(FDH)and - J-Stage

The OpeiationsReseaich The Operations Research Society Society

of of Japan Japan

Journal of the Operations Research Societ.v of Japan Vel. 39, No. 3, September 1996

MODELS AND MEASURES Part II:Iih]eeDisposal Hull

FOR

EFFICIENCY DOMINANCE IN DEA (FDH)and Russell Measure (RM) Approaches'

W. F. Bowlin University of Northern Iowa

1. Bardhan fhe University of

Tkxas at Austin

W. W. Cooper T7ieUnit:ersity of 7'exasat Austin

T. Sueyoshi ScienccUniversity of fbkyo

September 20,1994; RevisedApril12,1995) {Received Abstfuct Models and measures of eMc:iency Measure approaches te eMciency evaluatioR of EraciencyDominance) in DEA. (Measllres

dominance are

examined

as as

treat,edby Free DisposalHull and Russell they relate te additive medels and MED

1. Introduction Mixed integerversions of the additive associated measures "rere intromodels of DEA with duced in [4] as a way to dealwith dominance". Other models and measures are exainined in this paper. [I]hefocusison (a)the DisposalHull" (FDH)and (b)the "Russell Measure" (RM)approaches, euch of which has formed the basisformajor research efforts the first by H. M. Tulkens and his associates at the University of Louvain and the second by C. A. K, Lovelland hisassociates at the Universityof Georgia.See,e,g., [11] and See also [22j-[25]. l13]--[16]. "eMciency

C`Free

2. FDH

Ii}rbee Disposal Hull VLrestart with FDH after definingXJ-,and Ylas input and output vectors, respectively, with components xiJ･, i 1,･･･,m and y,j,r 1,･･･,s,for each of a collection of DMUj -･･ Making where Then, we use DMU, to designate the DMUj (Decision Units) j' 1,･ to be evaluated and sa}r that the eMciency ef itsobserved is dominated by performance DMUk ifX. 2 Xk and }'bS }'li with strict inequa!ity holding!n at leastone inputor output ==

=

=

,n.

component, "free As noted in Bardhan et al. [4], the assumptiQn of disposal"givesrise to a danger of identifying a dominatedDMU as efflcient when non-zero slack ispresent. This dangeris avoided in the FDH approach, however,by a first-stage algorithm which uses paired vector comparisons to identify nondominated optima DMUs.i The possible presence of alternate can nonetheless lead to problems likethose we now describe; Fbr simplicity we followLovell[16, pp. 38-40] and restrict attentioll to output･s by assuming that all DMUs use only a single input in the same amount. Then, using yrj to "Acknowledgment

versions

of

this paper

isowed to R. M. Thrall fornumereus comments and suggestions he offered on earlier itspredecessors. Support forthisresearch from t,heIC2 Institute ofThe University

and

Texas at Austin isalso gratefu11yacknowledged. ISee [[hilkens and Vanden Eeckaut [24, between input and pp. 3-5]who furtherdistinguish "weak dominanee--and introduceadditional categories such as dominance;;,etc,, which we do sue herebecausethese refinements do not Ieadto importantdifferences in our results, of

output not

pur-

333

NII-Electronic Library Service


of of Japan Japan

334

I,Barthan,LV.E Bo-clin, LM W. (lbolper & T.Sucyoshi

represent

output

r

forDMVj, the

ip* m,ax.=m,ln,. ==

Here J' E D

eMciency

( £

measure

in FDH isobtained

used

)L, 1, A, E

;E!i,l"')Lj

=

{o,1},vv E

D)

from

==

,Z,D

;:i (1)

that the indexesare selected from the set D of non-domina,t;ed DMUs, as determinedin the first stage algorithm, so tha,ty.kly..2 1, Vr,with equality achieved when the DMU. to be evaluated iseMcient-as indicated by the choice of Aj 1 from the means

==

A,

vector

The followingexarnple

can

help to show

what

isinvolved,

di

max

subject

to

(2)

12ip f{l4Ai+24)L2, 12gbSl 15Ai+14)L2, 12ipE{ l4A,+14A,, l == A,+A2,

Ai and A2 must b,eeither zero or ene,2 We then choose between DMUi and DMU2 "rhich both dominate DMU., the DMU to be evaluated, with 12 units recorded foreach of its three outputs. To put this in the form of (1), we divideeach constralnt in (2) by the corresponding output of DMU. and obtain, where

A:

ips{

ll/IAi+?l/1 Il lg/1)Li+ IS){2,

The

measure

==

qs s{

L2,

1 and

)Li+tl/i)L2, qsf{

defined by (1) isassociated 1 both give

)L: =

with

(3)

i=)Li+)L2J

alternate

optima

fbr(3) since

the

cheices

14

(4)

ip: ipl=Tt, =:

Thisvalue of ip* also givesthe proportioninwhich all of DMU.'s to be augmented. With the choice of either DMUi or DMU2 as the evaluator we thus findthat DMU. is inerncient-i,e.,ip' > 1-----and this same value of ip' indicatesthat DMU. shouid have produced 14 units in each of itsthree outputs. Althougheach of these alternate optima yield the same value of di*, the two solutions are associated with amounts of slack. This isdealtwith by Tulkens et al. by quitedifferent simply listing the slacks but this isinadequatewhen alternate optima are present (as in the ifip* is to be used forranking, In addition, simple listing of non-zero slacks above example) isalso questionablewhen alternate optima may be present. Seethe example in Bardhan et aL [4]. Nor isthis the end of possible troublesfor,as can be seen, we need to only replace the output values of 14 units by 12 units in (2) to obtain a value of ip' 1 which accords DMU. The same measure of efficiency as isaccorded to the two DMUs which dominate it. Figure 1 below will help to illuminatewhat isinvolved,The solid Iinerepresents the to as the Free DisposalHu]1, This FDH is derivedfrom graph of a step functionreferred the observed values of yi and fbr each DMIJ via the followingcensiderations, All points y2 on or below the solid lineto the left of R are dominated by the observations recorded at R as yi 8,y2 2g. This solid lineisnot extrapolated to the vertical axis, however,because Q with yi 4 and y2 3 isnot dominated by R, or vice versa and so on, PointsQ and R, which form the vertices firom which thi$ FDH isgenerated, are obtained from pairedcomparisons in the firststage. These vertices correspond to the set D of nondominatedvectors used in (1). to this set, the point P will have a value di Relative as

t･hemeasure

outputs

of efficiency.

are

=

=

=

=

==

=

2The measure

non-dominated of

DMUs

assigned

to D in the first pass

ef

the

algorithrn

are

all

given an

i

erncienc}'

unity.


The OpeiationsReseaichSociety The Operations Research

Society of ofJapanJapan

ET}i'ciency Dominancc-Ibec DisposallllJil

335

Y2

)

Yl

Figure 1:FDHPertraval u therefore be ineficient. This same

;

;(2,2)

value of ip' isused to obtain (3,3), P'. This projectionto P' therefore results in a point which is on the Ftee DisposalHull.However, the slack value of sl 4- 3 1 isunaccounted fbrin this measure of eficiency, Under the assumption of disposal"this slack might be ignored butT'ulkensand hiscolleagues to at leastlist these values. However, they generally proceed have been unable te includethem in a more comprehensive measure--as Tulkens notes [23, 15]: doubt tha,t seeking fora single numerical index of eficiency which p, includesthe slacks would be a usefuI extension, Additionalcolumns exhibiting the values of a}1 slacks would be more informative".Such a simple listingcreates diMcultieswith Tespect to the rankings that Tulkens and hisassociates use, however, and issuescan also be ra-ised with respect- to alternate optima with differing amounts of slack while, finally, problems alse arise becausea value ef e' 1 may be assigned to a dominated DMU.

a,nd

the

coordinates

=

of

=

=

==

"free

"I

=

3."leMID

and

approach

MED

Measures these problems via the following model

S} i=! Xio

÷

aL

[4],

2 r=1SrYro

n

to:

subject

Bardhan et

s

s,+

max

taken from

xi.

Aj+si-

=Exij

,

i=1,･･･,m,

j:,1

yro ==

2 yrjAj- s.-,

(5) r=l?-･･,s,

j;tl

1

-:

£ Aj. j'-m1

"iethegaln some evalua,tor

.

perspectlve ofDMU.

as

by on the case where DMUk isused inthis the fbllowingfocusing expression forthe ith input and rth output

on

model

as of


The Opeiations The Operations Research Reseaich Society Society

of of Japan Japan

336

LBaidhan, LV.Pl Bowiin,;U UL Cbqpcr&T Sucyoshi

DMUo,

-*

+*

S,

+S. Yro

Xio

The first term

(6)

=Xio-Xik+Yrh-Yro,

Yro

Xio

the input excess

input used and the second represent the outpvt shoftfa1} relative to the out･put produced by DMU. compared to DMUk. Both terms are stated in limeasure, DMU. to be as istrue for all the other evaluated in (5). Itistherefore meaningfu1 to sum these measures and interpret the result as a. measure of relative distance firomwhat isneeded to arrive at the fullDEA eficiency. This same measure of relative distanceisused for each DMUj in (5).IUIIeMciency for any one of these DMUj can be achieved only by traversing this distance,A ranking based on thismeasure of relative distancecan thereforebe used. There are, of course, other criteria that can be used for ranking such as tot･alcost, total profitab!lity, etc., but the above measure, however,isinvariantto the measures used in the =i. and y... SeeBardhan represents

to the

relative

amount

of

[4].

et al.

The rankings obtained from (5) in thisfashionmay differ from thoseobtained from (1). The latter with a value of 1 S di'. A normed measure can also be yields a radial measure derivedforthe inefficiency of DMU. relative to the eutput of DMUk by focusingonly on the output terms in (6) and writing '

-*

l+ To

possiblevalues

avoid

exceeding

Yrk

Sr

=-.

Yro

unity,

we

Yro the

use

Yro

unity

and

average

measure

which

over

all

may

of

relative

to

for output

eficlency

to

obtain

r

reciprocal

forDMU. which cannot exceed Qf ip' in (1), Moreover, we can

obtain

£ o'< -

r-iYrolYrk g1

'

(7)

s

ifDMU. isfu11y eficient, than We illustrate with P in Figure l where we observe that (5) designates R rather for its evaluation, as indicated by line from P to S and then from S the dotted Q going to R---torepresent as the relative the tidistancemeasure that isused,3 This gives measure of eMciency in the licomponent foryi and as the eficiency measure fory2, also in limeasure, and the average of these two ratios is with

unity

attained

ifand

this expression '

Yrk

be interpretedas the

also

outputs

of

.- Yre

Yro+Sr-'

We then have a

reciprocal

only

g t =

tilt; g =

E+g 2

This provides the nance) in Bardhan et

measure aL

which

with [4]

was

3Also

the discussion of this and called

`Ccity-block7'

other

metric

40

labeledas MED

a complementary

21 ]･--=40 '

21

in the

operations

measure

19 40

of EMciencyDomi(=Measure of

inefllciency given by

'

research

fbra literature. See Appendix A in [6]

metrics.



of of Japan Japan

337

Efl)'cicncy DominEince-FteeDisposnJ HbN which

was

to as MID

referred

of lneficiency Dominance). (Measure

These meftsures are obtained from the DMU which ismost dominant in this timetric, by the choice of R rath ¢ r than Q in Figure L However, these measures do not result in a scalar which will projectP inte R. Except in the special case of fullyeMcient to the components in the MED or performance, such identificationswill require recourse MID measures. as exhibited

4. RussellMeasures C, A. K. Lovell[l6, p. 29] has expressed measures.

the foIIowingconcerns

with

some

of

the

available

'

de

"I

not

combining

likethe ideaof slacks

aggregating

witharadial

slacks,

eMciency

I likeeven

and

lessthe idea

of

'

score,"

Possiblyforthese reasons,4 he turned (with Friedand others) to a use of "Russell measures in a series of papers evaluating the performance ef U.S. Credit Unions. We therefdre investigate this approach and relate itto our precedingdiscussions via the fo11owing mode] "5

m

s

2

max

iprtO

SUbjeCt

£ et i=1

r=1

n

! Z)YrjA,i, iprYro

1i''',s,

rb =

(8)

j:i

2xi,iAj,

eixi.}l "-1i 1

i

==

1,･･･,m,

=EAj

j'=1

A.iG {O, 1}, Vi,and add the m + s additional conditions O f{lei E{ 1, 1 S ip, to ensure that the requirements for dominance are satisfied. Unlikethe situat･ion for (5), however,it isnow necessary to assume that all data are positivesince the possibi}ityof infinite solutions must otherwise be admitted because recourse to the Ali-Seiford translation discussedin Bardhan et al, [4] is not available forthese ¢ and ei values, The objective in the above formulationdiffers from the one in Fare,Grosskopfand Lovell as well as the one in FEre and Lovell However, itsuits our immediatepurposes [12] [13].6 to treat these other formulationsof the above objective laterin this paper. We can t･hen begin our comparisons herewith the above effciency measures and models by presenting an optimal solution to (8) in the fo11owingform where

we

also

require

.

di:yro=yr" r==

when

AX

=

1 isan

optimal

Yro=Yrk-Sr-', when 4In

An

=

1 isan

optimal

1,･･･,s

choice, r=1i'''is

choice

and

Similarlywe and

el-xio xih ==

i= 1,,･.,m,

(9)

write

xio=xik+s"･*,

i=l7･･･,7n,

(10)

for(5).

a subsequent clevelop a measure paper, Lovelland Pastor[17] in the concluding sections of Bardhan et al. [4], 5See Fare and Lovell [13]. See also Russell[21], 6See also Russell[21].

which

isanalogous

to the ones

'

described



of ofJapanJapan

338

l.Bardhan,iMEBowlin, ;U iM Cbopet'&TlSuqyoshi

We

now

also

(9)to definenew

use

--

SF sl

the terms

where

the

on

variables

non-negative

ip:Yro-Yro=Yrl-Yro,r=

right

variables

are

Yrh7

=

=

eiXio

We (9).

from

obtained

iprYro Yro +STL'

r

(11)

to introducethe new (10)

use

also

1)'''7si

=

(12) (5).

i-- 1,...,m,

xi.-s,+･'=xik,

:=

1,''',s,

i=1,.･･im,

=xi.-e,'･xi.=xi.-xit,

O g e,S 1, 1 g ip. so satisfy (8)issatisfied and, similarly, the expressions (11) From (11) we see that these s.- and s,+･ are associated with Ai 1 as a solution to (5), forwhich Al 1 isoptimai, while from (12) we see that these ip. and eiare associated with Ak fbrwhich Ar isoptimal. Hence suppose we could have fbr(8), where

=

=

=1

=1

s

m

£ di;

s

-2e,'･

r=1

the choices

with

Ar

=

1

Ak

and

=

>

i--1

m

-Ee, di.

£

(i3)

i=1

ri=1

1. Using (ll) and

however,this would (12),

give,

(i4) t/l.,ii/i.+l..,iift>t9.,; the

contradicts

which

direction, assume

optimality

that the

for A#

assumed

choices

A2

1 and At

==

=1

in give

:1

(5).Turning in the

opposite

(is, tf.ll,;'ll+:.,/fi>t9.,:/l for(5). However, again,

using

and we ebtain (11) (12) m

s

s

Edir£ ei>Zip.' -

iul

r=1

J

the optimality

contradict･s

which

Theorem By

A

1.

virtue

of

this theorem

we

can

the

AI,as

used

of

and

the

other

and

alternate

(5)

are

optima

DMUo･

ofit is optimal fbr(5).

to the

(17)

i=: 1,･･-,m,

same

optimai

aTe

present. As

(17)

may

One minimizes eficiency be present in evaluating

complementary,

ineficiencies that

maximizes

oniy

xio-e,{xie=s,+-*i

are applicable ip:,e,*-

the

aboverwhen

in (8) and

objectives a,nd

slacks

ifand

therefore have the foIlowing

write

di:yro-yro=s.LX r=17･･･,s where

fbrAr in (8). We

assumed

(16)

e,i i--1

r=1

to (8) isoptimal

solution

m

T2

now

choice-either makes

clear,

Al or the two

accomplishments

the performance

With (17) in hand, we

ca.n do more than has heretofore been done with Russell MeaConsider,forinstance,how one might treat variables, Various approaches are describedin Banker et al, 152-153]. Here we focus only on the [2,pp. Banker-Morey [3] approach rather, itsextension intothe presentcase) foradded insight intothe assumption of disposal".To adapt this approach to a use of Russell measures, ene simply assigns values of unity to the pertinent ip. or ei and carries these values "nondiscretionary"

sures.

"free

(or,



of of Japan Japan

ErficiencJt Dominance-Jin]eDisposafHljli

intowhatever

339

isused. As describedin Cooper et al. [10], the ei are then interpretable as di. goods" which do not enter intothe overall eMciency scores except insofaras they can improve the values of other in the efliciency scores. Turning to (5), on the other hand,we reca.11 that ipJor e,"･ the slack values in the objective (but not in the constraints) enter intoour eMciency scores. The zeros associated with (17) under a choice of e,'･ or ip: 1 are then interpret･ed to mean that･this is the value te be used in MED or the other ratings and evaluations discussedin This isachieved by assigning a zere coeflicient to these slacks in the objective. Hence, [41. in addition to being a freegood in the constraints, non-zero slacks are also to be regarded as not involving any cost in their disposalin a manner analogous to th¢ case when free disposalisassumed. Puttingthis all together, this study can see that we have genera}izedthe Banker-Morey treatment of non-discretionary slacks t-ocases in which some not all) of the inputsand (but outputs are non-discretionary. This applies to the BCC' and CCR models used by Banker and Morey and extends to other models as well. Indeed,as describedin Banker et al. [2], even more situations involving and generaltreatments are available to accomniodate "ceilings" iNrith ranges fbr which some of the variables are discretionary and other ranges non-zero

aggregate

slacks

eMciency

associated

measure

these

with

"free

or

=

"fioors"

they

when

We the

are nondiscretionary. inake

now

our

form

promised

I.i k

ir

e +

minimize and

a

awkward

'n

objective

used

0:' E]/'=i

S

in Fare

et at.

which [12],

has

i/di;. Z)"T=,i

¢

os) + , that is (a) bounded by unity and (b)attains this beund provides a measure if DMU. is eMcient, As seen on the right of (18), the FGL objective is to weight･ed average of arithmetic and harmonic means. It isdifficult to interpret to use so we repiaced itby the simpler objective in (8), and we now justify ,.i.

This objective if and only

to the

return

this choice

by the foIlowing,

Theorem

2.

(18),

An

optimat

=

+]Ii

solutien

,.+

to (8)isalso

eptimal

when

the

objective

isreplaced

[[bprove t-histheorem, lete,'･,ip: be optimal for (8) and letbi,di, be optimtt1 in (8) isreplaced with Now assume that we could have (18).

when

6y the

objective

£

(ig)

.,bi+te.,S. O, Vr, so,

a

fortiori, m

Ebi i=:1

s

-Z&. ･r=1

s

m

- 1, Vr. All + 11ip.* ip. di.' the assumtion that these O,'･ and ipIwere where

::::

must

(19)

replace

with

di,

Hewever, bi, were

holdand

being satisfied, we havea contradiction for (8), avoid this contradiction,

constraints

so, as

to

rlb

optimal

we

(23) #-i":'t9-ii*r`i=iO te be

assumed

the theorem

optimal

asserts,

the

under

Hence, equality (18).

objective

have

we

must

S.,ei']l.li,2･= £

(2`)

.,bi't9.,21':,'

By

of

virtue

two theorems,

above

(fA:

1.

Corollary

the

have the foilowillg,

alse

then fbr(5),

1 is optimat

=

we

it is atso optimal

for(8)with (18)as

its

objective.

from the preceding developments. As back and forth, or we may directly by (17) proceed setting e,'･ xiklxi. and ip.'y.kfy,.,etc. In any case, (8), and iseasier to compute simpler to int･erpret.7It is also more general.For instance,iteliminates the need forthe rn + s additional constraints required foreach DMU. to be evaluated in (6)to insurethat dominance isnot violated, and there isno need te assume that all input-s and outputs are for every DMU, as is the case for Finally, the use of us in contact with positive (8). (5)puts the even more general model" of DEA with to still are related properties which other DEA models, as describedin Ahn, Charnes and Cooper [1], and this makes itpessib}e to exploit featuresand extension of these models-which iswhat we did when we showed how to handlenon-discretionary inputsand outputs in eur discussionof (17). [[bt-akefulladvantage of what has just been said, we complete the present development by removing the restrict･ion to integerselutions so that we can extend our results to the additive model as first With thisrestriction removed general given in Charneset al, [9]. from both (5)and (8), we then have We

are

already

in position to reap

now

noted,

can

we

=

several

to

use

advantages

move

=

LLadditive

Corollary 2. Let Ak reforesent Xu･tion to (li?. (fAZisoptimalfor

for(8)using

optimal

A

the

a

(5) with (18)

objective

by

so-

then it' isalso

removed,

the integrutityrestriction'removed.

n

ZxiJA;

A:.+si" }il

=2x,j

formedfroma

components

writing

n

x,.

non-negative

the integratity restriction

with

proofcan be obtained

simple

with

vector

,,:i

yro =EYriAj'

i= 1,''',m?

=m:k,

(25)

J:i

-s.-*

g2yrp AJ

j=1

=

yr*ki

'=

1,'''iS'

j=1

We then have '

Xio-Xlk=S,+' 7Fare,

i= 1i'''iM

and Grosskopf Lovell[13] also

(only)

objective

Reference to slacks,

';

which

which

(]7),however, means

simplify

matters

the ei and an eutput shows that this iseqnivalent that these approaches are suited minimizes

y.'k-yre -- s,r'i

and

fbrsome

(only)

applications

oriented

objective

1,'''is･

r=

by

using

which

an

input･ oriented

maximizes

to maximizing only the inpllt slacks only to very special situations.

(26)

or

the

Or.

the output



of of Japan Japan

EfficiencyDon]i"ancc--P}ecDisposafHuli

We

have

also

eixio==//k, i-- 1,･･･,m The

341

(27)

r=1,-･･,s. ip:y.'.=y.'k,

and

the proof followsthe same route as before.That is,we show that these e,*･ and ip: are optimal for (8)and that they remain optimal when the objective isreplaced by We can then conclude this extension of the precedingdevelopmentsby noting that the same route can be fo11owedto either of the two classes of additive models by omitting remainder

of

(18).

that the solutions must satisfy. not intendedto mean that further research on uses of Russellmeasures should be abandoned. Indeed,such research can buildon what has already beenaccomplished. The choice of suitable measures since the measure providesan example in is not the only Others inight be prescribed explored as follows, (18) possibility, The component sums in the objective fbr(8) sat･isfy or

imposmg

a convexity

constraint

The preceding developments are

'

E]Y-i ipr s

su, combining

the two,

we

have

O, EI);･II , Sl,

)1'

(28)

.

sli 3,: M

(,,)