NoK' l - HOI/AND
Linguis ic Measu es
Based on Fuzzy
Coincidence o Reaching
Consensus in G oup
Decision Making
F. He e a, E. He e a-Viedma,
and J. L. Ve degay
Depa men o Compu e Science and A i icial In elligence,
Uni e si y o G anada, Spain
ABSTRACT
Assuming a linguis ic amewo k, a model o he consensus eaching p oblem in
he e ogeneous g oup decision making is p oposed. This model con ains wo ypes o
linguis ic consensus measu es:
linguis ic consensus deg ees
and
linguis ic p oximi ies
o guide he consensus eaching p ocess. These measu es e alua e he cu en consensus
s a e on h ee le els o ac ion:
le el o he pai s o al e na i es, le el o he
al e na i es,
and
le el o he ela ion. They
a e based on a uzzy cha ac e iza ion o
he concep o coincidence, and hey a e ob ained by means o se e al conjunc ion
unc ions o handling linguis ic weigh ed in o ma ion, he LOWA ope a o o agg ega -
ing linguis ic in o ma ion, and linguis ic quan i ie s ep esen ing he concep o uzzy
majo i y. © 1997 Else ie Science Inc.
KEYWORDS:
Linguis ic modeling, g oup decision making, linguis ic
p e e -
ence ela ions, consensus
deg ees.
1. INTRODUCTION
Consensus
o
syn hesis
consis s in combining a da a se p o ided by
di e en in o ma ion sou ces wi h a iew o ob aining mo e elabo a e
Add ess co espondence o F. He e a, Depa men o Compu e Science and A.I., ETS de
Ingenieda In o m~ ica, Uni e si y o G anada, 18071 G anada, Spain.
Recei ed May 1, 1996; accep ed Oc obe 1, 1996.
In e na ional Jou nal o App oxima e Reasoning 1997; 16:309-334
@ 1997 Else ie Science Inc. 0888-613X/97/$17.00
655 A enue o he Ame icas, New Yo k, NY 10010 PII S0888-613X(96)00121-1
310 F. He e a e al.
in o ma ion [31, 32]. When he in o ma ion sou ces p o ide imp ecise
in o ma ion, he use o uzzy
se heo y
o deal wi h his ype o in o ma-
ion is mos ad isable. A usual si ua ion, in he eal wo ld, which p esen s
he app op ia e cha ac e is ics o apply consensus heo y and uzzy se
heo y oge he , is he
g oup decision making
(GDM) si ua ion.
In a classical GDM si ua ion he e is a p oblem o sol e, a se o
possible solu ion al e na i es, and a g oup o wo o mo e expe s, who
exp ess hei opinions abou he se o solu ion al e na i es and a emp o
each a collec i e decision wi h he maximum possible consensus on his
ques ion: wha is/a e he bes solu ion al e na i e(s) o he p oblem?.
Many pape s on consensus heo y applied o GDM make use o A ow's
wo k [1] as a s a ing poin and a basic guide. A ow p oposed a quali a i e
se ing composed by a se o axioms, which any accep able consensus ool
o GDM should sa is y.
A ow's impossibili y heo em
was an impo an
esul he eo . Acco ding o his heo em, i is impossible o agg ega e
indi idual p e e ences in o g oup p e e ence in a comple ely a ional way.
This is a p oblem ha disappea s in a ca dinal se ing in a uzzy con ex ,
on in oducing p e e ence in ensi ies, which p o ide addi ional deg ees o
eedom o any agg ega ion model [13, 9].
In a uzzy con ex , he applica ion o consensus heo y o GDM p ob-
lems p esen s wo ways o ela e o di e en decision schema a [6]. The
i s way, called
algeb aic consensus,
consis s in es ablishing a
g oup choice
p ocess
which ob ains a decision scheme as a solu ion o he GDM
p oblem. The second way, called
opologic consensus,
consis s in es ablish-
ing a
g oup consensus eaching p ocess,
which, guided by means o a
measu e o closeness among di e en decision schema a, called he
con-
sensus measu e,
a emp s o achie e he maximum possible deg ee o
consensus on solu ion al e na i e(s). Bo h consensus ypes may be com-
bined in a esolu ion scheme (see Figu e 1). Gi en ha he se o expe s
ini ially ha e di e ging opinions, i s ly, opologic consensus is applied, and
in each s ep, he deg ee o exis ing consensus among expe s' opinions is
measu ed. I he mode a o hinks ha he consensus deg ee is sa is ac-
o y, hen algeb aic consensus is applied in o de o ob ain a solu ion;
o he wise, he expe s a e pe suaded o upda e hei opinions. In his way,
a GDM p ocess may be de ined as a dynamic and i e a i e p ocess, in
which he expe s, ia he exchange o in o ma ion and a ional a gumen s,
ag ee o upda e hei opinions un il hey become su icien ly simila , and
hen he solu ion al e na i e(s) is/a e ob ained. He e, we shall ocus ou
esea ch on he opologic consensus.
As was men ioned ea lie , he opologic consensus is guided by means o
a consensus measu e. Assuming nume ical p e e ence ela ions o p o id-
ing he expe s' opinions, se e al au ho s in oduced
ha d consensus mea-
su es
a ying be ween 0 (no consensus o pa ial ag eemen ) and 1 ( ull
Consensus in G oup Decision Making 311
QUESTION
SET OF
ALTERNATNE$
TOPOLOGIC CONSENSUS
GROUP
OF
E)~PERTS
ecommendQ ions
consensus
1 opinions
I MODERAIOR [
COI~nsus
measu 'e
oplnions 1
ALGEBRAIC
CO NSENSUS
Figu e 1. G oup decision making p ocess.
CONSENSUS
$OLUT~N
consensus o comple e ag eemen ) [2, 3, 28, 29]. Howe e , consensus as a
ull and unanimous ag eemen is a om being achie ed in eal si ua ions,
and e en i i is, in such a si ua ion, he consensus eaching p ocess could
be unaccep ably cos ly. So, in p ac ice, a mo e ealis ic app oach is o use
"so e consensus measu es" [24], which assess he consensus deg ee in a
mo e lexible way, and he e o e e lec he la ge spec um o possible
pa ial ag eemen s, and guide opologic consensus un il widesp ead ag ee-
men (no always ull) is achie ed among expe s.
Along his line o easoning, bu in di e en uzzy GDM con ex s,
se e al al e na i e consensus measu es ha e been p oposed: in a nume i-
cal con ex , i.e., wi h nume ical assessmen s on he uni in e al [0, 1], by
Kacp zyk [24], Kacp zyk and Fed izzi [25, 26], and Fed izzi, Kacp zyk, and
Nu mi [15]; and in a linguis ic con ex , i.e., wi h linguis ic assessmen s on a
p ees ablished label se S, by Fed izzi and Mich [14], Mich, Gaio, and
Fed izzi [27], He e a, He e a-Viedma, and Ve degay [21, 23], and Bo -
dogna, Fed izzi, and Pasi [5]. In all hese cases, he au ho s ha e based
hei consensus measu es on he concep o
coincidence,
i.e., obse ing he
exis ing coincidence among expe s' opinions. Di e en coincidence mean-
ings ha e been conside ed, some based on
s ic coincidence,
i.e., accep ing
only he o al coincidence o null coincidence cases [24-26, 15, 27, 21, 23],
and o he s on
less s ic coincidence,
i.e., accep ing di e en coincidence
deg ees [24-26, 15, 14, 5].
He e, in a linguis ic con ex , we p opose o use a mo e lexible idea o
he concep o coincidence, i.e., using i as a uzzy concep . We p esen
uzzy coincidence
as a uzzy se de ined on he se o expe pai s and
cha ac e ized by closeness obse ed among hei espec i e opinions. In
pa icula , we assume a he e ogeneous linguis ic con ex o in oduce he
new uzzy coincidence concep , i.e., we de ine he g ada ion o he coinci-
dence deg ee exis ing among wo expe s om a label se , S, used o
312 F. He e a e al.
exp ess he expe s' opinions, o a new and mo e app op ia e (p ees ab-
lished) label se , in o de o exp ess he coincidence deg ees, G. In his
way, we p esen a ious ways o measu e he closeness obse ed among
expe s' opinions. Mo eo e , we s udy he uzzy coincidence among expe s
on h ee le els o ac ion: he
le el o he pai s o al e na i es,
he
le el o he
al e na i es,
and he
le el o he ela ion.
Then, using his new idea o uzzy
coincidence, we u he ad ance ou p e ious consensus models [21, 23] o
de i ing some new so e linguis ic consensus measu es, which a e applied
on he h ee coincidence le els. All consensus measu es a e ob ained using
di e en
conjunc ion unc ions
o manipula e weigh ed linguis ic in o ma-
ion [16], he
linguis ic o de ed weigh ing a e aging
(LOWA) ope a o [18,
22] o agg ega e linguis ic in o ma ion, and he
linguis ic quan i ie s
[42]
ep esen ing he uzzy majo i y concep .
In o de o do so, in he nex sec ion we p esen some p io conside a-
ions on some consensus measu es p oposed in he li e a u e wi h a iew
o cla i ying he con ibu ions in his pape . In Sec ion 3, we p esen b ie ly
he linguis ic se ing o he GDM p oblem conside ed. In Sec ion 4, we
p esen he new linguis ic consensus measu es. In Sec ion 5, 6, and 7, we
show he de i a ion model o he consensus measu es, and inally, some
conclusions a e poin ed ou .
2. BACKGROUND ON CONSENSUS MEASURES
As we said a he beginning, in a uzzy con ex , se e al al e na i e so e
consensus measu es ha e been p oposed. In his sec ion, we b ie ly analyze
hese measu es wi h a iew o be e cla i ying he new de elopmen s
p oposed in his pape .
In a nume ical con ex , Kacp zyk [24] p esen ed h ee nume ical consen-
sus measu es, which a e:
• assessed on uni in e al, [0, 1];
• de eloped in a simple GDM con ex wi h a homogeneous g oup o
expe s (all expe s' opinions ha e he same impo ance deg ee) and a
homogeneous se o al e na i es (all he al e na i es ha e he same
ele ance deg ee);
• calcula ed ac oss he global se o he al e na i es in a hie a chical
pooling p ocess om he expe s' opinions, p o ided by means o he
nume ical p e e ence ela ions, and using he uzzy majo i y concep
ep esen ed by a linguis ic quan i ie [42]; and inally
• ob ained: (1) he i s measu e, using a s ic idea o he concep o
coincidence, ha is, es ablishing a pa icula pai o al e na i es: i he
opinions o wo expe s a e equal hen hey a e in ag eemen ( alue
Consensus in G oup Decision Making 313
1), and o he wise hey a e in disag eemen ( alue 0); (2) he second
one, using a less s ic idea o he concep o coincidence, ha is,
es ablishing a pa icula al e na i e pai : i he opinions o wo expe s
a e mo e o less equal acco ding o a deg ee a (p ees ablished), hen
hey a e in ag eemen ( alue 1), and o he wise, hey a e in disag ee-
men ( alue 0); and (3) he hi d one, using ano he less s ic idea o
he concep o coincidence ep esen ed by a unc ion, s : [0, 1] ~ [0, 1]
de ined on he closeness be ween expe s' opinions.
Kacp zyk and Fed izzi [25, 26] ex ended Kacp zyk's measu es o GDM
con ex s wi h a he e ogeneous se o al e na i es and a he e ogeneous
g oup o expe s, espec i ely. Fed izzi, Kacp zyk, and Nu mi [15] modi ied
he de ini ion o Kacp zyk and Fed izzi's measu es and calcula ed hem
using he
o de ed weigh ed a e aging
(OWA) ope a o [34].
On he o he hand, in a linguis ic con ex , Fed izzi and Mich [14]
p esen ed a new nume ical consensus measu e, which is:
• de eloped in a homogeneous GDM con ex wi h mul iple c i e ia;
• calcula ed o each al e na i e, independen ly, om he expe s' opin-
ions p o ided by linguis ic labels (no p e e ence ela ions) by means
o compu a ion on a uzzy ep esen a ion o linguis ic labels
( apezoidal membe ship unc ions); and
• ob ained using a less s ic coincidence concep ep esen ed by means
o a euclidean dis ance d, which implemen s he linguis ic app oxima-
ion [30].
Mich, Gaio, and Fed izzi [27] modi ied his measu e and ob ained i by
applying a s ic coincidence concep , which di ided he expe g oup in o
subse s acco ding o hei e alua ions. He e a, He e a-Viedma, and
Ve degay [21] p esen ed wo ypes o linguis ic consensus measu es, one o
measu e he consensus deg ee and ano he o measu e he closeness
be ween expe s opinions. Bo h a e:
• assessed on he same label se , S, used o exp ess expe s' opinions;
• de eloped in a GDM con ex wi h a he e ogeneous g oup o expe s
and a he e ogeneous se o he al e na i es wi h impo ance and
ele ance deg ees assessed on [0, 1];
• calcula ed om he expe s' opinions, p o ided by linguis ic p e e -
ence ela ions, using linguis ic quan i ie s and a linguis ic agg ega ion
ope a o by di ec compu a ion on he labels ( he LOWA ope a o
[18, 22]) on h ee le els o ac ion: p e e ence on he pai s o al e na-
i es, p e e ence on he indi idual al e na i es, and p e e ence on he
global se o he al e na i es; and
• ob ained by applying a s ic coincidence concep , simila o Mich,
Gaio, and Fed izzi's concep , bu acco ding o an a e age consensus
policy, ha is, using e e y subse o expe s wi h o e wo expe s.

314
F. He e a e al.
In hei second pape [23], He e a, He e a-Viedma, and Ve degay
modi ied hei measu es o wo k in a GDM con ex wi h he e ogeneous
g oups o expe s wi h impo ance deg ees assessed on S, and homoge-
neous se s o al e na i es. The consensus measu es we e ob ained acco d-
ing o a s ic coincidence concep bu by means o a s ic consensus
policy, ha is, conside ing only he subse o expe s wi h maximum
ca dinali y. This consensus model inco po a ed a new de elopmen : i
in eg a ed wo ypes o
linguis ic a ionali y measu es
o achie e less dis-
o ed consensus solu ions. Finally, Bo dogna, Fed izzi, and Pasi [5] p e-
sen ed a linguis ic consensus measu e, which is:
• assessed on he same label se , S, used o exp ess he expe s'
opinions;
• de eloped in a linguis ic GDM con ex , simila o He e a, He e a-
Viedma, and Ve degay's, bu wi h a he e ogeneous se o c i e ia and
ha ing linguis ic impo ance deg ees assessed on S;
• calcula ed o each al e na i e independen ly, om he expe s' opin-
ions, p o ided by linguis ic labels, by means o he linguis ic e sion o
he OWA ope a o [35] and conside ing linguis ic quan i ie s; and
• ob ained using a less s ic coincidence concep ep esen ed by means
o a usual dis ance unc ion d de ined di ec ly on S and p oposed
ini ially by He e a, He e a-Viedma, and Ve degay [21].
Now, we p esen a consensus model wi h a s uc u e simila o [21, 23],
i.e., wi h wo ypes o linguis ic consensus measu es calcula ed on h ee
le els o ac ion, bu wi h he ollowing peculia i ies:
• i is designed o GDM si ua ions wi h he e ogenous g oups o expe s
and he e ogeneous se s o al e na i es 'using linguis ic weigh ing de-
g ees;
• i is de eloped in a he e ogeneous linguis ic con ex , i.e., using di e -
en linguis ic domains o exp ess he opinions, he impo ance, and
ele ance deg ees, as well as he consensus measu es;
• i s consensus measu es a e ob ained using a uzzy o mula ion o he
concep o coincidence.
3. LINGUISTIC SETI'ING OF THE GDM PROBLEM
As was men ioned ea lie , we assume a GDM p oblem de eloped in a
linguis ic con ex , i.e., he expe s use linguis ic e ms ins ead o nume ical
alues o exp ess hei p e e ences [10, 12, 18, 22, 30, 35, 40]. We conside
ini e and o ally o de ed e m se s on [0, 1], S =
{si},
i ~ H = {0 ..... T},
wi h an odd ca dinal, in which he middle label ep esen s an unce ain y
o "app oxima ely 0.5" and he emaining e ms a e placed a ound i
symme ically, as in [4]. Mo eo e , he e m se mus ha e he ollowing
Consensus in G oup Decision Making
315
cha ac e is ics:
1. The se is o de ed: s i >_ sj i i >_ j.
2. The e is he nega i e ope a o : Neg(s i) -- sj such ha j -- T - i.
3. Maximiza ion ope a o : Max(s i, sj) = si i s i >_ sj.
4. Minimiza ion ope a o : Min(s i, sy) = s i i s i <_ sj.
We conside ha he seman ic o he elemen s in he e m se is gi en by
uzzy numbe s de ined on he in e al [0, 1], which a e desc ibed by linea
apezoidal membe ship unc ions. This ep esen a ion is achie ed by he
4- uple (ai, bi, ai, li). The i s wo pa ame e s indica e he in e al in
which he membe ship alue is 1; he hi d and ou h pa ame e s indica e
he le and igh wid h.
Example 3.1. The ollowing se en label se , S, e i ies he a o emen ioned
p ope ies:
MA Maximum (1, 1, .25, 0)
1/34 Ve y_Much (.75, .75, .15, .25)
Mu Much (.6, .6, .1, .15)
M Medium (.5, .5, .1, .1)
L Li le (.4, .4,. 15,. 1)
VL Ve y_Li le (.25, .25, .25, .15)
MI Minimum (0, 0, 0, .25)
In his linguis ic con ex , he ma hema ical model o he GDM p oblem
conside ed is he ollowing. Le X = {xl,...,x n} be a he e ogeneous,
nonemp y, and ini e se o al e na i es o be analyzed by a he e ogeneous,
nonemp y, and ini e se o expe s E = {el,..., e a}. Assuming a label se ,
V = { i}, i ~ I = {0 ..... M}, o exp ess impo ance and ele ance deg ees,
o each al e na i e, x i ~ X, we suppose ha a linguis ic ele ance deg ee is
de ined, la.R(i)E V, om 30 s anding o "de ini ely i ele an " o M
s anding o "de ini ely ele an ," ac oss all he in e media e alues. Simi-
la ly, o each expe e k E E, we assume ha a linguis ic impo ance deg ee
is known, /. E(k)~ V, assigned by a dis inguished pe son, called he
mode a o , o each expe e k. Then, each expe e k p o ides his/he
opinions on X as a linguis ic p e e ence ela ion, pk c X >( X, wi h
membe ship unc ion /zek : X × X ~ S, whe e /zek(xi, xj) = pk deno es
he linguis ic p e e ence deg ee o he al e na i e x i o e xj. We assume,
wi hou loss o gene ali y, ha pk is ecip ocal in he sense ha p~ =
Neg(pk), and by de ini ion piki = s o ( he minimum label in S).
Gi en an expe e k, his impo ance deg ee, /zE(k), is in e p e ed as he
deg ee o which he expe is eally a decision make in ela ion o he
316 F. He e a e al.
decision p oblem. And gi en an al e na i e xi, i s ele ance deg ee, ~R(i),
is in e p e ed as he deg ee o which he al e na i e is eally an op ion in
ela ion o he p oblem domain.
EXAMPLE 3.2 Assume he ollowing nine label se V o exp ess he
impo ance and ele ance deg ees:
T To al
(1, 1, 0, 0)
EH Ex emely_High
(.98, .99, .05, .01)
VH Ve y_High
(.78, .92, .06, .05)
H High
(.63, .80, .05, .06)
M Medium
(.41, .58, .09, .07)
L Low
(.22, .36, .05, .06)
VL Ve y_Low
(.1, .18, .06, .05)
EL Ex emely_Low
(.01, .02, .01, .05)
N Null (0, O, O, O)
Le X = {xl, x2, x3,
X 4}
be a he e ogeneous se o ou al e na i es, o
which he espec i e linguis ic ele ance deg ees a e
/ZR(1) =
EH,
/ZR(2) = M, /xR(3) = VH, /xR(4) = VL.
Le E = {el, e 2, e3, e 4} be a he e ogeneous g oup o ou expe s, o which
he espec i e linguis ic impo ance deg ees a e
~e(1) =M, ~e(2) = VH, ~E(3) =M, ~e(4) = L.
Then, ollowing Example 3.1, linguis ic p e e ence ela ions o e X, in his
linguis ic con ex , may be conside ed as:
-- VL VM VL]
pl= VM -- M M
VL L -- VL '
VM L VM --
p2 ~
p3 = M -- VM p4 =
_
M VM
VL --
VM M VM
-- L VM VL]
M -- L
VL
VL M -- VL "
VM VM VM --
In he GDM p oblem, in o de o agg ega e linguis ic labels, we use he
LOWA ope a o [18, 22], which allows us o ep esen he concep o uzzy
majo i y in he agg ega ion p ocesses. The LOWA ope a o is based on
he
o de ed weigh ed a e aging
(OWA) ope a o de ined by Yage [34], and
on he
con ex combina ion o linguis ic labels
de ined by Delgado e al. [11].
Consensus in G oup Decision Making 317
DEFINITION 3.1
Le
A = {al,... , am}
be a se o labels o be agg ega ed.
Then he LOWA ope a o d~ is de ined as
~(a 1 .....
a m ) = ~'B T = ~m{wk, bk,
k = 1,..., m}
= WlQ)b 1 ~)
(1 - w 1) Q)~C~ n- 1{
lh, bh,
h
=
2 .... , m}
whe e
W = [w 1 ..... win],
is a weigh ing ec o such ha
(i)
w i ~
[0, 1]
and
(ii) Ei = 1,
lh = Wh/E~Wk,
h = 2,...,
m, and
B = {b 1 .....
b m} is a
ec o associa ed wi h A such ha
B = (A) = {a~(1) .... , a,~(,)},
whe e
a~( h < a~(i) Vi < j, wi h being a pe mu a ion o e he se o labels A.
~m is he con ex combina ion ope a o o m labels, and i m = 2, hen i is
de ined as
~2{wi,bi, i=l,2}=WlQ)Sj~(1-Wl)Q)si=sk, sj,siES
(j>i)
such ha ,
k = MIN{T, i + ound(w 1 • (j - i))},
whe e
ound
is he usual
ounding ope a ion, and b I = sj,
b 2 = s i. I wj =
1 and w i = 0 wi h i ~ j
Vi, hen he con ex combina ion is de ined as ~m {wi, bi ' i = 1 ..... m} =
bj.
O he app oaches o agg ega ion o linguis ic labels may be ound in [4,
11, 30, 35, 36, 38-40].
How o calcula e he weigh ing ec o o he LOWA ope a o , W, is a
basic ques ion o decide. Yage p oposed in [34, 37] an in e es ing way o
compu e he weigh s o he OWA agg ega ion ope a o using linguis ic
quan i ie s [42], ep esen ing he concep o uzzy
majo i y.
In ou case, we
use wo ypes o uzzy majo i y:
• Fuzzy majo i y o al e na i es,
used o quan i y he di e en uzzy
coincidence deg ees acco ding o one pai o expe s' opinions.
• Fuzzy majo i y o expe s,
used o quan i y he di e en consensus
measu es acco ding o e e y pai o expe s' opinions.
Acco ding o Yage [34, 37] he weigh s can be ob ained by means o he
ollowing exp ession:
Q(i) Q(i~nl )
w i---
- - , i= 1,...,n,
n
whe e Q is a nondec easing p opo ional quan i e ep esen ed by he
ollowing membe ship unc ion:
! i < a,
--a
Q( )= i
a< <b,
a
i
>b
wi h a, b, ~ [0, 1].
324 F. He e a e al.
5.2. Compu ing P ocess
In his i s s ep o he compu ing p ocess, o each pai o al e na i es,
(x i, xj),
he di e en
pai linguis ic consensus measu es
a e calcula ed
acco ding o he ollowing de ini ions:
DEFINITION 5.2
The pai linguis ic consensus deg ee, PCq, is de ined
acco ding o his exp ession:
PCij=dgQl(LC-'( Zcq(ekl), kl),k= l .....
m-1, l=k
+1 ..... m),
and k = dp( lzE(k) , lxe(l)) , wi h weigh ing ec o o he LOWA ope a o ,
w = [0.5, 0.5].
kl
is an
a e aging impo ance deg ee,
which ep esen s he impo ance
deg ee o he coincidence deg ee o he pai o expe s, e~l. I is ob ained
by means o he LOWA ope a o h wi h ha weigh ing ec o in o de o
achie e a mean agg ega ion o he impo ance deg ees.
LC -~
ep esen s a
amily o connec i es, i.e.,
linguis ic conjunc ion unc ions
[17]. We shall use
as linguis ic conjunc ion unc ions he ollowing -no ms, which a e mono-
onically noninc easing in he weigh s w, and sa is y he p ope ies e-
qui ed o any ans o ma ion unc ion o he weigh ed in o ma ion (a, w)
[16, 17]:
1. The classical Min ope a o :
LC(
(a, w) = Min(a, w).
2. The nilpo en Min ope a o :
Min(a,w) i w > Neg(a),
LCZ" (a, w) = ~ go
o he wise.
3. The weakes conjunc ion:
[ Min(a, w) i Max(a, w) = gM,
LCS
(a,
W)
[ go o he wise.
And ~Q1 is he LOWA ope a o o which he weigh ing ec o is
ob ained by means o he linguis ic quan i ie ,
01 ,
used o ep esen he
concep o uzzy majo i y o expe s.
REMARK 5.1 No e ha De ini ion 5.2 explici ly equi es his es ic ion,
G = IT, i.e., ha he linguis ic domain used o exp ess consensus measu es
is he same one used o exp ess impo ance and ele ance deg ees. This
limi a ion may be b idged i we use a me hod o ans o m labels among
di e en linguis ic domains, bu his is no ou goal in his pape .

Consensus in G oup Decision Making 325
EXAMPLE 5.3 Con inuing wi h he GDM con ex gi en in Example 3.2,
om impo ance deg ees o expe s, o each pai o expe s, eke, he
a e aging impo ance deg ees, kl ~ V, a e calcula ed,
{ 12 =
H,
13 =
M, 14 = M,
23 =
H,
24 ----- H, 34 =
M},
in which, o example, as /Ze(1) = M = u 4 and /ze(2) = VH = 6, hen
12
--- H = 5, since 5 = MIN{8, 4 + ound((6 - 4) × 0.5)}.
He e and in he nex examples, we assume he
nilpo en
Min
ope a o ,
LC~,
o manipula e linguis ic weigh ed in o ma ion, and as he linguis ic
quan i ie Q1 he quan i ie gi en in Example 3.3, "As many as possible,"
wi h he pai (0.5, 1).
Then, in his con ex , om he uzzy coincidence se s ob ained in
Example 5.2 and om he p e ious a e aging impo ance deg ees, on each
pai o al e na i es, (xi, xj), i 4= j, we calcula e he
pai linguis ic consensus
deg ee, PCij,
by means o he conjunc ion unc ion,
LCZ',
and o he
LOWA ope a o , ~bQ,, wi h he weigh ing ec o , W = [0, 0, 0, 0.32, 0.35,
0.33]:
{PC12 = EL,
PCI3 = M, PC14 =
M,},
{PC21 ---
EL, PC23 = N, PC24 = EL,},
{PC31 = M, PC32 = EL, PC34 = M,},
{ PC41 = L, PCa2 = N, PC43 = M,},
in which, o example,
PC41
is ob ained as
PC41
= ~bQ,(LC 2
(T, H), LC~' (H, M), LC Z' (N, M),
LC2 (H, H), LC 2 ( , H), LC Z' ( , M)) = L.
COMMENT 5.1 In gene al, he consensus deg ees a e low. Fo example, on
he pai s o al e na i es (x2, x3) and (x 4, x2), he e is no consensus among
he expe s' opinions, and on he se o pai s o al e na i es {(Xl,
x2), (x2,
xl), (Xa, x4), (x3, Xe)} he consensus is oo low. Only a maximum consensus
deg ee wi h a alue M is achie ed on some pai s o al e na i es. Howe e ,
i we obse e he se s o uzzy coincidences ob ained in Example 5.2, hei
membe ship unc ions p esen , in gene al, alues abo e he middle alue,
M, which should esul in high consensus deg ees. So, om his iewpoin ,
appa en ly, he e is a con adic ion. Howe e , we mus no o ge ha we
a e wo king implici ly in a he e ogeneous GDM con ex wi h di e en
meanings o uzzy majo i y. So, his si ua ion some imes is due o he
in luence o he chosen conjunc ion unc ion, and in o he s, i is due o he
in luence o he chosen linguis ic quan i ie . In ou case, bo h he conjunc-
326 F. He e a e al.
ion unc ion, LCS" and he linguis ic quan i ie "As many as possible"
induce a pessimis ic in luence o he consensus s a e. So, o example, i
LC( ~ is chosen as a conjunc ion unc ion, hen PC12 = M, and simila ly,
o he consensus deg ees will be highe . In he same way, i "A leas hal "
is chosen as a linguis ic quan i ie , main aining LC , hen PC12 = H.
The e o e, in sho , we mus choose bo h app op ia e conjunc ion unc-
ions and linguis ic quan i ie s in une wi h ou consensus idea.
Now, simila ly, om he uzzy coincidence on he pai s o al e na i es,
we de ine ano he pai linguis ic consensus measu e.
DEFINITION 5.3 The pai linguis ic p oximi y PPi~ o an expe e k is
de ined acco ding o his exp ession:
= 4'o,(LC-'(lZc,j(ek,),lx•(,)),, =1 ..... m,, ¢ k),
PP,
knowing ha when Zc,j(e~ ) ¢~ Cij hen Zc,(e~ ) = iXc,(e ~).
REMARK 5.2 No e ha in his de ini ion, as in De ini ion 5.2, he consen-
sus measu e is de ined by means o he LOWA ope a o 4'0' and he
conjunc ion unc ion LC-*, and using he se s o uzzy coincidence o
De ini ion 5.1, bu in his case conside ing only he impo ance deg ees o
he emaining expe s and no he a e aging impo ance deg ees. The e-
o e, in his sense, /xe(1) is used as he impo ance deg ee gi en o he
coincidence deg ee obse ed be ween he expe analyzed and ano he
expe e~ in he g oup.
EXAMPLE 5.4 As in Example 5.3, bu his ime assuming he impo ance
deg ees gi en in Example 3.2 ins ead o a e aging impo ance deg ees, on
each pai o al e na i es, (x/, xj), and o each expe e k, we calcula e pai
linguis ic p oximi ies PPi~ by means o he conjunc ion unc ion LC , and
he LOWA ope a o 4'Q, wi h he weigh ing ec o W = [0, 0.32, 0.68]:
1. Expe el:
{PP~e = L, PP13 = M, PP14 = M, PP11
= N, eel 3 = N, PPI 4 = N,
PP11 = M, PP~2 = M, PP134 = M, PP~I = M, PPg2 = N, PP~3 = M}
2. Expe e2:
{pp22 = L, PP~3 = N, PP214 = M, PP21 = L, PP~3 = N, PP~4 = L,
PP21 = M, PP 2 = M, PP 4 = M, pp2 = M, PP422 = N, PP423 = M}
3. Expe e3:
{ pp32 = L, pp33 = M, pp34 = M, Pp32, = L, pp33 = N, pp34 = L,
pp31 = M, pp32 = M, pp34 = M, pp3 = M, PP32 = N, pP433 = M}
Consensus in G oup Decision Making 327
4. Expe
e4:
{ PP42 = M, pea3 = M, PP44 = M, PP41 = M, PP43 = N, PP44 = M,
pp4 = M, PP~2 = N, pp4, = M, pp4 = M, PP442 = N, pp4 = M}
Fo example,
pp4
is ob ained as
pp4 = 4ao2(LC~ (L, M), LC~' (M, VII), LC~ (M, M)) = N.
COMMENT 5.2 Logically, as in Example 5.3 and o he same easons,
he e, all he expe s p esen e y low p oximi ies--in any case, ne e
highe han he middle alue M.
6. PHASE 2: WORKING ON THE ALTERNATIVES
6.1. Coincidence P ocess
In his s ep he concep o uzzy coincidence among expe s' opinions is
de ined, wo king on he le el o he al e na i es.
DEFINITION
6.1 The uzzy coincidence on an al e na i e, x i, is de ined as
a uzzy se C i in he non uzzy se o pai s o expe s, E 2, namely Ci =
{(ek ,
zG(ek ))}, cha ac e ized by a membe ship unc ion iZc~ : E 2 --~ G
indica ing he coincidence deg ee be ween expe s e k and e 's opinions on
pai s o al e na i es in which he al e na i e x i appea s:
iZci(ek ) = ~bQ2(LC~ ( Zcij(ek ), i'j),LC-' ( Zcji(ek ), 'ij),
j~i,j=
1,...,n),
and ;j = c~( zR(i) , lzR( j)) , wi h he weigh ing ec o o he LOWA ope a o
gi en by
W = [0.5, 0.5].
;y is an
a e aging ele ance deg ee,
which ep esen s he ele ance
deg ee o he coincidence deg ee achie ed on he pai o al ema i es
(x i, xi). I is ob ained in he same way as k in De ini ion 5.2. In his case,
he coincidence deg ee
iZc~(ek )
is ob ained by means o a LOWA ope a o
o which he weigh ing ec o is calcula ed by means o he linguis ic
quan i ie QZ used o ep esen he concep o uzzy majo i y o al e na-
i es. The es ic ion poin ed ou in Rema k 5.1 is applied he e oo and in
he nex de ini ions.
328 F. He e a e al.
EXAMPLE 6.1 Con inuing wi h he GDM con ex gi en in Example 3.2,
om he ele ance deg ees o he al e na i es, o each pai o al e na-
i es,
(xi, x:),
he a e aging ele ance deg ees
~: ~ V
a e
' =VII, ' =EH, ' =H, '
=H, ' =L, ' =M}.
{ 12 13 14 23 24 34
in which, o example, as /zR(1)=
EH
= 7 and p~R(2)= VH = 6, we
ha e
12 =
EH
= 7, since 7 = MIN{8, 6 + ound((7 - 6) × 0.5)}.
Assuming, like Q2, he linguis ic quan i ie gi en in Example 3.3, i.e.,
"A leas hal ," wi h he pai (0, 0.5), hen, om he uzzy coincidence se s
calcula ed in Example 5.2 and om he abo e a e aging ele ance deg ees,
by means o he LOWA ope a o ~bO2 wi h W = [0.33, 0.35, 0.32, 0, 0, 0],
and he conjunc ion unc ion
LC~'
o each al e na i e x i i s uzzy
coincidence se C i in E 2, is ob ained, esul ing in
C 1 = {(e12 , VH), (el3 , V/-/), (e14 ,
EH),
(e23 , VH), (e24 , VH), (e34 , EH)},
C2 = {(e12, H), (el3 , M), (el4, H), (e23,
H), (e24, H),
(e34 , H)},
C 3 = {(el2 ,
n),
(el3 , VH), (el4 ,
H),
(e23 ,
H),
(e24 ,
M),
(e34 ,
H)},
C 4 ~-- {(e12 , n), (el3 ,
H),
(el4 ,
M),
(e23 , n), (e24 , n), (e34 ,
H)}.
Fo example, /Xc,(e14) is ob ained as
Zc,(el,) = 6Qz(LC~ (N, H), LC~" (M, L), LC Z" (M, M),
LC~ (n, n), LC~" (N, L), LC~" (N, M)) = M.
6.2. Compu ing P ocess
In his second s ep, on each al e na i e, Xg, he di e en
al e na i e
linguis ic consensus measu es
a e calcula ed acco ding o he ollowing
de ini ions:
DEFINITION 6.2
The al e na i e linguis ic consensus deg ee, ACg, is de-
ined acco ding o his exp ession:
AC i = ¢kQl(LC-~( xC(ekl), kl); k= 1 .... ,m-
1,/= k + 1 .... ,m).
EXAMPLE 6.2 In he same way as we did in Example 5.3, on each
al e na i e, xg, we calcula e he al e na i e linguis ic consensus deg ee,
ACg,
bu his ime, conside ing he a o emen ioned uzzy coincidence se s
{AC 1 = M,
AC 2 =
M,
AC 3 =
M,
AC 4
---
M},
Consensus in G oup Decision Making 329
in which, o example,
AC 4
is ob ained as
hC 4
=
dpQ~(LC-~ (H, H), LC-" (n, M), LC-~ (M, M),
LC-~ (H, H), LC-~ (H, H), LC -~ (H, M)) = M
COMMENT 6.1 On his le el, he e ec poin ed ou in Commen 5.1 also
appea s, since he e is a maximum consensus deg ee wi h alue M on any
al e na i e, in spi e o he ac ha some obse ed coincidence deg ees a e
high. Besides, he e he e ec o he a e aging ele ance deg ees, used o
calcula e he coincidence deg ees, is included.
DEFINITION 6.3
The al e na i e linguis ic p oximi y APi k o an expe e k is
de ined acco ding o his exp ession:
APi k = dpQ~(LC-" ( Zc,(ek ),lze(l)),l : 1,...,m,l --/: k),
knowing ha when i. c(ek ) ~ C i, hen Xci(ek ) = ~c(e k).
EXAMPLE 6.3 In he same way we did in Example 5.4, he e, on each
al e na i e
x i
o each expe e k, we calcula e he al e na i e linguis ic
p oximi y AP~ k, bu his ime, conside ing he a o emen ioned uzzy coinci-
dence se s
{AP~ = L, AP~ = N, AP 1 = EL, AP41 = EL},
(AP? = L, = EL, AP 3 = EL, = EL},
{AP3a = L, AP~ = N, AP~ = EL, AP 2 = EL),
{AP~ = M, AP 4 = M, AP 4 = M, AP 4 = EL}.
He e, o example, AP 4 is ob ained as
AP 4 = d~Q~(LC -~ (EH, U), LC " (VH, VII), LC -' (EH, U)) = U.
7. PHASE 3: WORKING ON THE RELATION
7.1. Coincidence P ocess
In his las phase, he concep o uzzy coincidence among expe s'
opinions is de ined, wo king on he le el o he ela ion.
DEFIn ON 7.1
The uzzy coincidence on he ela ion is de ined as a uzzy
se
C = {(ek ,
lxC(ek ))} in he non uzzy se o pai s o expe s, E 2, cha ac-
e ized by a membe ship unc ion, ix c : E 2 ~ G, indica ing he coincidence

330 F. He e a e al.
deg ee be ween expe s e k and el's opinions on all he pai s o al e na i es:
I~C(ek,) = q~a2(LC-~(lXc,(ek,), lxn(i)),i= 1 ..... n).
REMARK 7.1 In his de ini ion, he coincidence deg ee, IxC(ekl), is ob-
ained as in De ini ion 6.1, i.e., by means o he LOWA ope a o ~bo2 and
o he conjunc ion unc ion LC -~ , bu in his case, using ele ance
deg ees Zn(i) ins ead o a e aging ele ance deg ees ~j ep esen ing he
ele ance deg ee o he coincidence deg ee achie ed on each al e na i e
X i •
EXAMPLE 7.1 Assuming he ele ance deg ees gi en in Example 3.2, as
was done in Example 6.1, o o e all al e na i es X, i s se o uzzy
coincidence, C, in E 2 is ob ained om ele ance deg ees and he uzzy
coincidence se s ob ained in Example 6.1, by means o he LOWA ope a-
o he2 wi h he weigh ing ec o W = [0.5, 0.5, 0, 0] and he same conjunc-
ion unc ion, LC{, esul ing in
C = {(el2, I/H), (el3,1,'7-/), (e14 , VH), (e23 , l/H), (e24 ,
H),
(e34 , VH)}.
Fo example, /. c(el4) is ob ained as
/zc(e14) = ~bQ2(LC-~ (EH, EH), LC ~ (H, M),
LC -~ (H, VH), LC -" (M, VL)) = VH.
7.2. Compu ing P ocess
In his las compu ing p ocess, on o e all opinions, i.e., on he ela ion,
he ela ion linguis ic consensus measu es a e calcula ed acco ding o he
ollowing de ini ions:
DEFINITION 7.2 The ela ion o linguis ic consensus deg ee, RC, is de ined
acco ding o his exp ession:
RC = q~QI(LC -~ (i~c(ek ), kl), k = 1,..., m - 1, l = k + 1 ..... m).
DEFINITION 7.3 The ela ion o linguis ic p oximi y Rpk o an expe e k is
de ined acco ding o his exp ession:
Re k = dpQI(LC -~ ( ~c(ekl), I~E(I)),
l = 1,...,
m, I :g
k),
knowing ha when zc(ek ) q~ C hen lzc(ek ) = I~c(e k).
EXAMPLE 7.2 Wo king as in Examples 6.2 and 6.3, hen
RC = M,
Consensus in G oup Decision Making 331
and
espec i ely.
{~1 =L,~2=EL,~3=L,~4=M},
COMMENT 7.1 In iew o he esul ing consensus measu es, which indi-
ca e a medium consensus cu en s a e, he consensus eaching p ocess
can s op o con inue. In he second case, hen, he mode a o has o make
some o he ollowing conside a ions:
• Ad ise all he expe s o change hei opinions on pai s o al e na i es,
and in pa icula on he se o pai s o al e na i es
{(X 1, X2),(X 2, Xl),(X 2,x3),(x
2,
X4),(X 3,x2), (X 4,xl), (X 4, X2)}.
• Ad ise he expe s {el, e2,
e3} o
diminish hei disag eemen among
hem.
• Decide on he use ulness o main aining he pessimis ic e ec o he
chosen linguis ic quan i ie o calcula e he linguis ic consensus mea-
su es and o he chosen conjunc ion unc ion in he ollowing s ep o
he consensus measu ing p ocess.
8. CONCLUSIONS
A consensus model is p oposed in o de o de elop a consensus eaching
p ocess in a GDM con ex wi h he e ogeneous g oups o expe s and a
he e ogeneous se o al e na i es in a he e ogeneous linguis ic amewo k.
This model con ains wo ypes o consensus measu es o guide he consen-
sus p ocess om wo di e en pe spec i es. The i s ype, called
linguis ic
consensus deg ees,
s udies he consensus s a e om a global pe spec i e,
conside ing all he expe s, and he second ype, called
linguis ic consensus
p oximi y,
s udies he consensus s a e om a pa icula pe spec i e, i.e.,
conside ing pa icula expe s. Fu he mo e, all he ypes o measu es a e
applied on h ee le el o ac ions o ep esen ing he cu en consensus
s a e. The e o e, he consensus model p esen s h ee consensus measu es
o each ype. The main ea u es o he consensus model a e he ollowing
ones:
• i s measu es a e based on a uzzy cha ac e iza ion o he concep o
coincidence de ined om an
ad hoc closeness able;
• i uses di e en linguis ic domains o exp ess he opinions and he
consensus measu es;
• i s measu es a e calcula ed by means o he LOWA ope a o , se e al
conjunc ion unc ions, and linguis ic quan i ie s ep esen ing he con-
cep o uzzy majo i y;
332 F. He e a e al,
• i p esen s a lexible s uc u e, which allows us o use di e en
linguis ic quan i ie s and di e en conjunc ion unc ions, wi h a iew
o inducing di e en consensus ideas.
In sho , a lexible consensus model has been p esen ed.
Finally, we poin ou wo aspec s ha a e ou side o ou objec i es in
his pape , bu a e o in e es in a decision p ocess oo. They a e: (1) he
nego ia ion p ocess be ween he expe s and he mode a o o eaching
and accep able consensus le el [33], and (2) how o model he possible
con lic s be ween expe s o goals and hei explici ep esen a ion [7, 8],
which can help in he nego ia ion p ocess and can enhance i s explana ion.
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