Robustness of the efficient DMUs in data envelopment analysis

t.. . . . . .

, o "

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

,

ELSEVIER

European Journal of Operational Research 90 (1996) 451-460

Theory and Methodology

Robustness of the efficient DMUs in data envelopment analysis Joe Zhu

*

Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, MA 01003, USA

Received June 1994; revised January 1995

Abstract By means of modified versions of CCR model based on evaluation of a decision making unit (DMU) relative to a reference set grouped by all other DMUs, sensitivity analysis of the CCR model in data envelopment analysis (DEA) is studied in this paper. The methods for sensitivity analysis are linear programming problems whose optimal values yield particular regions of stability. Sufficient and necessary conditions for upward variations of inputs and for downward variations of outputs of an (extremely) efficient DMU which remains efficient are provided. The approach does not require calculation of the basic solutions and of the inverse of the corresponding optimal basis matrix. The approach is illustrated by two numerical examples. Keywords: Data Envelopment Analysis (DEA); Efficiency; CCR model; Sensitivity analysis; Robustness

1. Introduction D a t a envelopment analysis (DEA), as developed by Charnes et al. (1978) (CCR), is a mathematical programming for characterizing efficiencies and inefficiencies of D M U s with the same multiple inputs and multiple outputs. As a databased approach, sensitivity and stability in D E A have been studied by many D E A researchers, e.g., Charnes et al. (1985) studied the sensitivity of C C R model and Charnes and Neralic (1990) studied the sensitivity of the additive model in D E A . Note that all these papers dealt with data changes via updating the inverse of the optimal basis matrix. T h o m p s o n et al. (1994) and Z h u (1996) utilized Strong Complementary Slackness Condition (SCSC) multipliers to analyze the sta-

bility of C C R efficiency. Charnes et al. (1992) employed the 1-norm and the oo-norm measures to figure out a particular region of stability. They also found that the sufficient conditions of Charnes and Neralic (1990) are not satisfied, yet the specified perturbation is permissible. In the current p a p e r we are interested in changes of inputs and outputs of an efficient D M U preserving efficiency. An efficient D M U is said to be robust to a given increase in input, or a given decrease in an output, if the D M U remains an efficient point after the change. Since either an increase of any output or a decrease of any input cannot worsen an efficient D M U , therefore we focus on upward (proportional) variations of inputs and downward (proportional)variations of outputs:

Xio=[~iXio [3i>_l,i=l,2,...,m, * The paper was finished while the author was in the School of Economics and Management, Southeast University, Nanjing, China.

(1) ~ro = aryro 0 < a r _ < l , r = 1,2 . . . . . s, (2) in which Xio ( i = 1 , 2 . . . . , m ) and Y~o ( r = l , 2 . . . . ,s) are respectively the inputs and outputs of

0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00054-2

452

J. Zhu / European Journal of Operational Research 90 (1996) 451-460

D M U o -- DMUio among n DMUs. It is supposed, consistent with other sensitivity analyses, that inputs and outputs of the remaining DMUs are unchanged. By Charnes et al. (1991), the set of all DMUs can be partitioned into four classes, E, E', F and N, where the first two classes are efficient and the second two are not. Note that the DMUs in class E' are the linear combinations of the DMUs in class E (Zhu and Shen, 1995). Therefore any increase of all inputs (1) or any decrease of all outputs (2) will let the class E' DMUs fall in class N (inefficient). Thus here we restrict our attention to the robustness of the DMUs in class E (extremely efficient). Modified CCR-type models with the D M U o under evaluation excluded investigate the increase of inputs (1) and the decrease of outputs (2) of the D M U o over which the D M U o remains efficient relative to the other DMUs. As noted by Lewin and Minton (1986), this kind of sensitivity analysis information is important for establishing the robustness of the efficiency scores. Note that our sensitivity analysis only requires the estimation of the modified CCR model. That is, it merely requires the optimal values rather than the reverse of optimal basis matrix. Note also that our approach can deal with proportionate change in any subset of inputs or outputs which may be of interest for analyses of DMUo'S behavior. This may constitute the unique feature of our approach. In fact, our approach - new variant of the CCR model, prescribes some polyhedrons of robustness. After the input and output changes of an efficient D M U o such that the changes are restricted in a particular polyhedron, the D M U o will remain efficient. Such polyhedrons can always be easily computed. Sufficient and necessary conditions for preserving efficiency of an extremely efficient D M U o are given in the next section. We illustrate the sensitivity analysis by two numerical examples in Section 3. It can be seen that the resulting radii of stability Obtained in Charnes et al. (1992) tend to be too restrictive. 2. Robustness of the extremely efficient D M U s

Consider an input-based modified CCR-type measure in a sense that the extremely befficient

D M U o under evaluation is not included in the reference set: Min

fl k n

s.t.

E

AjXkj <- i k X k o ,

j=l

AjXij <- Xio

i~

k,

(3)

]=1 n

E

Aiyrj>__Yro r = 1,...,s,

1=1 j~o ;tj, i k > 0. This minimization is completed for all t k (k = 1,... ,m). We denote each optimal value to (3) as ti* (i = k = 1. . . . ,m). By Thrall (1996), we know that all (i = 1. . . . . m) are greater than one. Theorem 1. In the event of an increase of the kth

input only, the extremely efficient DMU o stays efficient if and only if i k ~ {ikll <--ilk
0 n

s.t.

~ AiXkj + Aoi~Xko < Oi~xko, j=l

j~.o n

~_, Aixii + AoXio < OXio i ~ k, (4)

j=1

j~o

~_. AiYri + AoY,o >__yro

r= l,...,s,

j=l j~o

Aj,O~O. Let the optimal solution to (4) be (A~, Ao, 0 *) and assume, contrary to the assertion of the theorem,

J. Zhu / European Journal of Operational Research 90 (1996) 451-460

that 0 * < 1. This implies that h o = 0. Thus for any optimal solution to (4) is a feasible solution to (3). Hence 0* Ok --Ok. SO, 0* >_ 1. This leads to a contradiction and in turn establishes if part of the theorem. To establish the only if part we assume that D M U o remains efficient but Ok > 0F. Let (3) be set up to solve D M U o with its k t h input adjusted to OkXko (>O~Xko). The resulting model is as follows: Min

q~

s.t.

~ hiXkj <_~OOkXko, j=l j*o

~ AjXij~Xio i¢k, j=l j4~o n

(5)

Table 1 Points for the construction of m efficient hypothetical input points tl

t2

"'"

im

ill* 1 :

1 t~ :

.-. "" "..

1 1

i

i

...

h;,

remain unchanged. For the second point, the second of the observed inputs, the quantity Xzo, is replaced by O~X2o but all other inputs remain unchanged. And so on. In this m a n n e r a set of m hypothetical input points is constructed. Each one of these, together with the given outputs, is efficient. The hyperplane

BxO 1 + AjYry >Yro

r= 1,...,s,

j=l j÷o

Ay,~ > O.

453

"'"

(6)

d - B m O m ~-- 1

constructed by the O-points in Table 1 can be derived as follows. By substituting the rn E-points into (6), we have

O ( B 1 + B 2 + • .. +Bm = 1,

I

B1. + O~B 2 + . . . +nm = 1,

Let q~* be the optimal value to (5). Obviously, q~* = O k*/ O ko < 1, but by Thrall (1996) we can have that q~* > 1. This leads to a contradiction. [] T h e above t h e o r e m illustrates that minimizing the objective function in (3) provides the possible maximum increase factor for each individual input to allow D M U o to be efficient when the other inputs and all outputs are held at constant. We now turn to the case when more than one of the i = 1. . . . . m input are changed at the same time. Assume, then, that each input i is increased by a factor 0i > 1. We want to determine the largest possible increase of these input factors 0 i -~ 0 i * , i = 1. . . . . m while still preserving the efficiency of the observation D M U o. Subjecting each input to its maximal increase, one input at a time, one can then establish a list of i = l . . . . . m points as shown in Table 1. In association with the first of the points in Table 1, the first of the observed inputs, the quantity X~o, is replaced by 0i*Xlo but all remaining inputs

(7)

~ n 1 -~- n 2 d- * ' ' "~-Omnm = 1.

Thus, the coefficients Bi(i = 1 . . . . . m ) can be determined by (7). Further, hyperplane (6) can be determined. Moreover, on the basis of m hypothetical input points, we have an (efficient) hyperplane

Hlx I +

""

+HmX

m =

1,

(8)

when keeping mixed output constant. H i (i = 1. . . . . m) can be determined by the following m equations

fll x l o H t

+X2oH2

XloH 1 +O2x2oH2

XloH 1 +X2oH

+

" " "

+ ...

2 q- . . .

+XmoHm = 1, + X m o n m = 1,

(9)

+ O m X m o H m = 1.

On the basis of (6) and (7), we can obtain Bi

Hi=--

Xio

i = 1 . . . . . m.

(10)

J. Zhu / European Journal of Operational Research 90 (1996) 451-460

454

Thus, (8) turns into B1 --X

Bm 1 q- " " " +

Xlo

Xmo

(11)

X m = 1.

Theorem 2. In the case of change (1) o f inputs only, the extremely efficient D M U o stays efficient if and only if (/31 . . . . ,/3m) ~ F = {(/31,... ,/3m) I 1 < / 3 i <-~/3i*, i = 1,... , m , B1/31 + ' ' ' + Bm/3m <_ 1}.

To the case of change (2) of outputs only, we establish first the following modified (outputbased) CCR model: Max

ak

s.t.

~ AjYky > OlkYko , j=l j~o

~AjYrj

Proof. We shall suppose that (/3~',... ,/30) ~ F and D M U o with its input vector be equal to (/3~Xlo,... ,/3°Xmo) is inefficient. Then O* /3io Xio =

~

~ AjXij <~Xio i = l,...,m,

hj* Xij.

Aj, ot k >_ O.

B,

Bm

- - ' £ - 0 " /31)Xlo + " ' ' Xlo

0 */3°Xmo =

+

Xmo

1.

(12)

Further, we have B,/3~ + . . . +nm/3°m = ff~- > 1.

(13)

Therefore, (/3~ . . . . . /3° ) ¢~ F. This completes the proof of the if part. Next from T h e o r e m 1 we know that if D M U o remains efficient after input changes of the form (1), then 1 1 Then o

Bm q- " ' "

(16)

j=l j~o

Note that in this instance, the original D M U o is excluded from the reference set. Hence we can substitute 0 */3i°Xio, i = 1,. .. ,m into (11):

B1

r ~ k,

j~o

j=l js~o

--/31Xlo Xlo

>--Yro

j=l

-1" - - / 3 ° m X m o =

Xmo

This maximization is completed for all a k (k = 1,2 . . . . . s). We denote each optimal value to (16) asa r (r=k=l . . . . . s). T h e o r e m 3. output only, efficient if where a r is

In the event o f a decrease of the rth the extremely efficient D M U o stays and only if o~r ~ { a r [ O~r* __
Proof. The proof is analogous to the proof of T h e o r e m 1 and is omitted. Note that (16) gives the possible maximum decrease rate for each single output to allow DMUo to be efficient when keeping other outputs and all inputs constant. Analogous to Table 1, we have Table 2. Then analogous to T h e o r e m 2, we can get

1 ~ >

1 , ( 0 < 1). (14)

We can write (14) as Bm o n'--'LO/3~Xlo dr "'" "q- - - O / 3 r a X r a o = 1. Xmo

(15)

Xlo

This implies that after its inputs are adjusted to the levels Xio=/3i° X io, i = 1. . . . . m, D M U o is dominated by the hyperplane (11). I.e., the adjusted D M U o is inefficient. []

T h e o r e m 4. In the case o f change (2) o f outputs only, the extremely efficient D M U o stays efficient if Table 2 Points for the construction of s efficient hypothetical output points ~1

~2

"'"

~s

~ 1

1 a~

--. --.

1

1

455

J. Z h u / E u r o p e a n J o u r n a l o f O p e r a t i o n a l R e s e a r c h 90 (1996) 4 5 1 - 4 6 0

Table 3 Points for the construction of m + s efficient hypothetical points fit

f12

"""

tim

al

a2

"""

as

fl~ 1 :

1 ,8~ :

" ". "..

l l

1 1

1 1

". ""

1 1

i

i

...

~

i

i

.

i

1 1

1 1

"'"

1 1

a~ 1

1 a~

"'" ""

1 1

i

:

"'" "..

:

:

:

"..

:

i

...

i

i

i

...

~;

i=1

. . . . , m ; 0__
t~, + /3; ^

"'"

+t~" /3,; ^

al al

+ - 7 7 +

r = l . . . . ,s; "'"

as a;

/

+ - - < 1

)

.

Applying the similar translation in F and A, we have /

/~= {/~l . . . . ,lJm) l O <- ~i <- /3i* , i = l . . . . . m ,

/31" +

+-7-_<1/3; '

and o n l y if ( a l , a 2 . . . . , a s) ~ A = {(al,a 2. . . . . as) l a f <_a~ < 1, r = 1 , . . . , s , A l a l + " " + A s a s > 1} where the coefficients o f A r (r = 1 , . . . ,s) can be calculated by the equation A l a 1 + " " +Asas= 1 with the spoints in Table 2.

/i

It can be seen that the modified CCR-type measures (3) and (16) identify respectively two polyhedrons F and A. Either the input change of (1) restricted in F or the output change of (2) restricted in A will preserve the efficiency for the extremely efficient DMU o. Now we consider the case of simultaneous change of all inputs (1) and change of all outputs (2). Similar to Tables 1 and 2, we have Table 3. Substituting the m + s points in Table 3 into

Obviously, (/31. . . . . /3,,) ~ F ~ (/~l . . . . . ~m) ~/~, and ( a l , ; . . ,a s) ~ A ,~ (~1 . . . . . cG) ~ A. Thus

C1/31 + . . +

"'"

-.[-Crn/3m + C m + l a m + l

+Cm+ram+

r =

1

(17)

one can determine the coefficients C k ( k = 1 , . . . , m , m + 1 . . . . ,m + s). 5. The condition (/31. . . . . / 3 m , a l . . . . . a s ) / 3 m , a l . . . . . as) ll
Theorem

{(a 1

. . . . .

as) l0 _
al as } -77+"-+-7-7,_<1 . a I

(/~1 . . . . .

a s

/3m,al ....

,t~s) E / I

:=*' ( / 3 1 , - ' '

,/3m ) ~ F

and (a 1. . . . ,a s) E A. Therefore DMU o remains efficient. [] After the simultaneous change (1) of inputs and change (2) of outputs of DMU o such that the point (/31. . . . . /3,,, a 1. . . . ,a s) belongs to the polyhedron H determined by models (3) and (16), the efficiency of the extremely efficient DMU o is robust. Sometimes, one may have interest in the proportional changes of subsets of inputs and (or) outputs. We illustrate with the input change. Let I be the input index set relative to a given subset of inputs. We may restrict our attention to the following data change of DMU o

/ 7 = {(/31 . . . . .

-.~ . . .

+Cm/3m

+ Cm+lam+

1 .at- . . .

+fro+ram+

r

< 1} /s sufficient for extremely efficient D M U o to preserve efficiency after changes (1) and (2). Proof. I n / / , let fli = / 3 i - 1, i = 1. . . . . m and ~ r = 1 - a t , r = l . . . . . s. Then

a

=

..... <)


^"i ,

Xio

= [ dxio, d > l ~ Xio

i ~ I, i gi I.

(18)

Consider the following modified (input-based) CCR model: 1 For I = {1. . . . . m} the model was used for ranking efficient D M U s by A n d e r s e n and Petersen (1993). However, they did not recognize that the model can be used in sensitivity analysis for efficient D M U s and the model may fall into an infeasible case (see Appendix A).

J. Zhu / EuropeanJournal of OperationalResearch 90 (1996) 451-460

456

Min s.t.

p

Table 4 Data for the five DMUs with one output and two inputs

~ l~jXij <~PXio

i ~ I,

j=l j*o

•jXij <_~Xio

i q~I,

j=l j~o

(19)

DMU 1 2 3

Ylj

Xlj

XEj

2 4 2

4 12 8

6 8 2

4

3

6

6

5

2

2

8

~ AjYrj>__Yro r = 1 , . . . , S , j= 1

/,o Aj, p >_ 0. Let the optimal value to (19) be p*, t h e n on the basis of previous discussions we can easily obtain: 2

Similarly, we can obtain the m a x i m u m increase rate of x24 and the m a x i m u m decrease rate of Y14 respectively by (3) and (16). T h a t is,/3~ = 1.5 and al* = 0.8. N o w by substituting/31. = 1.5 a n d / 3 ~ = 1.5 into (7), we have 1.5B 1 + B 2 = 1, B 1 + 1.5B 2

Theorem 6. f f 1 < d < p *, then the extremely efficient DMU o remains efficient after the proportional change (18) of inputs.

1.

T h u s B 1 = B 2 = 0.4. T h e r e f o r e F~-" {(/31,/32)I1 ---
1 _
and A = {a110.8 < a I < 1}. Moreover, by substituting (/31,/32, a l ) = (1.5, 1, 1), (1, 1.5, 1) and (1, 1, 0.8) into (17), we have 1.5C 1 + C 2 + C 3 = 1,

3. Numerical example W e will first illustrate the a p p r o a c h developed in the above section by the example p r e s e n t e d in Table 4, taken f r o m C h a r n e s and Neralic (1990). Let us consider the robustness of the efficiency o f D M U 4 (in class E). F o r the m a x i m u m increase rate of X14 , we have Min

/31

s.t.

4A 1 + 12A 2 + 8A 3 + 2A 5 < 6/31, 6A 1 + 8A 2 + 2A 3 + 8A 5 _< 6,

C1 + 1.5C 2 + C 3

1,

C 1 + C 2 -Jr-0 . 8 C 3

1,

T h u s C 1 = C 2 = 2 and C a = - 5 . T h e r e f o r e //=

{(/31,/32,O'1)11 -
0.8 _ a 1 _< 1, 2/31 + 2/3 2 - 5 a I < 1}. By simple calculation we know that our results are identical to the results in C h a r n e s and Neralic (1990). F o r example, in Charnes and Neralic

(20)

2A~ + 4A 2 + 2A 3 + 2A 5 >_ 3, A1,A2,A3, A5,/31 -> O.

T h e optimal value to (20) is/31" = 1.5.

2 A similar theorem can be derived for the output ease.

Table 5 Data for the three DMUs with one output and two inputs DMU

Ylj

Xli

X2j

1

1

1

1

2 3

0.25 0.25

0.25 1

1 0.5

J. Zhu / European Journal of Operational Research 90 (1996) 451-460

(1990), the change interval for &t in )~14 ~-- Y14 -- t~l is [0,0.6]. If Y 1 4 - - ~ 1 = a l Y 1 4 , then a 1 ~ [ 0 . 8 , 1 ] ¢:~ ~1 t~ [0, 0.6].

Finally, let us take a look on the numerical example used in Charnes et al. (1992) (Table 5). In order to compare with the approach in Charnes et al. (1992), we have to focus on the change (18) of inputs of the extremely efficient DMU1 shown in Table 5 when I - - {1, 2}. That is, Min

p

s.t.

0.25A2 + A3 _
(21)

457

nal result in Charnes et al. (1992) with radius R~ = 0.75 tends to be too restrictive for the input changes when compared to our result obtained from (16) and (19).

Acknowledgements

This research was partly supported by National Natural Science Foundation of China, 79370017. The author is grateful to one anonymous referee for his helpful and perceptive comments and suggestions.

0.25A 2 + 0.25A 3 >__ 1, A E , A a , p -> 0. Appendix A

The optimal value to (21) is p * = 2.8. This indicates that DMU1 may increase its input vector proportionally up to a factor 2.8 and remains efficient. By substituting DMU1 into (16), we have the maximum decrease rate of Yll, Otl*= Or* = 5/14. Then the levels

r

1

IX21:

1+

~Yll:

1 -

x11:

(22) ^

are efficient for all (d, t ~ ) ~ O = { ( d , tDI0_
Min

p

~

s.t.

r 0.25A 2 + 0.25A 3 + 7 > 1, 0.25A 2 + A3 - ~"< 1,

As noted in the main text, the modified CCR model, e.g., model (19), is not always feasible. This appendix shows that the modified CCR model is infeasible if and only if certain pattern of zero inputs/outputs is involved. However one should note that this type of data is seldom to occur in a real world situation. We also show whether our results extend to other D E A models respectively under assumptions of variable returns to scale (VRS) (i.e., BCC model), nonincreasing returns to scale (NIRS) and nondecreasing returns to scale (NDRS). Without loss of generality, we assume I {1. . . . . m} in (19). That is, s

t~jXij <_PXio

i = 1,...,m,

j=l j~o

(23)

(A.1)

A 2 + 0.5A 3 - r _< 1,

~ Ajyrj > Yro r= l , . . . , s , j=l

A2, A3,~" >_ 0.

j*o

The optimal value to (23) is 9/19. Evidently, the cell with radius R~o--9/19 is contained in the p o l y h e d r o n / 2 . Further the origi-

Theorem

3 The calculation o f / 2 is similar to that of H. 4 There is a convex restriction in Charnes et al. (1992) for the additive model.

5 A similar model was used in the evaluation of technology and productivity change (see F~ire and Grosskopf, 1990).

Aj,p > O. A.1. Consider DMUo, let I o = {il Xio = 0} and Ij = {ilxii = 0}, j ~: o; II o = {r [ Yro ~ 0} and

458

J.

Zhu / European Journal of OperationalResearch 90 (1996) 451-460

IIj = {r I Y~j 4: 0}, j 4: o. I f the input-based modified C C R model (A.1) is infeasible, then there exists no k such that I o A I k = I o and II o f~ II k = II o. 6 Proof. If (A.1) is infeasible, we shall assume that I o n l k = I o and II o N I I k -- II o. Then we have

Min s.t.

P

E AjXij-l-AkXik<---PXio = 0 j~o j~k

ielo,

E AjXijq-AkXik<-~PXio -TbO j~o j÷k

iq~Io,

E AjYrj-l-AkYrk>---Yro ~ 0 j~o j~k

relIo,

E AjYrjq-AkYrk>--Yro = 0 j~o j*k

rq~IIo,

Aj>_0,

.

Aj=I,

pfree.

Note that Xik = 0 for i ~ I o and Yik 4:0 for i ~ II o. Let Aj -- 0 for j 4: o, k, then the restric-

~

Aj
Ai>l

j=l

j=l

y=l

j~o

j~o

j~o

respectively, three new modified D E A measures under VRS, NIRS and NDRS are obtained, s Look at the following linear programming problem: Max

g n

tions of (A.2) turn into:

AkYrk>Yro

It can be seen that under condition of constant returns to sacle (CRS), the new sensitivity analysis approach fails to work only when a certain pattern of zero inputs/outputs occurs. Now let us consider other modified D E A measures by imposing an additional constraint on the sum of lambda variables in (A.1). For instance, if one imposes n

(A.2)

t~kXik <-~DXio

where s i represent slack variables. If i ~ I o then X j o = 0 so that s i = 0 and h i = 0 for all j 4 : o , because s i>_O and x i / > O , for all j 4 : o . Thus (A.1) is infeasible. Similarly we can prove that if II o N IIy ~ II o for each j 4: o, then the input-based modified CCR model (A.1) is infeasible. []

s.t.

i ~ Io,

~

AjYrj

~---gYro

r =

1,...,s,

j=l

i~iio.

(A.3)

j~o

n Obviously, (A.3) has a feasible solution,

rn

T h e o r e m A.2. Consider DMUo, let I o = {il Xio = 0} and I / = { i l x i j = O } , j 4 : o ; I I o = { r l Y r o 4~O} and IIj = {r I yrj 4: 0},j * o. I f 1 o n lj 4:Io f o r each j 4: o or II o n IIj 4: II o f o r each j ~ o, then the inputbased modified C C R model (A.1)/s infeasible. 7

(A.5) Aj=I,

j=l j*o Aj > 0 ,

gfree.

This problem can be regarded as an output-based modified BCC envelopment model with equal inputs.

Proof. If I o n Ij 4: I o for each j 4: o, consider the

restriction in (A.1) PXio = E xijAj d- s i

i = 1,2 . . . . . m ,

(A.4)

T h e o r e m A.3. L e t g * be the optimal value to (A.5). / f g * < 1, then the input-based modified

j~o

6 The output-based version is always feasible with all zero variables. 7 See Footnote 6.

s Note that if the condition in Theorem A.2 is satisfied, then these modified D E A measures are also infeasible. So in the following text we assume that the condition is not satisfied.

J. Zhu / European Journal of Operational Research 90 (1996) 451-460 D E A measures respectively under conditions o f VRS and N I R S are infeasible.

Proof. N o t e t h a t p is free. T h e r e f o r e t h e i n p u t

459

Theorem A.4. Let h * be the optimal value to (A.6). I f h * > 1, then the output-based modified D E A measures respectively under conditions o f VRS and N D R S are infeasible.

constraint of

Proof. T h e p r o o f o f this t h e o r e m is a n a l o g o u s to

~ AjXij ~-~pXio j-1

t h e p r o o f of T h e o r e m A.3 a n d is o m i t t e d . []

jg-o always h o l d s in i n p u t - b a s e d m o d i f i e d D E A m e a sures. If there exists a feasible solution )tj (j ~ o) such t h a t t h e o u t p u t c o n s t r a i n t o f

~ Ajyrj >- Yro j=l j~o h o l d s w h e n t h e s u m m a t i o n o f Aj ( j :~ o) is e q u a l to o n e , t h e n g * < 1 c a n n o t b e t h e o p t i m a l v a l u e to (A.5). T h u s t h e o u t p u t c o n s t r a i n t d o e s n o t hold. C o n s e q u e n t l y , t h e o u t p u t c o n s t r a i n t c a n n o t h o l d u n d e r c o n d i t i o n of ~ A~_
modified

DEA

N o w we t u r n to t h e o u t p u t - b a s e d m o d i f i e d D E A m e a s u r e s . C o n s i d e r t h e following l i n e a r program formulation: Min

h

s.t.

~ j=l j~o

t~jXij ~_~hxio

i = l,...,m,

(A.6) j=l j~o h i>_0,

h free.

H o w e v e r , the i n p u t - b a s e d e x t e n d e d D E A m e a s u r e u n d e r N D R S a n d the o u t p u t - b a s e d ext e n d e d D E A m e a s u r e u n d e r N I R S a r e always f e a s i b l e 9, so o u r results can e x t e n d to t h e i n p u t based NDRS-DEA measure or the output-based NIRS-DEA measure.

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This p r o b l e m c a n b e r e g a r d e d as an i n p u t - b a s e d modified BCC envelopment model with equal outputs. 9 See Footnote 8.

J. Zhu / European Journal of Operational Research 90 (1996) 451-460

460

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