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European Journal of Operational Research 195 (2009) 563–574

O.R. Applications

Selecting non-zero weights to evaluate effectiveness of basketballplayers with DEA

W.W. Cooper a,*, Jose L. Ruiz b, Inmaculada Sirvent b

a The Red McCombs School of Business, The University of Texas at Austin, 1 University Station B6500, Austin, TX 78712-0212, USAb Centro de Investigacion Operativa, Universidad Miguel Hernandez, Avd. de la Universidad, s/n, 03202 Elche (Alicante), Spain

Received 19 March 2007; accepted 13 February 2008Available online 20 February 2008

Abstract

In this paper, we show how DEA may be used to identify component profiles as well as overall indices of performance in the contextof an application to assessments of basketball players. We go beyond the usual uses of DEA to provide only overall indexes of perfor-mance. Our focus is, instead, on the multiplier values for the efficiently rated players. For this purpose we use a procedure that werecently developed that guarantees a full profile of non-zero weights, or ‘‘multipliers.” We demonstrate how these values can be usedto identify relative strengths and weaknesses in individual players. Here we also utilize the flexibility of DEA by introducing boundson the allowable values to reflect the views of coaches, trainers and other experts on the basketball team for which evaluations are beingconducted. Finally we show how these combinations can be extended by taking account of team as well as individual considerations.Published by Elsevier B.V.

Keywords: Data envelopment analysis (DEA); Indexes of performance; Efficiency; Effectiveness; Weights

1. Introduction

In this paper, we address the problem of developingindexes of performance that result from aggregation of sev-eral indicators by using DEA. These indexes generally havethe form of a weighted sum of the variables. It sometimeshappens that experts, by virtue of their knowledge of theproblems, are able to prescribe a set of weights, which(unlike in DEA) are common to all DMUs. However, anissue often arises in practice because the weights that areto be assigned to different inputs and outputs are unknownfor such a priori specification. In addition, the weights, ifused, may be unsatisfactory in terms of their further effectson the overall evaluation.

As is well known, DEA yields efficiency scores in theform of a weighted sum of either several inputs or severaloutputs without any need for a priori information on the

0377-2217/$ - see front matter Published by Elsevier B.V.

doi:10.1016/j.ejor.2008.02.012

* Corresponding author. Tel.: +1 015124711822; fax: +1 015124710587.E-mail addresses: cooperw@mail.utexas.edu (W.W. Cooper), jlruiz@

umh.es (J.L. Ruiz), isirvent@umh.es (I. Sirvent).

relative values of these variables. In addition, DEA alsopermits incorporating ‘‘bounds” on the weights and thusrelaxes the need for knowledge of exact values for theweights to be used. This all suggests that DEA mightimprove upon presently used indexes of performance invery flexible ways.

In the development of DEA, the possibility of providingefficiency indexes without any need to have informationavailable about costs and prices or other preassignedweights has been emphasized and pointed up as one ofthe attractive features of this new methodology. In placeof preassigned weights, DEA determines values in a man-ner that maximizes the efficiency score of the entity beingevaluated. This allows great flexibility in the choice of theweights. As a consequence, the resulting weights maysometimes be unreasonable, and they may not be consis-tent with accepted views. See, e.g., Allen et al. (1997) andThanassoulis (2001) for purposes, motivations and usesof weights restrictions or bounds on admissible values.

Here we will use a new approach described in Cooperet al. (2007a) which provides another (new) way to choose

564 W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574

weights for use in DEA that has advantages like those wewill describe. This approach involves a two-step procedurethat effects a choice of weights that are associated with alter-native optimal solutions of the extreme efficient DMUsprovided by dual, ‘‘multiplier,” formulations of the DEAmodel. Among other things, for efficient units, the proce-dure we describe guarantees a set of strictly positive weightsfor all inputs and outputs and thus eliminates the need fordealing with zero values for these weights which has oftenconstituted a weakness in DEA (see Allen et al., 1997).

We here provide an example of the use of this approachin the form of an application to the assessment of basketballplayers. This kind of evaluation is already being used in theSpanish Basketball League (called the ACB league) whichuses an index of player assessments that has the form ofan unweighted sum of the classical indicators in basketballstatistics (points, rebounds, assists, fouls, etc.). The researchto be described was directed to examining DEA as a possi-ble alternative to presently used ways of specifying weights.To be more specific, in this application we exploit the abilityof DEA both to (1) set conditions on the relative value ofthe indicators based on expert opinion by means of restric-tions on the weights and (2) allow the weights to vary acrossplayers in order to reflect their different characteristics. Thispaper is thus in the line of papers like those by Melyn andMoesen (1991), with the so-called ‘‘benefit of the doubt”weighting, Takamura and Tone (2003) and Lauer et al.(2004). (The latter two papers also use AHP (the analytichierarchy processes), to specify bounds on the weights).

One point needs to be made in that we use weightrestrictions, such as those provided by the ‘‘AssuranceRegion” (AR) approaches of Thompson et al. (1986),which allow us to specify player profiles by means of upperand/or lower limits for the ratio of the weights that are tobe assigned to the performance attributes of the players.Thus, the players are assessed in a manner that is consistentwith the views of the coach and his staff and also takeaccount of the different positions the players occupy.

As stated in the above discussion, we also use DEA toanalyze the resulting indexes in terms of relative evaluationsin a manner that can provide additional useful informationin the context of applications to basketball. In particular,we here use DEA to provide insight into (a) how the efficientplayers achieved their efficiency and (b) which aspects oftheir game the deficient players can improve.

We also depart from the more customary ‘‘efficiency”analyses and focus instead on ‘‘effectiveness” in the sensethat in our analysis there is no reference to resources con-sumed, as in the ‘‘efficiency” evaluations used in micro eco-nomics. Thus, we follow Prieto and Zofio (2001) in theiruse of DEA to examine the effectiveness of municipal per-formance in Spain by examining outputs resulting fromgovernment subsidies but not the resources (or otherinputs) used, and which are needed for efficiency evalua-tions that depend on resource utilization. Here we confineattention to player outputs such as points scored and/orpercentage of free throw successes and leave out of consid-

eration such things as player salaries, etc. Similarly we donot include benefit measures such as revenues earned, orlike considerations. We also confine attention to playerqualities that take account of their positions on a teamwithout respect to audience appeal, etc.

DEA, as well as parametric techniques for efficiencyanalysis, have been used previously in the world of sports.For example, Sueyoshi et al. (1999) and Fried et al. (2004)apply DEA models to players of baseball and golf. Sextonand Lewis (2003) do the same with Major League Baseballteams in the US and Zak et al. (1979) analyze the efficiencyof basketball teams of the NBA in the US with Cobb–Douglas production functions. Relative efficiency in sportshas also been measured at country levels, as in Lozanoet al. (2002) where DEA is used to measure the perfor-mances of the nations participating at the Summer Olym-pics. Applications can also be found where efficiency isanalyzed from the perspective of managers of sport teams.For instance, Scott et al. (1985) utilize Cobb–Douglas fron-tiers to examine the salary and marginal revenue productsin professional basketball and Fizel and D’Itri (1999) carryout an analysis of managerial efficiency with DEA modelsin order to investigate the impact on organizational perfor-mance of practice like firing and hiring managers.

The paper is organized as follows: In Section 2 weinclude some of the theory underlying our use of DEA todevelop effectiveness indexes in the context of the problemswe address. We focus on weight specifications in which spe-cial attention is given to the choice of weights betweenalternative optimum solutions in DEA formulations. Weuse Section 2 to briefly describe the procedure proposedin Cooper et al. (2007a) for the selection of weights. InSection 3 we apply these procedures to the assessment ofbasketball players in the context of the Spanish PremierLeague. Section 4 concludes.

2. Theoretical aspects

As already stated, we propose a use of DEA to developindexes of performance. In general, if we have a set of n

decision making units (DMUs) that use m inputs toproduce s outputs, we can carry out an analysis of theirrelative efficiency by either of the following pair of dualproblems. First, we state the primal problem

max /0 þ eXm

i¼1

s�i0 þXs

r¼1

sþr0

!

subject toXn

j¼1

kixij ¼ xi0 � s�i0; i ¼ 1; . . . ;m

Xn

j¼1

kjyrj ¼ /0yr0 þ sþr0; r ¼ 1; . . . ; s

kj P 0 j ¼ 1; . . . ; n

s�i0 P 0 i ¼ 1; . . . ;m

sþr0 P 0 r ¼ 1; . . . ; s;

ð1Þ

W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574 565

where e > 0 is a non-Archimedean element, smaller thanany positive real number, and then we state the dual

min x0 ¼Xm

i¼1

mixi0

subject toXs

r¼1

lryr0 ¼ 1

�Xm

i¼1

mixij þXs

r¼1

lryrj P 0 j ¼ 1; . . . ; n

mi P e i ¼ 1; . . . ;m

lr P e r ¼ 1; . . . ; s;

ð2Þ

where the xij and yrj represent the input and outputamounts recorded for DMUj, j = 1, . . .,n, and xi0 and yr0

represent these input and output amounts for the DMU0

being evaluated. (Note: These are also referred to as the‘‘envelopment” and the ‘‘multiplier” models, respectively,and are output oriented, see Charnes et al., 1978.)

Remark. Model (1) is the so-called CCR model in envel-opment form. It is also referred to as the ‘‘constant returnsto scale model.” A referee asked us to consider extension tovariable returns to scale technologies. However, as shownby the theorem on p. 49 in Cooper et al. (2004) the CCRmodel in (1) can also be used to determine returns to scale,if desired, by reference to the following conditions,

1 The rest of the DMUs usually have unique optimal solutions for theirweights.

(i) Constant returns to scale prevail at ðx0; y0Þ ifPnj¼1k

�j ¼ 1 in any alternate optima.

(ii) Decreasing returns to scale prevail at ðx0; y0Þ ifPnj¼1k

�j > 1 for all alternate optima.

(iii) Increasing returns to scale prevail at ðx0; y0Þ ifPnj¼1k

�j < 1 for all alternate optima.

where ðx0; y0Þ represent the vectors of coordinates for theefficient point used to evaluate the performance ofDMU0. See pp. 46–52 in Cooper et al. (2004) which showshow this relates to the CRS model as a boundary formeasuring ‘‘returns to scale inefficiencies” in scale perfor-mances. Thus, our constant returns to scale model can beused to determine both returns to scale and technical inef-ficiencies. Here we concentrate on the latter using therecently developed methods in Cooper et al. (2007a) todevelop non-zero weights for (2).

In this paper, we do not carry out the further analysesneeded to accomplish these results. Instead, we proceedon the assumption of constant returns to scale because ofthe need for dealing with the weight restrictions we willbe using (see Thanassoulis, 2001). See also Fare et al.(1985) who use both the CRS and VRS models in theirreturns to scale analyses. We could also use the standardVRS model except the weight restrictions we later intro-duce generate new (artificial) variables in the envelopmentmodel (1), which can, inter alia, affect the returns to scaleresults. In any case the CRS region dominates the bound-aries of the corresponding VRS model with coincidencealways occurring in at least one point.

In this paper, the above models are modified to con-struct an index of performance and identify its componentsfor use in analyzing the components of this index. The thusweighted variables are applied only to the outputs. Noinputs are used. Formulations (1) and (2) are thereby sim-plified, because we include only one constraint in which theinput of each DMU is represented as a scalar with unitvalue. Thus, as already discussed, our measures are direc-ted to measures of ‘‘effectiveness” rather than ‘‘efficiency”.

We are particularly interested in formulation (2), sincethe chief purpose of this paper is the use of DEA modelsto select component weights. For this purpose, Cooperet al. (2007a) point up the need for addressing the problemof selecting weights between alternative optimal solutions ofthe dual multiplier formulation of the DEA model as part ofa procedure for choosing a set of weights. It is well knownthat the optimal solutions of (1) for extreme efficient unitsare usually highly degenerate, which (2) lead to the presenceof alternate optima. This creates a problem when interpret-ing the relative value of the variables in the efficiency assess-ment, because the optimal weights may result from selectingonly one of these alternatives (as is done by most of theDEA computer codes), and the weights may differ fromone optimum to another and thus lead to portrayals of per-formance that depend only on the software used. Moreover,it often occurs that only a few weights are non-zero in anoptimal solution. The DMU under analysis is then beingassessed on only a small subset of the inputs and outputs.

In Cooper et al. (2007a) a two-step procedure is proposedfor the selection of weights that is based on two general cri-teria of selection and is implemented by means of two mixedinteger linear programming (MILP) problems. We brieflydescribe this procedure in the following. First we focus ononly the ‘‘extreme efficient DMUs”, i.e., the DMUs thatare in the class E designated in Charnes et al. (1991).1 Formembers of this class, E, of extreme efficient DMUs, thefollowing MILP problem is solved in the first step

min b0 ¼Xj2E

bj

subject toXs

r¼1

lryr0 ¼ 1

�Xm

i¼1

mixij þXs

r¼1

lryrj þ tj ¼ 0 j 2 E

Xm

i¼1

mixi0 ¼ 1

tj �Mbj 6 0 j 2 E

bj 2 f0; 1g j 2 E

mi P 0 i ¼ 1; . . . ;m

lr P 0 r ¼ 1; . . . ; s

tj P 0 j 2 E:

ð3Þ

2 As a result, the e > 0 conditions in (1) and (2) are superfluous.

566 W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574

Here bothPs

r¼1lryr0 ¼ 1 and �Pm

i¼1mixij þPs

r¼1lryrjþtj ¼ 0; j 2 E correspond to (2), and

Pmi¼1mixi0 ¼ 1 reflects

the fact that the DMU0 being evaluated is efficient whileM is a big positive value. Interpreting the solution in geo-metric terms we say that in solving (3) we will be selectingoptimal weights from the dual (=multiplier model) formu-lation only those which are associated with hyperplanesthat are supported by the maximum number of extremeefficient DMUs. The so obtained weights will thereforebe the ones corresponding to the facets of the efficient fron-tier of highest dimension that the unit under assessmentcontributes to span. In this sense, these will have the max-imum possible support from the data from among thealternate optima associated with the current portion ofthe efficient frontier. In particular, if the assessed unit is lo-cated on a full dimensional ‘‘efficient (or effective) facet”(FDEF) then the weights will be selected for their associa-tion with such a facet of the frontier, see Cooper et al.(2007b).

In the second step, we solve the following MILPproblem

max z0

subject toXs

r¼1

lryr0 ¼ 1

�Xm

i¼1

mixij þXs

r¼1

lryrj þ tj ¼ 0 j 2 E

Xm

i¼1

mixi0 ¼ 1

tj �Mbj 6 0 j 2 EXj2E

bj ¼ b�0

mixi0 P z0 i ¼ 1; . . . ;m

lryr0 P z0 r ¼ 1; . . . ; s

bj 2 f0; 1g; tj P 0 j 2 E

z0 P 0

mi P 0 i ¼ 1; . . . ;m

lr P 0 r ¼ 1; . . . ; s;

ð4Þ

where b�0 is the optimal value of the objective in (3).Thus, from the weights chosen in the first step, this sec-

ond step MILP problem selects those which maximize therelative value of the variable with minimum value for thecorresponding ‘‘virtual” input or output that is representedby mixi0 and lryr0. In this step, we look for weights thathave associated programs of performance in which theinputs and outputs globally maximize their relative‘‘importance” (see Thanassoulis, 2001).

A weight represents the rate of decrease (increase) in theeffectiveness score with respect to the per unit decrease(increase) in the attribute of DMU0’s performance forwhich the weight serves as the coefficient. See Chapter 4in Cooper et al. (2004) for a more full discussion. Cooperet al. (2007a) show that, in the case of the efficient DMUs,

the weights provided by this two-step procedure are strictlypositive, so this procedure makes possible a portrayal ofthe efficiency of the unit under assessment in which no var-iable is completely ignored.2

We also note that this procedure can be adapted in astraightforward manner to situations in which efficiencyis evaluated by using AR approaches to bound the allow-able value. This is accomplished by simply introducingrestrictions on the weights both in (3) and (4). In addition,in order to use this two-step procedure with AR models, itis also necessary to replace the set E of extreme efficientDMUs with the set of extreme efficient DMUs that areAR efficient (see Cooper et al. (2007a) for discussions).

In our application to ‘‘effectiveness” evaluations, whereonly outputs are considered, restrictions on the weights areused to incorporate the views of the basketball experts inwhich a set of AR-I type constraints will take the form

lr;r0 6lr

lr06 ur;r0 ð5Þ

where lr0 and lr are the weights of outputs r0 and r, respec-tively, and lr;r0 and ur;r0 are the lower and upper bounds onthe allowable values of the ratios of these weights. There-fore, models (2)–(4) will include these constraints whenused in our application below.

3. Assessments of basketball players

As noted earlier, the Spanish Premier Basketball League(called ACB) utilizes a scalar index for assessment of play-ers that results from aggregating classical indicators of per-formance statistics (points, rebounds, assists, . . .). In theaggregation used by this league, the variables that representpositive aspects of the game of a player (e.g., points orrebounds) are weighted with +1, whereas those that repre-sent negative aspects (e.g., turnovers) have a weight of �1.

One can immediately see weaknesses in this index. Per-haps the most important one is the fact that all of the fac-tors are considered to have the same ‘‘importance” as aconsequence of assigning a weight of magnitude 1 to eachof them. However, people familiar with basketball wouldgenerally agree that fouls are not as important as eitherpoints or rebounds. This approach also does not takeaccount of the position that the player plays and hence failsto reflect the fact that, for example, rebounds are not asimportant for a playmaker as they are for a center.

By contrast, DEA allows us to set requirements on therelative value of the indicators (by using weights restric-tions) so that players playing in the same position will beassessed with reference to a previously specified profile(e.g., restrictions that incorporate preferences on the indi-cators for players that play in the position of center).And, in addition, DEA allows the weights to vary acrossplayers (within the previously established limits), whichmakes it possible that each player exploits his own charac-

3 To simplify the notation, we maintain /�0 as the optimal value of theAR models used. Obviously, this will not be the optimal value of (1), butrather it will correspond to the dual problem (2), once the AR-Iconstraints have been incorporated into this latter problem. (This isjustified by the dual theorem of linear programming, which gives/� ¼ min x�0.)

W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574 567

teristics in order to obtain a performance index in the bestpossible light.

Furthermore, DEA provides us with information onsources and amounts of inefficiency or ineffectivenessfor each player and thereby points to where improve-ments in performance can be made. The result is an eval-uation of each player with reference to the rest of theassessed players. For efficient players, the optimal weightswill provide insight into how they achieved their effective-ness. One purpose of this paper is to show how a DEAassessment of basketball players might be used in placeof the classical basketball indexes now being used and,more importantly, we want to show exactly how theproposed new methodology also provides additionalinformation.

The data have been taken from http://www.acb.com/and correspond to the 2003–2004 season. We have selecteda sample of 172 players consisting of those who haveplayed at least 17 games (half a regular season). We con-sider only players who have played a large enough numberof games to reflect their performances reliably. These 172players have been classified into the five following groupsaccording to their position: playmaker, guard, small for-ward, power forward and center. The idea is to havehomogenous samples when assessing the efficiency of theplayers. For this study, we have defined the following asindicators for the main aspects of the game: shooting,rebounding, ball handling and defense. In particular, theproposed summary of indicators to be included inthe model has made possible an important reduction ofthe dimensionality of the output space compared to thelarge number of factors used by the ACB index. Other lessrelevant variables, such as blocked shots, have not beenconsidered. To be specific, the following variables wereused for our analyses.

– Adjusted field goal (AFG) = (PTS � FTM) � AFG%,where PTS = points made (per game), FTM = freethrows made (per game) and AFG%, called ‘‘adjustedfield goal percentage,” is defined as PTS�FTM

2�FGA, where

FGA is the number of field goal attempts. AFG% isused in NBA statistics (see http://sports.espn.go.com/nba/statistics/) for the purpose of measuring ‘‘shootingefficiency by taking into account the total points a playerproduces through his field goal attempts. The intuitionbehind this adjustment is largely to evaluate the impactof ‘‘three-point shooting”. Therefore, AFG is a shootingindicator adjusted for opportunities. We could have sep-arately considered PTS-FTM and AFG% but we pre-ferred to aggregate both variables into AFG in orderto avoid mixing a percentage with a volume measure –see Dyson et al. (2001) for a discussion of the pitfallsthat can be encountered in DEA applications.

– Adjusted free throw (AFT) = FTM � FT%, where FT%is the free throw successes percentage. Our commentson the mix of percentages with volume measures is alsoapplicable to this variable.

– Rebounds (REB): the number of rebounds per game.– Assists (AST): the number of assists per game.– Steals (STE): the number of steals per game.– Inverse of turnovers (ITURN). We have used the inverse

of the number of turnovers per game in order to treatthe information regarding this indicator as an outputthat decreases with increases in turnovers, instead ofan input, This approach is used because it enables usto obtain an index with the same form as the one usedby the ACB league.

– Non-made fouls own (NFO) = 5 � FO, FO being thenumber of fouls made (per game) by the assessed player.The purpose of this transformation is the same as in theprevious variable, ITURN.

– Fouls opposite (FOPP): the number of fouls per gamethe opposite players have made on the player that isbeing assessed.

In this application, we fortunately had access to thetechnical staff of Etosa Alicante, the team of the ACB lea-gue of the Lucentum Basketball Club of the city of Ali-cante. This made it possible to employ the opinions ofexperts available for use in developing and responding toour analysis. To be specific, the members of the technicalstaff of Etosa Alicante provided us with informationregarding the relative value of the above defined outputs,at least in the form of bounds on the possible values –see (5) – for each of the different positions. In particular,the coach of Etosa was asked to give his opinion on theirrelative ‘‘importance.” For each position this resulted ina set of AR-I type restrictions.

Thus, the effectiveness of players was assessed by usingan output-oriented AR-I model, as in (2), with constantreturns to scale, using outputs in the constraints whileincluding only a nominal input constraint. Thus, the ARmodels used in this context provide an analysis of the effec-tiveness of players at least within limits obtained from apreviously specified profile. Finally, since the effectivenessscore we use for each player /�0 is greater than or equalto unity, the assessment index is defined by 0 6 I�0 ¼/��1

6 1,3 which is in the same form as the ACB measure.The results of the application will be shown below. As

representative cases, only the performances of playmakersand centers are reported in order to simplify the discussionand keep the length of this paper within reasonable bounds.

3.1. Assessment of the playmakers

The sample of playmakers consists of 41 players. In theopinion of the Etosa coach about the relative value of the

568 W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574

factors involved when assessing a playmaker, the followingare the most important ones: (a) field goal shooting and (b)the free throw points (and consequently the fouls made bythe opposing players on them) as well as (c) the assists and(d) the steals as aspects concerning ball handling anddefense. He also stated that the fouls made by a playmakerare a factor of little importance. This led us to include thefollowing set of constraints in (2), so that this became anAR-I type formulation,

lITURN 6 lAFG lREB 6 lAFG

lITURN 6 lAFT lREB 6 lAFT lNFO 6 lREB

lITURN 6 lAST lREB 6 lAST lNFO 6 lITURN

lITURN 6 lSTE lREB 6 lSTE

lITURN 6 lFOPP lREB 6 lFOPP

ð6Þ

Table 1 presents evaluations of player performances. In thefirst column we record I�0 ¼ /��1

0 as a measure of the degreeof effectiveness, while in the remaining eight columns weprovide profiles for the performance deficiencies that areobtained as explained in the following. It is to be noted thatwhen using AR models the coordinates yr0; r ¼ 1; . . . s; ofan efficient (=effective) projection point used to evaluateDMU0 are given by the corresponding envelopment formu-lation as yr0 ¼

Pnj¼1k

�j yrj, where k�j , j = 1, . . .,n, are the val-

ues of the intensities at an optimum. Therefore,

dþ�r0 ¼ yr0 � yr0 ¼Xn

j¼1

k�j yrj � yr0; r ¼ 1; . . . ; s ð7Þ

provides an estimate of the amount of inefficiency (=lackof effectiveness) in each output for DMU0.4 To better inter-pret these quantities we transform them into percentagesby computing

dþ�r0

yr0100%. These values for each output are

the quantities recorded under the corresponding column,and these columns are shown ordered from left to rightaccording to the relative importance of the variables andthe lines bordering them group them in this same manner.

Turning to column one, under ‘‘Index,” we note that thefirst four players listed as fully effective, with values ofI�0 ¼ 1, consists of Bennett, Bullock, Prigioni and Sanchez.As required for full effectiveness, these players also have allslacks equal to the zero values shown in the remaining col-umns. We therefore lay these players aside for analysis insubsequent tables and now deal with the lack of effective-ness in the different attributes considered for the otherplayers as recorded in Table 1.

As can be seen, these values supply a lot more informa-tion than is available only from the I�0 scores recorded foreffectiveness in column one. (This was also noted by theEtosa coach). For instance, we can see that the main weak-ness for Turner is that he scores relatively few of the pointshe should have made from the free throw line, as shown bythe value of 44.01% which is the measure of this shortfall

4 See, for instance, Cooper et al. (2007b, pp. 152–155) for details ondefinitions and calculations in AR models.

shown under AFT (adjusted free throws). This is partiallydue, however, to the fact that opposing players make veryfew personal fouls on this player as indicated by 70.82%under FOPP (fouls by opposing players).

We similarly find that Oliver and Rodrıguez should con-siderably improve their field goal scoring, as is indicated bythe values of 117.79% and 139.15% listed for these attri-butes under AFG. Comas, on the other hand, shouldmainly increase his AST (the number of assists per game)score by 86.25% while Montecchia should make an effortto improve both AFT (free throw scoring) by 242.63%and FOPP (fouls per game by opposing players) by246.51% as well as his ball handling since his shortfall inthis aspect of his game (AST) stands at 114.85%. Montanezalso falls short as a playmaker in this latter variable (AST)by 148.43%.

This information was found to be interesting and usefulby the ETOSA coach, as we previously remarked. How-ever, no such detail is available from the dual and theenvelopment formulations of the AR model used for thefully effective players. We therefore turn to models (3)and (4) to obtain information on these components of theirgames that are recorded in Table 2.

Table 2 shows the virtual values and records the weights(or multipliers) assigned to these output components, asobtained from (3) and (4) – once (6) is added to the corre-sponding sets of constraints immediately below them. Thisis done in the two rows labeled (4) – for model (4), above –for each player. A comparison with the corresponding val-ues obtained from the EMS DEA computer code are thenshown in the next two rows for each player. As can be seenthe former – i.e., the row labeled (4) – shows all of thesevalues to be positive whereas the latter – i.e., the EMS code– is replete with zeros so that the results from this code failto provide a full profile in every case. In fact, in three of thefour cases, the reported value simply confirms the alreadydetermined effectiveness of the performances with a valueof w�r yr0 ¼ 1 for only one output and all other w�r ¼ 0.

It should be noted that the virtual output values in eachrow sum to 1 for each player and represent the componentsof their effectiveness score. Thus, we conclude from theresults in the first row that Bennet is a very complete playerin the sense that he is good in all aspects of the game, sincehe exhibits high values in the most important factors. Inparticular, Bennet is a good scoring player, as is shownby his score of 0.50933 under AFG. For comparison wenote that the weights provided by the EMS software givea different picture of Bennet’s game by putting zero weightson AFT and STE and thereby indicate that this playerneglects such aspects of the game – which is in contrastto expert opinion. Unlike Bennet, Sanchez and Prigionican be classified as specialists. To be specific, Sanchezachieves his effectiveness because of his having a high valueof 0.92836 in AST (assists), while Prigioni achieves effec-tiveness because of his 0.93021 value under STE (steals).These results are consistent with, but add information inthe form of minimal values to what is well known in Span-

Table 1Effectiveness scores and improvement percentages (playmakers)

Player Index AFG (%) AFT (%) STE (%) AST (%) FOPP (%) REB (%) ITURN (%) NFO (%)

Bennett, Elmer 1 0 0 0 0 0 0 0 0Bullock, Louis 1 0 0 0 0 0 0 0 0Prigioni, Pablo 1 0 0 0 0 0 0 0 0Sanchez, Pepe 1 0 0 0 0 0 0 0 0Turner, Andre 0.8920 12.10 44.01 12.10 32.05 70.82 70.82 1.23 �8.89Oliver, Albert 0.8656 117.79 15.52 15.52 25.48 17.64 �17.38 �14.36 11.32Comas, Jaume 0.8616 19.83 18.16 16.06 86.26 16.06 25.58 18.24 18.23Rodrıguez, Javi 0.8460 139.15 18.21 18.21 25.08 21.30 �4.26 13.80 19.03Montecchia, A. 0.8089 23.63 242.63 23.63 114.85 246.51 27.00 �41.93 15.37Montanez, Roman 0.8028 24.56 39.97 24.56 148.43 24.56 67.41 6.02 42.96

Table 2Virtual outputs and weights (=multipliers) for playmakers

Player AFG AFT STE AST FOPP REB ITURN NFO

Bennett,Elmer

(4) 0.50933(0.08973)

0.08810(0.02591)

0.05026(0.02591)

0.15706(0.02591)

0.16413(0.02591)

0.01232(0.00419)

0.00940(0.02591)

0.00940(0.00419)

EMS 0.23370(0.04117)

0 (0) 0 (0) 0.35216(0.05811)

0.41414(0.06539)

0 (0) 0 (0) 0 (0)

Bullock,Louis

(4) 0.67699(0.08868)

0.09398(0.02553)

0.02321(0.02553)

0.04952(0.02553)

0.10832(0.02553)

0.01843(0.00668)

0.01478(0.02553)

0.01478(0.00668)

EMS 1 (0.13099) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0)Prigioni,

Pablo(4) 0.01501

(0.00632)0.00560(0.00632)

0.93021(0.43692)

0.02224(0.00632)

0.00898(0.00632)

0.005(0.00281)

0.00545(0.00632)

0.00708(0.00281)

EMS 0 (0) 0 (0) 1 (0.46970) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0)Sanchez,

Pepe(4) 0.01180

(0.00488)0.00590(0.00488)

0.00859(0.00488)

0.92836(0.14658)

0.01254(0.00488)

0.01811(0.00488)

0.00193(0.00488)

0.01277(0.00488)

EMS 0 (0) 0 (0) 0 (0) 1 (0.15789) 0 (0) 0 (0) 0 (0) 0 (0)

W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574 569

ish basketball: Sanchez is a very good playmaker directorwhereas Prigioni is probably the best defensive player.Finally, the weights for Bullock provided by the Cooper,Ruiz and Sirvent procedure confirm – as is again wellknown – that this playmaker is mainly a good scoringplayer but does not neglect the other aspects of his game.Again, this is in contrast to the EMS results, where weightis put solely on AFG (adjusted field goals). Results likethese underscore the need for selecting weights from amongthe alternate optimal solutions, which can bring out allaspects of a player’s performance – as is exhibited hereby the contrast between results from the EMS softwareand the Cooper, Ruiz and Sirvent procedures.

In order to avoid possible effects from scaling of thevariables, such as can occur when using AR-constraints –see Dyson et al. (2001) – the above analysis was repeatedby replacing the AR-I constraints with the ‘‘contingentweight restrictions” suggested in Pedraja-Chaparro et al.(1997). For this purpose each of the coupled variables in(6) – say, for instance, REB and AFG – the constraintlREB 6 lAFG must be replaced with the following groupof constraints,

lREByREB;j 6 lAFGyAFG;j; j ¼ 1; . . . ; 41; ð8Þ

which is equivalent to the constraint

lREB

lAFG

6 minyAFG;j

yREB;j

; j ¼ 1; . . . ; 41

( ): ð9Þ

When this was done the main conclusions we previouslydrew remained unaffected. Only very little change was evi-denced in the estimates of both the virtual values and theweights and in only a very few players.

The just described results can be viewed as beingobtained from an analysis in which the basketball playersare assessed with reference to a pre-specified profile thatis reflected in the weight restrictions that, as previouslynoted, set the preferences on the game in terms of a playerplaying in a given position. This might also be refined oraltered for other purposes. For instance, a basketball teammight be interested in a player with a specialized profile fora given position. In our experience with Etosa Alicante, itwas suggested, among other things, that there was a needfor assessing the playmakers according to a scoring profile.The basketball experts found this ability of the DEA meth-odology especially attractive for the assessment of playerswhen the managers are interested in hiring players as mem-bers of a team. To investigate the sample of playmakerswith respect to a scoring profile that accords with the opin-ion of the Etosa technical staff we therefore replaced (6)

Table 3Effectiveness scores and improvement percentages (shooting playmakers)

Player Index AFG (%) AFT (%) FOPP (%) AST (%) STE (%) ITURN (%) REB (%) NFO (%)

Bennett, Elmer 1 0 0 0 0 0 0 0 0Bullock, Louis 1 0 0 0 0 0 0 0 0Oliver, Albert 0.8631 120.18 15.87 15.87 20.46 12.30 �12.90 �17.58 11.26Rodrıguez, Javi 0.8417 144.08 18.80 18.80 19.67 14.25 17.80 �4.62 18.93Turner, Andre 0.8081 23.75 50.38 61.03 11.70 �2.66 11.12 20.05 �9.53Montanez, Roman 0.7990 25.16 40.15 25.16 144.46 23.15 6.82 67.23 42.93Santangelo, M. 0.7379 35.52 244.17 82.79 �8.40 4.39 �7.69 27.79 �18.10Sanchez, Pepe 0.7262 134.47 180.95 146.30 �4.31 10.07 �8.48 �20.86 �14.38Marco, Carles 0.7241 38.09 116.98 117.27 �8.86 29.31 20.59 39.88 �7.72Montecchia, A. 0.6675 49.81 261.07 164.34 �0.22 �26.06 �21.76 21.71 14.34... ..

. ... ..

. ... ..

. ... ..

. ... ..

.

Prigioni, Pablo 0.5400 139.21 283.83 346.21 72.37 �8.91 �57.89 51.87 �10.88

570 W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574

with the following set of constraints in the DEA model(which again leads to an AR-I type formulation),

lITURN6lAFG lSTE6 lAFG lAST6 lAFG lREB6lAST

lITURN6lAFT lSTE6 lAFT lAST6lAFT lREB6lSTE lNFO6 lREB

lITURN6lFOPP lSTE6 lFOPP lAST6lFOPP lREB6 lITURN:

ð10Þ

Table 3 records the new indexes I�0 and the associated direc-tions of improvement corresponding to the assessment ofplaymakers with the scoring profile in (10) (with the vari-ables again grouped between the solid lines correspondingto a decreasing order of relative importance). As columnone (under ‘‘Index”) shows, Sanchez and Prigioni are nolonger effective and Prigioni, in particular, is now very defi-cient. This was to be expected since, as concluded before,these players are specialists in ball handling and defense,respectively, as witness their very high virtual output valuesunder STE (steals) and AST (assists) in Table 2. They there-fore become deficient in effectiveness, as represented bytheir I�0 value, when the most important aspects of the gameare directed to the attributes of scoring. In fact, we see thatthey need to substantially improve the attributes repre-sented by AFG, AFT and FOPP in the first three columns,within the heavy lines, that are now regarded as mostimportant. This can also be concluded for some of the play-makers previously rated as ineffective. See, for instance,Montecchia, whose I�0 score has dropped from 0.81 in Table1 to 0.67 in Table 3 and this reemphasizes his need to im-prove his scoring (in particular from the free throw line).Nevertheless, regarding this new scoring profile, he showsbetter performances than before in other aspects of hisgame that are more concerned with teamwork, such as as-sists, steals or turnovers. Note also the negative value ofthe deficiencies recorded for these variables in the last twocolumns, which indicate that he exceeds the values consid-ered as effective for this player in these aspects of the game.

3.2. Assessment of centers

Evidently these profiles depend on the types of positionas well as the attributes to be employed, such as ‘‘playmak-

ers.” We now show this for the position of centers. Thesample of centers consists of 44 players. To assess theireffectiveness, we need to modify the set of weight restric-tions we used in the case of playmakers. Again, this wasdone in a manner that remains within the bounds of theopinions of members of the Etosa technical staff. For cen-ters, they believed that rebounds are one of the mostimportant player qualities. Moreover, in the opinion ofthe Etosa coach, the aspects concerning ball handling, suchas the assists and the turnovers, are not particularly impor-tant for a center. These types of considerations led us toincorporate the following set of restrictions on the weightsin (2) in order to specify the center profile

lSTE 6 lAFG

lSTE 6 lAFT lAST 6 lSTE lNFO 6 lAST

lSTE 6 lREB lITURN 6 lSTE lNFO 6 lITURN

lSTE 6 lFOPP:

ð11Þ

The results of the new AR formulations are shown in Table4. As a consequence of the new set of constraints (11), wehave moved REB (rebounds) from the third to the first setof bold lines in this table. Besides, we maintain only STE(steals) in the second set of bold lines and transfer ASTand ITURN to a third level of importance in the next tolast set of bold lines. From this table we can see that fiveplayers were found to be fully effective: David, Garces,Kambala, Scott, and Thompson. As with playmakers, Ta-ble 4 records the index I�0 of the top ten assessed playersand their corresponding possibilities for improvement inthe case of the not fully effective players. We can see thatReyes as a center has his main weakness in the free throwline as shown by a value of 30.90% under AFT, while heoutperforms the estimated levels for effectiveness in otheraspects of the game considered less important for a center(see the values �35.29%, �12.34% and �16.73% underSTE, ITURN and NFO, respectively). This same difficultyregarding the free throw line is evidenced even more in thecase of Tabak who does not seem to provoke fouls fromopposing players. The same is true for Bramlett, and evenmore so in the case of Tomasevic. In confirmation we maysay that it is well known that the latter player has very low

Table 4Effectiveness scores and improvement percentages (centers)

Player Index AFG (%) AFT (%) REB (%) FOPP (%) STE (%) AST (%) ITURN (%) NFO (%)

David, Kornel 1 0 0 0 0 0 0 0 0Garces, Ruben 1 0 0 0 0 0 0 0 0Kambala, K. 1 0 0 0 0 0 0 0 0Scott, Brent 1 0 0 0 0 0 0 0 0Thompson, Kevin 1 0 0 0 0 0 0 0 0Reyes, Felipe 0.9338 7.09 30.90 9.76 9.65 �35.29 16.85 �12.34 �16.73Tabak, Zan 0.8550 16.96 184.11 16.96 89.95 56.88 24.76 �23.29 �26.93Bramlett, A.J. 0.8349 19.78 183.17 19.78 51.56 4.94 �41.33 �1.57 50.19Reynolds-Dean 0.8309 34.37 20.36 31.83 43.18 �32.90 �6.67 �5.82 �23.39Tomasevic, D. 0.7935 81.48 221.10 26.02 53.46 �42.70 �60.08 �17.30 �16.24

Table 5Virtual outputs and multiplier weights for centers

Player AFG AFT REB FOPP STE AST ITURN NFO

David, Kornel (4) 0.05444(0.00751)

0.90004(0.34480)

0.02156(0.00421)

0.01202(0.00421)

0.00409(0.00421)

0.0026(0.00174)

0.00260(0.00421)

0.00260(0.00174)

EMS 0.00000 (0) 1.00000(0.38309)

0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0)

Garces, Ruben (4) 0.02132(0.00405)

0.00389(0.00405)

0.95403(0.09723)

0.01153(0.00405)

0.00367(0.00405)

0.00185(0.00349)

0.00185(0.00405)

0.00185(0.00108)

EMS 0 (0) 0 (0) 1 (0.10191) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0)Kambala, K. (4) 0.22280

(0.02413)0.68480(0.26894)

0.04510(0.00744)

0.03441(0.00744)

0.0048(0.00744)

0.00233(0.00744)

0.00335(0.00744)

0.00233(0.00162)

EMS 0.31267(0.03386)

0.68732(0.26993)

0 (0) 0.00000 (0) 0 (0) 0 (0) 0 (0) 0 (0)

Scott, Brent (4) 0.24283(0.02754)

0.60236(0.24803)

0.08595(0.00940)

0.05306(0.00940)

0.00663(0.00940)

0.00361(0.00189)

0.00278(0.00940)

0.00278(0.00189)

EMS 0 (0) 0.22332(0.09196)

0 (0) 0.77668(0.13754)

0 (0) 0 (0) 0 (0) 0 (0)

Thompson,Kevin

(4) 0.21513(0.02655)

0.64024(0.25424)

0.08340(0.00875)

0.04093(0.00875)

0.00824(0.00875)

0.00402(0.00370)

0.00402(0.00875)

0.00402(0.00214)

EMS 0 (0) 0.10699(0.04249)

0.89301(0.09371)

0 (0) 0 (0) 0 (0) 0 (0) 0 (0)

5 This also happens with other softwares. See the comparisons providedin Cooper et al. (2007a).

W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574 571

free throw percentages. Besides, this AR analysis reflectsanother well known fact of Spanish basketball expert opin-ion because, in spite of being a center, Tomasevic is a goodplayer in other aspects such as ball handling and defense,especially in the number of assists – note the negative per-centage of improvement under the variable AST, whichmeans that he exceeds the ability of his referent effectiveplayer in this aspect of the game.

For the five players with scores of I�0 ¼ 1 in Table 4, wealso provide the virtual outputs and the correspondingweights (or multipliers), that are recorded in Table 5. Thevirtual outputs associated with the weights provided bythe Cooper et al. (2007a) procedure show that Kambala,Scott and Thompson are centers with complete games,being particularly good scoring players. In addition to highvalues, in excess of 60% for AFT, they all have relativelygood values in all of the most important attributes for cen-ters. Garces, however, is a specialist in catching reboundswith a virtual value of 0.95403 under REB and, finally,David is at his best from the free throw line as shown bythe value of 0.90004 under AFT. Again, the weights pro-vided by the Cooper, Ruiz and Sirvent procedure are seen

to provide a more comprehensive portrayal as compared tothe optimal weights provided by EMS which, as in theother tables, exhibit numerous zeros.5 Thus, on the basisof the EMS weights, Kambala would simply be a goodscoring center as represented by his EMS scores underAFG and AFT, Scott would be a specialist in free throwpoints as shown under AFT and FOPP and, finally,Thompson would be very good in rebounds and wouldnot be a bad player from the free throw line. As comparedto the scores in the rows labeled (4) in Table 5, however,and, as was the case in Table 1, the procedure used bythe EMS computer code provided only partial views ofwhat is well known, qualitatively, about the games of theseplayers.

Finally, we would like to stress that all the results wehave obtained in this section are qualitatively consistentwith the conclusions we might draw from the ACB indexof player assessment. However, it is to be noted that such

572 W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574

a comparison by the latter index, if meaningful, could onlybe made in terms of rankings, since the ACB provides onlya scalar index for each player that is calculated by simplyaggregating his recorded values in the indicators used inthe statistics. Therefore, it is obvious that DEA, in theway we have used it in this application, provides muchmore information, since the ACB index does not take intoaccount (1) the position where the assessed player plays, (2)which aspects of the game of a player in that position mustbe analyzed more carefully or (3) the behavior of the rest ofplayers playing the same position in order to determineattainable values in the used indicators that can be usedas targets for the players being evaluated. Finally, the useof only (0,1) in the ACB indicator omits any informationon the degree of attainment for the attributes of interest.

Remark. A referee suggested that we provide a tableshowing the data for all 41 playmakers and centers, which

Table A1Data for playmakers

Player AFG AFT REB

Bennett, Elmer 5.68 3.40 2.94Victoriano, L. 0.83 0.87 1.54Hernandez, B. 1.21 0.63 1.75Sanchez, Pepe 2.42 1.21 3.71Gomis, Joseph 4.18 1.91 1.61Lewis, Danny 2.85 1.59 1.62Rodrıguez, Javi 2.42 2.89 3.06Larragan, Borja 1.36 0.40 0.34Comas, Jaume 3.16 1.66 1.88Rodilla, Nacho 2.35 1.47 1.55Galilea, J.L. 3.62 0.58 1.41Lopez, Ferran 2.12 0.50 1.53Santangelo, M. 5.44 1.06 2.18Cherry, Carlos 1.61 1.13 0.97Rodrıguez, N. 1.61 1.07 2.15Martınez, G. 2.25 0.84 2.10Reynes, P. 3.02 0.69 1.76Johnson, Sydney 2.37 0.91 2.68Jofresa, Rafa 2.13 1.02 1.17Montecchia, A. 4.96 1.01 2.28Llompart, Pedro 0.89 0.31 0.35Popovic, Marko 1.86 1.63 0.82Corrales, Ivan 3.08 1.41 1.50Gil, David 0.72 0.82 0.97Prigioni, Pablo 2.37 0.89 1.94Calderon, J.M. 3.57 1.30 2.82Brewer, Corey 3.85 1.32 1.94Azofra, Nacho 3.32 0.90 1.65Miso, Andres 2.09 0.54 1.00Marco, Carles 4.89 1.64 2.03Dumas, Stephane 1.91 0.85 1.41Guzman, J.M. 1.29 0.49 1.47Oliver, Albert 3.08 3.07 3.44Cistero, Maiol 0.49 0.42 0.70Martınez, Rafa 1.09 0.56 0.72Bullock, Louis 7.63 3.68 2.76Cabezas, Carlos 2.60 1.32 1.79Turner, Andre 4.97 2.31 2.41Montanez, Roman 5.28 2.52 1.71San Emeterio 1.88 0.97 2.76Mc Guthrie, C. 1.79 0.64 0.52

we do in the Appendix, where the sample consists ofplayers in the ACB from all of the teams in the league.

4. Conclusions

In this paper we have presented an application of DEAto the assessment of basketball players in the Spanish Bas-ketball League. The purpose of this application is twofold:on one hand, we want to show that DEA can be advanta-geously used as an alternative to the ACB index of playerassessment. Even when the results are similar, DEA pro-vides much more information. In addition, and in contrastto the ACB index, we have made a choice of weights byexploiting the flexibility of DEA in the sense that we pro-vide player-specific weights that profile player characteris-tics and also take into consideration the prior knowledgeof the experts about the relative importance of the indica-

AST STE ITURN NFO FOPP

6.06 1.94 0.36 2.24 6.331.88 1.04 0.89 2.38 1.791.97 0.66 0.76 2.94 0.846.33 1.76 0.40 2.62 2.571.91 0.76 0.52 2.39 2.881.85 1.15 0.57 2.71 2.854.65 1.59 0.33 1.88 5.121.10 0.34 1.53 3.55 0.902.47 1.76 0.55 2.03 3.031.73 0.88 0.94 3.39 2.792.91 0.81 0.57 2.66 1.812.12 0.85 0.63 3.47 1.682.71 1.00 0.60 2.71 2.471.24 0.85 0.87 3.41 1.882.24 1.38 0.87 3.53 1.973.80 0.67 0.64 2.93 1.832.38 0.65 0.60 3.29 2.032.71 1.21 0.56 2.76 2.881.13 0.42 0.73 3.21 1.292.38 1.38 0.71 1.94 1.690.65 0.12 2.43 3.82 0.881.36 0.55 0.88 3.32 2.644.38 0.97 0.35 2.00 2.381.58 0.21 0.92 4.06 1.303.52 2.13 0.86 2.52 1.422.18 1.30 1.03 3.09 1.941.35 0.76 0.54 2.65 2.882.88 0.94 0.79 1.71 1.440.63 0.46 1.71 3.79 1.384.15 1.06 0.40 2.41 2.382.06 0.76 0.71 3.29 1.471.74 0.53 0.95 3.11 1.533.09 1.21 0.56 2.00 4.441.06 0.33 1.83 3.06 0.670.63 0.47 2.67 3.91 1.311.94 0.91 0.58 2.21 4.241.29 0.68 0.87 3.24 1.764.53 1.74 0.37 2.47 3.621.68 1.18 0.44 1.56 4.261.29 0.85 0.92 2.88 2.031.43 0.52 1.21 3.78 0.74

W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574 573

tors used. It is to be noted that the basketball expertsappreciated the usefulness of this methodology for theassessment of players. In particular, they especially likedthe fact already mentioned that they are able to assess play-ers according to a previously specified profile and then havethe procedure go on to provide much more detailed (quan-titatively expressed) information.

Of particular interest was the fact that our approachadded to the customary uses of DEA by exhibiting wholeprofiles that underlie the effectiveness score. This makes itpossible, for instance, to distinguish between a good allaround player from another player with the same effective-ness score but which is attained because of a high virtualvalue for one attribute accompanied by low values on theother attributes. This not only shows weaknesses meriting

Table A2Data for centers

Player AFG AFT REB

Kambala, K. 9.23 2.55 6.06Bueno, Antonio 3.29 0.76 2.57De Miguel, I. 3.09 1.11 4.12Junyent, Oriol 4.52 1.57 4.80Garces, Ruben 5.26 0.96 9.81Gonzalez, R. 2.64 1.06 2.14Fernandez, P. 1.31 0.31 2.23Guardia, Salva 4.58 1.90 4.74Jackson, Robert 5.60 1.08 5.62Garcıa, Dani 1.60 0.11 1.82Bramlett, A.J. 5.73 0.65 8.06Alston, Derrick 5.56 1.45 6.73Scott, Brent 8.82 2.43 9.15Reynolds-Dean 5.90 2.11 6.53Horton, Steve 0.56 0.19 1.48Jones, Alvin 2.43 0.91 5.00Mikhailov, M. 1.31 0.08 3.60Femerling, P. 3.59 1.42 5.27Duenas, Roberto 3.54 0.67 5.19Varejao, A. 3.88 0.83 4.41Vazquez, Fran 3.78 0.88 4.18Burke, Pat 5.93 0.87 5.35Thomas, John 5.25 1.47 5.13Struelens, Eric 3.81 0.64 5.22Rogers, Paul 2.27 0.26 4.33Oberto, F. 7.05 0.50 5.35Tomasevic, D. 4.55 0.78 7.50Garcıa, Asier 2.21 0.57 1.67Toledo, S. 3.08 0.40 2.91Guillen, R. 2.41 1.00 2.48Savane, Sitapha 6.45 1.54 5.67David, Kornel 7.25 2.61 5.12Betts, Andrew 3.98 0.87 3.45Jelic, Dusan 1.52 0.50 2.53Reyes, Felipe 8.25 1.86 8.24Tabak, Zan 7.64 0.87 7.09Alzamora, Alf. 3.05 1.68 3.12Brown, John 6.83 1.02 6.06Llorens, Jordi 1.32 0.44 2.73Kornegay, Chuck 3.81 0.74 5.83Gabriel, German 3.38 0.79 3.06Weis, Frederic 0.82 0.06 2.70Thompson, Kevin 8.10 2.52 9.53Fernandez, G. 4.54 0.73 2.56

attention for particular players, it also provides informa-tion on team qualities that need to be strengthened. Stillfurther such information can be obtained by decomposingthe virtual output values into their weight and quantitycomponents as in, for instance, a substitution analysis.

In conclusion we would like to point out that themethodological aspects of DEA we have described herecan be extended to many other contexts, including casesin which the focus is not only on defining an index of per-formance but also on assessing components underlying theI�0 values.

Remark. A referee raised the question of how to choosebetween multiple optima. This is an under-researched topicof great importance that extends beyond DEA to linear

AST STE ITURN NFO FOPP

0.31 0.66 0.45 1.44 4.630.18 0.32 0.88 2.68 1.390.94 1.35 0.81 1.26 3.060.80 0.53 0.67 1.87 2.430.53 0.91 0.46 1.72 2.840.21 0.93 1.56 2.68 1.640.13 0.58 1.19 2.55 0.840.53 0.56 0.67 1.76 2.970.15 0.73 0.79 1.69 2.460.50 0.12 3.78 3.32 0.321.44 0.88 0.47 1.21 2.561.24 1.18 0.72 2.03 3.611.91 0.71 0.30 1.47 5.651.26 1.41 0.52 2.35 3.000.04 0.48 4.50 3.41 0.780.33 0.79 0.77 2.58 2.460.60 0.57 1.58 3.20 1.200.82 0.82 0.67 2.36 2.970.50 0.34 0.63 2.94 1.501.04 1.26 0.79 2.00 2.590.21 0.27 1.32 2.82 1.670.25 0.65 0.65 2.70 1.750.52 0.84 0.52 1.48 2.550.75 0.69 0.71 2.22 1.280.29 0.54 1.20 2.83 1.581.82 0.94 0.50 1.65 2.853.15 1.56 0.52 2.15 3.180.52 0.19 1.59 3.52 0.630.50 0.38 1.52 3.50 1.090.28 0.24 2.64 3.59 1.000.80 0.73 0.64 1.80 3.501.53 0.97 0.62 1.50 2.850.80 0.55 0.80 1.80 2.350.12 0.29 2.13 2.65 1.241.59 1.09 0.34 1.76 5.121.18 0.44 0.44 2.00 2.820.74 0.76 1.03 2.18 2.940.79 0.65 0.47 1.94 2.380.24 0.33 1.03 2.06 1.180.50 0.80 0.63 1.80 2.200.32 0.59 0.81 2.59 1.970.20 0.35 1.54 2.75 0.951.09 0.94 0.46 1.88 4.680.26 0.41 0.94 2.18 1.35

574 W.W. Cooper et al. / European Journal of Operational Research 195 (2009) 563–574

programming in general, so we call it to the attention ofpotentially interested readers.

Acknowledgements

We would like to thank the Lucentum Basketball Clubfor the opportunities of access to the expert opinions thatthey provided. We are also grateful to the Ministerio deEducacion y Ciencia (MTM2004-07473) for financial sup-port. Support from the IC2 Institute of the University ofTexas at Austin is also gratefully acknowledged.

Appendix

See Tables A1 and A2.

References

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