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Journal of Industrial Engineeringand Management
JIEM, 2013 – 6(4): 1124-1138 – Online ISSN: 2013-0953 – Print ISSN: 2013-8423
htt:!!"#$"%i$%r&!10$3926!'ie$899
An integrated approach for solving a MCDM problem, combination of
entropy fuzzy and F!"#ME$%EE techni&ues
Amin Shahmardan1, Mohammad Hendijani Zadeh2
1Department of Industrial Engineering and Management Systems, Amir Kabir University of Tecnology, Teran !Iran"
# Department of $inancial Management, Scool of Management, University of Teran, Teran !Iran"
amin%samardan&aut%ac%ir , endi'ani&ut%ac%ir
(eceived) *uly #+1 Accepted) September #+1
Abstract'
!urpose' The intention of this paper is the presentation of a new integrated approach for
solving a multi attribute decision maing problem b! the use of entrop! fu""! and
#$%&'M(TH(( )fu""! preference raning method for enrichment evaluation* techni+ues
Design(methodology(approach' -n these sorts of multi attribute decision maing problem, a number
of criteria and alternatives are put forward as input data raning of these alternatives according
to mentioned criteria is regarded as the outcome of solving these inds of problems initiall!,
weights of criteria are determined b! implementation of entrop! fu""! method according to
determined weights, #$%&'M(TH(( method is e.erted to ran these alternatives in terms of desirabilit! of /M )decision maer*
Findings' 0eing in an uncertain environment and vagueness of /Ms judgments, lead us to
implement an algorithm which can deal with these constraints properl! this techni+ue namel!
called entrop! fu""! as a weighting method and #$%&'M(TH(( is performed to fulfill this
approach more precisel! according to tangible and intangible aspects the main finding of
applied approach is the final raning of alternatives helping /M to have a more reliable
decision
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#riginality()alue' The main contribution of this approach is the giving real significance to /Ms
attitudes about mentioned criteria in determined alternatives which is not elucidate in former
approaches lie Anal!tical Hierarch! %rocess )AH%* furthermore, previous methods lie
Shanon entrop! do not pa! attention sufficientl! to satisfaction degree of each criterion in
proposed alternatives, regarding to /Ms statements comprehensive e.planations about these
procedures have been made in miscellaneous sections of this article
*ey+ords ) entrop! fu""!, #$%&'M(TH((, multi criteria decision maing, fu""! set, decision maer,
fu""! environment
1. Introduction
Mltile riteri* "eii%n *.in& (M/M) i * %er+l t%%l e" i"el +%r *r*ieent *n"
r*n.in& r%le %nt*inin& ltile, *ll %n+litin&, riteri* (ilel, .%.*n, 7*n,
2006)$ Seletin& * r%er eth%" reire *n ini&ht *n*li *%n& **il*le M/M
tehnie$ Mlti /riteri* eii%n M*.in& (M/M) eth%" h*e een ileente" +reentl
in ter %+ %lin& "i++erent r%le in %th %+ ert*in *n" nert*in enir%nent$
""iti%n*ll, it i inre*in&l e" in enir%nent*l %li e*l*ti%n * (*) it %++er the
%iilit t% "e*l ith intri*te ie, () it in%r%r*te riteri* th*t *re "i++ilt t% %netie,() it rereent * h%liti ie in%r%r*tin& t*n&ile * ell * int*n&ile (%r ;+ier<)
*et, %+ten ne&lete" %ther e*l*ti%n tehnie h * =P (Mn"*, 2004)$
One %+ the r%le hih h* *l* een %ntr%eri*l +%r * *n+*trer * * eii%n
M*.er (M), i the eleti%n %+ * r%'et *%n& * ner %+ r%'et *%r"in& t% t*n&ile *n"
int*n&ile (+ier) *et$ >hi r%le h* een t +%r*r" * * neri*l e#*le in thi
*er$ ?hen * *n+*trer i "eterine" t% h%%e * r%'et, "i++erent r%et %+ th%e
r%'et h*e t% e rtinie" t% elet the %t r%+it*le %ne$ @irt %+ *ll, *n+*trer h%l"
eli"*te the riteri* %+ *r*ieent *n" ei&ht the re&*r"in& t% their i%rt*ne "e&ree$ @%r
ei&htin& thee ei+i riteri*, * &re*t ner %+ tehnie h*e een tilie" li.e n*lti*l
=ier*rh Pr%e (=P) hih * "eel%e" S**t (1982) relt in *ttriti%n %+ ei&ht
t% the ei+i riteri* (M*h*ri, Srin&*el, e r.er Aere.e, 2004B >r.in, ern*r"ini
M*h*ri, 2011)$ In +%rer %r., /h*.r*%rt *n" *ni. (2006) el%e" =P tehnie in
eletin& the %ti*l *teri*l h*n"lin& eient n"er * ei+i h*n"lin& enir%nent *n"
er+%re" enitiit *n*li t% i"enti+ the %t riti*l *n" r%t riteri* in the eleti%n
r%e$ In *n%ther t" hel" *& *n" O"eir (2006), *n intelli&ent *r%*h i r%%e",
here %th tehnieB + l%&i *n" =P *re %e t%ðer, re+erre" t% * + =P, *n"
* e" +%r * *hine t%%l eleti%n r%le$ Cener*ll, e *re +*in& ith "i++erent nert*in
*et "e t% h*in& in%lete in+%r*ti%n *%t the$ In the l*t +e e*r, ner%
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t"ie *ttetin& t% h*n"le thi nert*int$ ireii%n, *n" 'etiene h*e een *rrie"
%t *i*ll e*n %+ + et the%r, * + et the%r i&ht r%i"e the +le#iilit
nee"e" t% rereent the ireii%n %r *&e in+%r*ti%n reltin& +r% * l*. %+ .n%le"&e %r
in+%r*ti%n$ >he *li*ti%n %+ + et the%r t% lti riteri* e*l*ti%n eth%" n"er the
+r*e%r. %+ tilit the%r h* r%en t% e *n e++etie *r%*h$ @%r int*ne * it enti%ne"
e+%re, + the%r *n" =P *re %ine" t% e%e the @ =P (@=P) eth%" (Dee,
/hen, /h*n&, 2008), hih i * + e#teni%n %+ =P, *n" * "eel%e" t% %le
hier*rhi*l + r%le$ >h, neeit %+ in& the%r %+ + intr%"e" *"eh (1965)
i ineit*le$ Een th%&h ert*in riteri* *re e#ree" in *ntit*tie re+erene, %e %+
the *re t*te" in * *lit*tie %er*ti%n hih h%l" e interrete" *ntit*tiel$ In %ther
%r", e*e %+ the *&ene *n" nert*int %+ "eii%n *.er< '"&eent, + ner
l%&i i *lie"$ In thi *er, Entr% ?ei&ht n"er Intiti%niti @ enir%nent eth%" i
el%e" +%r ei&ht *ttriti%n$ +ter %n"tin& +irt t*&e, e *re in nee" %+ the %ne %+ the
r*n.in& eth%" +%r %r Mlti /riteri* eii%n n*li (M/)$Pre+erene 7*n.in& Or&*ni*ti%n
Meth%" +%r Enrihent E*l*ti%n (P7OME>=EE) i %ni"ere" * %ne %+ the %t e++iient *n"
it*le r*n.in& eth%" (ilel et *l$, 2006)$ @le#iilit *n" iliit %+ thi %tr*n.in& eth%"
*.e it %re "eir*le +%r it er (Cel"er*nn, Sen&ler 7ent, 2000)$ P7OME>=EE
tehnie h* een ree*te"l in *ri% +iel" li.e %rt+%li% eleti%n r%le (Aether* "e
lei"*, 2011), t%. tr*"in& ie (/h*h*r%&hi, l*"i E+*h*ni%r, 2006), ->*le
*n" M*reh*l (1995) %n"te" *n *li*ti%n +%r the P7OME>=EE eth%" t% e*l*te *ter
re%re r%'et$ l-Fl% *n" ->*le (1998) e P7OME>=EE +%r r%'et %rt+%li% in the
*ter re%re %nte#t$ le*.*&e *n*&eent tr*te& %+ *ter "itriti%n net%r. h* een
*n*le" M%r*i *n" "e lei"* (200G) in& P7OME>=EE in * &r% "eii%n %nte#t$
@-P7OME>=EE (@ Pre+erene 7*n.in& Or&*ni*ti%n Meth%" +%r Enrihent E*l*ti%n) i
the e#teni%n %+ P7OME>=EE n"er "%in*ne %+ + enir%nent$ ?e e @-P7OME>=EE
eth%" t% "e*l ith the *&ene *e" +r% the lin&iti ter hih *re e" in
e#rein& the "i++erene *%n& the *ltern*tie$ e %+ @-P7OME>=EE, thee *ltern*tie
(r%'et) *re r*n.e" +r% et t% %rt t% *.e it e*ier "eii%n *.in& +%r M in nert*in
enir%nent (Hil* *"eiren, 2011)$
>hi *er e%"ie +%r re*r.*le eti%nB +irt %+ *ll, there *re %e e#l*n*ti%n *%t
Entr% @ eth%" *n" @ >he%r$ Ne#t e&ent %+ thi *rt i "e%te" t% %e
l*ri+i*ti%n *%t P7OME>=EE *n" @-P7OME>=EE tehnie *n" it nee*r reiite %+
the$ >he ne r%%e" *r%*h i "erie" in eti%n 3 %+ thi *er$ ?hile, ttin&
+%r*r" *n illtr*tie e#*le, *ll %+ the enti%ne" eth%" *re tilie" t% %le * M/M
r%le *n" it i *ttete" t% e %re +*ili*r ith r*ti*l e %+ thee tehnie in 4th
eti%n %+ thi *er$ lti*tel the %nli%n *%t e#l*ine" tehnie i *"e, in l*t *rt
%+ thi *er$
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2. Principles of Fuzzy Entropy and F-PROMETHEE
2.1. Fuzzy Entropy
%r"in& t% the e%n" l* %+ ther%"n*i, * te h* * ten"en t% e%e %re
"i%r"ere" %er tie, hih "el*re the entr% %+ te *nn%t i"e inherentl$ Sh*n%n(1948) * the +irt %ne h% "e%ntr*te" the %net %+ Entr% in te hi
%ni*ti%n the%r$ Entr% i %ne %+ the h*r*teriti %+ *n t%h*ti te *n" n%
"*, it i %ni"ere" * %ne %+ the %t re*lent *li*ti%n in *ri% +iel"$ M%t %+ the
tie, e#*t *le %+ *r*eter *re %re, hene, nert*int inre*e "r**ti*ll$
the e %+ + ner *n" *ritheti, e *n "e*l ith thi it*ti%n r%erl$ @
entr% * intr%"e" *"eh (1968), l*ter e D* *n" >erini (19G2) intr%"e" %e %+
it reireent hih hele" t% h*e * etter intiti%n +%r "e&ree %+ +ine$ In *"eh<
+ et, * "e&ree %+ eerhi *le eteen er% *n" %ne i *i&ne" t% e*h eleent$
C* *n" erher (1993) &&ete" the %net %+ *&e et$ tine *n" rill% (1996)
%inte" t% the %ini"ene %+ intiti%niti + et (I@S) ith %net %+ *&e et$ S%e
e*re %+ Entr% @ *n" "een"ent r%ertie %+ th*t h*e een &&ete" F*r
(199G), Sin&r*ll* *n" %%.er (2004), Et% (1981) et$ In thi *rt, e reie %e %+
thee "e+initi%n$
Definition 1. n IFS A in the niere %+ "i%re X i "e+ine" ith the +%ll%in& +%r
(t*n*%, 1986):
A={⟨ x ,μ A( x) , v A( x)⟩ | x∈ X } (1)
?here
μ A: X →[ 0,1] , v A : X →[0,1] (2)
?ith the %n"iti%n
0 ≤ μ A( x )+v A( x) ≤ 1 , ∀ x∈ X (3)
>he ner μ A(x) *n" v A(x) "en%te the eerhi "e&ree *n" n%n-eerhi "e&ree %+ xt% A, reetiel$
Oi%l, e*h %r"in*r + et * e ritten *: (=n& /hen, 2009)
{⟨ x , μ A( x ) ,1−μ A( x )⟩ | x∈ X } (4)
@%r e*h IFS A in X. ?e ill *ll (5) the intiti%niti in"e# %+ x in A. It i * e*re %+
heit*n "e&ree %+ x t% A. It i %i% th*t 0 ≤ π A(x) ≤ 1 +%r e*h x∈ X.
π A( x) = 1−μ A( x)−v A( x) (5)
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@%r %neniene %+ n%t*ti%n, IFSs( x) i "en%te" * the et %+ *ll IFSs( x) in X. (t*n*%,
1986)$
Definition 2 (e, i* 7%, 2000)$ @%r eer A ∈ IFSs( x), the IFS λA +%r *n %itie re*l
ner λ i "e+ine" * +%ll%:
λ A = {⟨ x ,1−(1−μ A( x ))λ , (v A( x ))λ ⟩ | x ∈ X } (6)
Sh*nn%n (1948) r%%e" the entr% +nti%n *%r"in& +%ll%in& +%rl*ti%n:
H ( p1 , p 2 , ... , pn)=−∑i=1
n
pi log( pi) (G)
( P 1, P 2,..., P n) rereent the r%*ilitie %+ r*n"% *ri*le *ll*te" +r% * r%*ilit
+nti%n P$ *e" %n the Sh*nn%n +nti%n, e D* *n" >erini (19G2) "e+ine" * n%n
r%*ilit entr% +nti%n %n * +init nier*l et$
X ={ x1 , x2 , ... , xn } (8)
E LT ( A) =−k ∑i=1
n
[μ A( x i) lnμ A( xi) + (1−μ A( x i)) ln(1−μ A( xi)) ] , k >0 (9)
@%ll%in& e*ti%n h% the e*re %+ intiti%niti + Al*h% *n" Ser&i*"i (200G)
th*t *ti+ie" the +%r enti%ne" reireent$
E IFS
LT ( A)=−1 /(nln2)∑
i=1
n
[ μ A( xi) ln μ
A( x
i)+v
A( x
i) lnv
A( x
i)−(1−π
A( x
i))ln (1−π
A( x
i))−π
A( x
i)ln2 ] (10)
In thi *er, %t*ti%n %+ ei&ht t %+ riteri* i "%ne the Entr% @ eth%"$ Det
W (!" ,!2 ,...,!n) e the ei&htin& et%r %+ riteri* ( ∑ #=1
n
! #=1 )$
! # $ 0.
% {% 1,% 2,...,% n} rereent the riteri*$
Ai= {⟨% # ,μ Ai(% #) , v Ai
(% #)⟩ ∣ % #∈% } , i=1,2 ,... ,& (11)
Ai rereent the *ltern*tie (r%'et)$ μ Ai *n" v Ai h% the "e&ree hih *ti+ie *n" n%t
*ti+ie the riteri%n % #$
μ Ai(% #)∈[0,1] , v Ai
(% #)∈[0,1] (12)
μ Ai(% #)+ v Ai
(% #)∈[0,1] (13)
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π Ai(% #) = 1 – μ Ai(%#) ' v Ai(%#) in"i*te th*t hi&her π Ai(% #) relt in %re heit*n *r&in %+ M
*%t i*t %+ Ai reet t% % #. Ne#t te i t% e +%ll%in& +%rl* t% e#hiit E IFS
LT (% #) %+
*n riteri%n$
?here # 1,2,...,n *n"1
(mln2) i * %nt*nt hih *re 0≤ E IFS
LT (% #)≤1 $
E IFS
LT (% #)=−1/( &.ln2)∑
i=1
&
[ μ i# (% #) lnμ i# (% #)+ vi# (% #) ln v i#(% #)−(1−π i#(% #))ln (1−π i#(% #))−π i#(% #) ln2 ] (14)
& i the ner %+ r%'et hih i 3 in %r r%le$ @in*ll # ("e&ree %+ "ier&ene) i
re*re" ei+i "e+initi%n %n riteri%n % #.
#=1− E IFS LT
(% #) , #=1,2 , ... , n (15)
>he *le %+ # rereent the %nherent %ntr*t intenit %+ riteri%n % #, then the entr%
ei&ht %+ the #th riteri%n i ! #=( # / ∑ #=1
n
( #
2.2. PROMEETHEE and F-PROMETHEE et!ods"
P7OME>==E * intr%"e" r*n (1982) *n" l*ter e#ten"e" *rn *n" M*reh*l
(1994)$ It i %ni"ere" * %ne %+ the %tr*n.in& eth%"$ >hi tehnie i re&*r"e" * *re*ti%n t% %lete *&&re&*ti%n (M>) eth%" (M*h*ri, Aere.e e r.er, 2004)$
P7OME>=EE tehnie i %ne %+ the intitie eth%" %+ M/M hih i % h intelli&ile +%r
M (*lli M*r%t*, 200G)$
@%r in& thi MM tehnie, %t +%ll%in& te *re *rrie" %t:
1$ =*in& * t*le inl"e" ei+i *ltern*tie * ell * ert*in riteri* +%r *eent$
2$ Pre+erene +nti%n h%l" e "e+ine" P #(,*)
hih t*te "ei*ti%n eteen t%
*ltern*tie (,*) %n * *rtil*r riteri%n + # int% * re+erene "e&ree r*n&in& 0 t%1$
3$ /h%%in& %ne %+ the i# %ile h*e %+ re+erene +nti%n t +%r*r" r*n,
Ain.e *n" M*reh*l (1986) (*l h*e, -h*e +nti%n, A-h*e +nti%n, leel
+nti%n, line*r +nti%n *n" C*i*n +nti%n)
P #( ,*) = # - #()− - # (*)) 0≤ P #( ,*)≤1 (16)
- #() '- #(*) e#ree the "ei*ti%n %re %+ t% *ltern*tie %n * ert*in riteri%n$
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4$ .n%in& the ei&ht %+ riteri* *ll*te" in rei% *rt (Entr% @) e "e+ine
the +%ll%in& +%rl*:
π ( ,*) = ∑ #=1
k
! # P #( , *) (1G)
ϕ ()=1/ (n−1)∑*
π ( ,*)(18)
ϕ ' ())=1/(n−1)∑*
π (* , )) (19)
ϕ ( )=ϕ ( )−ϕ
' () (20)
/() *n" / ' () rereent %itie *n" ne&*tie re+erene +l% +%r e*h *ltern*tie hih
e*re h% *n *ltern*tie (*) i %t r*n.in& (+%rl* 1G) %r %t r*n.e" (+%rl* 18)$
/() (+%rl* 19) "il* the *le +nti%n *n" *ltern*tie (*) ill e %re *ttr*tie, i+ it*le +nti%n h* * hi&her *%nt$
In *rti*l r*n.in&, e +*e ith three it*ti%n in ter %+ eri%rit %+ *ltern*tie (*) t%
*ltern*tie () (+%rl* 20), ein& in"i++erent t% *ltern*tie (+%rl* 21), ein&
in%*r*le t% *ltern*tie (+%rl* 22) (r*n Ain.e, 1985)$
{
φ .())>φ .(*) )n( φ
' ())<φ ' (*)φ
.())>φ .(*) )n( φ ' () )=φ
' (*)
φ
.
())=φ
.
(*) )n( φ
'
())<φ
'
(*)
(21)
{φ .())=φ .(*) )n( φ
' ())=φ ' (*) (22)
{φ .())>φ .(*) )n( φ
' ())>φ ' (*)φ
.())<φ .(*) )n( φ ' ())<φ ' (*)
(23)
On the %ther h*n" e %+ PROMETHEE II , e %nl e*re the 0n1t *n" it ill e %r *in
e*l*ti%n *li*ti%n +%r "eii%n *.in&$ Eer *ltern*tie h* * hi&her 0n1t it h* * etter
%iti%n in lti*te r*n.in& *n" e h*e * %lete r*n.in&$
?hen * M t*te hi e#eriene *n" %ni"er*ti%n in lin&iti ter * int "*t*, the
%rit *n" +ine *re t*.en l*e$ In h * irt*ne, r%*ilit %+ *.in& it*.e
in %r *eent *rie inre"il$ >he %lti%n i the e %+ @-P7OME>==E th*t i the
%in*ti%n %+ +ine *n" P7OME>=EE$
In thi *er @-P7OME>=E i ileente" * it * &&ete" C%* *n" D&er%
(2000)$ >he r%e"re %+ @-P7OME>=EE * *e * P7OME>=EE, t + l%&i &et
in%le" in thi eth%"%l%& * ell$ in& thi tehnie, it ill e e*ier +%r M t%
interret hi *lit*tie *ttit"e *n" in+%r*ti%n t% *the*ti*l e#rei%n
x (,&,*) i the reent*ti%n %+ * + ner hih i h%n in +i&re 1$
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x i * *ri*le el%n& t% the + et *n" it eerhi +nti%n - ( x) h* the *le K0,1L$
@%r x or x3* , x "%e n%t el%n& t% the et$
@%r K x 4 4 *] the eerhi "e&ree i in"i*te" eerhi +nti%n th*t *rie eteen
0 t% 1$
@i&re 1$ Preent*ti%n %+ + ner x(,&,*)
In %r *er, the *ti%n i th*t ei&ht %+ riteri* ere r%&ht Entr% @ *n"
re+erene threh%l" (5 n p) *re ri ner$
%r"in& t% the r%ertie %+ %r M/M r%le, line*r re+erene +nti%n (te5) ith
in"i++erene *n" trin&ent re+erene threh%l" (5 n p) i %r et h%ie t% e el%e"$
{ P () , *)=0
P () , *)=(( −5)/ ( p−5) P () , *)=1
-67 ( ≤5 -67 5≤( ≤ p
-67 ( ≥5
(24)
In @-P7OME>=EE ( ) e#hiit the "i++erene eteen t% + ner %+ (,*) hih relt
in * + ner h * (n,8, )$ %e e*ti%n *re tr*n+%re" t% the +%ll%in& e*ti%n
reetielB
{ P ( ,*)=0
P ( ,*)=( −5)/ ( p−5) P ( ,*)=1
-67 n−8≤5 -67 5≤n−8 n n+ ≤ p
-67 n+ ≥5
(25)
Eenti*l +%rl* +%r *i %t*ti%n ith + ner (%i Pr*"e, 19G8) *re
h*r*terie" in t*le 1$
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""iti%n (&,,*) L9 ⊕ (n,8, ) L9 (&n, 8, * ) L9
O%ite –(&,,*) L9 (– &, , *) L9
Str*ti%n (&,,*) L9 ' (n, 8, ) L9 (&'n, 8, * ) L9
Mltili*ti%n S*l*r (&,,*) L9 × (n, 0 , 0) L9 (&n, n, *n) L9 L9
Mltili*ti%n @
for m>0, n>0 (&,,*) L9 ⊗ (n,8, ) L9 ≈ (&n, 8&n, &*n) L9
for m<0, n>0 (&,,*) L9 ⊗ (n,8, ) L9 ≈ (&n, n'&, *n'8&) L9
for m<0, n<0 (&,,*) L9 ⊗ (n,8, ) L9 ≈ (&n, '*n'&, 'n'8&) L9
Inverse for (m>0) (&,,*) L9:" ≈ (&:" , *&2 , &:2) L9
>*le 1$ *i + %er*ti%n
Eent*ll, in *li*ti%n %+ @-P7OME>=EE, e *re &ien %e + ner$ %r"in& t%
the enti%ne" %t*ti%n, *le ill e *ll*te"$ 7i&ht n% thi + ner h%l"
e h*n&e" t% the "e+ +%r *%r"in& t% +%ll%in& +%rl*:
F () ,& ,*)=()+2m+*)/4 (26)
One *+ter th*t, le+t t*&e %+ P7OME>=EE *re er+%re" *n" 0 , 0 ' *n" 0n1t ill e e*re"
re&*r"in& t*te" +%rl* +%rerl$
#. Proposed Entropy $F-PROMETHEE inte%rated approac!
it h* enti%ne" e+%re, "eterin*ti%n %+ ei&ht rel*te" t% riteri* h%l" e %te"
intiti%niti + entr% hih * reente" Al*h% *n" Ser&i*"i (200G)$ >hi
te h%l" e t*.en *%r"in& in+%r*ti%n &ien t% the %ntr*t%r (M) (;,v). Seentl,
thee riteri* * ell * their ei&ht *re el%e" t% e*re 0 , 0 ' *n" 0n1t %+ eer
*ltern*tie (r%'et)$@in*l "eii%n i *"e %*ri%n %+ *ltern*tie 0n1t , ll "*t* *re
+ ner *n" the h%l" e h*n&e" t% "e+ +%r$ >here+%re e h*e t% %n"t
+%ll%in& t*&e:
1$ *t* /%lleti%n: >he eene %+ thi t*&e i "een"ent %n e#eriene *n"
%ni"er*ti%n %+ *n+*trer (M)$ Eer *ltern*tie h* * ; *n" v "e&ree hih
*ti+ %r n%t *ti+ reetiel * ei+i riteri%n$
2$ i&nin& ei&ht t% the riteri* intiti%niti + entr%$ >hee *ll*ti%n *re "%ne
re&*r"e" t% enti%ne" +%rl*$
3$ Me*rin& the 0 , 0 ' *n" 0n1t %+ eer *ltern*tie (r%'et)$
4$ 7*n.in& %+ r%'et *%r"in& t% r%&ht *%nt$
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&. 'uerical e(aple of proposed approac!
In thi t*&e, ith the r%e %+ h*in& * etter n"ert*n"in& *%t *ll"e" r%e"re, *
"eii%n-*.in& r%le i %le"$ In %r illtr*tie int*ne, e h*e * *n+*trer h% i
&%in& t% elet %ne %+ the three r%%e" r%'et *%r"in& t% i# enti%ne" riteri* (/%t
Per+%re", /%n"ti%n 7i., ret*ti%n i*t, eri%" %+ er+%rin& the r%'et, iil*rit
ith +%rer r%'et, *n" r*te %+ retrn)$ % # *n" Ai *re the %l %+ riteri* *n" r%'et
reetiel$
@%ll%in& "i*&r* (@i&re 2) "eit te--te t*&e tr*.e" +%r %lin& * M/M r%le
the e Entr% @ *n" @-P7OME>=EE eth%"$
@i&re 2$ She*ti "eiti%n %+ the ne r%%e" *r%*h +%r %lin& * M/M r%le
Entr% @ /%t*ti%n: e "ie" rei%l, e e Intiti%niti @ Entr%+%r ei&htin& the riteri*$ @%ll%in& t*le rereent the *%nt %+ (μ A# , v A# , π A#) %+ e*h r%'et
in eer riteri%n$
)1 )2 )# )& )* )+
μ A1 v A1 π A1 μ A2 v A2 π A2 μ A3 v A3 π A3 μ A4 v A4 π A4 μ A5 v A5 π A5 μ A6 v A6 π A6
1 0$6 0$1 0$3 0$8 0$1 0$1 0$G 0$2 0$1 0$55 0$25 0$2 0$6 0$1 0$3 0$8 0$1 0$1
2 0$8 0$1 0$1 0$G 0$15 0$15 0$65 0$25 0$15 0$5 0$3 0$2 0$65 0$2 0$15 0$G5 0$2 0$05
3 0$85 0$1 0$05 0$G5 0$1 0$15 0$G5 0$15 0$1 0$5 0$2 0$3 0$G 0$1 0$2 0$G 0$1 0$2
>*le 2$ e%tr*ti%n %+ "e&ree hih *ti+ *n" n%t *ti+ riteri%n % # in r%'et Ai
the e %+ Entr% @ r%e"re +%ll%in& ei&ht %+ riteri* *re *hiee"$
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)riteria )1 )2 )# )& )* )+
?ei&ht 0$238 0$221 0$131 0$042 0$162 0$206
>*le 3$ r%&ht *%nt %+ ei&ht %+ riteri*
7i&ht n% *%r"in& t% r%&ht ei&ht %+ riteri* *n" int "*t* (+ ner) rel*te" t%
ei+i r%'et rereente" in +%ll%in& t*le, @-P7OME>=EE eth%" i e#erte" t% h*e *
+in*l r*n.in& t% r%%e" r%'et +%r *n+*trer (M)$ *lit*tie *et %nerne" t%
*n+*trer< re+erene *re interrete" *%r"in& t% +%ll%in& + ner$ *lit*tie
t*teent *re lite" * AC: Aer C%%", C: C%%", M: M%"er*te, =: =i&h, A=: Aer =i&h, AD:
Aer D%, D: D%, ME: Me"i, =I: =i&h I*t, *n" A=I: Aer =i&h I*t$
,ualitatie stateents Euialent fuzzy nu/ers ,ualitatie stateents Euialent fuzzy nu/ers
AC (0,0$1,0$2) AD (0,0$1,0$2)
C (0$2,0$3,0$45) D (0$2,0$35,0$4)
M (0$45,0$5,0$6) ME (0$4,0$5,0$6)
= (0$6,0$G5,0$85) =I (0$6,0$G,0$8)
A= (0$85,0$9,1) A=I (0$8,0$9,1)
>*le 4$ Din&iti *le +%r i%rt*ne
)riteria )1 )2 )# )& )* )+
O/0ectie
FunctionMin Min Ma( Min Ma( Ma(
1 K800,1000,1200L C ME K15,1G,18L D K0$4,0$5,0$6L
2 KG00,800,1000L M =I K14,16,1GL ME K0$45,0$55,0$6L
3 KG50,G50,900L = =I K15,18,20L ME K0$35,0$45,0$55L
>*le 5$ @ ner *n" *lit*tie t*teent rel*te" t% eer % # in eer Ai
it * t +%r*r" +%rerl, re&*r"in& the *ll*te" 0 <1t %nerne" t% r%'et, the ill e
r*n.e" reetiel et t% %rt$
0 0 ' 0 <1t Ranin%
1 0$10G8G40 0$2G00310 -0$16215G 3
2 0$31005 0$013G445 0$2963055 1
3 0$1319045 0$266053 -0$1341485 2
>*le 5$ @-P7OME>=EE +l%
@i&re 3$ >%t*l 7*n.in&
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J%rn*l %+ In"tri*l En&ineerin& *n" M*n*&eent – htt:!!"#$"%i$%r&!10$3926!'ie$899
*. )onclusion
In thi *er, * ne *r%*h t% %le the M/M r%le h* een reente"$ In *n
nert*in *n" t%h*ti enir%nent, in& + the%r *n" it *li*ti%n *re 'ti+i*le$
ei&htin& eth%" (Entr% @) *n" *n %tr*n.in& tehnie (@-P7OME>=EE) ere
e#l*ine" th%r%&hl in rei% eti%n$ n illtr*tie e#*le * t +%r*r" t% e%e
%re **int ith r*ti*l e %+ inte&r*te" *r%*h$ M (*n+*trer) h%l" elet
%ne %+ the r%%e" *ltern*tie (r%'et) *%r"in& t% ert*in riteri* %+ th%e r%'et$ In
thi neri*l e#*le, the e %+ Entr% @ tehnie, ei&ht %+ riteri* ere
"eterine" *n" @-P7OME>=EE eth%" * ileente" t% h*e * +in*l r*n.in& %+ r%'et
re&*r"in& r%&ht ner$ *ntit*tie *n" *lit*tie *et hih *re %nerne" t% M<
re+erene *re etter ere" in *erte" eth%"B eentl * %re reli*le
%neene i *%lihe"$ I+ e *re %e" t% enti%n t% the *in %ntriti%n %+
r%%e" *r%*h, e *n &&etB ein& in *n nert*in enir%nent *n" *&ene %+
M< '"&ent, le*" t% ileent *n *l&%rith hih *n "e*l ith thee %ntr*int
r%erl$ >hi tehnie n*el *lle" Entr% @ * * ei&htin& eth%" *n" @-
P7OME>=EE i e#erte" t% +l+ill thi *r%*h %re reiel re&*r"in& t% t*n&ile *n"
int*n&ile *et$ On the %ther h*n", +%rer tehnie li.e Sh*nn%n Entr% * * ei&htin&
eth%" "i" n%t %rie the M< %ini%n *%t riteri* *n" r%'et th%r%&hl r*ther th*n
the r%%e" *r%*h$
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J%rn*l %+ In"tri*l En&ineerin& *n" M*n*&eent, 2013 ($'ie$%r&)
rtileQ %ntent *re r%i"e" %n * ttriti%n-N%n /%eri*l 3$0 /re*tie %%n liene$ 7e*"er *re *ll%e" t% %, "itrite
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