Some rankings for the Athens Olympic Games using DEA models with a constant input

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1 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Some ranking for the Athen Olympic Game uing DEA model with a contant input João Carlo Correia Baptita Soare de Mello Lidia Angulo Meza Brenda Branco da Silva Federal Fluminene Univerity Brazil jcmello@producao.uff.br lidia_a_meza@pq.cnpq.br brendabranco@yahoo.com Abtract There i no official method to etablih a final ranking for the Olympic game. The uual ranking i baed on the Lexicographic Multicriteria Method, the main drawback of which i to overvalue gold medal. Furthermore it doe not take in account that the variou port may be of different importance. Thi work propoe a ranking model to eliminate thoe drawback. Firt we ue a modified cro evaluation DEA (Data Envelopment Analyi) model with weighted retriction for each port. The output are the number of gold, ilver and bronze medal and the input i a unitary contant for all countrie. After obtaining a rank for each and every port we build a general ranking uing a weighted um. The weight are calculated taking in account the number of countrie that participated in each port. We ue our model with the reult of the Athen Olympic Game. Keyword: DEA, Olympic, Ranking, weight retriction, unitary input 1 Introduction The firt recorded Olympic Game occurred in Olympia, Greece in 776 ac (Ficher, 2003). The modern Game, were born from an initiative of Baron de Coubertin in 1892 and occurred in 1896 in Athen. The Game tried to maintain the initial pirit of individual competition. A noted by Lin et al (2003) the purpoe clearly failed and the Game have become a national competition. Depite their national character, the Olympic Committee ha never iued an official ranking to pick an overall Olympic winner country. However the IOC preent the medal data in a table that ugget a ranking. A a matter of fact, the ma media do ue thi table a a ranking. Thi quai-official ranking i baed on the Lexicographic Multicriteria method, a explained in Lin et al (2003). Thi ranking doe not deal properly with the 2008 Aociação Portuguea de Invetigação Operacional

2 78 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) poible exitence of countrie that have won a large number of ilver and bronze medal but no gold medal. In the literature there are ome tudie for alternative ranking for the Olympic Game. In almot all of them the ranking were obtained by comparing the number of medal earned with explicative variable. In all of them there i no difference in the value of medal earned in different competition. We have two main goal in thi tudy. The firt one i to provide a rank (different from the quai-official ranking) that take into account only the number of medal won and not the explicative variable. Thi ranking mut not over evaluate the gold medal. We mut mention that there are already at leat two ranking with thi characteritic. We believe that our ranking ha ome theoretical advance in the DEA field. The econd and mot important goal i to take into account that medal won in different competition do not have the ame value. A a matter of fact, the exiting ranking do not take into account that in ome dicipline there are more event than in other, and o there are more poibilitie of winning a medal. For intance, in gymnatic there are a lot of gold medal to be earned and in football there are only two poibilitie for a country to win a gold medal (one for men, the other for women). A far a we know thi the firt tudy on that matter at leat concerning Olympic ranking. To take into account the difference in winning value for different port, we aggregate competition into cluter, a done by the IOC ( each dicipline i a cluter. Our method i illutrated uing data from the 2004 Athen Olympic Game. In the next ection we review the literature about Olympic ranking and the literature of Data Envelopment Analyi ranking in port. In ection 3, we etablih the fundamental of our ranking, the ranking for each port and the two general ranking. At lat, ection 4 preent ome concluion concerning the reult we have obtained. 2 A Review on the Analyi of the Olympic Game reult The Lexicografic Method i not the ole method ued to rank countrie in the Olympic Game. Some newpaper produce a ranking determining the total number of medal earned by each country. They imply add up bronze, ilver and gold medal. The obviou drawback of thi method i to under-evaluate gold medal. An alternative approach i to make an arbitrary evaluation of each medal, for intance, 1 point for bronze, 2 for ilver and 3 for gold. Thi i a much unophiticated approach, a it aume all medal to be equally deired, albeit in proportion to their value. The previou approache follow contradictory aumption. It i important to tudy alternative way to rank competitor in the Olympic Game. Morton (2002) ued tatitical method to determine who win the Olympic. Other tatitical approach conidering ocio economical variable and the number of medal earned ha been performed by Bernard and Bue (2004). There are already ome approache uing DEA to etablih Olympic ranking. The very firt one wa propoed by Lozano et al (2002). They ued population and GNP a input and the medal a output. In a imilar approach, Lin et al (2003) built a new model taking in account one more contraint: the total amount of medal i a contant. Thi reulted in the development of a new model, the o-called Zero Sum Gain DEA model (ZSG-DEA). Churilov and Flitman (2006) ued DEA to etablih a ranking, the input of which were ome ocial economic variable. Intead of uing a output the number of gold, ilver and bronze medal, they ued four linear combination of thee figure. Thi approach eliminate the problem of nil valued weight. For each country, they determined which output ha the greatet weight, in order to divide the countrie into cluter. They alo made a claical cluter analyi uing ocio economical

3 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) variable, and compared the two claification. The author emphaized the importance of Olympic ranking and aked for new tudie on thi ubject. All the work mentioned hereabove take into account the reult in the Olympic and the ocio economical condition of each country. Balmer et al (2001); Balmer et al (2003), Soare de Mello et al (2004) and Soare de Mello et al (2008) ue only the reult themelve. The former tudy the exitence of home advantage and the latter want to etablih a ranking for both Summer and Winter Game. Another work comparing ummer and winter Game i the one of Johnon and Ali (2004). The uncertainty in Olympic Game wa tudied by Baimbridge (1998). In all the above mentioned paper there i no difference in the value of a medal in different port, i.e., the one hundred meter gold medal ha the ame value of a baeball gold medal. In thi paper we will propoe a method the take into account that the medal are obtained in different port. We hall alo mention that there are DEA baed ranking for ome other port. Among them we can mention Epitia-Ecuer and Garci-Cebrian (2006), Barro and Leach (2006), Haa (2003), Soare de Mello and Gome Junior (2006) and Calôba and Lin (2006). 3 Building the new ranking Our method i performed in two tep. The firt one i to make a ranking for each port independently. Thi i done to avoid the poibility of a country that ha a good performance in a port that ha a great number of different competition (athletic, gymnatic, wimming and o on) to be placed in a higher poition than a country with a imilar performance in a port with a few number of different competition (baeball, football, volleyball, and o on). In the econd tep, we mut aggregate the different ranking obtained in tep one. Thi i achieved uing a weighted um of the partial performance. We mention different form to determine thee weight and we carry out the calculation for two of them. For the firt tep, we will ue a DEA model for each port. The DMU (Deciion Making Unit that are the unit under evaluation in DEA) are all the countrie that won a medal in thi port. The three output are the number of gold, ilver and bronze medal each country earned. We do not ue input. According to Lovell and Pator (1999) thi lead to mathematical inconitencie and o we adopt a unitary input for each DMU. Owing to the exitence of a ingle contant input, we ue the Contant Return to Scale DEA model (DEA CCR) Charne et al (1978). In the particular cae of a contant input the CCR model become (1) in which i the DMU 0 efficiency; i the j-th output (j=1,...,) of the k-th DMU (k=1,...,n); i the i-th input (i=1,...,r) of the k-th DMU; and are the output and the input weight, repectively. A we have a unitary input it doe not appear in the formulation. Maximie h y ubject to j1 j 0 j j0 j1 y 1, k 1,..., n j jk 0, j 1,...,, (1) Although thi i an input oriented DEA model, thi model allow other interpretation owing to the abence of the equality contraint. If thi model wa to be ued with an input oriented interpretation, thi model will become meaningle owing to the preence of a unitary input. The dual for thi model i preented in (2)

4 80 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Minimize ubject to n k 1 0, k k n k=1 jk k jo y y, j=1,..., k (2) In thi model there i no input reduction. The minimization of the hare um interpretation make the model a meaningful one even in the preence of a contant input. Thi model ha already been derived by Caporaletti et al (1999). The author interpreted thi model a a multi-attribute one, in the pirit of DEA only with output. Thi i the ame a conidering a unitary and contant input. Foroughia and Tamiz (2005) ue an analogou model but they mied the theoretical conideration. A model with the ame objective function and different contraint i ued by Kao and Hung (2007). For the Olympic ranking, model (1) i tranformed into model (3), where g, and b refer to the gold, ilver and bronze medal in the Athen 2004 Olympic Game. Maximie h y y y ubject to y y y 1, k 1,...,n g gk k b bk j 0, j g,,b 0 g g0 0 b b0 (3) Obviouly the medal are not equally important. To take that fact into account we will ue weight retriction in our DEA model. For ure, a gold medal i more important than a ilver one and thi one i more important than a bronze one. However, the difference in their relative importance i not the ame. In oppoition to Baron de Coubertin ideal, victory i the main goal of the competitor. So the difference in importance between gold and ilver medal hould be greater than the difference between ilver and bronze medal. Having thee aumption in mind, the unitary input DEA model baed on Soare de Mello and Gome Junior (2006) model i hown in (4). Maximie h y y y ubject to y y y 1, k 1,...,n g gk k b bk g b g b j 0, j g,,b 0 g g0 0 b b0 (4) Thi i a weight retriction model with non-homogeneou retriction; uch model were tudied by Podinovky (2004) The non Archimedean contant i required to avoid a critical ditortion, i.e. in pecial condition the three medal may be equally valued. Such a ituation lead to nonuitable ranking. For intance, in table 1, medal for Baeball are hown.

5 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Table 1. Medal for Baeball Country Gold Silver Bronze Cuba Autralia Japan Source: International Olympic Commitee A hown in that table each one of the three countrie earned only one medal. If we applied the model with homogenou weight retriction, the three countrie will be equally efficient. Thi i not a deirable reult. Common ene will attribute the firt poition for Cuba, the econd one for Autralia and the third place to Japan. Even with the contraint impoed in thi model, there i a high degree of freedom for the weight. To avoid thi freedom, we ue a econdary model inpired on the Sexton et al (1986) Cro Evaluation model. A we have only one input, the cro evaluation model become a fixed weight model Anderon et al (2002). So, we ue average weight, which are eaier to calculate than uing the aggreive and benevolent model of Doyle and Green (1994). Average weight are ued in a model imilar to cro evaluation ued by Lin et al (2003). The ue of both weight retriction and cro evaluation combine two approache for improving dicrimination in DEA: the firt one with deciion maker value judgement and the econd one i a fully objective one (Angulo-Meza and Lin, 2002) On the other hand, it i common knowledge in DEA that for ome DMU the weight are not uniquely determined (ee for intance, Roen et al, (1998); Soare de Mello et al, (2002); Cooper et al, (2007)).We ue an auxiliary linear programming model to determine a unique et of weight for each DMU. The aim of thi model i to maximize the difference of the weight for gold and ilver medal auming that the efficiency previouly determined in model (4) remain the ame. Thi model i hown in (5). Maximie ubject to y y y h, y y y 1, , j g,,b g g g0 0 b b0 0 g gk k b bk g g g b j (5) The objective function guarantee the maximization of the difference between the importance of the gold and ilver medal. The invariance of the efficiency value i aured by the firt contraint. The remaining contraint are the ame a ued in model (4). In each port, to obtain the gold medal weight we calculate the average of all the weight attributed to the gold medal by the complete et of DMU. In a imilar way we obtain the weight for ilver and bronze medal. The performance of a particular DMU in a given port i calculated uing equation (6). P Y Y Y (6) 0 g g b b 3.1 Some reult for the firt tep Uing model (4) and (5) and equation (6), we obtain the reult for every port. In table 2 we how reult for Sailing.

6 82 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Table 2. Sailing reult DMU yg y yb g b P0 United Kingdom , , Brazil 2 0, , , Spain 1 2 0, , , Autria 1 1 0, , , Greece 1 1 0, , , USA 1 1 0, , , France 1 1 0, , , Irael 1 0, , , Norway 1 0, , , Ukraine 2 0, , ,29999 Canada 1 0, , , China 1 0, , , Czech Republic 1 0, , , Denmark 2 0, , , , Argentina 1 0, , , , Italy 1 0, , , , Japan 1 0, , , , Poland 1 0, , , , Slovenia 1 0, , , , Sweden 1 0, , , , Average weight 0, , , Table 3. The reult for Archery DMU yg y yb g b P0 Korea 3 1 0,3333 0, Italy 1 0,3333 0, , Chinee Taipei 1 1 0, , , , China 1 0, , , , Japan 1 0, , , , Autralia 1 0, , , , United Kingdom 1 0, , , , Ukraine 1 0, , , , Average weight 0, , , A hown in table 2, in the cae of ailing, the ranking obtained by our model and the lexicographic method i the ame. A cae where the two method lead to different ranking i Archery, whoe reult are hown in table 3. Table 4 how the average weight for all dicipline a well a the difference between average weight.

7 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Dicipline Table 4. Medal Average Weight for every dicipline. g b g b Archery Athletic Badminton Baeball Baketball Beach volleyball Boxing Canoe/kayak flatwater Canoe/kayak lalom racing Cycling mountain bike Cycling road Cycling track Diving Equetrian Fencing Football Gymnatic Artitic Gymnatic Rythmic Trampoline Handball Hockey Judo Modern Penthathlon Rowing Sailing Shooting Softball Swimming Sync wimming Table tenni Taekwondo Tenni Triathlon Volleyball Water polo Weightlifting Wretling The difference between the average weight for gold and ilver i larger than the difference between ilver and bronze, i.e., g b. Although thi i an obviou conequence of thi contraint, it can be een that for almot all the dicipline,. Thi mean that for the majority of countrie the gold medal i g b much more important that the other medal. On other hand, for a large number of dicipline, b i very mall. For 21 dicipline, thi difference i Thi i the value choen for the non-archimedean

8 84 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) contant. So we conclude that thi contant i very important to ditinguih the weight value for ilver and bronze medal. A a matter of fact, we may conclude that for a large number of countrie winning a ilver medal or one of bronze ha almot the ame value. 3.2 Aggregation of the partial reult A mentioned in the introduction of thi paper, the final ranking i obtained uing a weighted um of the performance of each country in each port a hown in equation (7) i i i=1 I = P n (7) In which P0i i the performance of the country 0 in port i and ni are the weight for port i, i = 1,, 37. Different method can be ued to etimate the weight for each port. In a firt model, we can uppoe that all port are equally important and o the weight are all the ame. In a econd model, we can meaure the importance of each port for their potential of attracting pectator, mainly when TV broadcating are concerned. A direct approach would need the figure for TV audience. Thi i a rather difficult tak, o we may ue a a proxy the number of countrie participating in each port. Depite the everal drawback of thi approach, we are jutified to believe that the greater the number of countrie participating in a port, the greater the number of potential pectator. Another method would weigh each port according to it competitivene Mitchell and Stewart (2007). In thi paper, we will ue both the firt and the econd model taking into account the number of participating countrie rather than TV audience. The number of participant countrie in each port i hown in table 5. After uing equation (7) to obtain the indexe, thee are normalized uing equation (8). I0 Index 0 * 100 I max (8) The reult for both model are hown in table 6 and 7.

9 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Table 5. Participant countrie for each port Sport n i Sport n i Archery 43 Handball 16 Athletic 196 Hockey 14 Badminton 32 Judo 94 Baeball 32 Modern Pentathlon 26 Baketball 18 Rowing 55 Beach volleyball 24 Sailing 61 Boxing 72 Shooting 106 Canoe/kayak flat-water 45 Softball 8 Canoe/kayak lalom racing 22 Swimming 154 Cycling mountain bike 34 Sync wimming 24 Cycling road 49 Table tenni 50 Cycling track 39 Tae-kwon-do 60 Diving 30 Tenni 53 Equetrian 68 Triathlon 33 Fencing 42 Volleyball 19 Football 22 Water polo 13 Gymnatic Artitic 42 Weightlifting 79 Gymnatic Rhythmic 21 Wretling 99 Trampoline 19 Table 6. Final Ranking uing identical weight for all port Ranking Country Index Ranking Country Index 1 USA 100,00 39 Belgium 4, Ruia 90,95 40 Uzbekitan 4, Germany 71,21 41 North Korea 3, Autralia 54,34 42 Azerbaijan 3, Italia 46,80 43 Irael 3, France 42,13 44 Ireland 3, Korea 37,37 45 Mexico 3, United Kingdom 37,01 46 Georgia 2, Japan 34,29 47 Slovenia 2, Ukraine 28,83 48 South Africa 2, Netherland 27,68 49 Etonia 2, Hungary 27,64 50 Cuba 1, Brazil 27,48 51 China 1, Greece 18,43 52 Ethiopia 1, Argentina 17,48 53 Venezuela 1, Canada 16,57 54 Spain 1, Romania 15,64 55 Portugal 1, Norway 14,82 56 Jamaica 1, New Zealand 11,39 57 Kenya 1, Czech Republic 10,49 58 U Arab Emirate 1, Bulgaria 10,42 59 Morocco 1, Belaru 10,41 60 Finland 1, Autria 10,29 61 Egypt 0, China Taipei 9,50 62 Hong Kong 0,990322

10 86 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Slovakia 9,22 63 Denmark 0, Turkey 7,67 64 Syria 0, Chile 7,43 65 Zimbabwe 0, Thailand 7,28 66 Bahrain 0, Switzerland 7,24 67 Colombia 0, Iran 6,28 68 Croatia 0, Poland 6,28 69 Mongolia 0, Latvia 6,22 70 Cameroon 0, Lithuania 5,70 71 Dominican Republic 0, SCG 5,43 72 India 0, Kazakhtan 5,15 73 Nigeria 0, Paraguay 4,95 74 Trinidad and Tobago 0, Indoneia 4,87 75 Eritrea 0, Sweden 4,75113 Table 7. Final Ranking uing different weight for all port Ranking Country Index Ranking Country Index 1 USA 100,00 39 Egypt 5,77 2 Ruia 88,08 40 Jamaica 5,37 3 China 65,79 41 Kenya 5,07 4 Germany 52,19 42 Slovakia 4,85 5 Autralia 41,67 43 Azerbaijan 4,68 6 Japan 35,75 44 Lithuania 4,55 7 France 35,12 45 Georgia 4,44 8 United Kingdom 34,67 46 North Korea 4,33 9 Italy 34,12 47 Morocco 4,31 10 South Korea 32,08 48 Switzerland 4,22 11 Cuba 30,53 49 Belgium 4,18 12 Ukraine 25,13 50 South Africa 4,10 13 Netherland 24,84 51 Latvia 3,89 14 Greece 18,26 52 Irael 3,68 15 Spain 18,08 53 Ireland 3,66 16 Hungary 17,98 54 Slovenia 3,24 17 Brazil 16,03 55 Mexico 3,22 18 Romania 14,93 56 Indoneia 3,16 19 Belaru 13,76 57 Etonia 2,72 20 Norway 12,20 58 Bahama 2,42 21 Bulgaria 11,39 59 U Arab Emirate 2,37 22 Canada 11,02 60 Portugal 2,18 23 Turkey 10,33 61 Venezuela 1,97 24 Autria 9,18 62 Zimbabwe 1,91 25 Thailand 8,91 63 Serbia/Montenegro 1,90 26 Poland 8,82 64 Cameroon 1,89 27 Sweden 8,81 65 Dominican Republic 1,89 28 Czech Republic 8,76 66 Paraguay 1,81 29 Chinee Taipei 8,66 67 Finland 1,70 30 Iran 7,57 68 Nigeria 1,06 31 Denmark 7,47 69 Syria 1,05 32 Argentina 7,44 70 Mongolia 1,01

11 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) New Zealand 7,27 71 Colombia 0,97 34 Kazakhtan 7,26 72 India 0,83 35 Chile 6,53 73 Hong Kong 0,82 36 Ethiopia 6,43 74 Eritrea 0,53 37 Uzbekitan 6,22 75 Trinidad and Tobago 0,47 38 Croatia 6,17 There are very few tie in both ranking, motly due to numerical preciion. In the quaiofficial ranking all the countrie are equally ranked that won only, for intance, one bronze medal. The larget difference between the quai-official ranking and the econd model of our propoed ranking concerned the Czech Republic that went up 14 rung. The reult of thoe two model are very imilar mainly for the firt poition. A comparion between the quai-official (lexicographic) ranking and the ranking obtained with model 1 how ome difference. The greatet one i Cuba, which wa the eleventh country in the lexicographic ranking and drop to the fiftieth poition in model 1 ranking. Another country that ha a coniderable change of rank i China that drop from the econd poition to the twenty eventh. On the other hand, Argentina that wa ranked thirty fourth goe up to fifteenth. A the quai-official ranking and the ranking obtained uing model 2 are very imilar, we expect to obtain almot the ame difference between model 1 and model 2. A a matter of fact, when uing model 2 China i ranked third and Cuba eleventh and they drop to the previouly mentioned poition when uing model 1. 4 Final Comment Two ranking were propoed: one of them take into account that all port are equally important. The other one take into account the impact of each port meaured by the number of countrie participating in each port. Thi meaure take into account a well that the chance to win a medal are not the ame in the different port. The two indexe obtained have two common characteritic. The firt i that they do not overrate gold medal. The econd i that they put on an equal footing port that give away a large number of medal - athletic or gymnatic, for intance - and team port (uch a baket ball or volley ball) in which a ingle medal reward a large number of athlete. The firt poition on the ranking uing model 2 and according to the quai-official ranking are almot the ame. The major difference appear in the middle and bottom of the table due to other factor including the abence of tie in our method. The imilaritie of thee two ranking can be explained by the approache being ued. In model 2 we ued a weighing cheme that take into account the number of countrie diputing medal. The lexicographic method ue the number of medal to rank the countrie. It i a fact that in the Olympic Game, the greater the number of medal, the greater the number of participant countrie. So, in a way model 2 ha a very imilar approach than the lexicographic method without over evaluating the gold medal, which might explain the mall difference that were found. The major difference appear when comparing model 1 and 2, i.e., model conidering equal and different weight for each port. A mentioned earlier, China and Cuba go up everal rung a we move from model 1 to model 2. Thi happen becaue both countrie concentrate their effort in port with a large number of medal. Thi how an invetment in higher impact port. On the other hand, Argentina drop from fifteenth to thirtieth between model 1 and model 2. Thi how an oppoite trategy, becaue Argentina concentrate effort in port with few medal and few participant uch a team port (football, baketball, and other).

12 88 J.S. MELLO et al. / Invetigação Operacional, 28 (2008) Further development hould include a well the tradition of a given port in the Olympic Game. 5 Reference Anderon, T. R., Hollingworth, K., e Inman, L. (2002) The fixed weighting nature of a croevaluation model, Journal of Productivity Analyi, Vol 17, pp Baimbridge, M. (1998) Outcome uncertainty in porting competition: The olympic game , Applied Economic Letter, Vol 5, No 3, pp Balmer, N. J., Nevill, A. M., e William, A. M. (2001) Home advantage in the winter olympic ( ), Journal of Sport Science, Vol 19, No 2, pp Balmer, N. J., Nevill, A. M., e William, A. M. (2003) Modelling home advantage in the ummer olympic game, Journal of Sport Science, Vol 21, No 6, pp Barro, C. P., e Leach, S. (2006) Performance evaluation of the englih premier football league with data envelopment analyi, Applied Economic, Vol 38, No 12, pp Bernard, A. B., e Bue, M. R. (2004) Who win the olympic game: Economic reource and medal total, Review of Economic and Statitic, Vol 86, No 1, pp Caloba, G. M., e Lin, M. P. E. (2006) Performance aement of the occer team in brazil uing dea, Pequia Operacional, Vol 26, No 3, pp Caporaletti, L. E., Dulá, J. H., e Womer, N. K. (1999) Performance evaluation baed on multiple attribute with nonparametric frontier, Omega, Vol 27, No, pp Churilov, L., e Flitman, A. (2006) Toward fair ranking of olympic achievement: The cae of ydney 2000, Computer and Operation Reearch, Vol 33, No 7, pp Cooper, W. W., Ruiz, J. L., e Sirvent, I. (2007) Chooing weight from alternative optimal olution of dual multiplier model in dea, European Journal of Operational Reearch, Vol 180, No, pp Doyle, J., e Green, R. H. (1994) Efficiency and cro-efficiency in dea derivation, meaning and ue, Journal of the Operational Reearch Society, Vol 45, No, pp Ficher, D The encyclopedia of the ummer olympic: Franklin Watt. Foroughia, A. A., e Tamiz, M. (2005) An effective total ranking model for a ranked voting ytem, Omega, Vol 33, No, pp Haa, D. J. (2003) Productive efficiency of englih football team - a data envelopment analyi approach, Managerial and Deciion Economic, Vol 24, No 5, pp Johnon, D. K. N., e Ali, A. (2004) A tale of two eaon: Participation and medal count at the ummer and winter olympic game, Social Science Quarterly, Vol 85, No 4, pp Kao, C., e Hung, H.-T. (2007) Management performance:an empirical tudy of the manufacturing companie intaiwan, Omega, Vol 35, No, pp Lin, M. P. E., Gome, E. G., Soare de Mello, J. C. C. B., e Soare de Mello, A. J. R. (2003) Olympic ranking baed on a zero um gain dea model, European Journal of Operational Reearch, Vol 148, No 2, pp

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