u ing he fluid
and e ol ing na u e o social ies.
Ad an ages o FL-based app oach
Gi en he limi a ions o adi ional app oaches, mo e so-
phis ica ed me hods a e needed o measu e TS in a way ha
cap u es he complexi y and unce ain y o human ela-
ionships. FL, wi h i s abili y o handle ambigui y and
ep esen in o ma ion on a con inuum, o e s a powe ul
al e na i e o TS assessmen .
Unlike adi ional models ha label ela ionships as
simply ”s ong”o ”weak”, FL allows o he ep esen a ion
o ela ionships wi h a ying deg ees o s eng h. This
flexibili y is c ucial when dealing wi h social ies. By ap-
plying FL, we can mo e accu a ely model he nuances o
hese ela ionships by conside ing hei inhe en unce -
ain ies and complexi ies.
Beyond imp o ing TS assessmen , FL also enhances
decision-making p ocesses. By p o iding a mo e nuanced
unde s anding o ela ionships, FL allows o be e -
in o med decisions in a ious con ex s, such as cus ome
ela ionship managemen , eam composi ion and a ge ed
ma ke ing s a egies.
Ou line o FL
The FL is a ma hema ical amewo k in oduced by Za-
deh,
30
designed o ex end adi ional logic by accommo-
da ing he concep o pa ial u h. Unlike classical bina y
logic, which igidly classifies s a emen s as en i ely ue o
en i ely alse (1 o 0), FL conside s di e en deg ees o
u h. This flexibili y makes FL pa icula ly use ul o
modelling and managing unce ain y and imp ecision in
complex, eal-wo ld si ua ions whe e dicho omous deci-
sions a e o en insu ficien .
Fundamen al concep s o FL
The ounda ion o FL lies in he concep o uzzy se s, an
ex ension o classical se heo y. In uzzy se s, each elemen
is assigned a membe ship alue anging be ween 0 and 1,
which ep esen s he deg ee o which he elemen belongs o
he se .
30,31
A membe ship alue o 0 means no membe -
ship, while a alue o 1 signifies ull membe ship. These
alues a e de e mined by a membe ship unc ion, which
co ela es inpu da a wi h he co esponding deg ee o
membe ship.
4In e na ional Jou nal o Enginee ing Business Managemen
Fo illus a ion, we conside he concep o ” allness”.In
adi ional bina y logic, an indi idual would be ca ego ised
as ei he ” all”o ”no all”, bu he e a e no in e media e
s a es. Howe e , he FL pe mi s a mo e nuanced assess-
men . A pe son’s heigh could ha e a membe ship alue
wi hin he uzzy se ” all”signi ying how all hey a e
pe cei ed o be, as shown in Figu e 2. Fo ins ance, a heigh
o 180 cm migh co espond o a membe ship alue o 0.7,
sugges ing he pe son is mode a ely all, while a heigh o
190 cm could ha e a membe ship alue o 0.9, indica ing a
highe deg ee o allness.
Membe ship unc ions
Membe ship unc ions a e in eg al o he unc ioning o
FL, as hey de e mine how inpu s a e ansla ed in o deg ees
o membe ship wi hin a uzzy se . Va ious ypes o mem-
be ship unc ions a e commonly employed in uzzy sys-
ems,
32
each sui ed o di e en scena ios:
·T iangula Membe ship Func ion: Cha ac e ised
by a simple iangula shape, his unc ion is defined
by h ee key pa ame e s: he lowe bounda y, he peak
(whe e he membe ship deg ee is 1) and he uppe
bounda y. Due o i s simplici y and ease o use, i is
one o he mos commonly u ilised membe ship
unc ions.
·T apezoidal Membe ship Func ion: Simila o he
iangula unc ion bu wi h a fla op, i is defined by
ou pa ame e s: he lowe limi , he s a o he peak,
he end o he peak and he uppe limi . This unc ion
is o en used when he e is a ange o alues ha a e
equally ep esen a i e o ull membe ship.
·Gaussian Membe ship Func ion: Rep esen ed by a
bell-shaped cu e, he Gaussian membe ship unc ion
is cha ac e ised by wo pa ame e s: he mean and he
s anda d de ia ion. I is o en employed o model
na u al phenomena whe e he ansi ion be ween low
and high membe ship is g adual and smoo h.
The selec ion o an app op ia e membe ship unc ion
depends on he specific na u e o he da a and he con ex o
he p oblem. Each unc ion o e s unique benefi s ega ding
simplici y, accu acy, and in e p e abili y, making i c ucial
o choose he one ha bes aligns wi h he equi emen s o
he applica ion.
Fuzzy ules and in e ence
The FL sys ems ope a e based on a se o uzzy ules, which
a e essen ially ”i - hen”s a emen s ha desc ibe he ela-
ionships be ween inpu a iables and he ou pu a iable.
These ules a e designed o encapsula e expe knowledge
o empi ical obse a ions in a o m ha can be p ocessed by
a FL sys em.
33
A ypical uzzy ule migh ake he ollowing
o m: IF p emise (an eceden ), THEN conclusion
(consequen ).
Fo example, a ule migh in ol e wo inpu a iables,
such as In e ac ion Time and Emo ional In ensi y, o in e an
ou pu a iable, like TS. The inpu a iables a e fi s p o-
cessed h ough hei espec i e membe ship unc ions o
de e mine hei deg ees o membe ship in ele an uzzy
se s. These membe ship alues a e hen combined using
uzzy ope a o s (e.g., AND, OR, NOT) o p oduce a uzzy
ou pu .
The p ocess o de i ing a uzzy ou pu based on a se o
uzzy ules is called uzzy in e ence. Fuzzy Logic Con-
olle s (FLC) in e p e s he inpu a iable alues and apply
uzzy in e ence o gene a e an ou pu . The FLC ypically
consis s o ou main componen s, as shown in Figu e 3.
·Fuzzifie : The Fuzzifie con e s c isp inpu alues
in o uzzy se s by de e mining he deg ee o which
each inpu alue belongs o he ele an uzzy se s.
This is done using p edefined membe ship unc ions.
·In e ence Engine: The In e ence Engine is he
decision-making co e o he FLC. I applies uzzy
ules o he uzzified inpu s o p oduce uzzy ou pu s.
The In e ence Engine uses logical ope a o s (e.g.,
AND, OR) o combine he inpu uzzy se s acco ding
o he ules s o ed in he sys em’s ule base.
·Fuzzy Rules: The Fuzzy Rules a e he i - hen
s a emen s ha define how inpu a iables ela e o
he ou pu a iable. The uzzy ules cap u e expe
knowledge o empi ical ela ionships, which he
sys em models. These ules a e s o ed in a Fuzzy Rule
Figu e 2. Boolean logic and FL. Figu e 3. FLC s uc u e.
Higashi e al. 5
Base (FRB), which he In e ence Engine e e ences
du ing compu a ions.
·De uzzifie : The De uzzifie is he final s ep in he
FLC p ocess, whe e he uzzy ou pu gene a ed by he
In e ence Engine is con e ed back in o a c isp alue.
Al hough he in e nal wo kings o he FLC in ol e
uzzy se s, he ou pu o en needs o be a specific,
ac ionable alue. Common de uzzifica ion me hods
include he Cen oid me hod, which compu es he
cen e o g a i y o he uzzy se , and he Maximum
Membe ship me hod, which selec s he ou pu co -
esponding o he highes membe ship alue.
O e all, FL p o ides a obus amewo k o handling
he complexi y and ambigui y inhe en in eal-wo ld da a. In
he con ex o TS measu emen , FL allows o a mo e ac-
cu a e and flexible assessmen , pa ing he way o mo e
in o med decision-making in social ne wo k analysis and
o he domains.
P oposed uzzy-based sys em
In his sec ion, we p esen ou p oposed uzzy-based sys-
em, e e ed o as he Fuzzy-based Sys em o Assessmen
o TS (FSATS). The p ima y objec i e o FSATS is o
p o ide a flexible and accu a e ool o e alua ing TS in
a ious con ex s, enabling mo e e ec i e decision-making.
The s uc u e o FSATS is illus a ed in Figu e 4. We im-
plemen wo models: FSATSM1 and FSATSM2.
FSATSM1 conside s h ee inpu pa ame e s: In e ac ion
Time (IT), Le el o In imacy (LoI) and Emo ional In ensi y
(EI), and he ou pu pa ame e is TS. In FSATSM2, he
Recip oci y (Rc) is in oduced as a new addi ional pa-
ame e . The ITcap u es in e ac ion equency and du a ion,
EI eflec s emo ional dep h, and LoI ep esen s closeness,
o ming a obus baseline o assessing TS. Rc is excluded
om FSATSM1 o simpli y he model and ocus on hese
co e aspec s. Howe e , in FSATSM2, Rc, which ep esen s
mu ual exchanges in a ela ionship, is included o enhance
he model’s abili y o p o ide a mo e comp ehensi e and
nuanced assessmen o TS, especially in scena ios whe e Rc
plays a c ucial ole in s eng hening in e pe sonal
ela ionships.
9,34
The conside ed pa ame e s a e explained in ollowing:
In e ac ion Time (IT)
IT e e s o he o al du a ion and equency o in e ac ions
be ween wo indi iduals o en i ies wi hin a gi en pe iod. I
is a key ac o in de e mining he s eng h o a ela ionship,
as equen and p olonged in e ac ions o en indica e a
close and mo e obus connec ion. Highe alues o IT
sugges a s onge ie, as hey eflec mo e oppo uni ies o
ela ionship building and mu ual unde s anding.
12
Le el o In imacy (LoI)
LoI measu es he deg ee o pe sonal closeness and emo-
ional dep h p esen in a ela ionship. I eflec s how
com o able indi iduals a e wi h sha ing pe sonal in o -
ma ion, discussing sensi i e opics, o p o iding emo ional
suppo . High le els o in imacy ypically indica e a s ong
ie, as hey sugges a deep, us -based ela ionship.
35
Emo ional In ensi y (EI)
EI e e s o he s eng h o emo ions expe ienced and ex-
p essed du ing in e ac ions be ween indi iduals. I cap u es
he emo ional in es men in he ela ionship, including
eelings o a ec ion, conce n, o e en conflic . High EI is
o en associa ed wi h s ong ies, as in ense emo ions can
signi y a deep connec ion and a high le el o engagemen .
36
Recip oci y (Rc)
Rc measu es he balance and mu ual exchange o esou ces,
suppo and ac ions wi hin a ela ionship. I eflec s he
ex en o which bo h pa ies con ibu e equally o he e-
la ionship, whe he h ough ime, e o , emo ional suppo ,
o o he o ms o exchange. High le els o ecip oci y a e
indica i e o a s ong ie, as hey demons a e a commi men
o main aining a balanced and mu ually beneficial
ela ionship.
37
Tie S eng h (TS)
TS is he ou pu pa ame e in bo h FSATSM1 and
FSATSM2, ep esen ing he o e all s eng h o he ela-
ionship be ween wo indi iduals o en i ies. TS is de i ed
om he combina ion o he inpu pa ame e s—IT, LoI, EI
and, in he case o FSATSM2, Rc. A highe TS alue in-
dica es a s onge , mo e esilien connec ion, while a lowe
Figu e 4. P oposed sys em s uc u e.
6In e na ional Jou nal o Enginee ing Business Managemen
TS alue sugges s a weake o mo e enuous
ela ionship. The assessmen o TS is c ucial o unde -
s anding he dynamics o social ne wo ks and hei impac
on in o ma ion flow, collabo a ion and o e all ne wo k
cohesion.
The pa ame e alues (IT, EI, LoI, and Rc) we e de-
e mined h ough a comp ehensi e su ey p ocess o el-
e an li e a u e, analysis o expe imen al da a, and
consul a ion wi h domain expe s. This app oach ensu es an
accu a e and obus ep esen a ion o TS in social
ne wo ks.
35,38–44
The membe ship unc ions o FSATS a e shown in
Figu e 5. They a e designed o bo h FSATSM1 and
FSATSM2 models. To acili a e he uzzifica ion p ocess
and ensu e flexibili y in applying he p oposed sys em
ac oss a ious scena ios, he alues o IT, LoI, EI and Rc
a e s anda dised be ween 0 and 100%. This s anda disa ion
allows he FSATS o be easily adap ed o di e en con ex s
and scena ios. Fo example, in a co po a e communica ion
scena io, he maximum alue o IT migh be se a 40 hou s
pe week (100%). Howe e , in mo e ime-sensi i e sce-
na ios, such as eme gency esponse o c i ical communi-
ca ions, he maximum alue o IT migh be se a 10 hou s
pe week o less, depending on he specific equi emen s o
esponsi eness and communica ion equency.
Figu e 6 illus a es he iangula and apezoidal
membe ship unc ions employed o he inpu pa ame e s.
These unc ions a e chosen o hei simplici y and
Figu e 5. Membe ship unc ions. (a) In e ac ion Time. (b) Le el o In imacy. (c) Emo ional In ensi y. (d) Recip oci y. (e) Tie S eng h
(FSATSM1). ( ) Tie S eng h (FSATSM2).
Higashi e al. 7
e ec i eness in modelling he g adual ansi ions be ween
di e en le els o each pa ame e . The specific e m se s o
each inpu pa ame e a e summa ised in Table 1.
TðITÞ¼ Sho ðShÞ,Medium ðMeÞ,Long ðLoÞg
TðLoIÞ¼ Ve y low ðVlÞ,Low ðLwÞ,Medium ðMdÞ,
High ðHgÞ,Ve y high ðVhÞg
TðEIÞ¼ Low ðLÞ,Medium ðMÞ,High ðHÞg
TðRcÞ¼ Weak ðWeÞ,Mode a e ðMoÞ,S ong ðS Þg
TðTSÞ¼ TS1, TS2, TS3, TS4, TS5, TS6, TS7, TS8, TS9g
The ma hema ical defini ions o he membe ship unc-
ions o he inpu pa ame e s a e shown in he ollowing
equa ions.
μShðITÞ¼gðIT;Sh0,Sh1,Shw0,Shw1Þ
μMeðITÞ¼ ðIT;Me0,Mew0,Mew1Þ
μLoðITÞ¼gðIT;Lo0,Lo1,Low0,Low1Þ
μVlðLoIÞ¼gðLoI;Vl0,Vl1,Vlw0,Vlw1Þ
μLwðLoIÞ¼ ðLoI;Lw0,Lww0,Lww1Þ
μMdðLoIÞ¼ ðLoI;Md0,Mdw0,Mdw1Þ
μHgðLoIÞ¼ ðLoI;Hg0,Hgw0,Hgw1Þ
μVhðLoIÞ¼gðLoI;Vh0,Vh1,Vhw0,Vhw1Þ
μLðEIÞ¼gðEI;L0,L1,Lw0,Lw1Þ
μMðEIÞ¼ ðEI;M0,Mw0,Mw1Þ
μHðEIÞ¼gðEI;H0,H1,Hw0,Hw1Þ
μWeðRcÞ¼gðRc;We0,We1,Wew0,Wew1Þ
μMoðRcÞ¼ ðRc;Mo0,Mow0,Mow1Þ
μS ðRcÞ¼gðRc;S 0,S 1,S w0,S w1Þ
The TS is he ou pu pa ame e o bo h models and i s
e m se is shown in Table 1.
TS ¼
Tie S eng h L1
Tie S eng h L2
Tie S eng h L3
Tie S eng h L4
Tie S eng h L5
Tie S eng h L6
Tie S eng h L7
Tie S eng h L8
Tie S eng h L9
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
¼
TS1
TS2
TS3
TS4
TS5
TS6
TS7
TS8
TS9
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
We define he membe ship unc ions o TS o
FSATSM1 as ollows.
μTS1ðTSÞ¼gðTS;TS10,TS11,TS1w0,TS1w1Þ
μTS2ðTSÞ¼ ðTS;TS20,TS2w0,WT2w1Þ
μTS3ðTSÞ¼ ðTS;TS30,TS3w0,TS3w1Þ
μTS4ðTSÞ¼ ðTS;TS40,TS4w0,TS4w1Þ
μTS5ðTSÞ¼ ðTS;TS50,TS5w0,TS5w1Þ
μTS6ðTSÞ¼ ðTS;TS60,TS6w0,TS6w1Þ
μTS7ðTSÞ¼gðTS;TS70,TS71,TS7w0,TS7w1Þ
While, he membe ship unc ions o TS o
FSATSM2 a e defined as ollows.
μTS1ðTSÞ¼gðTS;TS10,TS11,TS1w0,TS1w1Þ
μTS2ðTSÞ¼ ðTS;TS20,TS2w0,WT2w1Þ
μTS3ðTSÞ¼ ðTS;TS30,TS3w0,TS3w1Þ
μTS4ðTSÞ¼ ðTS;TS40,TS4w0,TS4w1Þ
μTS5ðTSÞ¼ ðTS;TS50,TS5w0,TS5w1Þ
μTS6ðTSÞ¼ ðTS;TS60,TS6w0,TS6w1Þ
μTS7ðTSÞ¼ ðTS;TS70,TS7w0,TS7w1Þ
μTS8ðTSÞ¼ ðTS;TS80,TS8w0,TS8w1Þ
μTS9ðTSÞ¼gðTS;TS90,TS91,TS9w0,TS9w1Þ
Figu e 6. T iangula and apezoidal membe ship unc ions.
Table 1. Pa ame e s and hei e m se s.
Pa ame e s Te m se s
In e ac ion ime (IT) Sho (Sh), medium (Me), long (Lo)
Le el o in imacy (LoI) Ve y low (Vl), low (Lw), medium (Md), high (Hg), e y high (Vh)
Emo ional in ensi y (EI) Low (L), medium (M), high (H)
Recip oci y (Rc) Weak (We), mode a e (Mo), s ong (S )
Tie s eng h (TS) TS1, TS2, TS3, TS4, TS5, TS6, TS7, TS8, TS9
8In e na ional Jou nal o Enginee ing Business Managemen
The FRB o FSATSM1 and FSATSM2 is shown in
Tables 2 and 3, espec i ely. The FRB is o med by a uzzy
se o dimensions (|T(TS)| = |T(IT)| × |T(LoI)| × |T(EI)| × |
T(Rc)|), whe e |T(x)| is he numbe o e ms on T(x). The
con ol ules ha e he o m: IF ”condi ion”THEN ”con ol
ac ion”. Fo example, o Rule 2 o FSATSM2: ”IF IT is
Sho , EI is Low, LoI is Ve y low and Rc is Mode a e,
THEN TS is TS1”. The FRB allows he FSATS o e alua e
he TS based on a combina ion o inpu condi ions, which
a e adap able assessmen s o a ious applica ion scena ios.
We uned he membe ship unc ions o bo h models by
many simula ions based on ou expe ience and he ela ed
wo k on TS. When we wan a s ong e ec o he pa ame e
o a peak alue, we use he iangula membe ship unc ion,
while in cases when we wan he s ong e ec o he pa-
ame e in a egion, we use he apezoidal membe ship
unc ion.
Simula ion esul s
In his sec ion, we p esen he simula ion esul s. The simu-
la ions a e pe o med on a compu e unning Linux Ubun u
OS wi h he ollowing specifica ions: 8 GB o RAM, an i5
(3.2 GHz x imes 4) p ocesso , and an SSD (650 GB). Fo
simula ions, we used ou de eloped FuzzyC sys em.
45
Simula ion esul s o FSATSM1
The simula ion esul s o FSATSM1 a e shown in Figu e 7.
They show he ela ion be ween TS and EI o a ious LoI
alues while conside ing IT as a cons an pa ame e .
In Figu e 7(a), we conside he IT alue 0.1. When LoI is
fixed a 0.7, inc easing EI om 0.1 o 0.5 esul s in a 13%
inc ease in TS. Fu he inc easing EI om 0.5 o 0.9 leads o an
addi ional 24% inc ease in TS. These esul s indica e ha as EI
inc eases, TS also inc eases, demons a ing he significan
impac o emo ional in ensi y on he TS. This is pa icula ly
ele an in si ua ions whe e decisions a e based on he
emo ional engagemen o he pa ies in ol ed, such as in
cus ome suppo scena ios, whe e s onge emo ional con-
nec ions can lead o highe cus ome sa is ac ion and loyal y.
We compa e Figu e 7(b) wi h Figu e 7(a) o de e mine
how IT has a ec ed TS. We change he IT alue om 0.1 o
0.5. The TS is inc easing by 23% when he LoI alue is
0.7 and he EI is 0.5. This compa ison highligh s he in-
fluence o IT on TS; as in e ac ion ime inc eases, he TS
becomes s onge , e en when o he ac o s such as LoI and
EI emain cons an . This is c i ical in decision-making
p ocesses ela ed o esou ce alloca ion in eam-based
p ojec s, whe e inc eased in e ac ion ime can enhance
collabo a ion and p ojec ou comes.
In Figu e 7(c), we inc ease he alue o IT o 0.9. We see
ha he TS alues ha e g own significan ly mo e han he
esul s o Figu es 7(a) and (b). This significan g ow h in TS
sugges s ha when in e ac ion ime is high, he e ec s o
LoI and EI a e amplified, esul ing in much s onge ies.
This is pa icula ly impo an in s a egic pa ne ships o
long- e m business ela ionships, whe e sus ained in e ac-
ion is essen ial o main aining s ong and p oduc i e ies.
Table 2. FRB o FSATSM1.
Rule IT LoI EI TS
1 Sh Vl L TS1
2 Sh Vl M TS1
3 Sh Vl H TS2
4 Sh Lw L TS1
5 Sh Lw M TS1
6 Sh Lw H TS3
7 Sh Md L TS1
8 Sh Md M TS2
9 Sh Md H TS4
10 Sh Hg L TS2
11 Sh Hg M TS3
12 Sh Hg H TS5
13 Sh Vh L TS3
14 Sh Vh M TS4
15 Sh Vh H TS6
16 Me Vl L TS1
17 Me Vl M TS2
18 Me Vl H TS4
19 Me Lw L TS2
20 Me Lw M TS3
21 Me Lw H TS5
22 Me Md L TS3
23 Me Md M TS4
24 Me Md H TS6
25 Me Hg L TS4
26 Me Hg M TS5
27 Me Hg H TS7
28 Me Vh L TS5
29 Me Vh M TS6
30 Me Vh H TS7
31 Lo Vl L TS2
32 Lo Vl M TS3
33 Lo Vl H TS5
34 Lo Lw L TS3
35 Lo Lw M TS4
36 Lo Lw H TS6
37 Lo Md L TS4
38 Lo Md M TS5
39 Lo Md H TS7
40 Lo Hg L TS5
41 Lo Hg M TS6
42 Lo Hg H TS7
43 Lo Vh L TS6
44 Lo Vh M TS7
45 Lo Vh H TS7
Higashi e al. 9
