ALS Components

8/10/2019 ALS Components

1/12

E L S E V I E R

European Journal of Operational Research 80 (1995) 462-473

EUROPE N

JOURN L

OF OPER TION L

RESE RCH

o m p o n en t s o f e f f i c i en cy ev a l u a t i o n

i n d a t a en v e l o p m en t a n a ly s is

gha Iqba l l i a , C a t h e r i n e S L e r m e L a w r e n c e M S e i f o rd c

a School of Management, The Un iversity of Massachusetts at Amherst, Amherst, M A 01003, USA

b C arroll School o f Management, Boston College, Chestnut Hill, )VIA 02167, U SA

c Industrial Engineering and Operations Research, The University of Massachusetts at Amherst, A mherst, M A 01003, USA

Abstract

This paper examines three essential components which comprise efficiency evaluation in data envelopment

analysis. The three components are present in each DEA model and determine the implicit evaluation scheme

associated with the model. These components provide a framework for classifying the various DEA models with

respect to (i) the form of envelopment surface, (ii) the orientation, and (iii) the pricing mechanism implicit in the

multiplier lower bounds. The discussion focuses on the standard DE A models, includes additional issues relating to

efficiency evaluation, and is illustrated by a computational example.

K e y w o r d s : Data envelopment analysis; Efficiency; Framework

I I n t r o d u c t i o n

Since the seminal paper by Charnes, Co op er and R hode s in 1978, a variety of data envelopment

analysis models has appea red in the literature as have numerou s studies employing the tec hnique

(Banker, Charnes, Cooper, Swarts, Thomas, 1989; Seiford, 1990). Each of the various models for data

developme nt analysis (DEA) seeks to determine which of n decision making units D M U s ) determine an

e n v e l o p m e n t s u r f a c e 1 (or e f f i c i e n t f r o n t i e r ) . Units that lie on (determine) the surface are deemed e f f i c i e n t

in DEA terminology. Units that do not lieon the surface are termed i n e f f i c i e n t and the analysis provides

measures of their relative efficiency. These measures of relative efficiency depend upon the particular

evaluation scheme implicit in the DEA model employed. As will be shown, the efficiency evaluation is

determined by three components: envelopment surface, orientation, and multiplier lower bounds. As will

be demonstrated, these components provide a unifying framework for classification of DEA models.

* Corresponding author.

I This envelopment surface can be interpreted as the empirical production surface which represents the state of the technology.

0377-2217/95/ 09.50 1995 Elsevier Science B.V. All rights reserved

S S D I 0377-2217(94)00131-U


2/12

A.L Al i e t al. / European Journal of Operat ional Research 80 1995) 462-473

463

T h r e e o f t h e D E A m o d e l s t h a t a r e m o s t o f t e n a ss o c i a t e d w i t h t h e D E A m e t h o d o l o g y a r e t h e C C R ,

B C C a n d a d d i t iv e m o d e l s 2 . T o i n t r o d u c e t h e c o m p o n e n t s w h i c h f o r m t h e f r a m e w o r k d e v e l o p e d i n t hi s

p a p e r w e b r ie f ly e x a m i n e t h e s e m o d e l s . O u r n o t a t i o n i s a s f o ll o w s . W e a s s u m e t h a t t h e r e a r e n D M U s t o

b e e v a l u a t e d . E a c h D M U c o n s u m e s v ar y i n g a m o u n t s o f m d i f fe r e n t in p u t s to p r o d u c e s d if f e re n t

o u t p u t s . S p e c i f i c a l ly , d e c i s i o n m a k i n g u n i t l , c o n s u m e s a m o u n t x a > 0 o f i n p u t i a n d p r o d u c e s a m o u n t

Yrl

0

o f o u t p u t r 3 . F i n a ll y , in t h e m o d e l f o r m u l a t i o n s ,

X t

a n d Y / d e n o t e , r e s p e c t i v e l y , t h e v e c t o r s o f

i n p u t a n d o u t p u t v a l u e s f o r

D M U t

w h i l e t h e s n m a t r i x o f o u t p u t s is d e n o t e d g a n d t h e m n m a t r i x

o f i n p u t s i s d e n o t e d X .

T h e p r im a l a n d d u a l l i n e a r p r o g r a m m i n g s t a t e m e n t s f o r t h e i n p u t o r i e n te d ) C C R m o d e l a r e:

C C R , p Y t , ) l ) ) C C R I D Y t , X l ) )

m i n O - s l e +

l e ) m a x /zY /

s t . YA - s = Yl , s . t .

v X t = 1 ,

O X - X A - e = O , t z Y - v X < O ,

A_>O, e>__O, s>__O, g _> e l , v > e l .

B C C I p Y t , X t ) )

m i n O - s l s +

l e

s. t . YA - s = Yl ,

O X I - X A - e = O ,

1 A = 1 ,

A_>O, e_>O , s>_O.

T h e p r i m a l a n d d u a l l in e a r p r o g r a m m i n g s t a t e m e n t s f o r th e i n p u t o r i e n t ed ) B C C m o d e l a r e :

B C C D Y 1 , ) l ) )

m a x t z Y t + oJ

s . t .

v X I

= 1 ,

I - * Y - v X + ~ o l < O ,

I z > e l , v > e l .

T h e p r i m a l a n d d u a l l in e a r p r o g r a r n r n i n g s t a t e m e n t s o f t h e a d d i t iv e m o d e l a r e :

A D D p Y t , X z ) ) A D D D Y t , X t ) )

m i n - l s + l e ) m a x ] Y /- -

1]X --~ .o

s . t .

Y A - s = Y t ,

s . t. / x Y - v X + ~ ol < 0 ,

- X A - e = - X I , / z ~> 1 , v ~> 1 ,

1 A = 1 ,

A > _ O , e _ > O , s > _ O .

E a c h o f t h e a b o v e m o d e l s s e e k s t o d e t e r m i n e t h e e f fi c ie n c y o f a p a r t i c u l a r D M U Y ~,

X t )

w i t h r e s p e c t

t o t h e e n v e l o p m e n t s u r f a c e d e t e r m i n e d b y t h e e f fi c ie n t b e s t p r a c t i c e ) D M U s . T h e s o l u t io n o f a D E A

m o d e l f o r D M U Y /,

X l )

r e s u l ts i n a m e a s u r e o f e f fi c i en c y a n d f u r t h e r i d e n t i f ie s a n e f f i c ie n t p o i n t

~ , ) l ) o n t h e e n v e l o p m e n t s u r f a c e. A s w e s h a ll se e , e ff i c ie n c y e v a l u a t i o n a n d t h e l o c a t i o n o f t h is

e f f i c ie n t p r o j e c te d ) p o i n t ~ , ) ~ t ) a r e d e p e n d e n t o n b o t h t h e f o r m o f t h e e n v e l o p m e n t s u r f a c e a n d th e

e v a l u a t i o n s y s t e m im p l i c it i n t h e p a r t i c u l a r D E A m o d e l . F u r t h e r m o r e , s u b t l e d if f e r e n c e s i n t h e m a t h e -

m a t i c a l f o r m u l a t i o n s o f t h e m o d e l s c a n p r o d u c e s i gn i fi c an t d i ff e r e n ce s i n o n e e v a l u a ti o n c o m p o n e n t y e t

l e a v e a n o t h e r u n c h a n g e d .

2 The acronymic designation o f the first two models arises from the authorship o f the original articles Charnes, Cooper, and

Rhodes, 1978, and B anker, Charnes, and Co oper 1984). The additive model is developed in Charnes et al . 1985).

3 The assumption o f posit ivity is for exposit ional convenience. Th ere are several wa ys in which i t can b e re laxed . Se e, for

example, All and S eiford 1990) and th e discussion therein.


3/12

4 6 4 A.L Ali et aL / European Journal of Operational Research 80 1995) 462-473

E x a m i n i n g t h e s t r u c t u r e o f t h e p r i m a l m o d e l f o r m u l a t io n s g i v e n a bo v e , w e n o t i c e t h a t t h e

C C R I e

a n d

B C C I e

m o d e l s d i f f e r o n l y i n t h a t t h e l a t t e r i n c l u d e s a c o n v e x i t y c o n s t r a i n t ( 1A = 1 ). A g a i n w i t h r e s p e c t

t o m o d e l s t r u c t u r e , o n e o b s e r v e s t h a t s u b s t i t u t i o n o f 0 = 1 ( a n d = 1 ) i n t h e

BCC~e

m o d e l p r o d u c e s t h e

a d d i t i v e m o d e l

A D D p ) .

T h i s s i m i l a ri t y i n m o d e l s t r u c t u r e p o i n t s t o a n u n d e r l y i n g f r a m e w o r k w i t h i n

w h i c h t h e s e a n d o t h e r D E A m o d e l s a r e e m b e d d e d . A s w i ll b e s e e n , t h e i d e n t if i c a ti o n o f s u c h a

f r a m e w o r k p r o v i d e s a b e t t e r u n d e r s t a n d i n g o f t h e s o l u t io n m e t h o d o l o g y a n d , a l s o, a i ds i n t h e i n t e r p r e t a -

t i o n o f r e s u l t s .

T h e c o n s e q u e n c e s o f d i f f e r e n c e s i n f o r m u l a t i o n a ll o w f u r t h e r c o m p a r i s o n s b e t w e e n m o d e l s. I t c a n b e

e a s il y s h o w n t h a t t h e e n v e l o p m e n t s u rf a c e s f o r t h e B C C a n d a d d i ti v e D E A m o d e l s a r e i d e n ti c a l a n d t h u s

t h e s e t s o f D M U s d e t e r m i n e d t o b e e f f i c i e nt a r e e x a c t ly t h e s a m e f o r b o t h m o d e l s . H o w e v e r , a s w e s h a ll

d e m o n s t r a t e , t h e t w o m o d e l s d i f f e r i n t h e i r i m p l i c it e v a l u a t i o n p r i n c i p l e s 4. I n c o n t r a s t , b o t h t h e C C R

a n d B C C m o d e l s o b e y th e s a m e e v a l u a t io n p r in c i pl e s. H o w e v e r , t h e f o r m ( g e o m e t r y ) o f t h e e n v e l o p m e n t

s u r f a c e is d i f fe r e n t f o r t h e s e t w o m o d e l s . T h e i n t e r re l a t io n s h i p s b e t w e e n e n v e l o p m e n t s u r f a ce , o r i e n t a -

t i o n , a n d l o w e r b o u n d s p e c i f i c a t io n a n d t h e i r e f f e c t o n e f f i c i e n c y e v a l u a t i o n a r e t h e s u b j e c t o f t h is p a p e r .

T h e f o r m a t f o r t h i s p a p e r i s a s f o l l o w s . I n S e c t i o n 2 , w e d i s c u s s t w o u n d e r l y i n g t y p e s o f e n v e l o p m e n t

s u r f a c e s f o r d a t a e n v e l o p m e n t a n a l y s i s . T h e n o t a t i o n i n t r o d u c e d f a c i l i t a t e s t h e s u b s e q u e n t d i s c u s s i o n o f

d i f f e r e n t e v a l u a t i o n s y s t e m s . T h e m a t h e m a t i c a l p r o g r a m m i n g m o d e l s i n t r o d u c e d i n t h i s s e c t i o n a r e

n o n o r i e n t e d m o d e l s a n d g i v e r is e t o t h e n o n - o r i e n t e d p r o j e c t i o n s d e s c r ib e d i n S e c t i o n 3. S e c ti o n 4

c o n s t r a s t s i n p u t a n d o u t p u t o r i e n t a t i o n s w h i l e S e c t i o n 5 d i s cu s s e s t h e e f f e c t o f c h a n g i n g t h e u n i t s o f

m e a s u r e m e n t o f th e d a t a . E f f i c ie n c y e v a l u a t io n c o m p o n e n t s a r e i d e n t if i e d a n d i l l u s tr a t e d w i t h a

c o m p u t a t i o n a l s t u d y i n S e c t i o n 6. C o n c l u d i n g r e m a r k s a r e p r e s e n t e d i n S e c t io n 7 .

2. Envelopment surfaces

T h e v a r i o u s m o d e l s p r o p o s e d i n t h e D E A l i t e r a tu r e e m p l o y s ev e r a l d i f f e r e n t t y p e s o f e n v e l o p m e n t

s u r f a c e s . O u r d i s c u ss i o n , h o w e v e r , w il l f o c u s o n t h e t w o b a s i c p i e c e w i s e l i n e a r e n v e l o p m e n t s u r f a c e s

c o m m o n l y r e f e r r e d t o a s

consta nt returns-to-scale C R S)

a n d

variable returns-to-scale V R S)

s u r f a c e s .

T h i s i s n o t i n a n y w a y r e s t r i c t i v e 5 :

O n e o f t h e t w o ty p e s o f e n v e l o p m e n t s u r fa c e s ( C R S o r V R S ) r e s u lt s f r o m t h e p a r t i c u l a r D E A m o d e l

e m p l o y e d , i .e ., t h e p a i r o f (d u a l ) li n e a r p r o g r a m s t h a t e f f e c t d a t a e n v e l o p m e n t . F o r e x a m p l e t h e C C R

m o d e l p r o d u c e s a C R S e n v e l o p m e n t su r f a c e w h i l e t h e B C C m o d e l a n d t h e a d d i t iv e m o d e l p r o d u c e a

V R S e n v e l o p m e n t s u r f a ce . T o f a c il it a te o u r d e v e l o p m e n t o f a f r a m e w o r k i n t h e s e c t io n s t o f ol lo w , w e

f ir st e x a m i n e t h e f o ll o w in g g e n e r a l n o n o r i e n t e d e n v e l o p m e n t m o d e l s .

T h e p r i m a l a n d d u a l s t a t e m e n t s o f th e n o n o r i e n t e d c o n s t a n t r e t u r n s -t o - s c a le

C R S )

m o d e l a r e :

x , , u , , , ) ) x , ,

. , , . ,

m i n

- u t s + v t e ) m a x tx Yt - v XI

s . t . YA - s = Y~, s . t .

t ~ Y - v X < _ O ,

- X A - e = - X t , / z >_ u l , v _> v l .

A_>O e_>O s_>O.

4 T h e e v a l u a t i o n p r in c i p l e s s y s t e m ) f o r a p a r t i c u l a r D E A m o d e l d e f i n e c r it e r ia t h a t d e t e r m i n e t h e m a n n e r i n w h i c h p r o j e c te d

p o i n t s a r e d e t e r m i n e d f o r t h e i n e f f i c i e n t D M U s . T h i s i s d i s c u s s e d i n S e c t io n s 3 a n d 4 .

5 F o r e x a m p l e , t h e m u l t ip l i c a ti v e m o d e l s C h a r n e s e t a l ., 1 9 82 , 1 9 83 ) p r o d u c e p i e c e w i s e l o g - li n e a r o r p i e c e w i se C o b b - D o u g l a s

e n v e l o p m e n t s . H o w e v e r t h e s e n o n - l i n e a r i ti e s c a n , by d a t a t r a n s f o r m a t i o n s , b e m a d e l i n ea r . S i m il a rl y , a l t h o u g h S e i fo r d a n d T h r a l l

1 9 9 0) p r o p o s e f o u r p i e c e w i s e l i n e a r e n v e l o p m e n t s u r f a c e s , t w o o f t h e i r s u r f a c e c l a s si f ic a t io n s r e s u lt f r o m h y b r i d c o m b i n a t i o n s o f

o u r t w o b a s i c s u r f a c e s . E x t e n s i o n o f o u r r e s u l t s t o t h e t w o a d d i t i o n a l h y b r i d s u r f a c e s i s s t r a i g h t f o r w a r d .


4/12

A.L A l i e t aL / E uropean Jou rna l o f Operat iona l Resea rch 80 (1995) 462-47 3

465

The primal and dual statements of the nono rien ted variable returns-to-scale VRS) model are:

V R S N p Y I , S l , U , v l ) )

w s N o Y , , x , , u , , v , ) )

min

- ( u t s + v l e ) m a x / Z Y l - u X t + 6 0

s.t.

Y A - - s = Y I ,

s.t.

~ Y - u X + 1 6 0 u t v > ~ v t.

1h = 1,

A_ > O e>_ O s_ >O.

We refer to the dual problem statements

( C R S N D o r V R S N D )

as

m u l t i p l i e r

or

p r i c i n g m o d e l s .

The

primal problems, on the other hand, can be characterized as

e n v e l o p m e n t

or

p r o j e c t i o n m o d e l s .

Of

course, only one of these probl ems primal or dual) needs to be solved; the solution to the other is easily

obtained by the duality theory of linear programming.

These nono rien ted progr ams directly address the underl ying characterizatio n of efficiency: A decision

making unit, l, is efficient if it lies on a facet-defining hyperplane of the envelopment surface;

specifically, a hype rpl ane of the form /xly - vtx = 0 for

C R S

envelopment or a hyperplane of the form

t z l y - - v l X

+ w l = 0 for FRS enve lopm ent,

The para mete rs u l and v z introd uced for notational convenience) are th e specific lower bounds on

the variables/x and u in the dual programs 6. Our discussion in the present section is with respect to the

general specification, u t and v t, of these vectors. Fur the r interp retati on and effect of specific lower

bounds are presented in Section 5.

Optimal values of variables for either of the primal nonorient ed envelopment programs for

D M U l

are

denoted by the s-vector

s t ,

the m-vector

e t ,

and the n-vector A . An optimal dual solution is given by the

s-vector/z , the m-vector

v t ,

and, for the VRS model, the variable

6 0 1 . 7

Identification of the envelopment surface requires the solution of a linear programming model for

each decision makin g unit l. Each of the n sets of values given by /z , v t, 60t), l = 1, 2, .. ., n, are the

coefficients of hyperplanes that define facets of the envelopment surface.

The

C R S e n v e l o p m e n t s u r f a c e

consists of particular facets that result from the intersection of

supporting) hyperp lanes in Em+s which pass throug h the origin) and the convex polyhedral cone

dete rmined by the vectors Yj, Xi) , j = 1, .. ., n; a hyperplane I z y - v x = ES=m/zryr - E i r n = l l . i X i = 0 inter-

sects the cone in a facet of the

C R S

envelo pment surface if and only if /.tY j- vXj = E~=l/.tryrj--

~m _

t = l 1 2 i X i j < O ,

f o r a l l j = 1 . . . . n

with equality for at least one j).

The

V R S e n v e l o p m e n t s u r f a c e

consists of particular facets that result from the intersection of

supporting) h yperpl anes in R m+s and the convex hull of the points Yj, X ) , j = 1 . . . . , n in Rm+s; A

hyperplane

I x y - v x + 60

= E S = l l . r Y r - - E i m = l b i X i - } - 6 0 = 0 intersects the convex hull in a facet of the

surface if an only if/zYj -

v X j

+ 60 = E~=l/z~y~j -

E i 1 v i x i j + oo


5/12

466 A.L Al i et al . / E uropean Journal of O perational Research 80 (1995) 462 -473

3 . N o n o r i e n t e d p r o j e c t i o n s

F o r t h e p r i m a l e n v e l o p m e n t p r o b l e m f o r D M U l , t h e o p t i m a l v e c t o r A d e f i n e s a p o i n t

{ n n /

x / = l j = l ]

o n t h e e n v e l o p m e n t s u r f a c e . T h i s p o i n t i s a l i n e a r c o m b i n a t i o n f o r t h e C R S m o d e l , o r a c o n v e x

c o m b i n a t i o n f o r t h e V R S m o d e l , o f u n i t s t h a t l ie o n t h e e n v e l o p m e n t s u r fa c e . T h e p o i n t ( ~ , X t ) is

r e f e r r e d t o a s t h e pro jec ted po in t . T h e f o l lo w i n g t h e o r e m e s t a b li s h e s t h a t t h e p r o j e c t e d p o i n t , i n fa c t , l ie s

o n a f a c e o f t h e e n v e l o p m e n t s u r f a ce .

T h e o r e m . Suppose A i s op t imal for the pr im al prob lem . Then the po in t (~ , J ( t ) = (g i = 1A j~t g j = , 1 A / Xt lies

on a face o f the enve lopment sur face de termined by a suppor t ing hyperp lane de f ined by the op t imal dua l

m ultiplie rs ix , v t, ogt).

Pr oof . F r o m d u a l i t y t h e o r y o f l i n e a r p r o g r a m m i n g , w e k n o w t h a t f o r e a c h j w i t h A~ > 0 , t h e c o r r e s p o n d -

i n g d u a l c o n s t r a i n t i s b i n d i n g , i .e . /x t Y ] -

v t x i

+ w = 0 . Thu s ea ch d ec i s io n m ak ing un i t j ~ A = { j I /x lY j -

v t x ] + o 9 = 0 } i s e f f i c i e n t a n d l i e s o n t h e h y p e r p l a n e i x t y - u t x + ogt = 0 . T h i s h y p e r p l a n e d e f i n e s a f a c e

( w h i c h m a y b e a f a c e t 8) o f t h e e n v e l o p m e n t s u r f a c e w i t h n o r m a l v e c t o r

ix ,

v t ). A s u f f i c i e n t , b u t n o t

n e c e s s a r y , c o n d i t i o n f o r m e m b e r s h i p i n A i s A ] > 0 . S i n c e e a c h p o i n t Y /, X i , f o r w h i c h A~ > 0 l i es o n t h e

p l a n e , w e h a v e

T h u s t h e p o i n t ( Y / , X 1) i s p r o j e c t e d o n t o t h e p o i n t ( ~ , J ( t) o n t h e h y p e r p l a n e , txty - v tx + o9 /= O. F o r the

C R S m o d e l , it c a n b e s i m i l a r l y s h o w n t h a t t h e p r o j e c t e d p o i n t l ie s o n t h e h y p e r p l a n e / x t y - v t x = 0. [ ]

T h e p r e c e d i n g t h e o r e m e s t a b l is h e d t h a t t h e e f f i c ie n t u n i t s (v e r ti c e s) w h i c h c o m p r i s e t h e p r o j e c t e d

p o i n t c o r r e s p o n d t o p o s it iv e A v a l u e s. A n a l t e r n a t e c h a r a c t e r i z a t i o n o f t h e p r o j e c t e d p o i n t i s a va i la b l e

f r o m t h e p r i m a l c o n s t r a i n t s .

( ~ , J f , ) = ( ~ A SY ],

~ A S X j ) = Y I + s , X t - e t ) .

j = l j = l

T h i s c h a r a c t e r i z a t i o n l e n d s i t s e l f t o r e f e r r i n g t o t h e v e c t o r s l a s t h e v e c t o r o f outpu t s lacks a n d t h e

m - v e c t o r e I a s t h e v e c t o r o f excess inputs. T h e r e l a t i o n s h i p b e t w e e n t h e v a l u e s o f o u t p u t s l a ck s a n d

e x c e ss i n p u t s a n d t h e r e l a ti v e p r i ce s i s e v i d e n t f r o m t h e c o m p l e m e n t a r y s l a c k n e ss c o n d i t i o n s f ro m l i n e a r

p r o g r a m m i n g d u a l i t y t h e o r y . S p e c i f ic a l ly

s l > O = i l r_ _ l

- U r r = 1 . . . S

e l > 0 = u [ = v /, i = 1 . . . . . m .

S i n c e p o s i t iv e v a l u e s o f t h e s l a c k o r e x c e s s v a r i a b l e s c o r r e s p o n d t o m u l t i p l i e r s w h i c h a r e a t l o w e r

b o u n d , t h e t e r m ix tsZ+ vie I i s e x a c t ly t h e o b j e c t i v e v a l u e u t s t + v i e t a t o p t i m a l i t y . T h i s e s t a b l i s h e s t h e

e a r l i e r c l a i m t h a t t h e s e l o w e r b o u n d s ( u l, v l ) d i r e c t l y e f f e c t t h e d i r e c t i o n o f p r o j e c t i o n . I n S e c t i o n 5 , w e

8 A f a c e t i s a f u l l d i me n s i o n a l f a c e .


6/12

A.L Ali et al. / E uropean Journal of Operational Research 80 1995) 462-4 73 4 6 7

f u r t h e r e x a m i n e t h e p a r t i c u l a r r e l a t io n s h i p b e t w e e n e f f ic i e n c y e v a l u a ti o n s o b t a i n e d f o r d i f f e r e n t s pe c if i-

c a t io n s o f t h e p a r a m e t e r s

u t

an d v t .

4. Oriented projections

T h e p r e v i o u s s e c ti o n e x a m i n e d n o n - o r i e n t e d p r o j e c t io n s ; th e a d d i ti v e m o d e l b e i n g t h e m o s t f r e q u e n t

e x a m p l e a p p e a r i n g i n t h e D E A l i t e r a tu r e o f a n o n - o r i e n t e d V R S m o d e l . I n c o n tr a s t, th e D E A m o d e l s

e x a m i n e d n e x t a r e o r i e n t a b l e .

I n p u t o r i e n t e d m o d e l s m a x i m i z e t h e p r o p o r t i o n a l d e c r e a s e i n t h e i n p u t v e c t o r w h i le r e m a i n i n g w i t h i n

t h e e n v e l o p m e n t s p a c e ) . C l e a rl y , a p r o p o r t i o n a l d e c r e a s e i s p o s s i b l e u n t il a t le a s t o n e o f t h e e x c e s s i n p u t

v a r i ab l e s is r e d u c e d t o z e r o . T h e i n p u t o r i e n t e d m o d e l s f o r C R S a n d V R S e n v e l o p m e n t a r e s ta t e d

b e l o w 9 .

x , , u , , v , ) )

C R S t D Y t , X t , u t , v t ) )

m i n

0 - e u l s + v i e ) m a x t ~ Y t ,

s . t . Y A - s = Y t , s . t . v X l = 1 ,

O X t - X A - e = O , t x Y - v X O , e > O , s -> O . ~ >__ e u t , v -> e v t .

x t , u , , v , ) )

V R S I D Y I , g l , I ll , v l ) )

m i n

0 - - e u l s + v l e ) m a x i . t Y + o 9,

s . t . Y A - s = Y t , s . t . v X l = 1 ,

O X t - X A - e = O , t z Y - v X + to l _< 0,

1 A = I , / z > e u

l , v - > e v t .

A_>O e ->O s_>O.

O u t p u t o r i e n t e d m o d e l s m a x i m i z e t h e p r o p o r t i o n a l i n c r e a s e in t h e o u t p u t v e c t o r p o s s ib l e w h i l e

r e m a i n i n g w i t h i n t h e e n v e l o p m e n t s p a c e . C l e a r l y , a p r o p o r t i o n a l i n c r e a s e i s p o s s i b l e u n t i l a t l e a s t o n e o f

t h e o u t p u t s l a ck v a r ia b l e s is r e d u c e d t o z e r o . T h e o u t p u t o r i e n t e d m o d e l s f o r C R S a n d V R S e n v e l o p -

m e n t a r e s t a t e d b e l o w :

( C R S o . ( r , , x , , u , v ) )

C R S o D Y I , X t , u l , v l ) )

m a x q b + 8 u l s + v l e )

m i n

v X t

s. t . ~bY~ - YA + s = 0, s . t . /xY = 1,

X A + e = X t , - t z Y + v X > O ,

A > O, e ~ O , s -> O. ].L >__ E u l, 1J -> E V l.

9 T h e l a b e l s f o r t h e s e m o d e l s u n d e r s c o r e t h e r o l e o f t h e e n v e l o p m e n t s u r f a c e i n o u r e f fi c ie n c y e v a l u a ti o n f r a m e w o r k . F o r C R S

e n v e l o p m e n t , C R S a n d C R S o w i t h u l = 1 , v I = 1 ) c o r r e s p o n d t o t h e i n p u t a n d o u t p u t o r i e n t e d C C R m o d e l s C h a r n e s , C o o p e r

a n d R h o d e s , 1 9 7 8) . F o r V R S e n v e l o p m e n t w i t h u l = 1 ,

v l =

1 ) i n p u t a n d o u t p u t o r i e n t a t i o n c o r r e s p o n d t o t h e

B C C

m o d e l s

B a n k e r , C h a r n e s a n d C o o p e r , 1 9 8 4) .


7/12

468 A . I . A l i e t a l. / E u r o p e a n J o u r n a l o f O p e r a t io n a l R e s e a r ch 8 0 1 9 9 5 ) 4 6 2 - 4 7 3

x t , u , v , X t , u , v

m a x ~ + e u t s + v t e ) m a x v X t + t o

s.t . ~bY~ - YA + s = 0, s. t. /,I(l = 1,

X A + e = X t , - I x Y + v X +

~o

>_ O ,

1 A = I , / . ~ > e u

l , v >_ e v t .

A > _ 0 e _ > 0 s _ > 0 .

O p t i m a l p r i m a l v a r i a b l e s f o r d e c i s i o n m a k i n g u n i t , l , a r e d e n o t e d 0 l ( o r ~ b t), A , s t , e t . W h i l e t h e

o p t i m a l v a l u e f o r 0 l ( o r ~b ) is u n i q u e t h e r e c a n b e m u l t i p l e o p t i m a l s o l u t i o n s f o r t h e o t h e r v a r i a b l e s . W e

d e n o t e o p t i m a l d u a l v a r ia b l e s b y /~ t , vt a n d , w h e r e a p p l ic a b l e,

to t .

F o r t h e s e o r i e n t e d m o d e l s , t h e p o i n t

(Y ~, X t ) i s p r o j e c t e d o n t o a f a c e o f t h e e n v e l o p m e n t s u r f a c e , i . e ., a s u p p o r t i n g h y p e r p l a n e d e f i n e d b y t h e

o p t i m a l d u a l m u l t i p l i e r s ~ l , v t , w t ) .

B y o r i e n t i n g t h e p r o j e c t i o n , f o r e i t h e r a C R S o r V R S e n v e l o p m e n t s u r f a c e o n e o b t a i n s d i f f e r e n t

p r o j e c t e d p o i n t s f o r i n e f f i c ie n t u n it s . T h e s e p r o j e c t e d p o i n t s r e f l e c t th e p a r t i c u l a r p r i o r i ty o f t h e

o r i e n t a t i o n . I n p u t - o r i e n t a t i o n s e e k s a p r o j e c t e d p o i n t s u c h t h a t t h e p r o p o r t i o n a l r e d u c t i o n i n i n p u t s i s

m a x i m i z e d . I m p l i c i t i n a n i n p u t o r i e n t a t i o n i s t h a t t h e p r i m a r y o b j e c ti v e o f t h e D M U b e i n g e v a l u a t e d i s

t o g a i n e f f i c i e n c y b y r e d u c i n g e x c es s i n p u t c o n s u m p t i o n w h i l e c o n t i n u i n g t o o p e r a t e w i t h i ts c u r r e n t

t e c h n o l o g y m i x ( c h a r a c t e r i z e d b y t h e a c t u a l i n p u t r a ti o s ). S i m i la r ly , o u t p u t - o r i e n t a t i o n s e e k s a p r o j e c t e d

p o i n t s u c h t h a t t h e p r o p o r t i o n a l a u g m e n t a t i o n i n o u t p u t s i s m a x i m i z e d . I n t h is s it u a t io n , t h e p r i m a r y

o b j e ct i ve i s t o r e a c h e f f ic i e n c y b y f o c u s i n g o n p r o d u c t i v i t y g a i ns w h i le p r e s e r v i n g t h e c u r r e n t o u t p u t m i x .

S a t i s f a c ti o n o f t h e p r i m a r y o b j e c t iv e , f o r e i t h e r o r i e n t a t i o n , m a y n o t b e s u f f ic i e n t t o r e a c h t h e

e n v e l o p m e n t s u r f a c e , i . e . a t t a i n e f f i c i e n c y .

O r i e n t e d p r o j e c t i o n s , h e n c e , r e q u i r e

f i r s t

d e t e r m i n i n g t h e m a x i m u m r a d i a l m o v e m e n t ( i n t e r m e d i a t e

p r o j e c t i o n ) i n t h e i n p u t o r o u t p u t d i r e c t i o n . ( I n t e r m s o f t h e m o d e l s s t a t e d a b o v e t h i s c o r r e sp o n d s t o a n

o p t i m a l v a l u e f o r 0 l o r ~ t , r e s p e c ti v e ly . ) T h i s m a x i m u m r a d i a l m o v e m e n t i d e n t i f i e s a n i n t e r m e d i a t e p o i n t

Y t , O t X t ) f o r t h e i n p u t o r i e n t a t i o n o r a p o i n t ( ~ bt Yl, X t ) fo r t h e o u t p u t o r i e n t a t i o n . T h i s i n t e r m e d i a t e

p o i n t r e f l e c t s t h e p r i m a r y e m p h a s i s o f t h e i n p u t o r o u t p u t o r i e n t a t i o n .

H o w e v e r r a d i a l m o v e m e n t b y it s e l f i s n o t s u f f ic i e n t t o a l w a y s g u a r a n t e e t h a t t h i s i n t e r m e d i a t e p o i n t

w i ll l ie o n t h e e f f i c ie n t f r o n t i e r. O n e m u s t f u r t h e r a p p l y a n o n - o r i e n t e d p r o j e c t i o n to t h e i n t e r m e d i a t e

p o i n t . T h e r e s u l t in g p r o j e c t e d p o i n t ( ~ , ) ~ t ) , w h i c h w i ll li e o n t h e e f f ic i e n t f r o n t i e r, i s ex p r e s s ed a s

Y l + s t , 0 1 X t - e t ) fo r in pu t o r i en ta t io ns an d as (c, tY l + s t , X t - e t ) f o r o u t p u t o r i e n t a t io n s .

T h e o r i e n t e d m o d e l s s t a t e d i n t h i s s e c t io n a c c o m p l i s h th e t o t a l p r o j e c ti o n . I. e ., t h e s o l u t io n o f t h e s e

m o d e l s p r o d u c e s a f i n a l p r o j e c t e d p o i n t ( ~ , X t ) w h i c h li e s o n t h e e f f i c ie n t f r o n t ie r . T h e p r e c e d i n g

c o n c e p t u a l d i s c u s s i o n i n t e r m s o f a n i n t e r m e d i a t e p o i n t o n l y il l u s tr a t e s a n e s s e n t ia l n u a n c e i n t h e

m e c h a n i s m o f e f fi c i e n c y e v a l u a t i o n f o r a n o r i e n t e d m o d e l . A l t h o u g h w e w i ll n o t d o s o , th i s c o n c e p t u a l

d e v e l o p m e n t c a n b e f o r m a l i z e d a s a t w o s t a g e p r o c e d u r e c h a r a c t e r i z e d b y t h e s o l u t i o n o f f ir s t s t a g e a n d

s e c o n d s t a g e m o d e l s . S e e A l i a n d S e i f o r d ( 1 9 9 3 ) f o r d e t a i l s .

5 . M u l t i p l i e r l o w e r b o u n d s a n d u n i t s o f m e a s u r e m e n t

A s d e s c r i b e d e a r l i e r, t h e m a n n e r i n w h i c h t h e p r o j e c t i o n ( o n t o t h e d e v e l o p m e n t s u r f a c e ) is e ff e c t e d in

d i f f e r e n t D E A m o d e l s d e p e n d s n o t o n l y o n t h e o r i e n t a t i o n b u t a l so o n t h e v a l u e s o f t h e v e c t o r s u l, v l,

i .e . t h e l o w e r b o u n d s o n m u l t i p l i e r s . Im p l i c i t in t h e c h o i c e o f v a l u e s f o r t h e v e c t o r s u t , v t a r e a s s u m p t i o n s

a b o u t m a r g i n a l w o r t h s o f o u t p u t s a n d i n p u t s . W e i l l u st r a te t h is d e p e n d e n c e w i t h t h r e e e x a m p l e s .

i ) E q u a l b o u n d s

t 1 , r = 1 . . . s , a n d v~ = 1 , i = 1 , . . . , m , i m p l i c i t ly a s s u m e t h a t t h e

o w e r b o u n d s g i v e n b y u r =

m a r g i n a l w o r t h o f e a c h u n i t o f t h e n o n z e r o o u t p u t s la c k s a n d n o n z e r o e x c e s s i n p u t s fo r a D M U a r e


8/12

A.I. Ali et al. / European Journal of Operational Research 80 1995) 462- 473 4 6 9

i d e n t i c a l ( a n d e q u a l t o 1 ). F u r t h e r , t h e m a r g i n a l w o r t h s o f n o n z e r o o u t p u t s l ac k s f o r a p a r t i c u l a r o u t p u t

o r n o n z e r o e x c e s s in p u t s f o r a p a r t i c u l a r i n p u t a r e t h e s a m e a c r o s s a ll d e c i s i o n m a k i n g u n i t s . T h i s p r ic i n g

m e c h a n i s m i m p l ie s t h a t a u n i t o f s l ac k f o r a n y m e a s u r e ( i n p u t o r o u t p u t ) is e q u a t e d t o a u n i t o f w a s t e ,

a n d t h a t t h e o b j e c t i v e f u n c t i o n o f t h e p r i m a l m o d e l s s i m p l y a c c o u n t s f o r g l o b a l w a s t e . T h i s p r i c i n g

m e c h a n i s m t h e r e f o r e l e a v e s u n c h a n g e d t h e p a r t i c u l a r c o n v e r s i o n s y s te m d e f i n e d b y t h e s c a le a n d u n i ts o f

m e a s u r e m e n t o f th e v a r i o u s in p u t s a n d o u t p u t s . F o r i n s ta n c e , i f i n p u t 1 m e a s u r e s a n n u a l w a g e s i n

m i l l io n s o f d o l l a r s, i n p u t 2 m e a s u r e s a n n u a l s a l a r ie s i n t h o u s a n d s o f d o l l a r s a n d i n p u t 3 m e a s u r e s t h e

n u m b e r o f h o u r s w o r k e d i n t h o u s a n d s o f h o u rs , t h e n w h e n e v e r t h e r e is sl a ck in t h e s e i n p u ts , o n e m i l li o n

d o l l a r s o f a n n u a l w a g e s is e q u i v a l e n t t o o n e t h o u s a n d d o l l a r s o f a n n u a l s a l a r ie s , a n d e q u i v a l e n t t o o n e

t h o u s a n d h o u r s o f w o r k . I t f o ll o w s t h a t c a u t i o n s h o u l d b e e x e r c i se d i n s e le c t in g u n i ts o f m e a s u r e m e n t i f

t h e s t a n d a r d m o d e l is t o b e u s e d w i t h o u t m a n i p u l a t i o n o f t h e d a t a . T o f ul ly s t a n d a r d i z e t h e m o d e l a n d

p r e v e n t t h e i n a p p r o p r i a te a g g r e g a ti o n o f n o n - c o m m e n s u r a b l e m e a s u re s , s o m e p r e p r o c e ss i n g o f d a ta m a y

b e r e q u i r e d . S u g g e s t e d a p p r o a c h e s h a v e i n v o l ve d s ca l in g o r n o r m a l i z i n g t h e d a t a u s i n g a v e r a g e ,

m i n i m u m , o r m a x i m u m v a l u e s. I n o u r o p i n i o n , t h e u s e o f a v e ra g e s s h o u l d b e d i s c o u r a g e d d u e t o th e l a c k

o f j u s t i f i c a t i o n / i n t e r p r e t a t i o n . F u r t h e r m o r e , t h e a v e r a g e v a l u es , t h em s e l v e s , a r e a g g r e g a t e s o f b o t h

e f f i c i e n t a n d i n e f f i c i e n t l e v e ls a n d t h e r e f o r e c a n o n l y b e r e p r e s e n t a t i v e o f a n i n e f f i c i e n t t h e o r e t i c a l

D M U 1 0

( 2 ) D M U s p e c i f i c b o u n d s

1 _ 1 / Y r l r 1 , . . , s , v [ = 1 / x i l , i = 1 , . m ,

i m p l i c it l y a s s u m e t h a t t h eo w e r b o u n d s g i v en by

u r - = . . . ,

m a r g i n a l v a l u e s o f n o n - z e r o o u t p u t s l a c k a n d e x c e s s in p u t v a r i a b l e s a r e n o t i d e n t i c a l. T h e s e m a r g i n a l

v a l u e s a r e c o n s i s te n t w i th a n e m p h a s i s o n p r o p o r t i o n a l r e d u c t i o n o f i n p u t s a n d p r o p o r t i o n a l a u g m e n t a -

t i o n o f o u t p u t s f o r e a c h D M U . T h e s e l o w e r b o u n d s a r e D M U s p e ci fi c, r e f l e c ti n g t h e f a c t th a t d i f f e r e n t

t e c h n i q u e s d e m a n d d i f f e r e n t in p u t a n d o u t p u t m i x es . C o n s e q u e n t l y , d i f f e r e n t r e l a ti v e v a l u e s a r e a s s i g n ed

t o t h e v a r i o u s i n p u t s a n d o u t p u t s .

( 3 ) B a r y c e n t r i c b o u n d s

L o w e r b o u n d s g i ve n b y t h e r e c i p r o c a l c o o r d i n a t e s o f th e b a r y c e n t e r o f a ll u ni ts , n a m e l y

l 1 Y rl , r = l . . . . , S ,

r --

l

a n d

v [ = l x i t , i = l . . . . , m ,

l

p o s s e s s d i f f e r e n t c h a r a c t e r i s ti c s . T h e y d i s t in g u i s h i n p u t s a n d o u t p u t s i n t e r m s o f a c o n s i s t e n t r e la t i v e

v a l u e ; i. e. , t h e b o u n d s a r e t h e s a m e a c r o s s a ll D M U s . P o s s i b le o b j e c t i o n s t o t h i s c h o i c e o f l o w e r b o u n d s

r e s t o n u n i f o r m i t y a c r o s s D M U s a n d a r t i fi c ia l it y . A u n i q u e s e t o f l o w e r b o u n d s f o r a ll D M U s i g n o r e s t h e

e v e n t u a l r e a l i ty o f s pa t ia l ly s e p a r a t e d m a r k e t s w h i c h a ll o w fo r d i f f e r e n t s u p p ly a n d d e m a n d c o n d i t io n s

a n d e q u i l i b r i a a n d , h e n c e , f o r m u l t ip l e , o r r a n g e s o f , r e l a t i v e p r i c e s . T h e b a r y c e n t e r o f t h e p o i n t s f r o m

w h i c h t h e l o w e r b o u n d s a r e d e r i v e d c o r r e s p o n d s t o a n a r t if i c ia l D M U t h a t i s d e f i n i t e l y i n e f f i c i e n t s i n c e i t

is in t e r i o r t o t h e p o l y t o p e d e f i n e d b y t h e s e t o f D M U s .

10 W h i l e w e d i s c o u r a g e t h e u s e o f a v e r a g e s t o normalize t h e d a t a t h e u s e o f t h e a v e r a g e a s a s u b s t i t u t e f o r a m i s s i n g d a t a v a l u e is

f r e q u e n t l y a p p r o p r i a t e .


9/12

470

Table 1

Example data set

A.I. Al i et aL /European Journal of Operational Research 80 1995) 462-473

DMU Output 1 Output 2 Input 1 Input 2

1 160 100 40 30

2 180 70 30 60

3 170 60 93 40

4 190 130 50 70

5 180 120 80 30

6 140 82 35 45

7 120 90 105 75

8 100 82 97 67

9 140 40 100 50

10 140 105 90 60

11 140 50 98 65

E a c h o f th e p r i c i n g m e c h a n i s m s r e f l e c t e d i n t h e a b o v e t h r e e l o w e r b o u n d s p e c if i ca t io n s c a n l e a d t o

d i f f e r e n t p r o j e c t e d p o in t s . T h i s is a c o n s e q u e n c e o f th e f a c t t h a t e a c h m e c h a n i s m r e f le c t s a d i f fe r e n t

e v a l u a t io n s y s t e m in t h e u n d e r l y i n g li n e a r p r o g r a m m i n g p r o b l e m .

I n t y p i c a l l i n e a r p r o g r a m m i n g p r o b l e m s , i f a c o n s t r a i n t is s c a l e d m u l t i p l i e d b y a n o n z e r o s c a l a r) , t h e

s o l u t i o n is n o t a l t e r e d . I n d a t a e n v e l o p m e n t a n a ly s is , w h e n t h e d a t a i s s c a l e d e a c h i n p u t o r o u t p u t v a l u e

is m u l t i p l ie d b y a n o n n e g a t i v e s c a l a r ), t h e r e s u l t s c a n c h a n g e a l. T h e m a n n e r i n w h i c h p r o j e c t i o n s a n d

e f f ic i e n c y e v a l u a ti o n s a r e a l t e r e d u n d e r d i f f e r e n t u n i ts o f m e a s u r e m e n t f o r d a t a d e p e n d s o n t h e

s p e c i f i c a t i o n o f t h e l o w e r b o u n d s .

A s s t a t e d e a r l i e r , t h e t e r m i z l s t v i e t i s e x a c t l y u t s Z e r e l a t o p t i m a l i t y . T h i s q u a n t i t y o b v i o u s l y

d e p e n d s o n t h e d e f i n e d l o w e r b o u n d s a s w e l l a s t h e d a t a . M o d e l s w i th D M U s p e c if ic b o u n d s a r e

u n i t s - i n v a r i a n t a s a r e t h e m o d e l s w i t h b a r y c e n t r i c b o u n d s . M o d e l s w i t h e q u a l l o w e r b o u n d s u l, v l ) - 1 , 1 )

a r e n o t u n i t s - in v a r i a n t u n l e s s t h e d a t a a r e u n i tl e ss ) . I f t h e d a t a a r e n o t u n i tl e s s, t h e n t h e o b v i o u s )

d e p e n d e n c e o f l s + l e o n u n i t s o f m e a s u r e m e n t a l te r s b o t h p r o j e c t e d p o i n t s a n d e f f ic i e n c y e v a l u a ti o n .

T h e r e a d e r w i ll n o t e t h a t f o r t h e o r i e n t e d m o d e l s d i f f e r e n t lo w e r b o u n d s p e c if i ca t io n s w il l n o t a l t er

t h e v a l u e o f 0 o r ~ b. T h a t i s , b o t h 0 a n d q5 a r e u n i t s i n v a r i a n t . T h i s r e f l e c t s t h e f a c t t h a t 0 a n d ~b d o n o t

e n c o m p a s s t h e e n t i r e d i s c r e p a n c y b e t w e e n t h e o b s e r v e d a n d p r o j e c t e d p o i n ts . I n o t h e r w o r d s , 0 a n d ~b

f a i l t o c a p t u r e a l l o f th e i n f o r m a t i o n a v a i l a b l e i n t h e m o d e l .

6 C o m p u t a t i o n a l i l lu s t r a t i o n

E a c h D E A m o d e l d e t e r m i n e s a n e n v e l o p m e n t s u r f a c e w i t h w h i c h i t e v a l u a te s e f f ic i en c y . A s s t a t e d

e a r l i e r , a D M U is e f f ic i e n t i f i t l ie s o n t h i s e f f i c ie n t s u r f a c e a n d i n e f f i c i e n t o t h e r w i s e . T h u s t h e f o r m o f

t h e e n v e l o p m e n t s u r f a c e a ff e c t s e f f i c i e n c y e v a l u a t i o n . W h e n u n i t l is e v a l u a t e d w e o b t a i n i ) a v a l u a t i o n

s y s te m , / ~ t , u l, o J ) , th a t i s r e p r e s e n t a t i v e o f a s u p p o r t i n g h y p e r p l a n e o f t h e e n v e l o p m e n t s u r f a c e , a n d i i)

a p r o j e c t e d p o i n t Y /, X t ) t h a t li e s o n t h e f a c e o f t h e s u r f a c e d e f i n e d b y t h is h y p e r p l a n e . T h e e f f i c i e n c y

e v a l u a t io n i s d e p e n d e n t u p o n t h e o r i e n t a t i o n a n d l o w e r b o u n d s p e c if i ca t io n o f th e m o d e l a s t h e s e c l e a rl y

a f f e c t t h e l o c a t i o n o f t h e p r o j e c t e d p o i n t . T h e i n t e r r e la t i o n s h i p b e t w e e n t h e s e t h r e e c o m p o n e n t s o f

e f f i c i e n c y e n v e l o p m e n t s u r f a c e , o r i e n t a t i o n , m u l t i p l ie r l o w e r b o u n d s ) i s i l l u s tr a t e d w i th t h e e x a m p l e

11 DE A mod els are atypical linear progra mmin g problems. They involve constrain t matrices that are 100 dense, are highly

degen erate, and exhibit cycling. A discussion of efficient computati onal construct s which address t hese pro blems is given in Ali

(1993).


10/12

A .L Al i e t al. / European Journal o f Operational Research 80 1995) 462-4 73

T ab l e 2

E f f ec t o f enve l opment su r f ace

471

D M U

Inp ut or ienta tion, equa l lower bou nds

Pr o j ec t ed po i n t ~ , X I ) w i th C R S

E n v e l o p m e n t s u r fa c e

Pr o j ec t ed po i n t ~ , Y t ) w i th

E n v e l o p m e n t s u r fa c e

V R S

1 160.00, 100.00, 40.00, 30.00) 160.00, 100.00, 40.00, 30.00)

2 180.00, 70.00, 30.00, 60.00) 180.00, 70.00 , 30.00, 60.00)

3 170.00, 112.00 , 67.81, 29.16) 174.88, 114.88, 69.75, 30.00)

4 190.00, 130.00, 50.00, 70.00) 190.00, 130.00, 50.00, 70.00)

5 180.00, 120.00 , 80.00, 30.00) 180.00, 120.00, 80.00, 30.00)

6 140.00, 82.00, 32.62, 41.94) 170.00, 85.00, 35.00, 45.00)

7 143.47, 90.00, 37.43, 26.73) 161.00, 101.00, 42.00, 30.00)

8 130.58, 82.00, 36.44, 24.29) 161.27, 101.27, 44.54, 30.00)

9 140.00, 91.00, 49.41, 24.71) 160.00, 100.00, 40.00, 30.00)

10 166.46, 105.00, 46.10, 30.73) 165.00, 105.00, 47.73, 31.82)

11 140.00, 88.00, 38.94, 25.83) 162.62, 102.62, 45.23, 30.00)

d a t a s e t p r e s e n t e d i n T a b l e 1 . I t c o n s i s t s o f 1 1 D M U s e a c h c o n s u m i n g 2 i n p u t s a n d p r o d u c i n g 2 o u t p u t s .

O u r d i s c u s s io n w i ll f o c u s e n t i r e l y o n t h e p r o j e c t e d p o i n t .

T a b l e 2 r e p o r t s t h e p r o j e c t e d p o i n t o b t a i n e d f o r C R S a n d V R S e n v e l o p m e n t s u r f a ce s w i t h a n in p u t

o r i e n t a t i o n . D M U s 1 , 2 , 4 , a n d 5 a r e e f f i c i e n t i n b o t h c a s e s . W h i l e i n t e n t i o n a l f o r o u r e x a m p l e , i t i s

u n u s u a l t o h a v e i d e n t ic a l s e t s o f e f f i c ie n t D M U u n d e r t h e t w o m o d e l s . H o w e v e r , n o t e t h a t e v e n w i th

i d e n t i c a l s e ts o f e f f ic i e n t D M U s t h e p r o j e c t e d p o i n t s a r e d i f f e r e n t fo r e a c h i n e f f i c ie n t D M U . I n a d d i t io n

t h e d i s ta n c e s b e t w e e n t h e o b s e r v e d a n d p r o j e c t e d v a l u e s a r e g r e a t e r f o r t h e C R S s u r f a c e a s t h e V R S

s u r f a c e m o r e c l o s el y fi ts th e d a t a .

T a b l e 3 r e p o r t s t h e p r o j e c t e d p o i n t s o b t a i n e d f o r t h e i n p u t o r ie n t e d , o u t p u t o r i e n t e d , a n d n o n o r i e n t e d

V R S m o d e l s w i t h e q u a l l o w e r b o u n d s . O f t h e s e v e n in e f f i c ie n t p o in t s , o n l y o n e D M U 6 ) i s p r o j e c t e d t o

t h e s a m e p o i n t i n a ll t h r e e c a se s . T h e p r o j e c t e d p o i n t f o r D M U 7 i s t h e s a m e f o r b o t h t h e o u t p u t

o r i e n t e d a n d n o n o r i e n t e d c a s e . F o r a l l o t h e r i n e f f i c i e n t D M U s d i f f e r e n t p r o j e c t e d p o i n t s a r e o b t a i n e d .

A s c a n b e s e e n f r o m t h e p r e c e d i n g t a b l e s th e e f f e c t o f t h e f i rs t t w o c o m p o n e n t s e n v e l o p m e n t s u r f a c e

a n d o r i e n t a t i o n ) o n e f f i c ie n c y e v a l u a t i o n c a n b e e x t e n s iv e . I n c o n t ra s t , t h e e f f e c t o f t h e t h i r d c o m p o n e n t

T a b l e 3

E f f ec t o f o r i en t a t i on

D M U V R S enve l opm ent su r face , equa l l ow er bounds

I n p u t o r i e n te d O u t p u t o r i e n te d N o n o r i e n t e d

2

3

4

5

6

7

8

9

10

11

160.00, 100 .00, 40.00, 30.00)

180.00, 70.00 , 30.00, 60.00)

174.88, 11 4.88, 69.75, 30.00)

190.00, 130 .00, 50.00, 70.00)

180.00, 12 0.00, 80.00, 30.00)

170.00, 85.00 , 35.00, 45.00)

161.00, 101 .00, 42.00, 30.00)

161.27, 101 .27, 44.54, 30.00)

160.00, 10 0.00, 40.00, 30.00)

165.00, 10 5.00, 47.73, 31.82)

162.62, 10 2.62, 45.23, 30.00)

160.00 100.00,

180.00 70.00,

182.50 122.50,

190.00 130.00,

180.00 120.00,

170.00 85.00,

190.00 130.00,

189.25 129.25,

185.00 125.00,

187.50 127.50,

188.75 128.75,

40.00, 30.00) 160.00,

30.00, 60.00) 180.00,

72.50, 40.00) 170.00,

50.00, 70.00) 190.00,

80.00, 30.00) 180.00,

35.00, 45.00) 170.00,

50.00, 70.00) 190.00,

54.25, 67.00) 187.75,

65.00, 50.00) 175.00,

57.50, 60.00) 182.50,

53.75, 65.00) 186.25,

100.00, 40.00, 30.00)

70.00, 30.00, 60.00)

110.00, 47.50, 40.00)

130.00, 50.00, 70.00)

120.00, 80.00, 30.00)

85.00, 35.00, 45.00)

130.00, 50.00, 30.00)

127.75, 51.25, 67.00)

115.00, 45.00, 50.00)

122.50, 47.50, 60.00)

126.25, 48.75, 65.00)


11/12

472 A. L Ali e t al. / European Journal of Operational Research 8 0 1995) 462-47 3

T a b l e 4

Effect of mul t ipl ier lower bounds

D M U V R S enve l opm ent su rf ace , i npu t o r i en t a t ion

E qua l l ow er bound s

Barycent r ic lower bounds

1 160.00, 100.00, 40.00, 30.00) 160.00, 100.00, 40.00, 30.00)

2 180.00, 70.00, 30.00, 60.00) 180.00 , 70.00, 30.00, 60.00)

3 174.88, 114.88, 69.75, 30.00) 170.00, 110.00, 60.00, 30.00)

4 190.00, 130.00, 50.00, 70.00) 190.00, 130.00, 50.00, 70.00)

5 180.00, 120.00, 80.00, 30.00) 180.00, 120.00, 80.00, 30.00)

6 170.00, 85.00, 35.00, 45.00) 170.00 , 85.00, 35.00, 45.00)

7 161.00, 101.00, 42.00, 30.00) 160.00, 100.00, 40.00, 30.00)

8 161.27, 101.27, 44.54, 30.00) 160.00, 100.00, 42.00, 30.00)

9 160.00, 100.00, 40.00, 30.00) 160.00, 100.00, 40.00, 30.00)

10 165.00, 105.00, 47.73, 31.82) 165.00, 105.00, 47.73, 31.82)

11 162.62, 102.62, 45.23, 30.00) 160.00, 100.00, 40.00, 30.00)

T a b l e 5

E f f ec t o f un i ts o f measur em en t

D M U

V R S env e l opmen t sur face , i npu t o r i en t a ti on , equa l l ow er bounds

U nsca l ed da t a Sca led da t a

scale fac tor 1, 1, 1, 1) sca le fac tor 100, 1, 1000, 1)

1 160.00, 100.00, 40.00, 30.00)

2 180.00, 70.00, 30.00, 60.00)

3 174.88, 114.88, 69.75, 30.00)

4 190.00, 130.00, 50.00, 70.00)

5 180.00, 120.00, 80.00, 30.00)

6 170.00, 85.00, 35.00, 45.00)

7 161.00, 101.00, 42.00, 30.00)

8 161.27, 101.27, 44.54, 30.00)

9 160.00, 100.00, 40.00, 30.00)

10 165.00, 105.00, 47.73, 31.82)

11 162.62, 102.62, 45.23, 30.00)

16000, 1 00.00, 40000 .00, 30.00)

18000, 70.00, 30000.00, 60.00)

17000, 1 10.00, 60000 .00, 30.00)

19000, 130 .00, 50000 .00, 70.00)

18000, 120 .00, 80000 .00, 30.00)

17000, 85.00, 35000.00, 45.00)

16000, 100 .00, 4000 0.00, 30.00)

16000, 100 .00, 4200 0.00, 30.00)

16000, 100.00, 40000.00, 30.00)

16500, 105.00, 47728.00, 31.82)

16000, 1 00.00, 40000 .00, 30.00)

i s l e s s p r o n o u n c e d , p a r t ic u l a r ly i n a n o r i e n t e d m o d e l . F o r t h e i n p u t o r i e n t e d V R S m o d e l r e s u l ts o f T a b l e

4 , w e s e e t h a t t h e p r o j e c t i o n s o b t a i n e d f o r D M U s 3 , 7 , 8 , 9 , a n d 1 1 a r e d i f f e r e n t f o r t h e t w o s e t s o f

m u l t i p l ie r l o w e r b o u n d s .

F i n a l l y , T a b l e 5 i l l u s t ra t e s t h e e f f e c t o f u n i t s o f m e a s u r e m e n t w h e n u s i n g e q u a l l o w e r b o u n d s . T h e

i n p u t o r i e n t e d V R S m o d e l w i th e q u a l l o w e r b o u n d s i s s o l v e d w i t h t h e d a t a a s in T a b l e 1 a n d a f t e r sc a l in g

t h e f i r s t o u t p u t b y 1 0 0 a n d t h e f i r st i n p u t b y 1 0 00 . T h e r e s u l t s a r e o n l y p r e s e n t e d f o r t h e e q u a l l o w e r

b o u n d s c a s e as t h e o t h e r t w o l o w e r b o u n d s p e c i f i c a t i o n s a r e u n i t s in v a r i a n t a n d p r o d u c e e q u i v a l e n t

p r o j e c t i o n s u n d e r s c a li n g .) D M U s 3 , 7 , 8 , a n d 1 1 h a v e d i f f e r e n t p r o j e c t e d p o i n t s w h e n s c a l e d . T h u s

s c a l in g t h e d a t a , p e r f o r m i n g t h e a n a ly s i s, a n d r e s c a l i n g t h e d a t a d o e s n o t p r o d u c e t h e s a m e r e su l t s a s

p e r f o r m i n g t h e a n a l y s is o n t h e o r i g in a l d a t a .

7 C o n c l u s i o n s

E a c h o f t h e D E A m o d e l s e v a l u a t e s e f f ic i en c y o n th e b a s i s o f t h r e e e s s e n t ia l c o m p o n e n t s : a f o r m o f

e n v e l o p m e n t s u r f a c e , o r ie n t a t i o n , a n d r e l a t i v e t r a d e o f f s i m p l i c i t i n t h e m u l t i p l i e r lo w e r b o u n d s .


12/12

A.I. Ali et al. /European Journal of Operational Research 80 1995) 462-473

473

i) For each type of envelopment surface different projected points for inefficient units) are obtained

for different evaluation systems, i.e. by effecting a particular orientation and/or by employing a

particular pricing mechanism.

ii) Models that are units-invariant produce efficiency evaluations and projected points that do not

depend on the units of measurement of the data. This property stems from the selection and

definition of multiplier lower bounds which adjust to mirror) changes in units of measurement.

iii) The suitability of a particular model for an application should be gauged with respect to the choice

of the form of envelopment and the evaluation system. The evaluation systems considered in this

paper allow only for lower bounds on the relative prices.

iv) Extensions to the basic DEA models considered in this paper either alter the form of the surface or

change the evaluation system. For example, both are altered for nondiscretionary variables as only a

subset of the input variables or only a subset of the output variables are used for effecting

orien tation Banker and Morey, 1986);. when additiona l restrictions are placed on the multipliers,

the evaluat ion system, and, possibly, the envelopment surface are altered Charnes, Cooper, Hua ng

and Sun, 1990).

e f e r e n c e s

A h n , T . , C h a rn e s , A . , a n d C o o p e r , W . W. (1 98 8 a) , I ' A n o te o n th e e f f i ci e n c y c h a ra c te r i z a tio n s o b ta in e d in d i f f e re n t D E A m o d e l s ,

Soeio-Eeonomic Planning Sciences 6 , 253-257 .

A h n , T . , C h a rn e s , A . , an d C o o p e r , W . W. (19 8 8b ) , U s in g d a ta e n v e lo p me n t a na ly si s t o me a s u re th e e f f i c i e n c y o f n o t- fo r -p ro f i t

o rg a n iz a t io n s : A c r i t i c a l e v a lu a t io n - C o m me n t , Managerial and Decision Economics 9 , 251-253 .

A l i , A . I . (1 9 93 ) , S t r e a m l in e d c o m p u ta t io n fo r d a t a e n v e lo p m e n t a n a ly s i s , European Journal of Operational Research 64 , 61-67 .

A l i , A . I . a n d S e i fo rd , L . M. (19 9 0) , T ra n s l a t io n in v a r i a n c e in d a ta e n v e lo p m e n t a n a ly si s , Operations Research Letters 9 / 5 ,

403-405 .

A l i , A . I . , a n d S e i fo rd , L .M. (1 99 3 ), T h e ma th e m a t i c a l p ro g ra mm in g a p p ro a c h to e f f i c i e n c y me a s u re me n t , i n : H . F r i e d , K . L o v e i l

and S. Schmidt (eds.), The

Measurement of Productive Efficiency: Techniques and Applications,

Oxford U nivers i ty Press , Oxford .

Banke r , R .D . , and Mo rey , R.C. (1986) , Eff i c ienc y analysis fo r exogeno us ly f ixed inpu ts and ou tp u ts , Operations Research 3 4 / 4 ,

513-521 .

B a n k e r , R . D . , C h a rn e s , A . , a n d C o o p e r , W . W. , (1 9 84 ), S o m e mo d e l s fo r e s t ima t in g t e c h n ic a l a n d s c a le in e f fi c i e n c ie s in D a ta

E n v e l o p m e n t A n a l y si s , Management Science 30/9 , 1078-1092 .

B a n k e r , R . D . , C h a rn e s , A . , C o o p e r , W . W. , S w ar ts , J . , a n d T h o ma s , D . A . (19 8 9) , A n in t ro d u c t io n to D a ta E n v e lo p m e n t A n a ly s i s

w i th s o me o f i t s mo d e l s a n d th e i r u s e s , Research in Governmental and Nonprofit Accounting 5, 125-163.

C h a rn e s , A . , C o o p e r , W . W. , a n d R h o d e s , E . (1 9 78 ), M e a s u r in g th e e f f i c ie n c y o f d e c is io n ma k in g u n i ts , European Journal of

Operational Research 2 /6 , 4 2 9 -4 4 4 .

C h a rn e s , A . , C o o p e r , W. W . , a n d R h o d e s , E . (1 9 81 ), E v a lu a t in g p ro g ra m a n d m a n a g e r i a l e f f ic i e n cy : a n a p p l i c a t io n o f D a ta

E n v e lo p m e n t A n a ly s is to p ro g ra m fo l low th ro u g h ,

Management Science

2 7 /6 , 6 6 8 -6 9 7 .

C h a rn e s , A . , C o o p e r , W. W. , H u a n g , Z . M. , a n d S u n , D . B . (1 99 0 ), P o ly h e d ra l c o n e - ra t io D E A mo d e l s w i th a n i l l u st r a t iv e

a p p l i c a t io n to l a rg e c o mme rc ia l b a n k s , Journal of Econometrics 46 , 73-91 .

C h a rn e s , A . , C o o p e r , W . W. , G o la n y , B ., S e i fo rd , L . , a n d S tu tz, J . (19 8 5) , F o u n d a t io n s o f D a ta E n v e lo p m e n t A n a ly s is fo r

P a re to -K o o p ma n s e f f i c i e n t e mp i r i c a l p ro d u c t io n fu n c t io n s , Journal of Econometrics 3 0 /1 2 , 9 1 -1 0 7 .

S e i fo rd , L . M. (1 99 0 ), A b ib l io g ra p h y o f d a t a e n v e lo p me n t a n a ly si s (1 9 7 8 -1 9 9 0 ) , Wo rk in g P a p e r (V e rs io n 5 .0 ) , D e p a r tm e n t o f

IE O R , U n iv e r s i ty o f Ma s s a c h u s e tt s a t A m h e rs t .

S e i fo rd , L . M. , a n d T h ra l l , R . M . (1 99 0 ), R e c e n t d e v e lo p me n t s in D E A : T h e m a th e ma t i c a l p ro g ra mm in g a p p ro a c h to f ro n t i e r

ana lys is , Journal of Econometrics 46 , 7 -38 .

ALS Components

Documents

Transcript of ALS Components