Else ie Edi o ial Sys em( m) o Fuzzy Se s and Sys ems
Manusc ip D a
Manusc ip Numbe : FSS-D-14-00513
Ti le: On he New on me hod o sol ing uzzy op imiza ion p oblems
A icle Type: Full Leng h A icle (FLA)
Keywo ds: Fuzzy op imiza ion; gene alized Hukuha a di e en iabili y; New on me hod.
Co esponding Au ho : D . Y. Chalco Cano,
Co esponding Au ho 's Ins i u ion: Uni e sidad de Ta apacá
Fi s Au ho : Y. Chalco Cano
O de o Au ho s: Y. Chalco Cano; Ge aldo Sil a; An onio Ru ián-Lizana
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In his a icle we conside op imiza ion p oblems whe e he objec i esa e
uzzy unc ions ( uzzy- alued unc ions). Fo his class o uzzy op imiza ion
p oblems we discuss he New on me hod o ind a non-domina ed solu ion. Fo
his pu pose, we use he gene alized Hukuha a diffe en iabili y no ion, which is
he mos gene al concep o exis ing diffe en iabili y o uzzy unc ions. This
wo k imp o es and co ec he New on Me hod p e iously p oposed in he li -
e a u e.
1
Abs ac
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On he New on me hod o sol ing uzzy op imiza ion
p oblems 4
Y. Chalco-Cano a,1,G.N.Sil a
b,2,A.Ru i
´
an-Lizana c,3
aIns i u o de Al a In es igaci´on, Uni e sidad de Ta apac´a, Casilla 7D, A ica Chile
bIns i u o de Biociˆencias, Le as e Ciˆencias Exa as, UNESP-Uni .Es adualPaulis a,
Cˆampus de S˜ao Jos´e do Rio P e o, Depa amen o de Ma em´a ica Aplicada, S˜ao Jos´e do
Rio P e o-SP, B asil.
cDepa amen o de Es ad´ıs ica e I.O., Uni e sidad de Se illa, Spain.
Abs ac
In his a icle we conside op imiza ion p oblems whe e he objec i es a e uzzy unc-
ions ( uzzy- alued unc ions). Fo his class o uzzy op imiza ion p oblems we discuss
he New on me hod o ind a non-domina ed solu ion. Fo his pu pose, we use he gen-
e alized Hukuha a diffe en iabili y no ion, which is he mos gene al concep o exis ing
diffe en iabili y o uzzy unc ions. This wo k imp o es and co ec he New on Me hod
p e iously p oposed in he li e a u e.
Key wo ds: Fuzzy op imiza ion, gene alized Hukuha a diffe en iabili y, New on me hod.
1In oduc ion
Fuzzy op imiza ion p oblems ha e been s udied by many esea che s in se e al
di ec ions wi h a lo o applica ions. The collec ion o pape s on uzzy op imiza ion
edi ed by Delgado e al. [11], Lodwick and Kacp zyk [19], Inuiguchi and Ram´ık
1Co esponding au ho , e-mail: ychalco@u a.cl
2e-mail: [email p o ec ed]
3e-mail: u i[email p o ec ed]
4The esea ch in his pape has been suppo ed by Fondecy -Chile, p ojec 1120665 and
1120674, by Minis e io de Ciencia y Tecnolog´ıa (Spain) h ough p ojec MTM 2010-
15383, by he B azilian Na ional Council o Scien i ic and Technological De elopmen -
CNPq (B azil) unde G an numbe s 309335/2012-4 and 479109/2013-3 and by Sao Paulo
S a e Founda ion (FAPESP) unde G an numbe 2013/07375-0 – h ough he Cen e o
Ma hema ical Sciences Applied o Indus y – CeMEAI/CEPID.
P ep in submi ed o Else ie 27 June 2014
*Manusc ip
Click he e o iew linked Re e ences
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[15], Rommel ange and Slowi´nski [26], Slowi´nski and Teghem [27], and he books
by Lai and Hwang [16,17] p o ide e iews abou his opic oma e yb oadpoin
o iew.
I is usually difficul o de e mine he coefficien s o an objec i e unc ion as a
eal numbe since, mos o en, hese possess inhe en unce ain y and/o inaccu-
acy. Gi en ha his is he usual s a e, we conside uzzy- alued objec i e unc ion
as one app oach o ackle unce ain y and inaccu acies in heobjec i e unc ion
coefficien s o ma hema ical p og amming models (see, e.g., Lodwick [18]). Op i-
miza ion p oblems wi h uzzy- alued objec i e unc ions we e s udied by many
esea che s. Fo ins ance, see [5,8,23,20,30–37,39]. In pa icula , in [5,9,33,37]
Ka ush-Kuhn-Tucke ype op imali y condi ions o his class o uzzy op imiza-
ion p oblems we e ob ained. Mo e ecen ly, Pi zada and Pa hak in [21] p oposed a
New on me hod o ind a non-domina ed solu ion o a uzzy op imiza ion p oblem
using he Hukuha a diffe en iabili y o uzzy- alued unc ions.
The concep o Hukuha a diffe en ibili y (H-diffe en iabili y, o sho ) o uzzy
unc ions is e y es ic i e. Fo ins ance, F(x)=C·x,whe eCis any uzzy in e -
al and xis a eal numbe , is no H-diffe en iable being ha Fis a gene aliza ion o
alinea unc ion.Ingene al,a uzzy unc ionde inedbyF(x)=C·g(x), whe e gis a
diffe en iable eal unc ion and Cis a uzzy in e al, is no always H-diffe en iable.
Howe e , i is always gH-diffe en iable. I is well-known ha he concep o gH-
diffe en iable uzzy unc ion (gene alized Hukuha a diffe en iable uzzy unc ion)
is a mo e gene al concep han le el-wise diffe en iabili y [37,38], Hukuha a di -
e en iabili y [14], and G-diffe en iabili y [1–3,6,7]. Thus, he mo e use ul concep
o diffe en iabili y o uzzy unc ions is gH-diffe en iabili y.
The condi ions imposed o implemen he New on me hod in oduced by Pi zada
and Pa hak in [21] in Theo em 4.1 a e e y es ic i e because heyask ha in he
neighbo hood o a nondomina ed solu ion, all poin s mus be compa able, bu he
o de ela ion used is only pa ial. Mo eo e , i is also equi ed ha he nondom-
ina ed solu ion is an ideal poin o he endpoin unc ions o heobjec i e uzzy
unc ions. The examples p esen ed in he same pape do no obey he condi ions
equi ed by Theo em 4.1. In addi ion, he objec i e unc ions o all examples hey
conside a e no H-diffe en iable. E en so, hey apply he New on me hod o he ex-
amples. No su p isingly, when hey apply o he Example 4.1, heyob ainapoin
ha is no a non domina ed solu ion, al hough he au ho s claim i is.
In his pape we o mula e he New on me hod o ind a non-domina ed solu-
ion o uzzy op imiza ion p oblems wi hou he equi emen ha all easibleso-
lu ions in he neighbo hood o a nondomina ed solu ion be compa able and use
gH-diffe en ibili y ins ead o H-diffe en iabili y. Finally, we co ec he examples
conside ed in [21].
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2No a ionand hespaceo uzzyin e als
A uzzyse onRnis a mapping u:Rn→[0,1]. Fo each uzzy se u,wedeno e
i s α-le el se as [u]α={x∈Rn|u(x)≥α} o any α∈(0,1]. The suppo o uis
deno ed by supp(u), whe e supp(u)={x∈Rn|u(x)>0}.Theclosu eo supp(u)
de ines he 0-le el o u,.i.e.[u]0=cl(supp(u)), whe e cl(M)means heclosu eo
he subse M⊂Rn.
De ini ion 1 A uzzyse uonRis said o be a uzzy in e al i :
(1) u is no mal, i.e. he e exis s x0∈Rsuch ha u(x0)=1;
(2) u is an uppe semi-con inuous unc ion;
(3) u(λx+(1 −λ)y)≥min{u(x),u(y)},x,y∈R,λ∈[0,1];
(4) [u]0is compac .
Le FCdeno e he amily o all uzzy in e als. So, o any u∈F
Cwe ha e ha
[u]α∈K
C o all α∈[0,1], whe e KCdeno es he space o all compac in e als in
R,and hus heα-le els o a uzzy in e al a e gi en by [u]α=!uα,uα",uα,uα∈R
o all α∈[0,1]. I [u]1is a single on hen we say ha uis a uzzy numbe . T ian-
gula uzzy numbe s a e a special ype o uzzy numbe s which a e well de e mined
by h ee eal numbe s a≤b≤cand we w i e u=(a,b,c)and
[u]α=[a+(b−a)α,c−(c−b)α],
o all α∈[0,1].
Fo uzzy in e als u, ∈F
C ep esen ed by !uα,uα"and ! α, α", espec i ely,and
o any eal numbe λ,wede ine headdi ionu+ and scala mul iplica ion λuas
ollows:
(u+ )(x)=sup
y+z=x
min{u(y), (z)}
(λu)(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
u'x
λ(,i λ!0,
0,i λ=0.
I is well known ha , o e e y α∈[0,1],
[u+ ]α=!(u+ )α,(u+ )α"=!uα+ α,uα+ α"(1)
and
[λu]α=!(λu)α,(λu)α"=!min{λuα,λ
uα},max{λuα,λ
uα}".(2)
Ac ucialconcep inob ainingause ulwo kingde ini iono de i a i e o uzzy
unc ions is de i ing a sui able diffe ence be ween wo uzzy in e als. Towa d his
end we ha e he ollowing de ini ion.
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De ini ion 2 ([29]) Gi en wo uzzy in e als u, , he gene alized Hukuha a di -
e ence (gH-diffe ence o sho ) is he uzzy in e al w, i i exis s, such ha
u⊖gH =w⇔⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
(i)u= +w,
o (ii) =u+(−1)w.
I is easy o show ha (i)and(ii)a ebo h alidi andonlyi wis a c isp numbe .
No e ha he case (i) is coinciden o Hukuha a diffe ence (see [14]) and so he
concep o gH-diffe ence is mo e gene al han H-diffe ence.
I u⊖gH exis s hen, in e ms o α-le els, we ha e
[u⊖gH ]α=[u]α⊖gH [u]α=!min{uα− α},max{uα− α}",
o all α∈[0,1], whe e [u]α⊖gH [u]αdeno es he gH-diffe ence be ween wo in e -
als (see [28,29]).
Gi en u, ∈F
C,we de ine he dis ance be ween uand by
D(u, )=sup
α∈[0,1]
H([u]α,[ ]α)
=sup
α∈[0,1]
max )***uα− α***,|uα− α|+.
So, (FC,D)isacomple eme icspace.
3Diffe en iable uzzy unc ions
Hence o h, Kdeno es an open subse o Rn.A unc ionF:K→F
Cis said o be
a uzzy unc ion.Fo eachα∈[0,1], we associa e wi h F he amily o in e al-
alued unc ions Fα:K→K
Cgi en by Fα(x)=[F(x)]α.Fo anyα∈[0,1], we
deno e
Fα(x)=! α(x), α(x)".
He e, he endpoin unc ions α, α:K→Ra e called uppe and lowe unc ions
o F, espec i ely.
Nex we p esen he concep o gH-diffe en iabili y o uzzy unc ions in he one
dimensional case.
De ini ion 3 ([3]) Le K ⊂Rwi h F :K→F
Ca uzzy unc ionandx
0∈Kandh
be such ha x0+h∈K. Then he gene alized Hukuha a de i a i e (gH-de i a i e,
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o sho ) o F a x0is de ined as
F′(x0)=lim
h→0
F(x0+h)⊖gH F(x0)
h.(3)
I F′(x0)∈F
Csa is ying (3) exis s, we say ha F is gene alized Hukuha a diffe -
en iable (gH-diffe en iable, o sho ) a x0.
The gH-de i a i e o an in e al- alued unc ion [28] is simila oDe ini ion3.
Mo e p ecisely, an in e al- alued unc ion F:K→K
Cis gH-diffe en iable a
x0∈K,wi hgH-de i a i e F′(x0)∈K
C,i (3)exis swi h espec o helimi in he
me ic space (KC,H), whe e he diffe ence is gi en by he gH-diffe ence be ween
in e als (see [28]).
Theo em 1 Le F :K→F
Cbe a uzzy unc ion. I F is gH-diffe en iable hen
he in e al- alued unc ion Fα:K→K
Cis gH-diffe en iable o each α∈[0,1].
Mo eo e
,F′(x)-α=F′
α(x).(4)
P oo . The p oo is a consequence o he de ini ion o gH-diffe en iabili y.
Example 1 Conside he uzzy mapping F :R→F
Cde ined by F(x)=C·x,
whe e C is a uzzy in e al and [C]α=[Cα,Cα]wi h Cα<Cα.No e ha Fisa
gene aliza ion o a linea unc ion and o each α∈[0,1] we ha e
Fα(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
!Cαx,Cαx"i x ≥0;
!Cαx,Cαx"i x <0.
Thus he endpoin unc ions αand αa e no diffe en iable a x =0.Howe e F
is gH-diffe en iable on Rand F′(x)=C o allx∈R.Ingene al,i F(x)=C·g(x),
whe e g :R→Ris a diffe en iable unc ion and C ∈F
C, heni ollows ela i ely
easy ha he gH-de i a i e exis s and i is F′(x)=C·g′(x),bu heendpoin
unc ions αand αa e no necessa ily diffe en iable.
In gene al we ha e he ollowing esul which connec s gH-diffe en iabili y o F
and he diffe en iabili y o i s endpoin unc ions αand α.
Theo em 2 Le F :K→F
Cbe a uzzy unc ion. I F is gH-diffe en iable a x0∈K
hen, o each α∈[0,1],oneo he ollowingcaseshold:
(a) αand αa e diffe en iable a x0and
[F′(x0)]α=!min )( α)′(x0),( α)′(x0)+,max )( α)′(x0),( α)′(x0)+";
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(b)( α)′
−(x0),( α)′
+(x0),( α)′
−(x0)and ( α)′
+(x0)exis and sa is y ( α)′
−(x0)=( α)′
+(x0)
and ( α)′
+(x0)=( α)′
−(x0).Mo eo e
[F′( 0)]α=!min )( α)′
−(x0),( α)′
−(x0)+,max )( α)′
−(x0),( α)′
−(x0)+"
=!min )( α)′
+(x0),( α)′
+(x0)+,max )( α)′
+(x0),( α)′
+(x0)+"
P oo . The p oo is a consequence o Theo em 9 in [6] and Theo em 1.
Rema k 1 No e ha he gH-diffe en iabili y is coinciden wi h he H-diffe en ia-
bili y (diffe en iabili y in he sense o Hukuha a in oduced by Pu i andRalescu
[22] as a gene aliza ion o he Hukuha a de i a i e o se - alued unc ions [14])
only when αand αa e diffe en iable and ( α)′(x)≤( α)′(x) o all α∈[0,1].
Thus, he gH-diffe en iabili y is a mo e gene al concep o diffe en iabili y o uzzy
unc ions han he H-diffe en iabili y. The gH-diffe en iabili y concep is also mo e
gene al han G-diffe en iabili y, see [1].
We a e now going o de ine he pa ial de i a i e o a uzzy unc ion Fde ined
on K⊂Rn,i.e.,F(x)=F(x1, ..., xn)∈F
C o each x=(x1, ..., xn)∈K.Fo
his, gi en a uzzy unc ion F:K→F
C,wedeno e he uzzyin e alF(x)by
F(x)=! (x), (x)"and, o each α∈[0,1],
Fα(x)=! α(x), α(x)".
De ini ion 4 Le F be a uzzy unc ion de ined on K ⊂Rnand le x0='x(0)
1, ..., x(0)
n(
be a ixed elemen o K. We conside he uzzy unc ion hi(xi)=F(x(0)
1, ..., x(0)
i−1,xi,x(0)
i+1, ..., x(0)
n).
I hiis gH-diffe en iable a x(0)
i, henwesay ha Fhas hei hpa ialgH-de i a i e
a x0(deno ed by (∂F/∂xi)(x0))and(∂F/∂xi)(x0)=(hi)′(x(0)
i).
De ini ion 5 Le F be a uzzy unc ion de ined on K and le
x0='x(0)
1, ..., x(0)
n(∈Kbe ixed.Wesay ha FisgH-diffe en iable a x0i all he
pa ial gH-de i a i es (∂F/∂x1)(x0),..., (∂F/∂xn)(x0)exis on some neighbo hood
o x0and a e con inuous a x0.
No e ha i Fis gH-diffe en iable a x0, hen(∂F/∂xi)(x0)isa uzzyin e al.So,
o each α∈[0,1], we deno e
.∂F
∂xi
(x0)/α
=∂Fα
∂xi
(x0)=⎡⎢⎢⎢⎢⎣
∂Fα
∂xi
(x0),∂Fα
∂xi
(x0)⎤⎥⎥⎥⎥⎦.
We ob ain '∂Fα/∂xi((x0)and'∂Fα/∂xi((x0) omTheo em2.
Nex we p esen an in e es ing p oposi ion which will be used oob ainou main
esul s.
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P oposi ion 1 Le F :K→F
Cbe a uzzy unc ion. I F is gH-diffe en iable a
x0∈K hen, o eachα∈[0,1], he eal- alued unc ion
α+ α:K→Ris
diffe en iable a x0.Mo eo e ,
∂Fα
∂xi
(x0)+∂Fα
∂xi
(x0)=
∂' α+ α(
∂xi
(x0).(5)
P oo . The p oo is a consequence o Theo em 2.
F om p e ious de ini ion we can de ine he g adien o a uzzy unc ion as ollows.
De ini ion 6 Gi en he uzzy unc ion F :K→F
C, heg adien o Fa x
0,deno ed
by ˜
∇F(x0),isde inedby
˜
∇F(x0)=66∂F
∂x17(x0), ..., 6∂F
∂xn7(x0)7,(6)
whe e (∂F/∂xj)(x0)is he j h pa ial G-de i a i e o F a x0.
No e ha ˜
∇F(x)isan-dimensional uzzy ec o . Fo he g adien o a uzzy unc-
ion we use he symbol ˜
∇,whe eas o heg adien o a eal- alued unc ionweuse
he symbol ∇.
De ini ion 7 Le F :K⊂Rn→F
Cbe a uzzy unc ion, whe e K ⊂Rnis an open
se . Suppose now ha he e is x0∈Ksuch ha g adien o F,˜
∇F, is i sel gH-
diffe en iable a x0, ha is, o eachi, he unc ion∂F
∂xi:K→F
Cis gH-diffe en iable
a x0.Deno e hegH-pa ialde i a i eo ∂F
∂xiby
D2
ijF(x0)o ∂2F
∂xixj
(x0),i i !j,
and
D2
iiF(x0)o ∂2F
∂x2
i
(x0),i i =j.
I F is wice gH-diffe en iable a each x0in K, we say ha F is wice gH-diffe en iable
on K, and i o each i,j=1,2, ..., n, he c oss-pa ial de i a i e ∂2F
∂xixjis con inuous
unc ion om K o FC,wesay ha Fis wicecon inuouslygH-diffe en iable on K.
We de ine a m- imes con inuously gH-diffe en iable uzzy unc ion in way simila
o De ini ion 7, ha is, F:K→F
Cis m- imes con inuously gH-diffe en iable
on Ki and only i all o he pa ial gH-de i a i es o o de m∈Nexis and a e
con inuous (in he sense o uzzy unc ion).
I Fis gH-diffe en iable we ha e ha he endpoin unc ion αand αa e no
necessa ily diffe en iable. Howe e , om P oposi ion 1, we ha e ha α+ αis
7
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