How to Write Abstract-FMCDM

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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%&ether, 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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