PERFORMANCE TEAM EVALUATION IN 2008 BEIJING OLYMPIC GAMES

XV INTERNATIONAL CONFERENCE ON INDUSTRIAL ENGINEERING AND OPERATIONS MANAGEMENT The Idustrial Egieerig ad the Sustaiable Developmet: Itegratig Techology ad Maagemet. Salvador, BA, Brazil, 06 to 09 October - 2009 PERFORMANCE TEAM EVALUATION IN 2008 BEIJING OLYMPIC GAMES João Carlos Correia Baptista Soares de Mello (UFF) jcsmello@pesquisador.cpq.br Lidia Agulo Meza (UFF) lidia_a_meza@yahoo.com Fábio Gomes Lacerda (UFF) fagolac@gmail.com Luiz Biodi Neto (UERJ) luizbiodi@terra.com.br I recet years, a lot of wor has bee doe dealig with alterative performace raigs for the Olympic Games. Almost all of these wors use Data Evelopmet Aalysis (DEA). Geerally speaig, those wors ca be divided ito two categorries: Pure raigs with uitary iput models ad relative raigs with classical DEA models; both output orieted. I this paper we itroduce a approach taig ito accout the umber of athletes as a proxy to the coutry ivestmet i sports. This umber is a iput for a DEA model, ad the other iput is the populatio of the coutry. We have three outputs, the umber of gold, silver ad broze medals ear by each coutry. Cotrary to the usual approach i the literature, our model is ot output orieted. It is a o-radial DEA model orieted to the iput umber of athletes. The Decisio Maig Uits (DMU) are all the coutries participatig i the Beijig Olympic Games, icludig those which did ot eared ay medal. We use a BCC model ad we compare the target of each DMU with the umber of athletes that eared, at least, a sigle medal. Palavras-chaves: DEA, Sport Evaluatio, Olympic Games

. Itroductio Performace evaluatio has bee widely studied i sport for the last 25 years (NEVILL et al., 2008). I the case of the Olympic Games the ra of the atios is traditioally carried out with the so-called Lexicographic Multicriteria Method (LINS et al., 2003). I this very paper the drawbacs of the Lexicographic Method are poited out ad a ew raig is suggested. There are already some other approaches usig Data Evelopmet Aalysis (CHARNES et al., 978). The very first oe was proposed by Lozao et al (2002). They used populatio ad GNP as iputs ad the medals as outputs. I a similar approach, Lis et al (2003) built a ew model taig i accout oe more costrait: the total amout of medals is a costat. This resulted i the developmet of a ew model, the so-called Zero Sum Gais DEA model (ZSG- DEA). Churilov ad Flitma (2006) used DEA to establish a raig, the iputs of which were some social ecoomics variables (populatio, GDP, DEL idex ad IECS idex). The outputs were some liear combiatio of the umber of medals eared by each coutry. Li et al (2008) use GDP per capita ad populatio as iputs ad impose differet weight restrictios for each DMU, accordig to a previous coutry categorizatio. Wu et al (2009) have used the cocept of Nash equilibrium i a BCC-DEA Cross Evaluatio for the Olympic Raig They came across the same problem described i Soares de Mello et al (2002), the existece of egative efficiecies i the BCC-DEA model. They solved the problem with the same approach used by Soares de Mello et al (2008) ad Agulo-Meza et al (2004). All the wors metioed hereabove tae ito accout the results i the Olympics ad the socio ecoomical coditios of each coutry. Models usig DEA ad othig but medals wo by each coutry were preseted by Soares de Mello et al (2008; 2009) ad Soares de Mello et al (SOARES DE MELLO, GOMES et al., 2008). Despite their importat differeces, the papers cited hereabove have some commo poits. The first oe is that those papers evaluate oly coutries that have eared medals. I other words the uits evaluated by DEA (Decisio Maig Uits - DMU) are the coutries that have oe at least oe medal. The secod i commo i all those papers is that the DMU s objective is to maximize some combiatio of the umber of medals eared. I this paper we are iterested i studyig if the umber of competitors of each coutry is a reasoable oe whe compared to all the other coutries. Moreover, we will evaluate all participatig coutries i the Beijig s Olympic Games icludig those oes that have eared o medal. This is possible because of a mathematical property of the DEA model with variable returs to scale, the so-called BCC DEA model (BANKER et al., 984). I such a model, it is possible for a DMU to be efficiet with all its outputs il valued. Our model will have three outputs as a majority of the models cocerig Olympic evaluatio, umber of gold, silver ad broze medals. The model will have two iputs: the populatio ad the umber of competitors for each coutry. As we ited to evaluate the dimesio of each coutry s team we will oriet the DEA model to oly iput. This leads to the study of o radial DEA models. Such model will be summarized i sectio 2. Sectio 3 presets the DEA model used i our study. The results are preset i sectio 4 ad Sectio 5 has some fial commets of this study. 2. Data Evelopmet Aalysis models with o discritioary variables 2

The classical BCC iput oriet model (BANKER et al., 984) i the evelopmet formulatio is preseted i (). Mi h0 subject to h x - x λ 0, i 0 i0 i = -y + y λ 0, j j0 j λ = λ 0, Where h 0 is the efficiecy of the DMU 0, the DMU uder evaluatio; x i is the iput i of DMU ; y j is the output j of DMU ; λ is the share of DMU for the DMU 0 target. Accordig to the model whe a iefficiet DMU achieves efficiecy, all its iputs are proportioally reduced. If we wat to achieve efficiecy chagig the value of a sigle iput, we shall use model (2). This model was itroduced by Baer e Morey (986) ad ca be see as a particular case of MORO model (LINS et al., 2004; QUARIGUASI FROTA NETO e ANGULO-MEZA, 2007), whe all but oe factor are ot allowed to be chaged. Mi h0 subject to h x - x λ 0, 0 v0 v x - x λ 0, i v i0 i -y + y λ 0, j j0 j = λ = λ 0, I model (2) x v is the variable allowed to be chaged. 3. Modellig As metioed earlier the mai goal of our study is to evaluate the performace of the Olympic teams i Beijig 2008 Games, taig ito accout the umber of athletes for each ad every coutry. We use as outputs the umber of gold, silver ad broze medals wo by each coutry. The cotrolled iput is the umber of athletes i each team, which is a proxy for the coutry ivestmet i sports. As a ucotrolled iput we use populatio of each coutry. We have restrait the weight of each medal as i Soares de Mello et al (2008; 2009) ad Soares de Mello et al (2008), i.e. the weight of the gold medal is ot lower tha the weight of the silver medal, which is ot lower tha the weight of the broze medal. Moreover, the differece of () (2) 3

weights betwee the gold ad the silver medals is ot lower that the differece of weights betwee the silver ad broze medals. Model (3) is the liear programme for a give coutry. Mi h st 0 x λ x 0 POP0 j POPj 0 ATHL0 j ATHLj hx λ x 0 y G j Gj 3 S j Sj 2 3 B j Bj 2 3 j λ y γ γ y λ y + γ γ + 2γ y λ y + γ γ λ = j λ 0, j I model (3), h o is the efficiecy of the observed coutry; λ j is the share of coutry j for the observed coutry o; x POP0 is the populatio of the observed coutry; x ATH0 is the umber of athletes of each coutry; y G, y S e y B are the umber of gold, silver ad broze medals of the observed coutry; y Gj, y Sj ad y Bj are the umber of gold, silver ad broze medals of coutry j. Variables γ, γ 2 ad γ 3 are the dual variables correspodig to the weight restrictios i the primal problem (multipliers formulatio). 4. Results Table summarizes the results for some coutries usig model (3). I this table the ideal umber of Athletes is the o radial target for each coutry, it is obtaied by multiplyig the efficiecy by the umber of athletes. The Number of Medal Wiers represets how may athletes did a coutry eed to wi all its medals. For istace, if a coutry wis eight medals with oly oe athlete it would have umber of medal wiers equal to oe. O the other had if a coutry wis a medal i a collective sport, for istace, soccer, it would lead at least eleve athletes to wi that medal. Coutry Number of Athletes Medals Number of medal wiers (3) "Ideal" umber of Athletes DEA Efficiecy Australia 439 46 85 439,000000 Belize 0 0,000000 Burudi 0 0,000000 Cetral Africa Republic 0 2,000000 4

Chia 564 00 59 564,000000 Domiica 0 0,000000 Gabo 0 0,000000 Jamaica 56 3 56,000000 Keya 52 4 4 52,000000 Nauru 0 0,000000 Niger 0 0,000000 Russia 4 72 34 4,000000 Togo 2 2,000000 Uited Kigdom 297 47 59 297,000000 Uited States 629 0 272 629,000000 Belarus 23 9 28 4 0,922794 Bahrai 0 0,886249 Suda 8 4 0,474044 Netherlads 230 6 63 83 0,3690 Rumaia 95 8 9 35 0,36662 Demar 85 7 8 30 0,358455 Triidad ad Tobago 28 2 4 28 0,350909 Bagladesh 3 0 0 0,333333 Icelad 9 4 6 0,305579 Brazil 265 5 74 58 0,28692 New Zealad 95 9 4 42 0,25002 Idia 48 3 3 8 0,736 Polad 230 0 20 39 0,68983 Mexico 67 3 4 0 0,54229 Chile 23 23 0,49759 Argetia 37 6 53 9 0,408 Nigeria 82 4 24 2 0,40244 Serbia 88 3 5 9 0,06003 5. Fial Commets Table Results for model (3) I Table, we otice that all but seve coutries has the Ideal umber of athletes larger tha the Number of athletes. This is a very desirable situatio, it meas that eve i the ideal case, the coutry will have some athletes that will ot wi ay medal at all, preservig the De Couberti Olympic spirit. For some coutries this does ot happe. This is the case of Brazil, Nigeria, Serbia, Argetia ad Icelad. If those coutries succeed to reduce their umber of athletes, they will ot be able to wi the same umber of medals. For those coutries the ivestmets policy i sports is probably ot the best oe. Future studies may explore this coclusio ad try to idetify which are the most profitable sport ivestmets for each coutry. Refereces ANGULO-MEZA, L., SOARES DE MELLO, J. C. C. B., GOMES, E. G. & BIONDI NETO, L. Eficiêcias egativas em modelos DEA-BCC: como surgem e como evitá-las. VII Simpósio de Pesquisa Operacioal e Logística da Mariha - SPOLM. Niterói, 2004. p. 5

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