An iterative approach for estimation of efficiency by weighted distance

Int. J.IndustrialMathematisVol. 3,No. 1(2011)25-33

An Iterative Approah for Estimation of

EÆieny by Weighted Distane

R. Saneifard

Departmentof Mathematis,OroumiehBranh, IslamiAzadUniversity,Oroumieh,Iran.

Reeived26April2010; revised11Deember2010; aepted 26Deember 2010.

|||||||||||||||||||||||||||||||-Abstrat

In group deision analysis, numerous approahes have been suggested in an attempt to

solve theproblemofaggregation ofindividualfuzzyopiniontoforma grouponsensusas

the basis of a group deision. If the inputs and outputs are fuzzy quantity, the deision

making unitsan notbe easilyevaluated andranked usingtheobtainedeÆienysores.

Inthisartile,akindofmodiedideabasedoninterative methodisintroduedfor

rank-ingof deisionmaking unitswith fuzzy databy weighted distane.

Keywords: Fuzzynumber;EÆieny;Dataenvelopmentanalysis;Weighteddistane.

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1 Introdution

More and more, modeling tehniques, ontrol problems and operation researh

algo-rithmshavebeendesignedto fuzzydatasinetheoneptoffuzzynumberandarithmeti

operationswiththese numbers wasintroduedand investigatedrst byZadeh. Data

en-velopment analysis was suggested by Charnes, Cooperand Rhodes [2 ℄, and builton the

ideaofFarrell[3 ℄whihisonernedwiththeestimationoftehnialeÆienyandeÆient

frontiers. Insomeases,wehavetouseimpreiseinputandoutput. Todealquantitatively

withimpreisionindeisionprogress,BellmanZadeh[1 ℄introduethenotionoffuzziness.

Some researhers have proposed several fuzzy models to evaluate deision making units

with fuzzy data and introdue a ranking approah with eÆieny measure of the model

[9 ℄. In thispaper, the researher rst introdues an approah with weighteddistane for

ranking of deisionmaking unitswith risp data. Seond, thismodel is usedfor ranking

of deisionmaking unitswith fuzzydata.

This paperisorganized asfollows. In Setion2, theresearherrealls some fundamental

results on fuzzy numbers. The proposed model is introduedin Setion 3. An approah

for ranking by usingweighted distane is introduedin setion 4. Interative method is

introduedinSetion5.

2 Preliminaries

The basidenitionof a fuzzynumbergiven in[5 ,11 , 12,13 , 14 ,15 ℄is asfollows:

Denition2.1. Afuzzy number isa mapping : <![0;1℄withthe followingproperties:

1. isan upper semi-ontinuousfuntion on <,

2. (x)=0 outside of some interval [a

1 ;b

2 ℄ <,

3. There are real numbers a

2 ;b

1

suh as a

1 a

2 b

1 b

2 and

3.1 (x) isa monotoniinreasingfuntion on [a

1 ; a

2 ℄;

3.2 (x) isa monotonidereasing funtion on [b

1 ; b

2 ℄;

3.3 (x)=1 for all x in [a

2 ;b

1 ℄.

Denition 2.2. A fuzzy number is a fuzzy setA on the real line < suh that

(x)= 8

>

>

<

>

>

: f

A

(x) if x2[a

1 ;a

2 ℄;

1 if x2[a

2 ;a

3 ℄;

g

A

(x) if x2[a

3 ;a

4 ℄;

0 otherwise :

(2.1)

Suhthatf

A

(:)isinreasingfuntionon[a

1 ;a

2 ℄andg

A

(:)isdereasingfuntionon[a

3 ;a

4 ℄.

The -ut of a fuzzy number A is dened as [A℄

= fx j u(x) g. Sine (:) is

uppersemi-ontinuousthen-uts arelosed andboundedintervals and we representby

[A℄

=[f 1

A ();g

1

A ()℄.

Denition2.3. Afuzzy number A in parametri form isa pair (A ;A)of funtionsA()

and A() that 01, whih satises the following requirements:

1. A() is a bounded monotoni inreasing left ontinuous funtion,

2. A() is a bounded monotoni dereasing left ontinuous funtion,

3. A)A ();01.

Denition 2.4. The symmetri triangular fuzzy number A =(x

0

;), with defuzzierx

0

and fuzziness isa fuzzyset wherethe membership funtion isas

(x)= 8

>

>

>

>

<

>

>

>

>

: 1

(x x

0

+) x

0

xx

0 ;

1

(x

0

x+) x

0

xx

The parametri form ofsymmetri triangularfuzzy numberis

A()=x

0

+ ; A()=x

0

+ :

Denition 2.5. For fuzzy set A Support funtion isdened as follows:

supp(A)=fxj(x)>0g ;

wherefxj(x)>0g is losure of set fxj(x)>0g:

Denition 2.6. [16℄, A funtion f : [0;1℄![0;1℄ symmetri around 1

2

, i.e. f( 1

2

)=

f( 1

2

+) for all 2 [0; 1

2

℄, whih reahes its minimum in 1

2

, is alled the bi-symmetrial

weighted funtion. Moreover, the bi-symmetrial weighted funtion isalled regular if

(1) f( 1

2 )=0,

(2) f(0)=f(1)=1,

(3) R

1

0

f()d= 1

2 :

Denition2.7. [14 ℄,FortwoarbitraryfuzzynumbersAandBwith-uts[f 1

A ();g

1

A ()℄

and [f 1

B ();g

1

B

()℄ respetively, the quantity

d(A;B) =[ Z

1

0

f()(f 1

A

() f 1

B ())

2

d+ Z

1

0

f()(g 1

A

() g 1

B ())

2

d℄ 1

2

(2.2)

is the weighted distane betweenA and B.

Denition 2.8. [4, 16 ℄, Let A is an arbitrary fuzzy number, the expeted interval and

expeted value of a fuzzy number A are noted by EI(A) and EV(A) respetively, and

onsidered as follows,(with f()=),

EI(A)=[E A

1 ;E

A

2 ℄=[2

R

1

0 f

1

A

()d;2 R

1

0 g

1

A

()d℄;

EV(A)= E

A

1 +E

A

2

2 =

R

1

0 f

1

A

()d+ R

1

0 g

1

A ()d

(2.3)

If A=(a

1 ;a

2 ;a

3 ;a

4

) isa trapezoidal fuzzy number then:

EI(A)=[ a

1 +2a

2

3 ;

2a

3 +a

4

3

℄; (2.4)

EV(A)= 1

3 (a

1 +2a

2 +2a

3 +a

4

): (2.5)

Proposition 2.1. If A and B are two fuzzy numbers and ; 2R then:

EI(A+B)=EI(A)+EI(B);

The addition and salar multipliationof fuzzynumbers aredened by theextension

priniple and an be equivalently represented as follows. For arbitrary fuzzy numbers

A =(A;A ) and B = (B;B), this artiledenes addition (A+B) and multipliationby

salark >0 as

(A+B)()=A()+B() ; (A+B)() =A()+B(); (2.7)

(kA )()=kA () ; (kA )()=kA(): (2.8)

Toemphasistheolletionofallfuzzynumberswithadditionandmultipliationasdened

by(2.2) and (2.3) denoted byF, whihisa onvex one.

3 The proposed model

Inthissetion,theresearherintroduerankingmodelbasedonweighteddistaneindata

envelopment analysis. This artile assumes that the DMU

p

is extreme eÆient [12 , 6 ℄.

Byomitting(X

p ;Y

p

) fromT

(PPSofCCRmodel),theresearherdenestheprodution

possibilityset T 0

asfollows,[7 ℄:

T 0

=f(X;Y)j X n

X

j=1;j6=p w

j X

j ; Y

n

X

j=1;j6=p w

j Y

j ; w

j

0;j =1; ;n;j 6=pg: (3.9)

Where

T

=f(X;Y)j X n

X

j=1 w

j X

j

; Y n

X

j=1;j6=p w

j Y

j ; w

j

0;j=1; ;ng: (3.10)

To obtaintheranking sore ofDMU

p

, thisartileonsiders thefollowingmodel:

min p

(X;Y)= P

m

i=1

f()(x

i x

ip )

2

+ P

s

r=1

f()(y

r y

rp )

2

s:t P

n

j=1;j6=p w

j x

ij x

i

; i=1 ;m;

P

n

j=1;j6=p w

j y

rj y

r

; i=1 ;s;

x

i

0; i=1 ;m;

y

r

0; r =1 ;s;

w

j

0; j =1 ;n;j6=p:

(3.11)

Where X =(x

1 ;:::;x

n

) , Y = (y

1 ;:::;y

n

) and =(

1 ;:::;

n

) are thevariables of the

model (3.11) and p

(X;Y) is the weighted distane (X

p ;Y

p

) from (X;Y) by weighted

distane, also f() is regular weighted funtion. Quadrati programming represents a

speial lass of nonlinear programming in whih the objetive funtion is quadrati and

the onstraints are linear. The KKT onditions of a quadrati programming problem

4 Comparison In Fuzzy DEA

Inthissetion,theresearhersupposesthat inputsand outputsof DMUsarefuzzy

num-bers. Therefore,

~

T 0

=f(X;Y)j X n X j=1;j6=p w j ~ X j ; Y n X j=1;j6=p w j ~ Y j ; w j

0;j=1; ;n;j6=pg (4.12)

Weighteddistane model withEqs. (2.2) an beextended to thefollowingmodel:

min p

(X;Y)= P m i=1 R 1 0 f()(f 1 x i () f 1 x ip ()) 2 d+ R 1 0 f()(g 1 x i () g 1 x ip ()) 2 d + P s r=1 R 1 0 f()(f 1 yr () f 1 yrp ()) 2 d+ R 1 0 f()(g 1 yr () g 1 yrp ()) 2 d

s:t (X;Y)2

~ T 0 : (4.13)

Where X =(x

1 ;:::;x

n

) , Y = (y

1 ;:::;y

n

) and =(

1 ;:::;

n

) are thevariables of the

model(4.13), thatallomponentsofvetorsX and Y forallDMUs arenon-negativeand

eah DMUhasat least onestritly positiveinputand output.

Forsolvingthemodel (4.13), we have some followingdenitions:

Denition 4.1. [8℄, For any pair of fuzzy numbers A and B the degree in A is bigger

than B has the following form:

M

(A;B)= 8 > > > > > < > > > > > :

0 if E

A 2 E B 1 <0; E A 2 E B 1 E A 2 E A 1 +E B 2 E B 1

if 02[E A 1 E B 2 ;E A 2 E B 1 ℄;

1 if E

A 1 E B 2 >0: (4.14) Where[E A 1 ;E A 2

℄and[E B

1 ;E

B

2

℄aretheexpetedintervalsofAandB. When

M

(A;B)= 1

2 ,

we will saythat Aand B are dierent. When

M

(A;B), we will saythat Ais bigger

than, or equal toB at least in degree and we will represent it by A

B.

Denition 4.2. Given a produtionpossibility(X;Y)2 ~

T 0

,we will say that it is produt

in degree in ~ T 0 if: min 8 > > < > > : M (x i ; P n j=1;j6=p w j ~ x ij ) ; M ( P n j=1;j6=p w j ~ y rj ;y r ) M (x i ;x~

ip ) ; M (~y rp ;y r )

i=1;:::;m r =1;:::;s

9 > > = > > ;

=: (4.15)

That is to say

x i P n j=1;j6=p w j ~ x ij

; i=1;:::;m;

y r P n w j ~ y rj

; r=1;:::;s:

Thereis: x i P n j=1;j6=p w j (E x ij 2

+(1 )E x

ij

1

) ; i=1;:::;m;

y r P n j=1;j6=p w j (E y rj 1

+(1 )E y

rj

2

) ; r =1;:::;s:

(4.17)

(For more detailssee[12℄).

Denition 4.3. A prodution possibility (X

o ;Y o ) 2 ~ T 0

is an -aeptable optimal

solu-tion of model (4.13) if it is an optimal solutionof the following model:

min p

(X;Y)

= P m i=1 R 1 0 f()(f 1 x i () f 1 x ip ()) 2 d+ R 1 0 f()(g 1 x i () g 1 x ip ()) 2 d + P s r=1 R 1 0 f()(f 1 yr () f 1 yrp ()) 2 d+ R 1 0 f()(g 1 yr () g 1 yrp ()) 2 d

s:t (X;Y)2

~ T 0 : (4.18) Where ~ T 0

=f(X;Y)j X

n X j=1;j6=p w j ~ X j ; Y n X j=1;j6=p w j ~ Y j ; w j

0;j =1; ;ng (4.19)

Proposition 4.1. If

1 < 2 then ~ T 02 ~ T 01 .

We writemodel(4.18) as follows:

min p

(X;Y)

= P m i=1 R 1 0 f()(f 1 x i () f 1 x ip ()) 2 d+ R 1 0 f()(g 1 x i () g 1 x ip ()) 2 d + P s r=1 R 1 0 f()(f 1 yr () f 1 yrp ()) 2 d+ R 1 0 f()(g 1 yr () g 1 yrp ()) 2 d s:t x i P n j=1;j6=p w j (E x ij 2

+(1 )E x

ij

1

) i=1;:::;m;

y r P n j=1;j6=p w j (E y rj 1

+(1 )E y

rj

2

) r =1;:::;s;

y

r

0 r =1;:::;s;

w

j

0 j=1;:::;n:

(4.20)

Model (4.20) is a risp -parametri model. Therefore this artile an solve it by the

interativemethod. Now thisstudyisgoing to explaintheinterative method.

DMUs Input -ut Output -ut

A (11,12,14) [11+,14-℄ (10,10,10) [10,10℄

B (30,30,30) [30,30℄ (12,13,14,16) [12+,16-2℄

C (40,40,40) [40,40℄ (11,11,11) [11,11℄

D (45,47,52,55) [45+2,55-3℄ (12,15,19,22) [12+3,22-3℄

Table 2

Theoptimalvaluesof-parametrimodel.

A

B

C

D

(R anking)

0.0 25.5 3.5 0 55.2 (C B AD)

0:0

0.1 27.0 3.5 0 55.2 (C B AD)

0:1

0.2 28.5 3.5 0 55.2 (C B AD)

0:2

0.3 30.0 3.5 0 55.2 (C B AD)

0:3

0.4 31.6 3.5 0 55.2 (C B AD)

0:4

0.5 33.3 3.5 0 55.2 (C B AD)

0:5

0.6 35.0 3.5 0 55.2 (C B AD)

0:6

0.7 36.7 3.5 0 55.2 (C B AD)

0:7

0.8 38.6 3.5 0 55.2 (C B AD)

0:8

0.9 40.4 3.5 0 55.2 (C B AD)

0:9

1.0 42.3 3.5 0 55.2 (C B AD)

1:0

4.1 Interative Method

Regardingtoproposition(4.1),toobtainthenearest(X;Y)of ~

T 0

impliesalesserdegree

ing Kaufman [11 ℄ the researher onsiders 11 sales, whih allow for dierent hoie of

deision-makeridea in(4.20) model.

1: =0:0 unaeptablesolution

2: =0:1 Pratiallyunaeptablesolution

3: =0:2 Almost unaeptablesolution

4: =0:3 Very unaeptablesolution

5: =0:4 Quiteunaeptablesolution

6: =0:5 Neitheraeptable norunaeptable solution

7: =0:6 Quiteaeptable solution

8: =0:7 Very aeptable solution

9: =0:8 Almost aeptablesolution

10: =0:9Pratially aeptable solution

11: =1:0Completelyaeptable solution

We hoie the

0

is the minimum aeptable degree with deision-maker idea. Then,

thisartilesolvingthe(4.20)-parametri modelforeah

k

thatk =1;:::;(10 10

0 ).

We obtainthe

k

-aeptable optimal fuzzy value of objetive funtionof original model

(4.13) with

k

-aeptablesolutionof model(4.20) inmodel(4.13).

4.2 Example

Example 4.1. We will onsider a simple example was introdued in [10 ℄ with its data

listed in Table1. TheseDMUs areevaluated by proposed modelin 4.13 with dierent

k .

The-parametri model isas follows:

min R

1

0

(x 11 )d+2 R

1

0

(y 10)d+ R

1

0

(x 14+2)d

s:t x30w

B +40w

C +w

D

(53:5 7:5)

yw

B

(15 2:5)+11w

C +w

D

(20:5 7)

x0;

y0;

w

B ;w

C ;w

D 0:

(4.21)

The-parametri modelfor B,C andD anbeshowed similarly. Theresults isshown

in Table 2.

5 Conlusion

In thepresentartileamodiedapproahbasedon weighteddistaneis introduedfor

rankingofdeisionmakingunitswithfuzzy data. Themethod isbasedontheinterative

method. -aeptable optimal solution of proposed model for 1

2

is an aeptable

-Referenes

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