objective measurement of efficiency: applying single price model to rank hospital activities

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Computers & Operations Research 31 (2004) 515–532www.elsevier.com/locate/dsw

Objective measurement of e$ciency: applying single pricemodel to rank hospital activities

Enrique Ballesteroa ;∗, Jos.e A. Maldonadob

aEscuela Tecnica Superior de Alcoy, Edi�cio Ferrandiz, Plaza Ferrandiz y Carbonell, Alcoy 03801, SpainbEscuela Universitaria de Informatica, Universidad Politecnica de Valencia, Camino de Vera s/n,

Valencia 46022, Spain

Received 01 October 2000; received in revised form 01 March 2001

Abstract

Single price model is a recent approach to measurement of e$ciency from assumptions and developmentswhich widely di3er from data envelopment analysis. Its purpose is to rank the activities objectively withindependence of the decision-maker’s opinions and preferences as subjective factors of ranking. In this paper,the previous version of the model is reformed in critical aspects, extended and applied to a case of 27 hospitalunits with real world information on two inputs and four outputs. In this revised version, the axiomatic basis isreduced to a single assumption, and new general formal proofs are provided. As objectively implemented, thecomplete ranking leads to the most e$cient activity and the most productive scale size in multiple output/inputproblems.

Scope and purpose

Potential users of this paper are boards of directors, independent consultants and government advisers, andcompany and non-pro;t organisation managers in so far as they are interested in e$ciency-based decision-making problems. E$ciency stresses the ability of an agent to produce large and high quality outputs withsmall inputs, namely, at low cost. “Agent” often means a ;rm or an organisation, but not only this, sincee$ciency also refers to workers, countries and projects. Generally speaking, the agent produces actions ormakes decisions to be assessed in terms of e$ciency. Each action or decision is called activity. Other namesfor the same notion are “alternative” and “decision-making unit”, both terms being etymologically relatedto the idea of determining the best choice among the activities. In decision analysis, a notable and rapidlygrowing ;eld of application involves the measurement of the e$ciency with which the outputs are produced.A book that is widely used and well-known to practitioners in the ;eld is by Charnes et al. (Data EnvelopmentAnalysis: Theory, Methodology, and Application, Kluwer Academic Publishers, Dordrecht, 1995). The modelwe develop and apply to hospitals is a departure from data envelopment analysis, which does not provide a

∗Corresponding author. Dpto. Econ. y Ciencias Social Agrar., E.T.S.I. Agronomos, Ciudad Universitaria, s/n, Madrid28040, Spain. Tel.: +34-96-6528472; fax: +34-96-6528409.

E-mail addresses: [email protected] (E. Ballestero), [email protected] (J.A. Maldonado).

0305-0548/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0305-0548(02)00231-9

mailto:[email protected]

mailto:[email protected]

516 E. Ballestero, J.A. Maldonado / Computers & Operations Research 31 (2004) 515–532

complete e$ciency ranking but only a criterion to distinguish between e$cient and ine$cient activities. Incontrast, the model herein provides an activity ranking from an impersonal and objective perspective. This isappealing to many decision-makers for purposes such as selecting either the best or the second best activity,and establishing incentive payments from the e$ciency ranking.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Complete ranking; E$ciency models; Health care production; Technical and scale indexes

1. Introduction

Measurement of e$ciency entails theoretical and practical di$culties in problems with multipleinputs and outputs. Firstly, in the context of e$ciency, “input” has the standard signi;cance ofresources such as labour, capital and land insofar as these resources are required by an organisationto enable it to provide goods and services. Secondly, “input” is used in the broad sense of loss,sacri;ce or penalty incurred especially in gaining something. E$ciency decreases when the resourcesthat are put in as components of production (for example, land, labour, or raw materials) increase,if the outputs remain the same. “Output” not only refers to industrial and mental production butalso to any kind of tangible and intangible bene;ts, favourable or helpful outcomes, as well as anyconsequence that brings advantage to the decision-maker.

Indeed, measuring and analysing the e$cient use of resources are critical objectives for anyorganisation. Users of e$ciency models vary from pro;t-oriented industrial manufacturing companyall the way to non-pro;t public and private service agencies. E$ciency has been the subject ofresearch in a wide range of production activities. Most relevant applications deal with activitiesin which some outputs and/or inputs are intangible, and therefore, almost impossible to specify inmonetary units. For example, hospital and school outputs as well as environmental costs of industrialplants are di$cult to measure in monetary terms; therefore, the e$ciency analysis of such activitiescan hardly be performed by aggregating bene;ts and costs in accounting terms. Hence, a key problemto be solved by the e$ciency models is aggregation.

If market prices could reliably be assigned to every bene;t and cost, then, aggregations in dollarswould be achieved for outputs and inputs. This occurs if and only if:

(i) all the inputs correspond to expenditure usually of money required for the purchase of goodsand services used in the production process; and

(ii) all the outputs correspond to amounts of money received by selling the production.

If conditions (i) and (ii) are simultaneously met, the e$ciency of each activity could be estimatedby a ratio such as aggregate output to aggregate input, both aggregations in dollars. However, ifaggregation at market prices is an unsolvable problem, the analyst will attempt to derive e$ciencyindexes from plausible axiomatic bases.

The non-stochastic e$ciency model in this paper is a departure from data envelopment analysis(DEA), the widely known non-parametric non-stochastic frontier-based approach to e$ciency thatwas proposed by Charnes et al. [1] with a huge body of literature. The principal DEA versionssuch as Charnes–Cooper–Rhodes’ (CCR) model and Banker–Charnes–Cooper’s (BCC) model are

E. Ballestero, J.A. Maldonado / Computers & Operations Research 31 (2004) 515–532 517

carefully explained by Cooper et al. [2]. The aim of DEA is to identify ine$ciencies to achieve thehighest attainable bene;t levels and to avoid the excessive consumption of inputs. To accomplish thisaim, DEA objectively classi;es the activities in two groups, e$cient and ine$cient alternatives, byresorting to linear programming models. The following characteristics of DEA should be pointed out:

(i) DEA is not based on parametric stochastic frontiers, in which e$ciency is estimated from acomponent of the skewed regression residual for each activity. The methodology of stochasticfrontiers was proposed by Aigner et al. [3], Battese and Corra [4].

(ii) DEA does not use linear programming models to select the most e$cient alternative. In contrast,a major contribution of DEA consists in the application of linear programming models to assesspast management accomplishments in terms of relative e$ciency. Simply put, DEA does notprovide the e$ciency ranking of the activities but only the set of e$cient DMUs versus theset of ine$cient ones.

(iii) Objectivity in the e$ciency assessment is a relevant characteristic of DEA. Ine$ciencies areestimated with independence of individual thought, opinion and preference.

Unlike DEA, the model we use is capable of providing the complete e$ciency ranking of theDMUs objectively. This means that the e$ciency ranking is obtained from observable facts un-coloured by opinions, feelings or preferences. The proposed model aggregates the inputs and theoutputs by a single weight system, namely, aggregation is performed by weights that do not dependon the particular DMU to be evaluated (Ballestero [5]). Hence, this approach is called single pricemodel (SPM).

The idea of a single set or common set of weights is not new in the DEA literature. However,the current paper puts forward a new mechanism for determining a “common set of weights”, as itis often referred to in the literature. We summarise some recent contributions as follows:

(a) Approaches developed in the late 1980s such as cone-ratio (Charnes, Cooper, Huang and Sun[6]) and assurance regions (Thompson et al. [7]; Wong and Beasley [8]; Dyson and Thanassoulis[9]). Their common aim is to restrict the solution space, namely, to reduce the set of e$cientactivities. To increase discrimination, some subjective constraints are introduced into DEA byboth techniques. These models either rule out less preferred solutions or establish discriminatingrelationships amongst the variables.

(b) Approaches developed in the early 1990s. This research is found in the Cook et al. [10] paper,which gives various criteria to prioritise e$cient activities, the results changing in functionof the di3erent assumptions in the article; in the Andersen and Petersen [11] paper, whichsuggests a ranking procedure from comparison of BCC technically e$cient activities regardinga reference technology spanned by all other activities; and in the Doyle and Green [12] paper,which describes four types of cross evaluation with its implementation as secondary objectivesto the standard DEA maximisation. The results obtained from these techniques are signi;cantlydi3erent.

SPM is a departure from the above (a) and (b) techniques as they are based on DEA, and therefore,on weight systems changing from activity to activity. In this paper, the SPM original version isextended. Firstly, the axiomatic basis is simpli;ed since one of the two assumptions required in theoriginal version is now removed. A new general proof of the main theorem underpinning the model


is given. Secondly, we provide ranking scores to measure scale e$ciency and to determine the mostproductive scale size in the SPM context; this approach was not developed in the previous paper[5]. Thirdly, the method is applied to a large scale case with information elicited from hospitals inreal world environments; in the light of this information, a comparison of DEA and SPM resultsis performed. Finally, a consistency analysis leading to “mountain-shaped” diagrams is undertaken,and the results commented.

2. E�ciency indexes and the complete ranking

We start with a special case of extreme simplicity, an e$ciency scenario with one input and oneoutput. In spite of its triviality, this scenario is a helpful preface to develop SPM in the multi-criteria(m input, s output) general case. It allows us not only to make clear the principles of measurementbut also to establish the basis of the multi-criteria ranking model. For one input and one output,both DEA and SPM substantially coincide in their results. However, SPM moves a step forwardto solving the ranking problem with a multiplicity of bene;ts and costs. Since SPM is capable ofaggregating the inputs and the outputs by a common weight system, the general case can be reducedto the (m= 1; s= 1) special case.

The notions of global, technical and scale e$ciency are de;ned by Banker et al. [13]. We will usehere these de;nitions with minor changes to simplify the speci;cs of the presentation of the DEAmodel. Consider N activities with the corresponding input–output vectors (xk ; yk); k=1; 2; : : : ; N . Inthis one input–one output setting, e$ciency can be measured either by an output-oriented or by aninput-oriented SPM model.

Firstly, we deal with the output-oriented case. Here, global e$ciency is measured by the SPMratio:

EG = yk=xk : (1)

Eq. (1) requires input and output to be measured in the same scale or at least in scales whosedivision makes sense. This is even more so when several inputs and several outputs are aggregated.

From (1) the EG values can straightforwardly be ranked. The best value, namely, max EG issometimes reached by more than one DMU.

If an activity is dominated by a linear convex combination of other DMUs, then, it is calledtechnically ine$cient (in DEA language) or simply ine$cient (in SPM language). The followinglinear programming model allows us to determine technical ine$ciencies. For the p activity, westate:

min’p; (2)

subject toN∑k=1

’kyk¿yp; (3)

N∑k=1

’kxk6 xp (4)


with the sum of weights equal to 1 and the non-negativity constraints. At least, one of the aboveconditions must be a strict inequality. If the above linear programming model leads to ’p=0, then,the p activity is technically ine$cient. If it leads to ’p = 1, such DMU is technically e$cient (ornon-ine$cient in SPM language). The technically e$cient DMUs lie on the production possibilityfrontier of the activity convex set. This frontier is a polygon in the bi-dimensional context of thisparagraph. This result is obtained in [13] by using another, but equivalent, linear programming model(see also [5]).

Let us de;ne the production scale by the input level, that is, we say that the production processis performed at the scale c if x= c. Suppose that the k = u activity is performed at the c scale. Letycmax be the output corresponding to the c input on the production possibility polygon, namely:

ycmax = max y for x = c; (5)

subject to the production possibility polygon. It is not necessary that the point (c; ycmax) coincideswith an existing activity on the polygon. An index of technical e$ciency for the u activity is givenby the SPM ratio:

ET(u) = yu=ycmax: (6)

Notice that the ratio (6) is at a maximum (equal to 1) when the u activity is technically e$cient.On the other hand, an index of scale e$ciency is expressed by the SPM ratio:

ES(c) = ycmax=c: (7)

All the activities with input x = c have the same index of scale e$ciency according to Eq. (7).From (6) and (7) we get:

ET(u)ES(c) = yu=c = EG(u): (8)

In other words, the index of global e$ciency is equal to the index of scale e$ciency times theindex of technical e$ciency.

Input-oriented measurement is performed in a similar way, but the activity size is measured bythe output, not by the input. The SPM indexes are the following:

E′G = xk=yk ; (9)

E′T(u) = xu=xcmin; (10)

E′S(c) = xcmin=c; (11)

where xu is the input of the k = u activity while the scale is measured by the corresponding outputyu = c. Symbol xcmin denotes the input corresponding to the c output on the production possibilitypolygon.

The above indexes are ranged over the intervals shown in Table 1.In output-oriented measurement (see Table 1) the greater indexes the better e$ciency whereas in

input-oriented measurement the greater indexes the worse e$ciency.So far we have dealt with the one input–one output scenario where the substantial, but not formal,

coincidence between DEA and SPM is worth remarking upon. In this case, the complete ranking of


Table 1Ranges of SPM indexes

Output oriented Input oriented

Global e$ciency 0–max EG Min E′G–∞

Technical e$ciency 0–1 1–∞Scale e$ciency 0–max EG Min E′

G–∞

DMUs is obtained either by the SPM ratios (1), (6) and (7) with output-oriented measurement orby the ratios (9), (10) and (11) with input-oriented measurement.

Can DEA and SPM extend the above meaningful results to the general case of multiple inputsand outputs? As noted, SPM can do it by aggregating the inputs and the outputs by a single weightsystem. As a result of this aggregation, we face a two-dimensional space with the aggregate inputXk and the aggregate output Yk . Therefore, either the output-oriented Eqs. (1), (6) and (7) or theinput-oriented Eqs. (9), (10) and (11) can be used to ;nd the e$ciency indexes and to rank theDMUs.

In consequence, the achievement of an aggregation model characterised by a common system ofweights (or prices) is the main feature of SPM. As a previous step, the DMUs are classi;ed inine$cient and non-ine$cient activities by a linear programming model as shown in [5]. These SPMine$cient and non-ine$cient activities are proven to be DEA technically ine$cient and technicallye$cient, respectively, and vice versa. Therefore, this ;rst step leads to the same technical frontier asDEA-BCC. The second step (formally developed in Section 3, below) is the principal contribution ofthe proposed model: the single weight system and the complete ranking of the activities are derivedfrom it.

With regard to scale e$ciency, DEA determines scale ine$ciencies. However, it does not providethe complete ranking. In other words, the analyst cannot ;nd the best DMU, the second best, etc.either in the sense of global e$ciency or in the sense of scale e$ciency, by using DEA.

In Section 5 a comparison of DEA and SPM results is presented from the numerical informationprovided in Section 4.

3. General formal statement

As explained above, the ;rst step in SPM is to classify the DMUs in two groups, namely, domi-nated and non-dominated activities. In certain cases, all the activities are non-dominated; however,this special circumstance is not relevant to develop the model.

Assumption 1. All the activities must be evaluated by using the same weight system, namely, theweights should not change from one activity to another. For every non-ine$cient DMU, the marginbetween the aggregate output (or bene;t) and the aggregate input (or cost) must be greater than orequal to zero (the aggregations being made by the single weight system).


Justi�cation: Advantages of using a single weight system (that is, weights independent of theactivity to be evaluated) have been shown in the preceding sections. On the other hand, bene;tsmust cover costs as a plausible condition of non-ine$ciency. Hence, the margin must not be negative.

Theorem 1. There exists one weight system satisfying Assumption 1.

Proof. Special case to rule out (it will be considered in Remark 1 below): at least, one output orinput reaches equal value for all the non-ine$cient activities. Except for this special rare case, theproof is organised as follows:

(i) From Assumption 1, we have

s∑i=1

�iyij¿m∑h=1

�hxhj (12)

for all non-ine$cient alternatives (j = 1; 2; : : : ; n) where yij is the i-bene;t of the j-alternative, xhjis the h-cost of the j-alternative, �i and �h are the weights attached to the i-bene;t and the h-cost,respectively.

By appropriate algebraic manipulation (see [5, p. 619]) Eq. (12) becomes:

s+m∑�=1

w�z�j¿ 1 for j = 1; 2; : : : ; n (13)

where

z�j = yij for �= 1; 2; : : : ; s and i = 1; 2; : : : ; s;

z�j = xhmax − xhj for �= s+ 1; s+ 2; : : : ; s+ m and h= 1; 2; : : : ; m;

w� = �i

/m∑h=1

�hxhmax for �= 1; 2; : : : ; s and i = 1; 2; : : : ; s;

w� = �h

/m∑h=1

�hxhmax for �= s+ 1; s+ 2; : : : ; s+ m and h= 1; 2; : : : m;

xhmax is the greatest h-cost, that is: xhmax = max xhj (j = 1; 2; : : : ; n).As noted, the non-ine$cient alternatives are non-dominated points on an e$cient frontier. As

usual, we assume that the feasible set of alternatives in the z� space is convex. Thus, the e$cientfrontier is bounded by the points:

E� = (z1∗; z2∗; : : : ; z�−1∗; z∗� ; z�+1∗; : : : ; zs+m∗); (14)

where z�∗ denotes the � anti-ideal or nadir value (i.e. the absolute minimum of the z�j values) whilez∗� denotes the � ideal or anchor value (i.e. the absolute maximum of the z�j values).


(ii) Establish the following system of (s+ m) linear equations:

w�z∗� +∑�

w�z�∗ = 1 for all �; (15)

where � = 1; 2; : : : ; �− 1; �+ 1; : : : ; s+ m.By solving system (15) we obtain the following single solution:

w� =1

(z∗� − z�∗)[1 +

s+m∑�=1z�∗=(z∗� − z�∗)

] for �= 1; 2; : : : ; s+ m: (16)

(iii) From well-known properties of convexity, every z frontier point dominates a linear convexcombination of the E� vectors. Therefore, the weighted value of every z frontier vector is greaterthan (or equal to) the weighted value of the corresponding dominated linear convex combination.From Eqs. (15) all the E� vectors have their weighted value equal to 1. Since any linear convexcombination of numbers 1 is also equal to 1, we obtain that the z weighted value expressed by eachleft-hand side of Eq. (13) is greater than or equal to 1. Therefore, Eq. (13) is satis;ed by every zfrontier point, namely, by all the non-ine$cient activities.

Thus, Theorem 1 is demonstrated.

Remark 1. Consider the special case mentioned at the top of the above proof, namely, z∗� = z�∗ forone value of �. Then, we ;nd the indeterminate form 0/0 in Eq. (16). This indeterminate form mustbe rewritten by calculating the limit of the weight when (z∗� − z�∗) → 0, namely:

lim(z∗�−z�∗ )→0

1

(z∗� − z�∗)[1 +

s+m∑�=1z�∗=(z∗� − z�∗)

] =1

0 + (0 + · · ·+ z�∗ + · · ·+ 0)=

1z�∗: (17)

This result is easily extended to the case of various exceptional attributes.

Corollary 1. Another way of computing weights (16) is to solve the equation system (15) by usinga linear programming model as follows:

mins+m∑�=1

w�z�j for j = 1; 2; : : : ; n (18)

subject to conditions (13) and (15) with the non-negativity constraints w�¿ 0 for all �. Obviously,the result is the same if we maximise instead of minimising the objective function.

Proof. As the weights (16) simultaneously satisfy relationships (13) and Eqs. (15), the minimisation(or the maximisation) of each objective function must yield the single solution derived from the linearequation system (15). Thus, Corollary 1 is demonstrated.

Notice the usefulness of the above linear programming model to check Theorem 1 numerically.


Remark 2. From Corollary 1, the single weight system (16) does not overestimate any particularweighted value, namely, the weight system is not biased in favour of any particular activity. Anal-ogously, the weight system does not underestimate any particular weighted value, namely, it is notbiased against any particular activity.

Both the aggregate input and the aggregate output are calculated by the single weight system (16).In the special case de;ned above, the weights are modi;ed from Remark 1. By the aggregation,the multiple output/multiple input e$ciency problem is converted into the (Xk; Yk) bi-dimensionale$cient problem, where

Xk =s+m∑�=s+1

w�x�−s; k ; (19)

Yk =s∑�=1

w�y�;k ; (20)

for all the activities, that is, k = 1; 2; : : : ; N .

Remark 3. Input-oriented SPM technical, scale and global e$ciency scores for thepth activity areobtained as follows. Firstly, solve the linear programming model:

Xpmin = minN∑k=1

Xk’k (21)

subject toN∑k=1

Yk’k = Yp; (22)

N∑k=1

’k = 1; ’k¿ 0: (23)

From (21)–(23) we have

Technical score = Xp=Xpmin; (24)

Scale score = Xpmin=Yp; (25)

Global score = Xp=Yp: (26)

4. Case study

To fortify our analysis with more evidence, we will compute the global and scale indexes for 27hospital units from a Spanish hospital. Data were obtained from the regional health care authority forthe year 1996. Four outputs were identi;ed: number of outpatient consultations, number of emergencyattendances, case-mix weighted acute medical in-patient discharges, and case-mix weighted acute


Table 2Information from 27 hospital units

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

1 1 35.3 4822 2137 242 1038 1339 158.3 57292 0 34.5 8088 2555 53 791 10443 1 193.6 10551 5997 8224 9500 2006 0.0 04 0 39.8 4262 2613 353 546 9345 0 35.3 2959 1508 540 1333 2526 1 120.9 9138 8182 5989 5256 594 72.7 14137 1 9.4 2533 2885 382 389 27 184.2 80188 0 20.6 3055 2498 367 903 979 0 109.0 5894 3665 2577 3599 79910 0 7.6 1575 1915 150 224 19811 1 61.7 3671 5769 1253 1863 70 131.9 688012 1 4.0 730 1782 36 225 99 189.6 982113 1 22.8 1555 1523 1069 1447 127 170.8 899614 1 10.0 767 3325 79 334 6 183.6 978415 1 6.0 365 1184 69 268 15 187.6 1018616 0 16.0 1463 1093 289 1000 3417 1 75.7 7508 3364 7006 5167 775 117.9 304318 1 31.1 2437 4588 1405 1701 61 162.5 811419 1 53.3 4417 6370 5738 2598 318 140.3 613420 1 16.0 2954 4923 468 732 301 177.6 759721 0 47.0 2683 1918 1987 1764 18622 1 2.0 1132 941 50 126 36 191.6 941923 0 14.0 2927 1504 321 492 2124 0 14.3 1954 1134 29 592 36325 1 39.3 3750 3108 4277 1755 174 154.3 680126 0 6.0 1497 1216 65 169 10527 0 17.0 1460 1875 21 801 138Ideal values 8182 8224 9500 2006 191.6 10186Anti-ideal values 941 36 126 6 0.0 0

(1) Code; (2) Non-ine$cient (DEA technically e$cient) activity = 1; ine$cient (DEA technically ine$cient)activity = 0; (3) x1 = average number of beds; (4) x2 = cost of non-nursery medical sta3 in money units; (5) z1 =y1 = outpatient consultations; (6) z2 = y2 = emergency attendances; (7) z3 = y3 = medical weighted discharges; (8)z4 = y4 = surgical weighted discharges; (9) z5 = 193:6− x1 = 193:6− (3); (10) 10551− x4 = 10551− (4).

surgical in-patient discharges. Two inputs were speci;ed: the average number of sta3ed beds, andthe cost of non-nursing medical sta3 expressed in money units (10,000’s Spanish pesetas). Theseoutputs and inputs are the most commonly used in literature on hospital e$ciency; see for example,Hollingsworth and Parkin [14]. Other inputs such as the costs of nursing and administrative sta3have not been introduced into our model. This is justi;able since the nursing sta3 is highly correlatedwith the number of beds whereas the administrative sta3 is a common input of the hospital, namely,its cost is not charged to the hospital units.

As noted in Section 2, a previous step consists in classifying the DMUs in two groups, ine$cientand non-ine$cient, or equivalently, DEA technically ine$cient and technically e$cient, respectively.The result of this classi;cation is shown in both Tables 2 and 3.


Table 3SPM and DEA e$ciency indexes

(1) (2) (3) (4) (5) (6) (7) (8) (9)

14 1 3.980 0.251 1.000 0.251 1.000 1.000 1.00012 1 3.501 0.286 1.022 0.279 1.000 1.000 1.00015 1 3.098 0.323 1.000 0.323 1.000 1.000 1.00019 1 2.833 0.353 1.000 0.353 1.000 1.000 1.00020 1 2.585 0.387 1.225 0.316 0.895 1.000 0.89518 1 2.533 0.395 1.236 0.319 0.994 1.000 0.99425 1 2.138 0.468 1.413 0.331 1.000 1.000 1.00010 0 2.088 0.479 1.816 0.264 0.957 0.959 0.99813 1 2.057 0.486 1.844 0.264 1.000 1.000 1.00017 1 1.995 0.501 1.275 0.393 1.000 1.000 1.0003 1 1.881 0.532 1.000 0.532 1.000 1.000 1.00027 0 1.792 0.558 2.128 0.262 0.813 0.818 0.9946 1 1.779 0.562 1.249 0.450 0.751 1.000 0.75111 1 1.734 0.577 1.753 0.329 0.670 1.000 0.6701 1 1.680 0.595 1.833 0.325 1.000 1.000 1.0007 1 1.680 0.595 2.389 0.249 0.883 1.000 0.88321 0 1.550 0.645 2.152 0.300 0.808 0.814 0.99324 0 1.519 0.658 2.495 0.264 0.912 0.915 0.9974 0 1.484 0.674 2.153 0.313 0.861 0.864 0.9979 0 1.398 0.715 2.078 0.344 0.743 0.743 1.00026 0 1.382 0.724 2.381 0.304 0.652 0.684 0.95322 1 1.378 0.726 2.061 0.352 1.000 1.000 1.00016 0 1.364 0.733 2.590 0.283 0.949 0.981 0.9678 0 1.312 0.762 2.952 0.258 0.678 0.709 0.9565 0 1.143 0.875 3.366 0.260 0.639 0.642 0.9952 0 0.991 1.009 3.195 0.316 0.820 0.840 0.97623 0 0.860 1.163 4.111 0.283 0.539 0.544 0.991

(1) Code; (2) Non-ine$cient (DEA technically e$cient) activity=1; ine$cient (DEA technically ine$cient) activity=0;(3) Output-oriented SPM global e$ciency index; (4) Input-oriented SPM global e$ciency index; (5) Input-oriented SPMtechnical e$ciency index; (6) Input-oriented SPM scale e$ciency index; (7) Input-oriented DEA global e$ciency index;(8) Input-oriented DEA technical e$ciency index; (9) Input-oriented DEA scale e$ciency index.

Henceforward, we develop SPM to aggregate both outputs and inputs, namely, to obtain

Aggregate output = Yk = w1y1k + w2y2k + w3y3k + w4y4k ; (27)

Aggregate input = Xk = w5x1k + w6x2k (28)

for k = 1; 2; : : : ; 27, where Yk and Xk are the aggregate output and input, respectively, while(w1; w2; : : : ; w6) is the vector of prices (or weights) to be determined by SPM.

From (27) and (28), we obtain the SPM global e$ciency indexes:

EGk = aggregate bene;t=aggregate cost = Yk=Xk ; (29)

E′Gk = aggregate cost=aggregate bene;t = Xk=Yk (30)

which permit us to rank the DMUs with output orientation and input orientation, respectively.


To determine the (w1; w2; : : : ; w6) single price system the following steps are required.First step: We make the change

z1 = y1; z2 = y2; z3 = y3; z4 = y4; z5 = (x∗1 − x1); z6 = (x∗2 − x2) (31)

for every one of the 15 non-ine$cient DMUs, where x∗1 and x∗2 are the maximum x1 and x2,respectively, among these 15 alternatives. This change is meaningful. In fact, Eq. (31) convert every“more is worse” cost into a “more is better” variable.

Second step: From the set of non-ine$cient DMUs, we apply Corollary 1 to compute the weights(16) and check Theorem 1 numerically.

First set of constraints: This group of conditions includes 15 relationships corresponding to thenon-ine$cient DMUs, with the following general form:

z1jw1 + z2jw2 + z3jw3 + z4jw4 + z5jw5 + z6jw6¿ 1; (j = 1; 2; : : : 15): (32)

From Table 2 the 15 constraints (32) are displayed as follows:

2137w1 + 242w2 + 1038w3 + 1339w4 + 158:3w5 + 5729w6¿ 1; (33)

......................................................................................................................................................................

3108w1 + 4277w2 + 1755w3 + 174w4 + 154:3w5 + 6801w6¿ 1: (34)

Second set of constraints: This group of conditions includes six constraints, the coe$cients ofwhich being the maximum and the minimum values of each z variable. Their numerical expressionsare written as follows:

8132w1 + 36w2 + 126w3 + 6w4 + 0w5 + 0w6 = 1; (35)

941w1 + 8224w2 + 126w3 + 6w4 + 0w5 + 0w6 = 1; (36)

941w1 + 36w2 + 9500w3 + 6w4 + 0w5 + 0w6 = 1; (37)

941w1 + 36w2 + 126w3 + 2006w4 + 0w5 + 0w6 = 1; (38)

941w1 + 36w2 + 126w3 + 6w4 + 191:6w5 + 0w6 = 1; (39)

941w1 + 36w2 + 126w3 + 6w4 + 0w5 + 10186w6 = 1; (40)

Third set of constraints: including the six non-negativity conditions for the weights.Objective function: From Corollary 1, we minimise (or maximise) any left-hand side (32). For

example:

Min(2137w1 + 242w2 + 1038w3 + 1339w4 + 158:3w5 + 5729w6): (41)

As a unique solution to the above linear programming model, we obtain

w1 = 0:00012;w2 = 0:00011;w3 = 0:00009;w4 = 0:00043;w5 = 0:00454;w6 = 0:000085: (42)


Readers interested in numerical computing can check that solution (42) and the weight system (16)exactly coincide, although this checking would be an undue burden to them.

By using the single weight system (42), the aggregate output (20) and the aggregate input (19)are obtained for every DMU. From these values, Eqs. (29) and (30) yield the output-oriented andinput-oriented e$ciency indexes, respectively. Also, the technical and scale scores are computed asindicated in Section 2. All these indexes along with the DEA scores with input orientation are shownin Table 3.

5. Comparison of results

We turn now to the analysis of numerical results derived from our case study. The point at issuehere is whether it is justi;able to test SPM in the light of DEA numerical results, and vice versa.The foundations of DEA and SPM are substantially di3erent. Apart from the common determinationof dominated and non-dominated activities, the ethos of SPM is the property of single price thatrelies on Assumption 1 and Theorem 1. This frame is dissimilar from the DEA framework, andno relationship between both models can be described. Therefore, if a high correlation betweenDEA and SPM results is persistently observed, then, this correlation seems to involve a su$cientguarantee of soundness for both approaches. Pursuing the idea of testing by comparison of results,some critical conclusions from the numerical case study are stated as follows:

(i) Correlation of global e=ciency indexes: Considering the DEA results, there are 10 globale$cient DMUs, since each of them scored 1 as shown in Table 3. These DEA e$cient activitiesare placed at the top of the SPM ranking with few exceptions. First, second, third, fourth,seventh, ninth and tenth places in the SPM ranking correspond to DEA e$cient activities.Fifth, sixth and eighth places correspond to activities which are almost DEA e$cient withscores 0.90, 0.99 and 0.96, respectively

(ii) Correlation of scale e=ciency indexes: Considering the DEA results, there are 11 scale e$cientDMUs, since each of them scored 1 as shown in Table 3. The 10 activities at the top of theSPM scale ranking have high or very high DEA scale e$ciency indexes. Second, seventh andninth places in the SPM scale ranking correspond to DEA scale e$cient activities. First, third,fourth, ;fth, sixth, eighth and tenth places correspond to activities which are almost DEA scalee$cient with scores 0.88, 0.96, 0.99, 0.99, 0.99, 0.99 and 0.96, respectively. However, the 10activities at the bottom of the SPM scale ranking include eight DEA scale e$cient activitiesand two which have the lowest DEA scale e$ciency indexes, 0.67 and 0.75.

Regarding examples analysed elsewhere [5], these empirical conclusions seem to strengthen si-multaneously the robustness of DEA as a methodology to estimate ine$ciencies and the validity ofSPM as a ranking model.

There is an ongoing issue that the validity of DEA to determine the most productive scale sizeappears doubtful in problems with multiple inputs and outputs. Regarding this issue, two consistencyconditions should be examined.

First class condition (Size measurement): Suppose we measure the scale size either by the ag-gregate input or by the aggregate output, the aggregation being made by the DEA weight system


Table 4First class condition of consistency: results

(1) (2) (3)

3 3 317 6 176 17 69 1 919 19 191 9 12 2 2525 4 24 25 1111 20 421 11 2118 21 1820 18 135 24 513 5 2024 13 248 10 827 8 2716 27 1610 7 107 14 714 12 2323 26 1412 23 1226 16 2615 15 1522 22 22

(1) Output aggregated by the k = 1 DEA weight system; (2) Output aggregated by the k = 12 DEA weight system;(3) Output aggregated by the DEA average weights.

corresponding to the kth activity. A necessary condition of consistency is the invariant ordering ofthe N activities on the scale, namely, the DMUs should be ordered with independence of the DEAweight system used.

Second class condition (Optimal size): If the input-oriented (or output-oriented) scale e$ciencyindex of an activity (Banker et al. [13]) is related to the activity size, then, the resulting settingshould provide the activity of most productive scale size.

The ;rst class condition is not satis;ed by DEA, as shown in Table 4.In fact, one can see that the scale ordering widely changes when the DEA weight system does.

Changes in the ordering seem to be wider as the number of inputs (or outputs) increases. The secondclass condition is not met either as Fig. 1 shows.

This ;gure graphs the curve describing the relationship between the activity size and the corre-sponding input-oriented DEA scale e$ciency index. The activity size is computed by aggregatingthe outputs by the DEA weight system associated with a particular DMU such as k=12. As be;tting


0.6

0.7

0.8

0.9

1

0 4 8 12 16

Output aggregated by the DEA weightsystem of activity k=12

DE

A s

cale

effi

cien

cy in

dex

Fig. 1. Mountain-shaped diagram (input-oriented). Output aggregated by a particular DEA weight system.

its shape, we shall refer to that curve as the mountain-shaped diagram. Instead of providing theactivity of most productive scale size, the mountain-shaped diagram displays a number of apexes orpeaks spread over the graph, each apex corresponding to a di3erent activity size. In other words,there are di3erent uppermost points, all of them associated with a DEA scale index equal to 1. Theseuppermost points lie far from one another, and therefore, they are projected on the horizontal axis atactivity size points also far from one another. Hence, the optimum scale size is not given by DEA.These last remarks together with (ii) seem to corroborate that DEA is a suitable tool to determinescale ine$ciencies rather than scale e$ciencies.

We could be tempted to use other weight systems derived from the DEA weighting (see Jessop[15]). One possible solution among many others is obtained by minimising the sum of squaredeviations:∑

i

∑k

(wik − w∗i )

2 +∑h

∑k

(whk − w∗h)

2: (43)

Minimisation of (43) is easily proven to yield the average weight system (w∗i ; w

∗h). In this system,

the weight attached to each output (input) is the mean of the di3erent weights given by DEA forthis output (input). However, the average weights as well as other systems related to DEA meetneither the second class condition nor a plausible requirement similar to the ;rst class condition (seeTable 4 and Fig. 2).

6. Conclusions

The purpose of SPM is not to classify the activities in two groups by tracing a boundary betweenglobal e$ciency and ine$ciency. However, SPM establishes the frontier between technically e$cientand ine$cient activities (de;ned as non-dominated and dominated DMUs, respectively), althoughthis classi;cation is only a previous step in SPM, not the core of the model. The SPM main step


0.6

0.7

0.8

0.9

1

0 21 3 5 7 94 6 8 10

Output aggregated by the DEA average weights

DE

A s

cale

effi

cien

cy in

dex

Fig. 2. Mountain-shaped diagram (input-oriented). Output aggregated by the DEA average weights.

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0.550

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500

Output aggregated by the single weight system

Inpu

t-O

rient

ed S

PM

sca

le e

ffici

ency

inde

x

Fig. 3. SPM chasm-shaped diagram (input-oriented).

consists in ;nding a single weight system to aggregate inputs and outputs, this system being derivedfrom Assumption 1 as a plausible postulate.

Aggregating the inputs and outputs by the single weight system is a crucial stage to reduce themultiple input–output problem of e$ciency to a simple manageable one input–one output setting,where the complete ranking of the activities is workable. In this one input–one output setting thefrontier of technical e$ciency can easily be plotted. From this frontier, both the scale and globale$ciency indexes (either input or output-oriented) are straightforwardly determined by simple ratios.Thus, the most productive scale size as well as the optimal global e$cient activity are obtainedby SPM (see Fig. 3 where the most productive scale size corresponds to the minimum of thechasm-shaped curve).


Comparison with DEA is fruitful. As for global e$ciency, the DEA e$cient activities (all ofthem with score equal to 1) are placed at the top of the SPM ranking, with some exceptions. As forscale e$ciency, DEA seems to be capable of estimating ine$ciencies. However, the activities whichscored 1 in DEA are distributed over the scale axis in the mountain-shaped diagram, and therefore,no estimation on the most productive scale size can be made.

Acknowledgements

Thanks are given to the referees for their helpful suggestions which have greatly improved thepresentation and accuracy of the paper. The English editing by Mr. Keith Stuart is appreciated.

References

[1] Charnes A, Cooper WW, Rhodes E. Measuring the e$ciency of decision making units. European Journal ofOperational Research 1978;2(6):429–44.

[2] Cooper WW, Seiford LM, Tone K. Data envelopment analysis. A comprehensive text with models, applications andreferences and DEA-solver software. Dordrecht: Kluwer Academic Publishers, 2000.

[3] Aigner DJ, Lowell CAK, Schmidt P. Formulation and estimation of stochastic frontier production function models.Journal of Econometrics 1977;6(1):21–37.

[4] Battese GE, Corra GS. Estimation of a production frontier model: with application to the pastoral zone of easternAustralia. Australian Journal of Agricultural Economics 1977;21(3):169–79.

[5] Ballestero E. Measuring e$ciency by a single price system. European Journal of Operational Research 1999;115:616–23.

[6] Charnes A, Cooper WW, Huang ZM, Sun DB. Polyhedral cone-ratio DEA models with an illustrative applicationto large commercial banks. Journal of Econometrics 1990;46:73–91.

[7] Thompson RG, Singleton FD, Thrall RM, Smith BA. Comparative site evaluations for locating a high-energy physicslab in Texas. Interfaces 1986;16(6):35–49.

[8] Wong Y-HB, Beasley JE. Restricting weight Texibility in data envelopment analysis. Journal of the OperationalResearch Society 1990;41(9):829–35.

[9] Dyson RG, Thanassoulis E. Reducing weight Texibility in data envelopment analysis. Journal of the OperationalResearch Society 1988;39(6):563–76.

[10] Cook WD, Kress M, Seiford LM. Prioritization models for frontier decision making units in DEA. European Journalof Operational Research 1992;59:319–23.

[11] Andersen P, Petersen NC. A procedure for ranking e$cient units in data envelopment analysis. Management Science1993;39(10):1261–4.

[12] Doyle JR, Green RH. Cross-Evaluation in DEA: improving discrimination among DMUs. Information Systems andOperational Research Journal 1995;33(3):205–22.

[13] Banker RD, Charnes A, Cooper WW. Some models for estimating technical and scale ine$ciencies in dataenvelopment analysis. Management Science 1984;30(9):1078–92.

[14] Hollingsworth B, Parkin D. The e$ciency of Scottish acute hospitals: an application of data envelopment analysis.IMA Journal of Mathematics Applied in Medicine 1995;12:161–73.

[15] Jessop A. Entropy in multiattribute problems. Journal of Multicriteria Decision Analysis 1999;8:61–70.

Enrique Ballestero is a researcher in the MCDM ;eld. His contributions have been published in international journalssuch as: Operations Research Letters (1991, 1993), Theory and Decision (1994), Journal of Multi-Criteria DecisionAnalysis (1993, 1994, 1997), Journal of the Operational Research Society (1996, 1998, 2000), European Journal ofOperational Research (1997, 1999, 2001), Decision Science (2002), and Journal of Environmental Management (2002).


He has published the handbook Multiple Criteria Decision Making and its Applications to Economic Problems, Kluwer,with Professor Carlos Romero.Jos+e A. Maldonado is a computer engineer from the Technical University of Valencia and B.Sc in Computer Science

from John Moores University, Liverpool, UK. He is currently pursuing a Ph.D. degree at the Technical University ofValencia. His research interests include healthcare e$ciency and representation of healthcare information.

objective measurement of efficiency: applying single price model to rank hospital activities

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