Aggregate efficiency measures and Simpson's Paradox

Aggregate efficiency measures andSimpson's Paradox*

ABRAHAM MEHREZ Kent State University

J. RAIWALL BROWN Kent State University

MOUTAZ KHOUJA University of North Carolina at Charlotte

Abstract. Much work has been directed to develop aggregate efficiency measures forfirms or decision-making units (DMUs) in which we are able to observe only the outputsand inputs. Assuming that each DMU has the same type of observed outputs and inputsand using only this information, Farrell's technical efficiency and the CCR ratio canbe used to assign an aggregate measure of efficiency to each DMU, which can thenbe used to compare the efficiency of the DMUs. This paper considers a subset of thegeneral aggregate efficiency problem called the matched output/input case in whicheach output is matched to exactly one input, forming a subunit. Dividing the output bythe input for each subunit within a DMU yields a subunit ratio that is the output perutiit input. For a particular subunit, the subunit ratios for two DMUs can be compareddirectly. If all the subunit ratios of one DMU exceed the corresponding subunit ratios inanother DMU, then we should reasonably expect that any aggregate efficiency measurehas the efficiency of the first DMU greater than the efficiency of the other DMU. Thisrequirement is defined as the Matched Output/Input Axiom, which is then shown tobe violated for certain data sets satisfying Simpson's Paradox. Both Farrell's technicalefficiency and the CCR ratio are then shown to violate the Matched Output/Input Axiom,which raises questions about the overall validity of both procedures.

Resume. Les travaux visant 1'Elaboration de mesures globales du rendement des unitesdecisionnelles ou des etitreprises, dans lesquellcs il n'est possible d'observer que lesextrants et les intrants, sont nombreux. En supposant que le meme type d'extrants etd'intrants est observ6 pour chaque unite ddcisionnelle et que cette information est la seulequi soit utilisee, le rendement technique de Farrell et le ratio CCR (Chames, Cooper etRhodes) peuvent etre utilises pour attribuer une mesure globale de rendement a chaqueunit^ decisionnelle, mesure qui peut ensuite permettre de comparer le rendement desdifferentes unites. Les auteurs etudient un sous-ensemble du probleme general de ren-dement global, le cas de concordance extrant-intrant, dans lequel chaque extrant estassocie a exactement un intrant, pour former un sous-ensemble. En divisant 1'extrant parrintrant pour chaque sous-ensemble d'une unite decisionnelle, on obtient, pour chacund'eux, un ratio repr6sentant 1'extrant par unite d'intrant. Pour un sous-ensemble parti-culier, les ratios de deux unites decisionnelles peuvent faire Tobjet d'une comparaisondirecte. Si la totalite des ratios des sous-ensembles d'une unite decisionnelle exceide la to-talite des ratios des sous-ensembles correspondants d'une autre unite decisionnelle, on esten droit de s'attendre a ce que l'application d'une mestire globale du rendement, quelle

* The authors would like to thank the anonymous reviewer of this paper's original draft for theconstructive criticisms and conunents which led to a revised paper.

Contemporary Accounting Research, Vol. 9, No. 1 (Fall 1992) pp. 329-342 ®CAAA

330 A. Mehrez J.R. Brown M. Khouja

qu'elle soit, indique que le rendement de la premiere unite est supedeur a celui de laseconde. Les auteurs demontrent que ce pdncipe, defini comme 6tant Taxiome de concor-dance extrant-intrant, est transgresse pour certains ensembles de donnees repondant auxcdteres du paradoxe de Simpson. Us demontrent egalement que le rendement techniquede Farrell ainsi que le ratio CCR derogent a Taxiome de concordance extrant-ititrant, cequi les amene a s'interroger sur la validite globale des deux procedes.

IntroductionDevising aggregate measures of efficiency for multiple-input, multiple-outputorganizations is an important problem that has received much attention in thelast thirty years. The ideal situation is to be able to measure efficiency withoutknowing the precise relationship between the inputs and the outputs. In otherwords, we want to be able to rate the efficiency of each organization by justobserving the inputs and outputs while remaining ignorant of how the observedinputs are used to produce the observed outputs. To do this, the inputs andoutputs of many organizations are observed, the most efficient organizations areidentified, and then each organization is given an efficiency rating relative to theset of most efficient organizations. Data envelope analysis (DEA) accomplishesthis task by defining the efficient organizations as those that are Pareto optimaland then uses these efficient organizations to draw an envelope of efficiency.Each organization's efficiency rating is then based on how far it is from theefficiency envelope.

Banker, Chames, and Cooper (1984) show that DEA can be derived by usingShephard's distance function with four postulates: convexity, inefficiency, rayunboundedness, and minimum extrapolation. These postulates do not put anyconditions on the relationships between the inputs and outputs within a partic-ular organization (the interorganization output/input relationships), and thereforeDEA applies to any set of organizations without regard to the input/output rela-tionships within each organization. Thus, in using DEA, the practitioner needsonly to observe the inputs and outputs and nothing else, which is very conve-nient.

Data envelopment analysis (DEA) has been used in a number of accountingapplications (see Seiford, 1990). Chames and Cooper (1990) and Chames,Cooper, and Rhodes (1981) suggested that DEA can be used to direct "compre-hensive audits" efforts of govemment. In the absence of profit, as a measure ofeffectiveness and efficiency, in nonprofit and public sector organizations, DEAcan be used to identify less efficient organizations (decision-making units orDMUs). These organizations (DMUs) can be subject to an audit; DEA can thusbe used to promote efficiency in nonprofit organizations. Sherman (1981, 1984)and Sherman and Gold (1985) also suggested that DEA can be used to allocatemanagerial audit resources in nonprofit and for-profit organizations. Shermanapplied DEA to evaluate the efficiency of seven teaching hospitals in the non-profit sector and 14 bank branches in the for-profit sector. The results weremore revealing than those obtained by the financial ratio and analytic reviewtechnique. DEA was specifically useful in the public sector since it can simulta-

Aggregate Efficiency Measures 331

neously consider multiple outputs and inputs and the production function neednot be known. He justified the use of DEA in the profit sector on two bases.First, financial ratio and the analytic review techniques can be biased by inflationand different accounting practices. Since DEA uses physical units for inputs andoutputs, it does not have those biases. Secondly, current profit and financial ratiomay penalize a DMU for expenditures on training and R&D that would generatefuture profit. The output of such expenditures, however, can be incorporated inthe efficiency measure as outputs if DEA is used. Bowlin (1986) used DEA toevaluate air force real property maintenance activities in 16 bases using data forthe fiscal year 1983. He concluded that in the absence of market structure andprofit as a performance indicator, DEA can act as a cost accounting system inpromoting and measuring efficiencies, analyzing variances, and controlling andreducing costs. Epstein and Henderson (1989) found DEA to be a useful toolfor managerial control and diagnosis. As a diagnosis tool, the ability of DEAto provide data reduction, bases for classifications, better understanding of rela-tionships, and comparison standards was pointed out as its main strengths. As aperformance measure, the ability of DEA to provide defensible standards basedon highly perforniing organizations (DMUs), handle situations with high prefer-ence ambiguity, completeness, objectivity, and low cost of computations makesit attractive. Similar studies relating DEA to accounting can be found in Nuna-maker (1983), Bedard (1985), Thomas (1985), and Greenberg and Nunamaker(1987).

Banker (1985) developed a modified DEA approach by considering the inputcosts and by adding the assumption of output separability (the input requirementsfor one output are not dependent on the levels of any other output). Since theinput costs are used in conjunction with DEA's efficiency envelope to determinerelative efficiencies. Banker's approach can jdeld results different than those ofthe general DEA approach. Since almost all DEA literature uses the generalapproach and does not consider the input costs, we will focus our attention onthe general DEA approach.

This paper develops a particular interorganizational output/input relationshipcalled the Matched Output/Input Case for which an independent measure ofefficiency can be derived. Since DEA applies to any set of organizations withoutregard to the interorganizational output/input relationships, DEA should applyto organizations whose interorganizational output/input relationships satisfy theMatched Output/Input Axiom and should yield the same relative efficienciesas the independently derived efficiency measure. Using Simpson's Paradox, weshow that CCR ratio model of DEA for which the Matched Output/Input Axiomdoes not violate any postulates need not produce the correct efficiency measures.

Aggregate efficiency measuresThe development of aggregate efficiency measures is a problem that has re-ceived much attention in the last 30 years. Farrell (1957) and Farrell and Field-house (1962) developed measures of productive efficiency for a firm or decision-


making unit (DMU) that assumed the various inputs and outputs are measuredwith a ratio scale. These measures can then be used to compare the efficiencyof one DMU versus another. When each DMU dED where D = {1,2 , . . . , ^}has just one input Xd and one output yd, the efficiency measure pj is to simplydetermine the output per unit input or pd = yd/xd. Then, the p^'s for each DMUcan be compared to determine the relative efficiencies of the DMUs.

When each DMU dED where D = {1 ,2 , . . . , ^} has one output yd but manyinputs Xid for i E I where / = {1 ,2 , . . . , ^} , then Farrell (1957) recommendsfirst dividing each input Xid by the corresponding output yd to get an inputvector per unit output Xd = {xid/yd,X2d/ydi---iX^/yd}. These input vectorsper unit output X^ for each DMU dED can then be used to determine anefficient production function that is a linear combination of the most efficientvectors (similar to a Pareto optimal function of the vectors). Each DMU's Xd iscompared to the efficient production function to determine the relative efficiencyPd where 0 < p^ < 1. Any DMU with pd = 1 is an efficient firm and is on theefficient production function.

When each DMU dfordED where D = {1,2 , . . . , 4} has many outputs y^dfor G e 0 where 0 = {1 ,2 , . . . , TI} as well as many inputs Xid for i E I where/ = {1,2 , . . . , ^ } , it is not clear how the outputs and inputs should be weightedto get an overall measure of efficiency (note that our notation differs from mostauthors in an attempt to develop a more nmemonic notation). The problem iscompounded by the fact that many real-world problems have data only about aDMU's inputs and outputs but have no data on how the inputs are used to getthe outputs. In this case, Farrell (1957) recommends treating the outputs sinwlarto the inputs and compute a technical efficiency measure Pd by determining anefficient production function that is a linear combination of the most efficientDMUs and points at infinity. Let the vector Yj represent the outputs for DMUd E D, Yd = {yid,y2d, • • -jy^d} and let the vector X4 represent the inputs forDMU d,Xd = {xid,X2j,...,X(^d}- Each DMU's outputs Y^ and inputs X^ arecompared to the efficient production function to determine the relative efficiencyPd where 0 < p^ < I. Any DMU with pj = 1 is an efficient DMU and is onthe efficient production function. To compute a specific p,. Boles (1972) showsthat the following linear programming problem (1) can be solved to yield p, foreach DMU t E D where \|/, is the reciprocal of the efficiency measure.

1— = max \|/,P/

s.t.

< 0

<x,0 for all ^ e D

Vt>0


This approach has much intuitive appeal because a DMU is compared to alinear combitiation of the most efficient DMUs as determined by the data. Inaddition, no other information such as how the inputs are combined to producethe outputs (the interorganizational output/input relationships) is needed to getthese efficiency measures. Chames, Cooper, and Rhodes (1978) showed thatproblem (1) is equivalent to the following fractional programming problem (2)by converting the dual of (1) to a fractional programming problem. Note thatthe weights UQ, for 8 e 0 and v,, for i e / are determined by the problem (2).

s.t.

SeeeyedMer ^ for alM e Z)(2)

«ef, Vi, > 0 for all 6 € 0 , i El

In effect, Chames, Cooper, and Rhodes (1978) showed that Farrell's technicalefficiency in problem (1) is equivalent to a weighted sum of the outputs dividedby a weighted sum of the inputs. This output/input ratio that is the objectivefunction of (2) is called the CCR ratio. The weights are determined by thestructure of the problem and specifically by the outputs and inputs of all theother DMUs. Again, this approach has much intuitive appeal because the datadetermine the weights and a user does not have to provide any additional in-formation. Before we study in depth Farrell's technical efficiency and the CCRratio defined above, let's consider just the output/input ratio in (2) without theconstraints.

Consider a subclass of DMUs, which produces | 0 | = ri outputs by using ^ =|/j = | 0 | = Tl inputs where each input is used to produce one output. In otherwords, for each DMU d E D, input Xjd is used to produce output yjj and no otherinput is used to produce output yjj. In this case, we say the inputs and outputs arematched and will call each matched pair a subunit. However, Farrell's technicalefficiency and the CCR ratio still apply to this case because it does not violateany of the assumptions in Farrell (1957) or any of the assumptions of the CCRratio model in Banker et al. (1984). Before we relate the matched output/inputcase to the CCR ratio, we will investigate some examples in which it can occur.Then, simpler alternatives instead of the CCR ratio will be investigated andrelated to Simpson's paradox to show that certain sets of outputs and inputscan yield efficiency measures that are wrong. Finally, this development willbe extended to generate output and input sets for which the CCR ratio yieldsincorrect measures of efficiency.

Matched output^nput caseFor the matched output/input case, the number of outputs T\ by definition mustequal the number of inputs ^, r; = ^, which is the number of subunits. Thus,


the index set for the outputs 0 is the same as the index set for the inputs / ,0 = /. To make the notation easier for the matched output/input case, we willuse index set / to refer to the subunits. Thus, each DMU d ^D has ^ matchedoutput/input pairs, Xi^ and y^, called subunits for every i 6 /. For each subunit/ e / and DMU d £ D, however, the output per unit input can be found by thesubunit ratio, p,d = yid/xij.

The matched output/input case can obviously occur when a number of firms(DMUs) manufacture the same set of products (outputs) and each product(output) is produced from exactly one input. However, the matched output/inputcase can occur in a variety of other ways. For example, consider an inventorymanagement system that keeps records on the quantities of each type of equip-ment (subunit) in the system. Of course, the number of each equipment typethe inventory system thinks is present (recorded quantity) is not necessarily thenumber that is physically present (physical quantity). Like a signal noise ratio,an output/input ratio for each equipment type (subunit) can be defined by consid-ering the physical quantity as the output and the sum of the physical quantity andthe "noise" (the absolute deviation between the physical and recorded quantity)as the input. The "noise" is obviously due to human error and the performanceof the subunit is perfect if the absolute deviation is zero (the subunit output/inputratio is one). Otherwise, the value of the subunit ratio is a fraction number be-tween 0 and 1 and is a measure of efficiency. The subunits (equipment type) aregrouped into DMUs (radar units, communication units, etc.), all of which havethe same set of equipment types (subunits). The problem that one of the authorsencountered in the real world is to develop an aggregate efficiency measure foreach DMU so the DMUs can be compared.

Another example of a matched output/input case is found in multinational cor-porations that have established a set of subunits (perhaps a set of manufacturinginstallations that make different products or a set of retail establishments thatsell different lines of goods) in different countries or regions. A DMU wouldconsist of all the subunits in one country or region in which each DMU (country)has a similar set of subunits. A subunit output/input ratio may be defined as thebenefit/(benefit plus cost) ratio for each subunit.

Still another example occurs in production and maintenance systems in whichthe subunit output/input ratio for each piece of equipment (subunit) is the(operational time)/(operational time plus maintenance time) ratio. The DMUswould correspond to departments with the same equipment (subunits). Again,the problem is to develop an aggregate measure of efficiency for each DMU.

The examples above describe systems consisting of different DMUs that aredivided into many subunits, each of which has its own output/input ratio. Acommon problem is to develop a way to combine the information in each sub-unit output/input ratio into an overall measure for the entire DMU. These ag-gregate DMU measures can then be used to compare the different DMUs in thesystem. For example, the relative efficiency of two different maintenance poli-cies in a maintenance system can be determined by comparing the aggregate


DMU measures of two DMUs with the same equipment (subunits) but differentmaintenance policies. Thus, we want to develop aggregate DMU measures thatare comparable for the matched output/input case.

Since yu and xy measure the same thing in every DMU d £ D,\he subunitratios pw = yid/xtd can be compared across all the DMUs d € D. Specifically,if Pij > pik, then we can say that DMU 7 is better than DMU k for subtmit i. Ifall the subunit ratios for DMU j are better than all the subunit ratios for DMUk, p,j > p,t for all i € /, then we should expect that any efficiency measures pjfor DMU 7 would be better than the corresponding efficiency measure for DMUk, pj > pk. This will be stated as an axiom.

Matched output/input axiom: For the matched output/input case, if all the subunitratios for DMU j are greater than the corresponding subunit ratios for DMU k,Pij > pik or

^ > ^ foralUG/,Xij Xik

then any measure of efficiency p; for DMU j must exceed the correspondingmeasure of efficiency pt for DMU k, pj > pk.

For the maintenance system example above, suppose we are comparing theeffect of two different maintenance policies where DMU j has used one policywhile DMU k has used the other policy. If the two DMUs have the same typeof machines (subutiits), then we have a matched output/input case in which theoutput for each machine (subunit /) yy is the operational time and the inputXid for each machine i is the operational time plus the maintenance time. ThenPtd = yid/xid is the proportion of time machine i is operating in DMU d. IfPij > pit. then machine / is operating a greater proportion of time in DMU7 ascompared with DMU k. If this was tme for every machine, p,; > p,t for everyi G /, then the Matched Output/Input Axiom simply says that the efficiencymeasure p; for DMU f must exceed the efficiency measure pt for DMU k,pj > pifc. In other words, if the operational time for every machine in DMU j isbetter than the operational time for the corresponding machine in DMU k, thenthe maintenance policy for DMU f must be better than the maintenance policyfor DMU jfc.

The Matched Output/Input Axiom does not say anything about how the mea-sure of efficiency pd for a DMU rf € D is to be constmcted. It simply says anyvalid measure must satisfy some reasonable conditions. The next section willinvestigate some simple measures and show that they do not satisfy the MatchedOutput/Input Axiom.

Simpson's ParadoxThe simplest aggregate efficiency measure for DMU d would be to ratio thesum of the outputs to the sum of the inputs. Namely, let


for all rf e Z). (3)

Although it might appear that the Matched Output/Input Axiom should hold forthis simple measure, it unfortunately does not as the following example withtwo DMUs {t, = \D\ = 2) and two subunits (^ = j / | = 2) shows. Let

i ^ Zii > zii = 1223 xu xi2 400

100 y2^^y22 \500 X21 A;22 6 '

Then

101 >'ii+>'2i ^ yn+yii 101

503 X\\+X2\ JC12+JC22 4 0 6

The proposed aggregate efficiency measure a, is sensitive to the magnitudesof the yy's and the ACy's. For the last example, let us divide the magnitude ofyn, x\2, y2\, and X2\ by 100, which does not change the output/input ratios as^12^12 = 1/4 instead of 100/400 and >'2i A21 = 1/5 instead of 100/500. Usingthese new values,

1 1-=002 = -

which reverses the direction of the inequality. These new values now satisfy theMatched Output/Input Axiom, which is further proof that q, is a poor choicefor an aggregate efficiency measure.

The counter example above is an illustration of Simpson's Paradox as pre-sented by Simpson (1951) and refined by Blyth (1972) in the statistics literature.Sunder (1983) applied it to cost allocation while Ijiri (in Sunder, 1983) devel-oped necessary and sufficient conditions for the existence of Simpson's Paradox.Fbr two DMUs and two subunits, Simpson's Paradox is a reversal of the MatchedOutput/Input Axiom. The following definition of Simpson's Paradox uses thenotation developed in this paper.

Simpson's Paradox: For the matched output/input case with two DMUs, t, —|£>| = 2, and two subunits, l,— \I\ — 2, there exist outputs y^ and inputs Xidin which all the subunit ratios for DMU j exceed the corresponding subunitratios for DMU fc, ̂ > — and ^ > — (p,, > pu and p2, > P2t), but the

Xij Xik X2j X2k ^

efficiency measure Oj is less than Oic,

In other words, Simpson's Paradox states that it is possible to find values of theoutputs and inputs that violate the Matched Output/Input Axiom. Assume we are


comparing DMU7 with DMU k where each DMU has two subunits ^ = |/ | = 2and without loss of generality, assume that py > p2/. Ijiri's necessary andsufficient conditions (in Sunder, 1983) for the existance of Simpson's Paradoxcan be revised to the following form.

Necessary and sufficient conditions for Simpson's ParadoxFor two DMUs j and k with two subunits apiece ^ = |/ | = 2 where py > p2j, thefollowing are necessary and sufficient conditions for the existence of Simpson'sParadox.

NecessityPly > Pu > P2j > P2A. which Can also be written as

yij ^ yik , y2j - y2k

X\j Xu X2j X2k

SufficiencyDefine a^ = — for any <i G D. Sufficiency is fulfilled if either one of the

X + Xld 2d

following two conditions is satisfied.

1 Let p; = P2; + (pi; - p2j)Cl.j,

then

and

2 Let 7t = p2k + (Pu - P2k)o.k,

then

Pu>Yt>p2, and «,> J*~ p^^

To illustrate these conditions, let us see if the two data sets used earlier inthis section satisfy them. Since the first set is an example of Simpson's Paradoxbut the second set is not, we would expect the first set to satisfy the conditionswhile the second set should violate at least one condition. The data sets aresummarized in Table 1 in which the first set satisfies the necessary conditionas well as both sufficient conditions (only satisfying one sufficient condition isnecessary); the second set satisfies the necessary condition but does not satisfyeither sufficient condition.

Ijiri's necessary and sufficient conditions can be used to derive an infinitevariety of outputs and inputs that satisfy Simpson's Paradox. Specifically, weused them to derive a series of four DMUs with two subunits apiece, whichsequentially satisfy Simpson's Paradox with efficiency measures Oi < 02 <03 < 04. This data set is listed in Table 2 and will be used in the rest of thepaper to test prospective aggregate efficiency measures.


TABLE 1Data set examples for Simpson's Paradox

Data set 1: Data set 2:An example of Not an example ofSimpson's Paradox Simpson's Paradox

Description

Subunit 1 outputSubunit 1 inputSubunit 2 outputSubunit 2 input

Subunit

Efficiency measures

Aggregate efficiency

Satisfy condition pn > p

Test for condition:

Pl2 > Pi > P21

„ ^ Pi ~ P22P12 - P22

Test for condition:

P12 > 72 > P21

1*1 *^

pu -P21

Valuecalculated

i

y\d+yid^'' Xu+X2tl

12 > P2I > P22?X\d

'' ~ XU+ X2d

Pi=P2i+(Pn-P2i)a,

Pi - P22P12 - P22

Satisfy condition?

72 = P22+(Pl2-P22)a272 - P21Pn - p 2 i

Satisfy condition?

DMUd= 1

13

100500

13

15

101503

3503

101503

2978T2

Yes

DMU

100400

16

14

6

101406

Yes

400406

204503

Yes

101406

DMUd= 1

1315

13

I

14

38

14

0

No

DMUd = 2

1416

14

16

15

Yes

4To

1

No

Weighted aggregate output/input ratiosIf simply taking the ratio of the sum of the outputs to the sum of the inputsviolates the Matched Output/Input Axiom, perhaps we can weight the individualoutputs and inputs so that the ratio of the weighted sum of the outputs to theweighted sum of the inputs will satisfy the axiom. Let us define the weightedratios

for all d 6 (4)

where the weights are positive, a, > 0 and fc, > 0 for all i € /. Note that the setof weights is the same for every DMU d. In this case, it is easy to prove that theMatched Output/Input Axiom is not satisfied. For the example in Table 2, let theoutputs and inputs be y-̂ = yid/a, and Xy = Xid/bi for i E I and d E D. Thenno matter what weights are used, an example can be generated which violatesthe Matched Output/Input Axiom.


TABLE 2Simpson's Paradox example for four DMUs

Description

Subunit 1 output

Subunit I input

Subunit 2 output

Subunit 2 input

Subunit

Efficiency measures

Aggregate efficiency

Satisfy condition pu >

Test fo condition:

Pl,d+l Pd P2d

^ P i - P2,<i+1

Pl,d+] ~ P2,d*l

Test for cotidition:

Pl,d+1 > Yrf+l > P2</

^ Yii+l "~ P2d

Pld-P2d

Valuecalculated

yu

Xld

yu

X-U

P w = —X\d

P2. = gy\d + yiA

^^ x\i + X'ia

P\4^r\ >P2d> P2,d+l ?

Pd = P2</ + (PW - P2<i>

P</ - P2,d+l

Pl,d+\ - P2,rf4.1

Satisfy condition?

Yd-PM + (Pt«-P2d)cYd+1 - P2/(

Pld - Pld

Satisfy condition?

Data set

DMUd=\

4

1

6

3

4

2

5

14

To

Yes

DMU

12

4

1

1

3

1

13

Yes45

13

3

Yes

13

3340

Yes

DMU

^ = 52.25

19

I = 0.75

1114

34

5353

Yes

1950

5350

3740

Yes

5335

953T8B5

Yes

DMU

1323 _ J32 3

49

1=0.5

1

27•TO

1

1328

Yes

4950

4344

Yes

332IS

A variant of the efficiency measure in (4) was actually used by one of theauthors to evaluate and compare a multiechelon inventory system as describedin the third section. In each echelon (DMU), there are the same ^ types ofequipment (subunits) where y/, is the physical quantity and Xij is the sum of thephysical quantity and the absolute deviation between the physical and recededquantity. In the actual application, the weights were specified by managementwhere a, = fo, > 0 for all i G / and 2,g/a, = 1. The (o/s were then used toevaluate the efficiency of each echelon (DMU). Unfortunately, this is a matchedoutput/input case and the reasoning in the last paragraph shows that this effi-ciency measure violates the Matched Output/Input Axiom.

CCR ratioSince weighting the outputs and inputs with a common set of weights cannotproduce an efficiency measure that satisfies the Matched Output/Input Axiom,

340 A. Mehrez

TABLE 3Matched Output/Input

Problem

FarreU's technicalefficiency

Problem (1)

CCR ratio

Problem (2)

J.R. Brown M. Khouja

Axiom counter example for FarreU's technical efficiency and the CCR ratio

Valuecalculated

¥<(Efficiency pd

X2

A4Slack s*^JSlack sliSlack SidSlack J2rfEfficiency PdU\d

U2d

V\d

V2.

DMU

d= X

111000000010.107643310.094904460.287261150.2375796

DMU

rf = 2

110100000010.077629770.068442810.207165820.1713367

DMU

d=-i

110010000010.018899580.016662940.050436140.04171326

DMU

d = 4

110001000010.007529780.007621510.020028440.01860657

the only other way to weight the outputs and inputs to get a ratio that wiUhopefully satisfy the axiom is to let every DMU have a different set of weights.But this is exactly what the CCR ratio in problem (2) does because the weights.Her and v,,, are different for each DMU t € D. Using the data set in Table 2,FarreU's technical efficiency is found by solving the linear programing problemdefined in (1) for each DMU d & D. Following the notation in Chames etal. (1978), let s^, for i e / represent the slack variables for the constraintsinvolving the outputs, Y ,̂ and let j,7 for / e I represent the slack variables forthe constraints involving the inputs, X^. Since the dual solutions to the linearprograms in (1) are also the optimal solutions to the corresponding problem in(2), we also determine the optimal CCR ratios. For the Matched Output/InputAxiom to be satisfied for the data set in Table 2, the resulting efficiency measuresmust satisfy pj > p2 > p3 > p4. The results of the Unear programing problemsin Table 3 clearly do not satisfy the Matched Output/Input Axiom because allthe aggregate efficiency measures p^ equal I. In other words, each of the DMUsis on the efficiency frontier. Thus, FarreU's technical efficiency and the CCRratio do not satisfy the Matched Output/Input Axiom.

Chames et al. (1978) point out that any DMU t that is on the efficiencyfrontier must have an aggregate efficiency measure of one p, = 1 and not haveany positive slacks, s'l, = 0 and 5,7 = 0 for all 1 € / . If the aggregate efficiencymeasure equals one p/ = 1 and some of the slack variables s*, correspondingto the outputs were positive, then some of the outputs in Y, could be increasedand still retain an efficiency of 1. Similarly, if p, = 1 and some of the slackvariables s^^ corresponding to the inputs were positive, then some of the inputs inX, could be reduced and still retain an efficiency of 1. For a DMU t G D, if the


aggregate efficiency measure is one Pf = 1 and some of the slack variables arepositive, Chames et al. (1978) recommend solving a "varied problem," whichis a modification of problem (1). Since the results in Table 3 have all the slackvariables equal to zero for every DMU d G D, tine "varied problem" does nothave to be constructed and every DMU is on the efficiency frontier.

Chames, Cooper, and Rhodes (1979) further stipulate that all the weights inproblem (2) should be strictly positive. My > 0 and v̂ ^ > 0 for all d 6 D andJ G /. This positivity requirement (non-Archimedean formulation) is necessaryto avoid classifying a DMU as on the efficient frontier when it obviously isn't.Since the results in Table 3 for the data set in Table 2 have all positive weights,Mjd > 0 and vy > 0 for all fi? e D and i G /, the positivity requirement is satisfiedand problem (2) does not have to be rerun with the positivity requirement. Sincethe solution to (2) is simply the solution to the dual of primal (1), duality theoryshows that having the slack variables in the primal (1) all zero is a necessary butnot sufficient condition for the dual variables to be positive. Requiring the dualvariables in (2) to be strictly positive, however, will produce a primal solutionwhere all the slack variables are zero. Thus, Banker et al. (1984) concentrateon solving (2) with a positivity requirement (each variable is constrained to begreater than or equal to a small "non-Archimedean" quantity, e). However, thecounter example in Tables 3 and 2 show the positivity requirement is not enoughto prevent DMUs from being incorrectly placed on the efficiency frontier.

DiscussionThe previous sections showed that both Farrell's technical efficiency and theCCR ratio do not satisfy the Matched Output/Input Axiom for the matchedoutput/input case. For the practitioner, the matched output/input technology doesnot violate any postulates of the CCR ratio. Therefore, the CCR ratio maybe used to measure efficiency of Matched Output/Input DMUs. The results,as shown in this paper, may be wrong. A new DEA model that satisfies theMatched Output/Input Axiom has been developed by Banker (pp. 343-355). Asmore models are developed to deal with different technologies, caution mustbe exercised so that the assumptions under which each model is applicable aremade clear. In this manner, practitioners can apply the correct model for theirorganization.

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Aggregate efficiency measures and Simpson's Paradox