Multiobjective centralized DEA approach to Tokyo 2020 Olympic Games

Annals of Operations Research https://doi.org/10.1007/s10479-022-05085-5 ORIGINAL RESEARCH Multiobjective centralized DEA approach to Tokyo 2020 Olympic Games Sebastián Lozano1 ·Gabriel Villa1 Accepted: 16 November 2022 © The Author(s) 2022 Abstract There exist two types of Data Envelopment Analysis (DEA) approaches to the Olympic Games: conventional and fixed-sum outputs (FSO). The approach proposed in this paper belongs to the latter category as it takes into account the total number de medals of each type awarded. Imposing these constraints requires a centralized DEA perspective that projects all the countries simultaneously. In this paper, a multiobjective FSO approach is proposed, and the Weighted Tchebychef solution method is employed. This approach aims to set all output targets as close as possible to their ideal values. In order to choose between the alternative optima, a secondary goal has been considered that minimizes the sum of absolute changes in the number of medals, which also renders the computed targets to be as close to the observed values as possible. These targets represent the output levels that could be expected if all countries performed at their best level. For certain countries, the targets are higher than the actual number of medals won while, for other countries, these targets may be lower. The proposed approach has been applied to the results of the Tokyo 2020 Olympic Games and compared with both FSO and non-FSO DEA methods. Keywords Fixed-sum outputs ·Centralized DEA ·Target setting ·Weighted Tchebychef method ·Tokyo 2020 1 Introduction Data Envelopment Analysis (DEA) is a non-parametric methodology for the assessment of the relative efficiency of a set of homogeneous operating units for which the term Decision- Making Unit (DMU) is generally used. These DMUs are assumed to consume inputs in order to produce outputs. Data Envelopment Analysis can compute the relative efficiency of a DMU by looking if it is feasible to reduce its input consumption (without reducing its outputs), or whether it is feasible to increase its outputs (without increasing its inputs). If that is the case, then the DMU is said to be inefficient, and an efficiency score that takes into account the excess input consumption and the shortfall in output production is computed. BSebastián Lozano [email protected] 1Department of Industrial Management, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Seville, Spain 123 Annals of Operations Research Furthermore, a target that reflects the estimated input and output improvements is provided for each DMU. This is the basic DEA methodology, which, depending on the application, may have specific refinements. In particular, in the case of applying DEA to the Olympic Games, the inputs usually considered are the Population and the Gross Domestic Product (GDP) of the participating nations. These inputs are considered non-discretionary and hence cannot be reduced (see Banker & Morey, 1986). Another feature that needs to be taken into account in DEA Olympics applications is the higher value of gold medals with respect to silver medals and of silver medals with respect to bronze medals. In this respect, note that the official ranking of participating National Olympic Committees (NOCs) is of lexicographic type whereby NOCs are ranked based on the number of gold medals won, using silver medals and bronze medals, in that order, to break ties. This lexicographic approach is noncompensatory and implicitly means that a gold medal is worth more than any number of silver medals and that a silver medal is worth more than any number of bronze medals. Another drawback of the official ranking is that it is based on the absolute results, not on the relative efficiency of the countries, that is, it fails to take into account the size and the wealth of the countries. Data Envelopment Analysis approaches carry out a more objective assessment of NOC performance by bearing these factors in mind, thereby estimating their efficiency in relative terms. As can be observed in the literature review presented in the next section, a number of DEA approaches have been proposed to analyse the efficiency of NOCs in Olympic Games. They can be divided into two categories: those that take into account that the total number of medals is fixed and those that do not. The latter generally result in DEA projections (i.e., output targets) that greatly exceed the total number of medals available. This problem is not unique to Olympic Games and affects not only DEA. Collier et al., (2011) argue that all deterministic methods that measure technical efficiency in sports using the distance to the efficient frontier of the production technology inferred from the observations suffer from serial correlation, that is, the efficiency estimates are biased downward. This also happens in football and basketball competitions, for example. Assuming that certain teams are inefficient, the removal of their inefficiency would result in increasing their number of wins, but if that happened, then the most efficient teams would be unable to maintain their total number of wins that were partly due to the inefficient performance of their rivals. Collier et al., (2011) proposes a correction that downwardly adjusts the efficient frontier to account for this serial correlation. They argue that additive DEA models are preferable to multiplicative or proportional adjustment approaches. Their model, however, considers that only one of the outputs (e.g., number of wins) is serially correlated while the other outputs are not. In Olympic Games, however, all three outputs generally considered, namely number of gold, silver, and bronze medals won, are fixed-sum outputs (FSO). The imposition of these three FSO constraints leads to centralized DEA models (when an envelopment formulation is used) or to DEA models with a common set of weights (CSW) (in the case of multiplier formulations). In this paper, a multiobjective FSO DEA approach is proposed. Recall that in multiobjective optimization, the different objective functions are in conflict with each other, that is, improving one generally implies worsening another. By definition, this is what happens in the corresponding Pareto optimal set. Hence, in the case of Olympic Games efficiency analysis, a multiobjective perspective seems appropriate since all countries want to increase their number of medals but, since the total number of these is fixed, if one country wins more medals then another must win fewer. Moreover, we consider each type of medal won by a country as a different objective function. In this regard, note that the number of medals won of each type are also in conflict with each other. Thus, it is assumed that a country aspires to win a medal in a certain sport. If it wins a gold medal, then it cannot win a silver or a bronze 123 Annals of Operations Research medal and vice versa. Therefore, if a country wins more gold medals, it can win fewer silver and bronze medals and vice versa. There are many multiobjective DEA approaches in the literature (e.g., Ebrahimnejad & Tavana, 2014; Estellita Lins et al., 2004; Korhonen et al., 2003; Malekmohammadi et al., 2011). The proposed multi-objective FSO DEA approach is of centralized DEA type in the sense that all the DMUs are jointly projected in a single model that imposes the three FSO constraints. In order to solve this multiobjective optimization model, the Weighted Tchebychef Method (WTM), is used, which has been used in several multiobjective DEA approaches (e.g., Arana-Jiménez et al., 2020; Despotis et al., 2016; Gutiérrez & Lozano, 2016; Lozano & Adenso-Díaz, 2018). The WTM is a common solution method in multiobjective optimization. It is based on the concept of the ideal point, that is, the point whose components are the optimal value of each of the objective functions when optimized separately. The idea behind WTM involves finding a solution whose objective function vector is as close as possible to the ideal point, using the Tchebychef metric. The WTM only guarantees a weak Pareto optimal solution but there are several extensions, such as the Augmented WTM, Modified WTM, and Lexicographic WTM (see Marler & Arora, 2004). Aiming to get all outputs as close as possible to their respective ideal values involves a fairness principle that acknowledges that all countries can aspire to reach their maximum potential in every type of medal. However, this has to be carried out taking into account the FSO constraints as well as the integrality of the targets. The Tchebychef metric used by the WTM implies that it focuses on the cases that remain most distant from their ideal values, in an effort to prevent any country from being left behind in any type of medal. Since WTM can generally only guarantee a weak Pareto optimal solution, it is common to use a secondary objective function that maximizes the sum of the different objective functions. However, since in our application that sum is constant, we use another, more appropriate, secondary goal, namely that of minimizing the sum of the absolute value of the changes in the number of medals of the different NOCs. The idea is that, although the targets of some countries will involve more medals and those of other countries will involve fewer, the absolute sizes of those changes are minimized, subject, of course, to the optimal value of the maximum distance to the ideal values computed by the WTM. The main contribution of the paper is, therefore, the introduction of an innovative centralized multiobjective perspective that focuses on setting fair FSO output targets from which performance indices and corresponding efficiency ranking are derived ex post. Almost all existing DEA approaches to the Olympic Games, regardless of whether they consider FSOs or not, use an output-oriented radial metric and focus on computing efficiency scores, not on output targets. The output targets computed by the proposed approach lie on the FSO efficient frontier, which is below the conventional efficient frontier and represent a fairer assessment of the performance of each NOC since this is what each country could aspire to obtain if all countries were efficient. Obtaining better results than these targets is possible (because in reality not all countries perform efficiently) but that should be interpreted as a sort of super-efficient performance. As regards the novelty of the proposed approach, note that there is no FSO DEA approach that provides targets for the DMUs in this Olympic Games application. Thus, on one hand, non-FSO DEA approaches (such as those found in Lozano et al., 2002 &Wuetal.,2010) can compute efficiency scores as well as targets but experience the problem that the total number of medals of those targets exceeds the actual number of medals that are awarded in the Games. On the other hand, existing FSO DEA approaches take into account the FSO character of the problem, but they aim at only efficiency estimation. They use variables that represent the changes in the number of medals of the DMUs (they have to, in order to 123 Annals of Operations Research impose the FSO constraints) but they do not compute targets as such. The proposed approach, instead of focusing on the computation of efficiency scores, focuses on computing sound and fair targets, and determines efficiency scores ex-post based on those targets. Note also that, although a FSO DEA target setting approach based on Nash Bargaining has been recently proposed (Lozano, 2023), it cannot be used to compute integer targets as required in this application. The structure of the paper is as follows. In Sect. 2, a literature review of DEA applications to Olympic Games is presented. Subsequently, in Sect. 3, the proposed approach is formulated and discussed. In Sect. 4, the proposed approach is applied to the recent Tokyo 2020 Olympics, and its results are compared with those of other methods from the literature. Finally, in Sect. 5, conclusions are drawn and further research is outlined. 2 Literature review The first paper applying DEA to Olympic Games was that by Lozano et al., (2002), which considered the aforementioned basic features (non-discretionary inputs and weight restrictions). It set the default of using a radial output orientation. Certain refinements, such as the imposition of the integrality of the output targets, have also been proposed (e.g., Wu et al., 2010). Intended for ranking the NOCs more than for target setting, Zhang et al., (2009) proposed a lexicographic approach that ranks the NOCs first by their relative efficiency in winning gold medals, while using the relative efficiency in winning silver and bronze medals (in that order) to break ties. Their approach was output-oriented but non-radial in that only one output was increased at a time. Leietal.,(2015) proposed a shared-input, parallel network DEA (NDEA) approach with two stages corresponding to the Summer and Winter Olympic Games, respectively. This approach used a multiplier formulation, weight restrictions, and an output orientation. Li et al., (2012) also used a multiplier DEA formulation, weight restrictions, and an output orientation but on a two-stage configuration that considered two inputs (Population and GDP per capita), an intermediate product (Number of athletes integrating the NOC team) and the usual three outputs (Number of gold, silver, and bronze medals won). Jablonsky (2018)used a similar approach, except for the inclusion of an additional input (Number of medals won in the previous Olympics). In the same way as Lei et al., (2015), Jablonsky (2018) computes efficiency scores but not output targets. Although not a strictly DEA approach, Calzada-Infante & Lozano (2016)usedaDominance Network (DN) for the efficiency analysis of Olympic Games. The nodes of a DN are the DMUs, and an arc exists from DMU i to DMU j if the latter dominates the former, that is, if DMU j consumes less input and produces more output than DMU i. Using complex network analysis tools, a number of local and global indicators can be computed to assess and visualize the relative performance of the different countries. Recently, Sekitani and Zhao (2021) have presented a complex restricted multiplier DEA approach in which the minimum sum of the absolute deviation of an affine function fitting of the total number of medals is imposed as a constraint. Apart from estimating and decomposing the efficiency scores, the method provides a regression function that predicts the total number of medals based on three inputs (Population, GDP, and Total number of medals won at the previous Olympic Games). Furthermore, from the corresponding dual formulation, a production possibility set is implied and can be employed to predict the number of medals won by a certain country. Indeed, the number of medals predicted for Japan in Tokyo 2020 123 Annals of Operations Research (26 gold, 15 silver, 22 bronze) is very close to the actual number of medals won (27 gold, 14 silver, 17 bronze). Note, however, that this type of prediction differs from the concept of efficient FSO targets considered in this paper, which assumes that all countries perform efficiently. Other input variables, apart from GDP and Population, have also been considered. Thus, for example, Soares de Mello et al., (2012) uses Population and Number of athletes of each NOC as input variables (as a proxy for the country investments in sports). The latter input is considered discretionary, and an input orientation is used in that paper which means that it does not suffer from the drawback of all the output-oriented approaches mentioned so far, which is that they compute efficiency scores (and corresponding rankings) for each NOC separately which leads to output targets whose sum exceeds the total number of medals of each type to be awarded in the Games. Flegl & Andrade (2018) use three inputs (GDP, Economic Active Population, and Corruption Perception Index) and as outputs, apart from the weighted number of gold, silver, and bronze medals won, a fourth input is considered that represents the aggregated weighted number of 4th to 8th positions. The DEA model used was of a Constant Returns to Scale (CRS) cross-efficiency approach. Wu et al., (2009b)also proposed a CRS cross-efficiency approach, using an aggressive formulation and Assurance Region (AR) constraints to reflect the different valuation of the medals. They utilized a hierarchical clustering algorithm to identify similar groups of countries based on their crossefficiency scores. Wu et al., (2009a) also computed cross-efficiency scores but under a game theory approach and assumed Variable Returns to Scale (VRS). Other multiplier DEA formulations include the context-dependent ARapproach of Li et al., (2008), the separation vector approach of Chiang et al., (2011), and the interval efficiency approach of Azizi and Wang (2013). Churilov & Flitman (2006) used a standard CRS DEA ratio formulation (without AR constraints), and considered four inputs (Population, GDP per capita, Disability Adjusted Life Expectancy Index, and Index of Equality of Child Survival) and four outputs, corresponding to aggregating the medals won by a given country using four different sets of weights. Soares de Mello et al., (2009) used a pure-output multiplier DEA formulation with AR constraints, and averaged the computed weights of the medal across the different countries that won medals in each cluster of sports. These average weights are employed to compute the efficiency index (and rank) of the participants in each cluster. The efficiency in the different clusters is aggregated to obtain an overall performance index. The idea behind considering these clusters is to distinguish between sports that have numerous different competitions (e.g., athletics, gymnastics, swimming) and those with a fewer number (e.g., baseball, football, volleyball). Of all the approaches reviewed above, the only one that takes into account the FSO character of the output variables is that of Soares de Mello et al., (2012). They do so, however, in an implicit way, that is, by using an input-oriented approach. Another approach that takes FSO into account in an implicit way (by projecting each DMU assuming a certain output reduction strategy on the part of the other DMUs) is the Zero-Sum Gains (ZSG) model of Lins et al., (2003). Although Lins et al., (2003) considers a single output that is a weighted aggregation of the number of gold, silver, and bronze medals won, the ZSG approach has been extended in Bi et al., (2014) to handle multiple outputs. These ZSG approaches use envelopment DEA formulations and do not impose the FSO constraints explicitly. Indeed, Bouzidis & Karagiannis (2022) have argued that the ZSG efficiency scores of Lins et al., (2003) are not really comparable across DMUs and proposed an alternative performance metric that is consistent with the FSO constraints (which they label as output interdependency). The explicit imposition of FSO constraints that establish that the total number of medals of each type is fixed requires a centralised DEA approach or its equivalent 123 Annals of Operations Research dual, a CSW DEA model. The Equilibrium Efficient Frontier (EEF) approach used in Yang et al., (2011), despite considering the FSO constraints explicitly does not consider CSW when computing the EEF and hence the efficiency assessment is not carried out using a common platform. This issue was corrected in Yang et al., (2015), whose Generalized Equilibrium Efficient Frontier (GEEF) approach uses FSO constraints as well as CSW. Moreover, GEEF has also been extended by Li et al., (2021) to a two-stage NDEA configuration similar to that in Li et al., (2015), although applied to the Winter Olympic Games. Although there are other GEEF approaches (e.g., Amirteimoori et al., 2017; Zhu et al., 2020) they have not been applied to the Olympic Games. Note, finally, that neither the ZSG nor the GEEF approaches take into account the integrality of the Olympic Games output variables or compute output targets, but instead compute only efficiency scores and rankings. This is the research gap we identified: a DEA approach able to compute fair, FSO-compliant targets for the DMUs. Table 1summarizes the main features of the aforementioned DEA applications to the Olympic Games. It can be observed that no FSO-explicit envelopment approach, such as the one proposed in this paper, aimed at computing output targets, has been proposed. Furthermore, although this is, in some respect, a secondary consideration, instead of the usual radial approach, the proposed centralized DEA approach is multiobjective and uses a non-radial solution method that includes a secondary goal. 3 Proposed multiobjective FSO approach Before formulating the proposed approach, let us introduce the notation to be used. 3.1 Data j,r∈{1,2, ..., n}indices on the set of DMUs. xj=x1j,x2j, ..., xmjinputs of DMU j. yj=yGj,ySj,yBjnumber of medals won by DMU j(G =Gold, S =Silver, B = Bronze). ytotal k=n j=1ykj total amount of medals of type k∈{G,S,B}. w=(wG,wS,wB)relative weight/value of medals of type k. 3.2 Decision variables (λ1r,λ 2r, ..., λnr )intensity variables utilized to compute the target of DMU r. ˆ yr=ˆyGr,ˆySr ,ˆyBrtarget of the number of medals to be won by DMU r. sr=(sGr,sSr ,sBr)shortfall (i.e., output slacks) in the number of medals won by DMU r. σrinefficiency score of DMU r. Assuming VRS, taking into account the non-discretionary character of the input variables and the integrality of the output variables, and considering a weighted additive (WA) metric (e.g., Cooper et al., 2011; Lovell & Pastor, 1995), the following conventional DEA model can be employed to compute an efficient target (and associated inefficiency score) for a certain DMU r 123 Annals of Operations Research Table 1 Summary of DEA Olympic Games applications Reference Envelop Multiplier Radial AR Integer Targets VRS Eff. Scores Ranking FSO Targets NDEA Remarks Lozano et al., (2002)✓✓✓✓✓✓ Technical/Scale efficiency Wu et al., (2010)✓✓✓✓✓✓ CRS Zhang et al., (2009)✓✓✓✓Lexicographic Soares de Mello et al., (2012) ✓✓ ✓✓✓ Input-oriented Churilov and Flitman (2006) ✓✓ ✓ ✓ CRS Li et al., (2008)✓✓✓ ✓✓ ✓ Context-dependent AR Wu et al., (2009a)✓✓✓ ✓✓ ✓ Game XE Wu et al., (2009b)✓✓✓ ✓ ✓ XE aggressive, CRS Soares de Mello et al., (2009) ✓✓✓ ✓ ✓ Pure output, clusters, CRS Chiang et al., (2011)✓✓ ✓✓ CSW, separation vector, CRS Azizi and Wang (2013) ✓✓ ✓✓ ✓ Interval efficiency Flegl and Andrade (2018) ✓✓ ✓ ✓ XE, Original inputs/outputs Calzada-Infante and Lozano (2016) ✓✓✓✓✓✓ ZSG single output Lei et al., (2015)✓✓✓ ✓✓ ✓Two parallel subsystems Li et al., (2015) ✓✓✓ ✓✓ ✓Two-stage series system 123 Annals of Operations Research Table 1 (continued) Reference Envelop Multiplier Radial AR Integer Targets VRS Eff. Scores Ranking FSO Targets NDEA Remarks Jablonsky (2018)✓✓✓✓✓ ✓Two-stage series, SBM Sekitani and Zhao (2021) ✓✓ ✓✓✓✓✓ Eff. decomp., prediction Lins et al., (2003)✓✓✓✓✓✓ ZSG single output Bi et al., (2014)✓✓✓✓✓✓ ZSG multiple outputs Yang et al., (2011)✓✓ ✓✓ ✓ EEF, single output Yang et al., (2015)✓✓✓ ✓✓ ✓ GEEF Li et al., (2021)✓✓✓ ✓✓ ✓ ✓GEEF, Two-stage series Bouzidis and Karagiannis (2022) ✓✓ ✓✓✓ ZSG single output Proposed approach ✓✓✓✓✓✓Multiobjective, WTM, secondary goal 123 Annals of Operations Research σr=Max  k wkskr s.t.  j λjrxij ≤xir ∀i  j λjr ykj ≥ˆykr =ykr +skr ∀k  j λjr =1 λjr ≥0∀js kr ≥0∀k ˆykr integer ∀k⇔skr integer ∀k (1) This VRS DEA model maximizes the weighted sum of the output shortfalls. The corresponding input constraints take into account their non-discretionary character. The output targets are forced to be integers. Indeed, the integrality of the output variables can be imposed either on the output targets ˆykr themselves or, alternatively, on the associated slack variables skr. Moreover, note that the latter are assumed to be non-negative. This is a key feature of conventional DEA models of technical efficiency: the DMUs are always projected onto efficient targets that dominate them. The problem with this approach is that, in the case that the sum of the outputs is fixed, the targets computed in this way violate the corresponding FSO constraints, that is, n r=1ˆykr >ytotal k, rendering them invalid. As regards the inefficiency scores σr, they are non-negative, whereby a zero value indicates relative efficiency. Let E={r:σr=0}be the set of efficient DMUs. For the inefficient DMUs, σr>0 indicates technical inefficiency and represents the potential improvement in the number of medals of DMU r, and values the corresponding medals according to the given weights w=(wG,wS,wB). The aim of this paper is to develop an alternative approach that, by taking into account the FSO constraints, would lead to valid targets. Since these constraints involve the targets of all the DMUs, this means that the proposed approach must project all the DMUs at the same time and, hence, it must use a centralized DEA perspective. The approach must also enable the number of medals to be won by a country to be increased or decreased. This is clear. Since the total number of medals is fixed, the only possibility for some countries to win more medals involves assuming that other countries win fewer medals. Hence, in what follows, the non-negative slack variables sr=(sGr,sSr ,sBr)are not used. Instead, we will consider output change variables δr=(δGr,δSr ,δBr )that are taken as free variables (i.e., unrestricted in sign). The main difficulty, however, lies in defining the objective function to be used by this centralized DEA model. To this end, a multiobjective optimization approach is proposed so that all DMUs can aim to maximizing the number of medals of every type, that is, 123 Annals of Operations Research Fig. 1 Abstract representation of the proposed FSO_WT approach Therefore, summarizing, a two-step lexicographic approach is proposed so that the first step aims at getting all outputs of all DMUs as close to their ideal values as possible (using a Minmax criterion) thereby implicitly defining a subset of the FSO efficient frontier (which we call the FSO_WT efficient frontier). The second step computes specific FSO_WT efficient targets by minimizing the magnitude of the total weighted effort required to reach them from the set of observed DMUs. Note that the two steps of the proposed approach use a centralized DEA perspective which, although not shown in Fig. 1, always maintains the FSO and the integrality constraints. 4 Application to Tokyo 2020 In this section, the application of the proposed approach to the results of the Tokyo 2020 Olympics is presented. The 93 NOCs that won at least one medal according to the Tokyo 2020 official website (https://olympics.com/en/olympic-games/tokyo-2020/medals) have been considered. Table 2shows the number of medals won by each NOC as well as their corresponding Population and GDP inputs. The sources of the input data include the 2019 Revision of World Population Prospects of the UN Population Division (https:// population.un.org/wpp/) and The World Factbook (https://www.cia.gov/the-world-factbook/ field/real-gdp-purchasing-power-parity/country-comparison), respectively. 123 Annals of Operations Research Table 2 Inputs and outputs for Tokyo 2020 NOC Population GDP (103US$) Gold (#) Silver (#) Bronze (#) Argentina (ARG) 45,195,777 991,523 0 1 2 Armenia (ARM) 2,963,234 40,384 0 2 2 Australia (AUS) 25,499,881 1,264,514 17 7 22 Austria (AUT) 9,006,400 498,780 1 1 5 Azerbaijan (AZE) 10,139,175 144,374 0 3 4 Bahamas (BAH) 393,248 14,450 2 0 0 Bahrain (BRN) 1,701,583 73,870 0 1 0 Belarus (BLR) 9,449,321 181,286 1 3 3 Belgium (BEL) 11,589,616 596,414 3 1 3 Bermuda (BER) 62,273 5,228 1 0 0 Botswana (BOT) 2,351,625 40,928 0 0 1 Brazil (BRA) 212,559,409 3,092,216 7 6 8 Bulgaria (BUL) 6,948,445 161,654 3 1 2 Burkina Faso (BUR) 20,903,278 44,266 0 0 1 Canada (CAN) 37,742,157 1,843,053 7 6 11 China (CHN) 1,439,323,774 22,526,502 38 32 18 Chinese Taipei (TPE) 23,816,775 1,143,277 2 4 6 Colombia (COL) 50,882,884 741,099 0 4 1 Côte d’Ivoire (CIV) 26,378,275 134,048 0 0 1 Croatia (CRO) 4,105,268 126,625 3 3 2 Cuba (CUB) 11,326,616 137,000 7 3 5 Czech Republic (CZE) 10,708,982 435,987 4 4 3 Denmark (DEN) 5,792,203 336,335 3 4 4 Dominican Republic (DOM) 10,847,904 197,735 0 3 2 Ecuador (ECU) 17,643,060 197,631 2 1 0 Egypt (EGY) 102,334,403 1,180,890 1 1 4 Estonia (EST) 1,326,539 48,987 1 0 1 Ethiopia (ETH) 114,963,583 248,972 1 1 2 Fiji (FIJ) 896,444 12,178 1 0 1 Finland (FIN) 5,540,718 268,662 0 0 2 France (FRA) 65,273,512 3,097,061 10 12 11 Georgia (GEO) 3,989,175 55,776 2 5 1 Germany (GER) 83,783,945 4,482,448 10 11 16 Ghana (GHA) 31,072,945 164,640 0 0 1 123 Annals of Operations Research Table 2 (continued) NOC Population GDP (103US$) Gold (#) Silver (#) Bronze (#) Great Britain (GBR) 67,886,004 3,118,396 22 21 22 Greece (GRE) 10,423,056 319,334 2 1 1 Grenada (GRN) 112,519 1,908 0 0 1 Hong Kong (HKG) 7,497,000 449,299 1 2 3 Hungary (HUN) 9,660,350 321,869 6 7 7 India (IND) 1,380,004,385 9,155,083 1 2 4 Indonesia (INA) 273,523,621 3,196,682 1 1 3 Iran (IRI) 83,992,953 1,027,238 3 2 2 Ireland (IRL) 4,937,796 428,825 2 0 2 Israel (ISR) 8,655,541 363,448 2 0 2 Italy (ITA) 60,461,828 2,562,135 10 10 20 Jamaica (JAM) 2,961,161 28,779 4 1 4 Japan (JPN) 126,476,458 5,231,066 27 14 17 Jordan (JOR) 10,203,140 101,738 0 1 1 Kazakhstan (KAZ) 18,776,707 487,868 0 0 8 Kenya (KEN) 53,771,300 227,638 4 4 2 Kosovo (KOS) 1,767,881 20,396 2 0 0 Kuwait (KUW) 4,270,563 209,738 0 0 1 Kyrgyzstan (KGZ) 6,524,191 33,918 0 2 1 Latvia (LAT) 1,886,202 59,102 1 0 1 Lithuania (LTU) 2,722,291 103,756 0 1 0 Malaysia (MAS) 32,365,998 906,239 0 1 1 Mexico (MEX) 128,932,753 2,525,481 0 0 4 Mongolia (MGL) 3,278,292 39,723 0 1 3 Morocco (MAR) 36,910,558 279,295 1 0 0 Namibia (NAM) 2,540,916 24,040 0 1 0 Netherlands (NED) 17,134,873 986,847 10 12 14 New Zealand (NZL) 4,822,233 210,877 7 6 7 Nigeria (NGR) 206,139,587 1,032,048 0 1 1 North Macedonia (MKD) 2,083,380 34,333 0 1 0 Norway (NOR) 5,421,242 340,303 4 2 2 Philippines (PHI) 109,581,085 963,121 1 2 1 Poland (POL) 37,846,605 1,261,433 4 5 5 Portugal (POR) 10,196,707 358,344 1 1 2 Puerto Rico (PUR) 2,860,840 110,238 1 0 0 123 Annals of Operations Research Table 2 (continued) NOC Population GDP (103US$) Gold (#) Silver (#) Bronze (#) Qatar (QAT) 2,881,060 255,010 2 0 1 Republic of Moldova (MDA) 51,269,183 34,680 0 0 1 Romania (ROU) 19,237,682 579,549 1 3 0 Russia (ROC) 145,934,460 3,968,180 20 28 23 San Marino (SMR) 33,938,000 2,008 0 1 2 Saudi Arabia (KSA) 34,813,867 1,609,323 0 1 0 Serbia (SRB) 8,737,370 116,339 3 1 5 Slovakia (SVK) 5,459,643 178,513 1 2 1 Slovenia (SLO) 2,078,932 81,614 3 1 1 South Africa (RSA) 59,308,690 730,913 1 2 0 South Korea (KOR) 51,269,183 2,211,315 6 4 10 Spain (ESP) 46,754,783 1,925,576 3 8 6 Sweden (SWE) 10,099,270 547,595 3 6 0 Switzerland (SUI) 8,654,618 588,472 3 4 6 Syria (SYR) 17,500,657 50,280 0 0 1 Thailand (THA) 69,799,978 1,285,287 1 0 1 Tunisia (TUN) 11,818,618 125,783 1 1 0 Turkey (TUR) 84,339,067 2,371,374 2 2 9 Turkmenistan (TKM) 6,031,187 86,858 0 1 0 Uganda (UGA) 45,741,000 96,838 2 1 1 Ukraine (UKR) 43,733,759 538,388 1 6 12 United States (USA) 331,002,647 20,524,945 39 41 33 Uzbekistan (UZB) 33,469,199 235,021 3 0 2 Venezuela (VEN) 28,435,943 269,068 1 3 0 Total - - 340 338 402 The first thing one may do is to compute the conventional WA DEA targets ˆyGr,ˆySr ,ˆyBr and inefficiency scores σras per Model (1). These are shown in Table 3, where it is apparent that, for each type of medal, the sum of the targets greatly exceeds the total number of medals awarded. As regards the medal weights utilized to obtain these results, (wG,wS,wB)= (0.5714286,0.2857143,0.1428571), they correspond to the assumption that wG=βwS= β2wB,whereβ=2, that is, one gold medal is worth two silver medals and one silver medal is worth two bronze medals. Note that, according to the conventional DEA approach, there are 16 efficient NOCs, namely AUS, BAH, BER, CUB, GEO, GBR, GRN, HUN, JAM, JPN, NED, NZL, ROC, SMR, UKR, and USA. The most inefficient NOCs are those of IND, INA, MEX, TUR, KSA, BRA, and THA. Interestingly, in spite of the high number of medals won by CHN, it is still labelled as inefficient according to the conventional DEA approach. 123 Annals of Operations Research Table 3 Targets and efficiency scores computed by WA DEA model (1) NOC rˆyG,rˆyS,rˆyB,rσrRank Argentina (ARG) 14 7 17 11.857 82 Armenia (ARM) 3 2 3 1.857 22 Australia (AUS) 17 7 22 0.000 1 Austria (AUT) 9 6 10 6.714 68 Azerbaijan (AZE) 6 4 5 3.857 43 Bahamas (BAH) 2 0 0 0.000 1 Bahrain (BRN) 3 2 2 2.286 28 Belarus (BLR) 7 4 6 4.143 44 Belgium (BEL) 10 6 11 6.571 67 Bermuda (BER) 1 0 0 0.000 1 Botswana (BOT) 3 1 3 2.286 28 Brazil (BRA) 21 21 22 14.286 88 Bulgaria (BUL) 6 5 5 3.286 38 Burkina Faso (BUR) 4 1 4 3.000 33 Canada (CAN) 18 11 22 9.286 72 China (CHN) 39 41 33 5.286 55 Chinese Taipei (TPE) 15 7 20 10.286 78 Colombia (COL) 12 6 14 9.286 72 Côte d’Ivoire (CIV) 6 3 5 4.857 51 Croatia (CRO) 5 4 5 1.857 22 Cuba (CUB) 7 3 5 0.000 1 Czech Republic (CZE) 9 6 10 4.429 48 Denmark (DEN) 7 6 7 3.286 38 Dominican Republic (DOM) 7 5 6 5.143 53 Ecuador (ECU) 7 5 6 4.857 51 Egypt (EGY) 16 7 20 12.571 86 Estonia (EST) 2 1 3 1.143 20 Ethiopia (ETH) 7 6 7 5.571 58 Fiji (FIJ) 1 1 2 0.429 17 Finland (FIN) 7 6 7 6.429 65 France (FRA) 21 20 22 10.143 77 Georgia (GEO) 2 5 1 0.000 1 Germany (GER) 23 22 22 11.429 81 Ghana (GHA) 7 4 5 5.714 60 Great Britain (GBR) 22 21 22 0.000 1 Greece (GRE) 8 6 8 5.857 61 Grenada (GRN) 0 0 1 0.000 1 Hong Kong (HKG) 8 6 8 5.857 61 Hungary (HUN) 6 7 7 0.000 1 123 Annals of Operations Research Table 3 (continued) NOC rˆyG,rˆyS,rˆyB,rσrRank India (IND) 26 31 26 25.714 93 Indonesia (INA) 22 21 22 20.429 92 Iran (IRI) 14 7 18 10.000 76 Ireland (IRL) 7 6 7 5.286 55 Israel (ISR) 8 6 9 6.143 63 Italy (ITA) 20 17 21 7.857 71 Jamaica (JAM) 4 1 4 0.000 1 Japan (JPN) 27 14 17 0.000 1 Jordan (JOR) 6 2 4 4.143 44 Kazakhstan (KAZ) 9 7 10 7.429 69 Kenya (KEN) 7 6 7 3.000 33 Kosovo (KOS) 2 1 3 0.714 19 Kuwait (KUW) 6 5 6 5.571 58 Kyrgyzstan (KGZ) 3 2 3 2.000 24 Latvia (LAT) 3 1 3 1.714 21 Lithuania (LTU) 4 3 4 3.429 40 Malaysia (MAS) 13 7 16 11.286 80 Mexico (MEX) 20 16 22 18.571 91 Mongolia (MGL) 4 1 4 2.429 30 Morocco (MAR) 7 6 7 6.143 63 Namibia (NAM) 3 1 3 2.143 26 Netherlands (NED) 10 12 14 0.000 1 New Zealand (NZL) 7 6 7 0.000 1 Nigeria (NGR) 14 7 18 12.143 85 North Macedonia (MKD) 3 1 3 2.143 26 Norway (NOR) 7 6 7 3.571 41 Philippines (PHI) 14 6 17 10.857 79 Poland (POL) 16 8 20 9.857 75 Portugal (POR) 8 6 9 6.429 65 Puerto Rico (PUR) 4 3 4 3.143 36 Qatar (QAT) 4 3 4 2.429 30 Republic of Moldova (MDA) 4 1 4 3.000 33 Romania (ROU) 10 6 12 7.714 70 Russia (ROC) 20 28 23 0.000 1 San Marino (SMR) 0 1 2 0.000 1 Saudi Arabia (KSA) 17 10 21 15.286 89 Serbia (SRB) 6 2 5 2.000 24 Slovakia (SVK) 6 5 6 4.429 48 Slovenia (SLO) 3 2 3 0.571 18 South Africa (RSA) 12 6 14 9.429 74 123 Annals of Operations Research Table 3 (continued) NOC rˆyG,rˆyS,rˆyB,rσrRank South Korea (KOR) 19 14 22 12.000 83 Spain (ESP) 18 12 22 12.000 83 Sweden (SWE) 9 7 10 5.143 53 Switzerland (SUI) 8 7 9 4.143 44 Syria (SYR) 4 2 3 3.143 36 Thailand (THA) 17 7 22 14.143 87 Tunisia (TUN) 6 3 5 4.143 44 Turkey (TUR) 19 16 21 15.429 90 Turkmenistan (TKM) 5 2 4 3.714 42 Uganda (UGA) 5 3 4 2.714 32 Ukraine (UKR) 1 6 12 0.000 1 United States (USA) 39 41 33 0.000 1 Uzbekistan (UZB) 7 6 7 4.714 50 Venezuela (VEN) 7 6 7 5.286 55 Total 912 687 968 – – In order to obtain valid FSO targets and associated performance indices of the DMUs, the proposed FSO_WT approach is used. This implies: 1. Solving Model (3) for each NOC rand each medal type k, thereby obtaining the corresponding ideal values δideal krand yideal kr=ykr+δideal kr. 2. Solving FSO_WT model (8) using weights as per (4), thereby obtaining the Minmax distance to the ideal point μmin max. 3. Solving the secondary-goal model (9)-(10), thereby obtaining the FSO_WT efficient targets (11) and the corresponding performance indices (12). As regards the solution method of all the models in this paper, including Models (1), (3), (8), and (9), these have been solved using Gurobi Optimizer (version 9.1.2), which uses Branch and Cut to solve such MILP problems. Non-centralized DEA models, such as Model (1), have very few (in fact, just three) integer variables and hence are solved very easily. Centralized DEA models, such as Models (3), (8), and (9), have more integer variables (namely three times the number of DMUs) and hence can take somewhat longer to compute. In any case, the computation times in a conventional laptop (with 8 Gb RAM and an i7 processor) are negligible (less than 2 s). Another important point, raised by one of the reviewers, is the uniqueness of the targets computed. We knew beforehand that the FSO_WT model (8) can have alternative optima, which is the reason why a secondary goal or phase-II model was proposed. However, although unlikely, it is theoretically possible that the secondary-goal model (9)-(10) might also have multiple optimal solutions. Fortunately, this has not been the case, and it could therefore be verified (ex post) that the optimal solution of Models (9)-(10) reported below was unique. Table 4shows the ideal values yideal kr for all types of medals of all NOCs, the FSO_WT targets ˆykr∗, performance indices ξr, and the corresponding ranking. The optimum of the FSO_WT model (8) is μmin max =0.4155844. Note that, as indicated in Corollary 1, the ideal values never fall below the observed values. Indeed, for most countries, the ideal value 123 Annals of Operations Research Table 4 Ideal point, FSO_WT targets, and performance indices computed by proposed approach NOC rˆyideal Gr ˆyideal Sr ˆyideal Br ˆyG,r∗ˆyS,r∗ˆyB,r∗ξrRank Argentina (ARG) 14 12 18 4 0 2 0.139 67 Armenia (ARM) 4 3 4 2 2 2 0.286 89 Australia (AUS) 17 13 22 5 6 22 −0.425 2 Austria (AUT) 9 8 10 3 0 5 0.091 58 Azerbaijan (AZE) 6 5 5 2 0 4 0.019 45 Bahamas (BAH) 21 21 0 0−0.286 9 Bahrain (BRN) 3 2 3 1 0 0 0.048 50 Belarus (BLR) 7 5 6 2 3 3 0.082 55 Belgium (BEL) 10 9 11 3 1 3 0.000 13 Bermuda (BER) 1 1 2 1 0 0 0.000 13 Botswana (BOT) 3 3 3 1 0 1 0.190 83 Brazil (BRA) 21 23 23 7 6 8 0.000 13 Bulgaria (BUL) 6 5 6 3 1 2 0.000 13 Burkina Faso (BUR) 4 4 4 2 0 1 0.286 89 Canada (CAN) 18 15 22 6 8 11 0.006 39 China (CHN) 39 41 33 38 32 18 0.000 13 Chinese Taipei (TPE) 15 12 20 5 4 6 0.114 63 Colombia (COL) 12 10 14 4 4 1 0.190 84 Côte d’Ivoire (CIV) 6 5 5 2 0 1 0.190 84 Croatia (CRO) 5 5 5 3 3 2 0.000 13 Cuba (CUB) 7 5 5 2 3 5 −0.408 3 Czech Republic (CZE) 9 7 10 3 6 3 0.018 44 123 Annals of Operations Research Table 4 (continued) NOC rˆyideal Gr ˆyideal Sr ˆyideal Br ˆyG,r∗ˆyS,r∗ˆyB,r∗ξrRank Denmark (DEN) 7 6 7 3 4 4 0.000 13 Dominican Republic (DOM) 7 6 6 2 3 2 0.163 77 Ecuador (ECU) 7 6 6 2 1 0 0.000 13 Egypt (EGY) 16 13 20 5 1 4 0.143 69 Estonia (EST) 2 2 3 1 0 1 0.000 13 Ethiopia (ETH) 7 6 7 2 0 2 0.034 47 Fiji (FIJ) 1 1 2 1 0 1 0.000 13 Finland (FIN) 7 6 7 2 0 2 0.163 77 France (FRA) 21 20 23 6 20 11 0.005 38 Georgia (GEO) 4 5 4 2 5 1 0.000 13 Germany (GER) 23 22 23 7 17 16 0.003 37 Ghana (GHA) 7 5 6 2 0 1 0.163 77 Great Britain (GBR) 22 21 23 7 21 22 −0.390 7 Greece (GRE) 8 6 8 3 0 1 0.024 46 Grenada (GRN) 0 0 1 0 0 1 0.000 13 Hong Kong (HKG) 8 7 8 3 0 3 0.061 52 Hungary (HUN) 87 83 7 7−0.214 11 India (IND) 30 32 26 9 2 4 0.152 75 Indonesia (INA) 22 23 23 7 1 3 0.156 76 Iran (IRI) 14 12 18 4 1 2 0.017 43 Ireland (IRL) 7 6 7 2 0 2 0.000 13 Israel (ISR) 8 7 9 3 0 2 0.071 53 Italy (ITA) 20 18 22 6 18 20 0.013 40 Jamaica (JAM) 42 42 0 4−0.429 1 Japan (JPN) 27 26 24 8 14 17 −0.402 5 Jordan (JOR) 6 5 5 2 0 1 0.133 65 Kazakhstan (KAZ) 10 8 11 3 0 8 0.171 80 123 Annals of Operations Research Table 4 (continued) NOC rˆyideal Gr ˆyideal Sr ˆyideal Br ˆyG,r∗ˆyS,r∗ˆyB,r∗ξrRank Kenya (KEN) 7 6 7 3 6 2 0.014 41 Kosovo (KOS) 2 2 3 2 0 0 0.000 13 Kuwait (KUW) 6 5 6 2 0 1 0.190 84 Kyrgyzstan (KGZ) 4 3 4 2 2 1 0.286 89 Latvia (LAT) 3 2 3 1 0 1 0.000 13 Lithuania (LTU) 4 3 4 2 0 0 0.190 84 Malaysia (MAS) 13 11 17 4 0 1 0.150 74 Mexico (MEX) 20 20 22 6 0 4 0.171 80 Mongolia (MGL) 4 3 4 2 0 3 0.190 84 Morocco (MAR) 8 6 8 3 0 0 0.143 69 Namibia (NAM) 3 2 3 1 0 0 0.048 50 Netherlands (NED) 12 12 15 4 12 14 −0.286 10 New Zealand (NZL) 76 72 6 7−0.408 3 Nigeria (NGR) 14 12 18 4 0 1 0.139 67 North Macedonia (MKD) 3 3 3 1 0 0 0.095 59 Norway (NOR) 7 6 7 2 5 3 0.000 13 Philippines (PHI) 14 11 17 4 0 1 0.071 53 Poland (POL) 16 13 21 5 5 5 0.036 48 Portugal (POR) 8 7 9 3 0 2 0.102 62 Puerto Rico (PUR) 4 3 4 2 0 0 0.143 71 Qatar (QAT) 4 3 4 2 0 1 0.000 13 Republic of Moldova (MDA) 4 3 4 2 0 1 0.286 89 Romania (ROU) 10 8 12 3 3 0 0.114 63 123 Annals of Operations Research Table 5 Inverse efficiency scores and ranking of different methods versus those of proposed approach NOC rξr(Rank) FSO_WT (Efficiency)−1BCC_AR (Efficiency)−1BCC_AR_Int (Efficiency)−1ZSG− Lins (Efficiency)−1ZSG−Bi (Efficiency)−1ZSG− BK JAM −0.429 (1) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) AUS −0.425 (2) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) CUB −0.408 (3) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) NZL −0.408 (3) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) JPN −0.402 (5) 1.000 (1) 1.000 (1) 1.085 (13) 1.000 (1) 0.985 (13) USA −0.396 (6) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) GBR −0.390 (7) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) ROC −0.310 (8) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) BAH −0.286 (9) 1.000 (1) 1.000 (1) 1.000 (1) 1.000 (1) 0.984 (1) NED −0.286 (10) 1.016 (13) 1.000 (1) 1.036 (11) 1.000 (1) 0.984 (11) HUN −0.214 (11) 1.169 (15) 1.688 (22) 1.174 (15) 1.000 (1) 0.986 (15) UKR −0.005 (12) 2.311 (32) 2.250 (34) 2.277 (31) 1.000 (1) 0.998 (31) CHN 0.000 (13) 1.026 (14) 1.026 (17) 1.141 (14) 1.026 (16) 0.985 (14) BRA 0.000 (13) 3.133 (39) 1.000 (1) 3.024 (36) 2.829 (55) 1.006 (36) KOR 0.000 (13) 3.060 (38) 2.333 (36) 2.955 (35) 2.253 (44) 1.005 (35) NOR 0.000 (13) 1.822 (22) 2.000 (27) 2.188 (29) 1.822 (34) 0.997 (28) SUI 0.000 (13) 2.214 (30) 3.000 (38) 2.186 (28) 1.614 (30) 0.997 (29) BEL 0.000 (13) 3.424 (42) 3.333 (43) 3.786 (43) 3.424 (61) 1.014 (43) BER 0.000 (13) 1.000 (1) 1.000 (1) 1.079 (12) 1.000 (1) 0.985 (12) BUL 0.000 (13) 2.210 (30) 1.000 (1) 2.538 (33) 2.210 (43) 1.000 (33) 123 Annals of Operations Research Table 5 (continued) NOC rξr(Rank) FSO_WT (Efficiency)−1BCC_AR (Efficiency)−1BCC_AR_Int (Efficiency)−1ZSG− Lins (Efficiency)−1ZSG−Bi (Efficiency)−1ZSG− BK CRO 0.000 (13) 1.660 (19) 1.667 (21) 1.763 (19) 1.577 (28) 0.992 (19) ECU 0.000 (13) 3.710 (46) 3.500 (45) 4.491 (48) 3.709 (63) 1.021 (48) EST 0.000 (13) 2.992 (37) 1.000 (1) 3.111 (37) 2.402 (45) 1.006 (37) FIJ 0.000 (13) 1.855 (23) 1.111 (18) 1.946 (24) 1.682 (32) 0.994 (23) GEO 0.000 (13) 1.198 (16) 1.958 (26) 1.361 (16) 1.000 (1) 0.988 (16) GRN 0.000 (13) 1.000 (1) 2.000 (27) 1.000 (1) 1.900 (35) 0.984 (1) IRL 0.000 (13) 3.528 (44) 4.000 (49) 4.661 (50) 3.527 (62) 1.023 (50) KOS 0.000 (13) 1.449 (18) 2.000 (27) 1.889 (22) 1.448 (25) 0.993 (22) LAT 0.000 (13) 3.491 (43) 2.000 (27) 3.834 (45) 3.047 (59) 1.014 (45) QAT 0.000 (13) 2.404 (34) 2.083 (32) 3.295 (39) 2.404 (46) 1.009 (39) SLO 0.000 (13) 1.278 (17) 2.026 (31) 1.475 (17) 1.278 (19) 0.989 (17) SMR 0.000 (13) 1.000 (1) 2.115 (33) 1.000 (1) 1.000 (1) 0.984 (1) SRB 0.000 (13) 1.866 (24) 2.275 (35) 1.854 (21) 1.088 (17) 0.993 (21) SWE 0.000 (13) 2.113 (29) 2.500 (37) 2.526 (32) 1.428 (23) 1.000 (32) DEN 0.000 (13) 2.070 (28) 1.000 (1) 2.048 (26) 1.618 (31) 0.995 (26) UGA 0.000 (13) 2.806 (36) 3.000 (38) 2.946 (34) 2.750 (53) 1.005 (34) GER 0.003 (37) 2.041 (26) 1.000 (1) 1.975 (25) 1.472 (26) 0.995 (25) FRA 0.005 (38) 1.985 (25) 1.156 (19) 1.929 (23) 1.711 (33) 0.994 (24) CAN 0.006 (39) 2.311 (32) 1.556 (20) 2.249 (30) 1.942 (38) 0.997 (30) ITA 0.013 (40) 1.720 (21) 1.938 (25) 1.680 (18) 1.134 (18) 0.991 (18) UZB 0.014 (41) 2.623 (35) 3.000 (38) 3.403 (41) 2.623 (49) 1.010 (41) KEN 0.014 (41) 1.695 (20) 1.789 (24) 1.826 (20) 1.548 (27) 0.993 (20) 123 Annals of Operations Research Table 5 (continued) NOC rξr(Rank) FSO_WT (Efficiency)−1BCC_AR (Efficiency)−1BCC_AR_Int (Efficiency)−1ZSG− Lins (Efficiency)−1ZSG−Bi (Efficiency)−1ZSG− BK IRI 0.017 (43) 4.534 (53) 5.875 (63) 4.917 (55) 4.371 (74) 1.026 (55) CZE 0.018 (44) 2.041 (26) 1.750 (23) 2.163 (27) 1.938 (37) 0.996 (27) AZE 0.019 (45) 3.964 (48) 3.700 (48) 3.916 (46) 1.439 (24) 1.015 (46) GRE 0.024 (46) 4.168 (49) 4.000 (49) 4.725 (53) 4.167 (70) 1.024 (53) ETH 0.034 (47) 6.133 (63) 4.727 (58) 6.031 (60) 3.790 (66) 1.038 (60) POL 0.036 (48) 3.149 (40) 7.000 (67) 3.252 (38) 2.694 (51) 1.008 (38) TUN 0.038 (49) 5.390 (60) 15.000 (78) 6.128 (62) 4.836 (77) 1.039 (62) BRN 0.048 (50) 8.726 (72) 3.077 (41) 9.691 (73) 2.740 (52) 1.077 (73) NAM 0.048 (50) 7.591 (66) 4.250 (53) 9.342 (70) 2.639 (50) 1.074 (70) HKG 0.061 (52) 4.943 (58) 5.273 (61) 4.857 (54) 2.980 (58) 1.025 (54) PHI 0.071 (53) 8.748 (73) 8.500 (73) 9.439 (71) 0.169 (83) 1.075 (71) ISR 0.071 (53) 4.261 (51) 4.000 (49) 5.425 (58) 4.260 (72) 1.032 (58) BLR 0.082 (55) 3.369 (41) 4.375 (55) 3.327 (40) 1.937 (36) 1.009 (40) RSA 0.086 (56) 7.591 (66) 17.500 (82) 9.140 (69) 0.199 (78) 1.072 (69) TUR 0.090 (57) 6.317 (64) 9.800 (74) 6.040 (61) 0.398 (47) 1.038 (61) AUT 0.091 (58) 5.321 (59) 3.667 (47) 5.220 (57) 2.007 (39) 1.029 (57) ESP 0.095 (59) 3.540 (45) 33.000 (92) 3.439 (42) 0.488 (41) 1.010 (42) MKD 0.095 (59) 7.740 (69) 5.000 (59) 9.000 (68) 3.077 (60) 1.070 (68) SVK 0.095 (59) 4.560 (54) 8.000 (71) 4.695 (52) 2.945 (56) 1.024 (52) POR 0.102 (62) 6.872 (65) 20.000 (83) 6.742 (64) 4.562 (75) 1.046 (64) ROU 0.114 (63) 5.470 (61) 7.500 (69) 6.542 (63) 0.336 (57) 1.044 (63) TPE 0.114 (63) 4.429 (52) 3.429 (44) 4.296 (47) 0.364 (53) 1.020 (47) 123 Annals of Operations Research Table 5 (continued) NOC rξr(Rank) FSO_WT (Efficiency)−1BCC_AR (Efficiency)−1BCC_AR_Int (Efficiency)−1ZSG− Lins (Efficiency)−1ZSG−Bi (Efficiency)−1ZSG− BK JOR 0.133 (65) 11.154 (75) 6.625 (66) 11.016 (74) 4.071 (68) 1.092 (74) THA 0.134 (66) 17.056 (80) 29.333 (89) 20.087 (80) 0.059 (93) 1.189 (81) ARG 0.139 (67) 22.308 (82) 17.000 (81) 21.554 (82) 0.119 (89) 1.205 (82) NGR 0.139 (67) 30.475 (88) 27.333 (88) 29.408 (88) 0.082 (91) 1.290 (88) EGY 0.143 (69) 9.948 (74) 6.250 (65) 9.595 (72) 0.192 (79) 1.077 (72) MAR 0.143 (69) 8.262 (71) 20.000 (83) 12.437 (75) 8.260 (88) 1.107 (75) PUR 0.143 (71) 4.662 (56) 7.000 (67) 7.066 (65) 4.662 (76) 1.049 (65) VEN 0.143 (71) 4.232 (50) 4.000 (49) 4.934 (55) 0.454 (42) 1.026 (56) KSA 0.148 (73) 45.464 (93) 44.000 (93) 54.007 (93) 0.068 (92) 1.554 (93) MAS 0.150 (74) 28.206 (87) 21.750 (86) 27.292 (87) 0.088 (90) 1.267 (87) IND 0.152 (75) 16.174 (78) 16.000 (79) 15.050 (77) 0.151 (86) 1.136 (77) INA 0.156 (76) 16.961 (79) 16.889 (80) 16.017 (79) 0.130 (87) 1.146 (79) DOM 0.163 (77) 5.697 (62) 4.273 (54) 5.611 (59) 0.495 (40) 1.033 (59) FIN 0.163 (77) 24.490 (84) 5.200 (60) 24.027 (84) 0.266 (64) 1.232 (84) GHA 0.163 (77) 41.993 (91) 5.500 (62) 41.299 (91) 0.160 (84) 1.417 (91) KAZ 0.171 (80) 7.748 (70) 7.500 (69) 7.580 (66) 0.725 (22) 1.055 (66) MEX 0.171 (80) 34.162 (89) 25.500 (87) 32.399 (89) 0.176 (81) 1.322 (89) TKM 0.171 (80) 13.007 (77) 30.000 (91) 15.174 (78) 5.233 (80) 1.136 (78) BOT 0.190 (83) 20.022 (81) 3.500 (45) 19.867 (80) 3.768 (65) 1.187 (80) CIV 0.190 (84) 38.536 (90) 4.400 (56) 37.952 (90) 5.733 (82) 1.381 (90) COL 0.190 (84) 7.642 (68) 6.143 (64) 8.185 (67) 0.394 (48) 1.061 (67) KUW 0.190 (84) 42.142 (92) 8.000 (71) 41.443 (91) 6.423 (85) 1.419 (92) 123 Annals of Operations Research Table 5 (continued) NOC rξr(Rank) FSO_WT (Efficiency)−1BCC_AR (Efficiency)−1BCC_AR_Int (Efficiency)−1ZSG− Lins (Efficiency)−1ZSG−Bi (Efficiency)−1ZSG− BK LTU 0.190 (84) 11.813 (76) 11.000 (76) 13.509 (76) 3.790 (66) 1.118 (76) MGL 0.190 (84) 4.713 (57) 12.000 (77) 4.677 (51) 1.369 (21) 1.023 (51) ARM 0.286 (89) 3.836 (47) 3.231 (42) 3.809 (44) 1.329 (20) 1.014 (44) BUR 0.286 (89) 24.433 (84) 4.400 (56) 24.200 (85) 4.254 (71) 1.233 (85) KGZ 0.286 (89) 4.561 (54) 10.667 (75) 4.528 (49) 1.607 (29) 1.022 (49) MDA 0.286 (89) 22.927 (83) 20.000 (83) 22.723 (83) 4.096 (69) 1.217 (83) SYR 0.286 (89) 25.378 (86) 29.000 (89) 25.126 (86) 4.353 (73) 1.243 (86) 123 Annals of Operations Research Fig. 7 NOCs ranked in increasing order of performance index ξr is not too close to unity (which would correspond to giving the same weight to all medal types), the results are not very sensitive. 5 Conclusions In this paper, an innovative way is proposed of assessing the efficiency, ranking, and setting targets in the Olympic Games. It uses a centralized DEA perspective that takes into account the FSO character of the total number of medals won by all the NOCs. The problem is formulated in a multiobjective optimization framework, and utilizes the ideal-point concept and the WTM solution. Since the latter generally has alternative optima, a secondary goal is proposed to project the DMUs onto their closest targets. In this second step, a rectangular distance is applied. The final targets thus computed involve winning more medals in the case of inefficient NOCs and, in order to compensate, winning fewer medals in the case of other NOCs that have performed super-efficiently. The proposed performance index is simply a weighted sum of the changes in the number of medals, and it can be positive, negative, or zero, depending on the NOC. As strengths of the proposed method, its FSO character should be highlighted (i.e., the total number of medals available is respected), as should the fairness principle implied by the WTM solution (i.e., for all NOCs and all medal types, the targets become as close as possible to their ideal values) and the closest-target principle implied by the secondary goal (i.e., from among the alternative FSO_WT efficient targets, the target that minimizes the rectangular distance to the observed outputs is determined). Another feature of the proposed approach is that, unlike all previous approaches, it computes not only performance indices (akin to a type of FSO super-efficiency score) but also, and more importantly, FSO_WT efficient targets. Furthermore, the computed performance indices enable the NOCs to be classified into three categories corresponding to underperforming, neutral, and super-efficient DMUs. The latter category refers to those NOCs that have won more medals than they would have done if all the NOCs had performed at their best level. 123 Annals of Operations Research Apart from the mathematical modelling aspects, the proposed approach represents a significant departure from existing FSO DEA approaches, whose sole objective is that of efficiency estimation. These approaches employ variables that represent the changes in the number of medals of the DMUs (a necessity in order to impose the FSO constraints) but they fail to report the corresponding targets. This is due to the existence of many alternative FSO efficient targets and, for those only interested in computing efficiency scores, unrealistic and indefensible targets may result. The proposed approach follows a different path whereby, instead of ignoring the existence of alternative FSO efficient targets, it tackles the challenging problem of computing reasonable targets for the DMUs. In other words, instead of focusing on computing efficiency scores, we focus on computing targets, and determine the performance indices of the DMUs ex post, based on those targets. These performance indices ξrare derived as a subproduct of the computed targets and represent an empirical way of measuring the distance of a DMU to its corresponding target and, as such, of the effort to reach said target. Therefore, ξr>0 implies winning more medals or medals of a more valuable type: more effort in either case. In contrast, ξr<0 means that the number and/or value of the medals won exceeds what could be expected given the inputs (i.e., the resources) of the country. In other words, such a strong performance is unsustainable in the event that all DMUs perform at their best level. Finally, ξr=0 indicates a neutral performance, in line with expectations. The proposed approach has been applied to the Tokyo 2020 Olympic Games and compared with conventional (i.e., non-FSO) DEA approaches as well as with other FSO approaches. Although there is a certain ranking similarity with the alternative approaches (higher in the case of BCC_AR, BCC_AR_Int, ZSG-Lins, and ZSG-BK and lower in the case of ZSG-Bi, Lexicographic ranking, and WA DEA), it is clear that, due to its differentiated character, the proposed approach provides a ranking distinct from the other methods. The NOCs that are ranked highest by the proposed approach are JAM, AUS, CUB, NZL, JPN, USA, GBR, ROC, BAH, NED, HUN, and UKR. All of them are super-efficient and have to reduce the number of medals won by a significant percentage which, for example, in the case of USA amounts to 27 fewer gold medals, and in the case of JPN and GBR, this entails 15 and 13 fewer gold medals, respectively. These reductions in the number of medals won corresponds to correcting the serial correlation that complicates efficiency assessment in this type of competition. As regards the countries that exhibited the worst relative performance in Tokyo 2020 according to the proposed approach, these are ARM, BUR, KGZ, MDA and SYR. In addition to these two extremes of most- and least-performing countries, it is important to identify a set of neutralperformance countries that can be said to have matched their expectations, while taking into account their Population and GDP as well as the FSO character of the competition. As regards the managerial implications of the proposed approach, information regarding the ideal number of medals a country could aspire to win based on its size and wealth is of interest. Moreover, the rationale that states that the attainment of the ideal number of medals is generally impossible leads to targets of a more realistic nature being set, which take into account the FSO character of the Games. Furthermore, since the proposed approach uses the current situation as a reference, the computed target could be considered as a first step towards the ideal target. As a continuation of this research, we envisage its extension to include NDEA approaches towards the Olympic Games. Furthermore, other multiobjective optimization methods, particularly interactive multiobjective optimization methods (e.g., Soltani & Lozano, 2020), can be applied to this centralized DEA scenario. 123 Annals of Operations Research Acknowledgements This research was carried out with the financial support of the Spanish Ministry of Science and Innovation, and the European Regional Development Fund (ERDF), grant PID2021-123515NB-I00. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix This appendix contains the list of the acronyms used in the text. AR Assurance Region. BCC_AR Banker, Charnes and Cooper +Assurance Region. BCC_AR_Int Banker, Charnes and Cooper +Assurance Region +Integer data. CRS Constant Returns to Scale. CSW Common set of weights. DEA Data Envelopment Analysis. DMU Decision-Making Unit. DMU Decision-Making Unit. EEF Equilibrium Efficient Frontier. FSO Fixed-sum output. EEF Equilibrium Efficient Frontier. FSO_WTFixed-sum output +Weighted Tchebychef Method. GDP Gross Domestic Product. GEEF Generalized Equilibrium Efficient Frontier. MILP Mixed-Integer Linear Program. NDEA Network DEA. NOC National Olympic Committee. VRS Variable Returns to Scale. WTM Weighted Tchebychef Method. ZSG Zero-Sum Gains approach. ZSG-Bi Zero-Sum Gains approach proposed by Bi et al., (2014) ZSG-BK Zero-Sum Gains approach proposed by Bouzidis and Karagiannis (2022) ZSG-Lins Zero-Sum Gains approach proposed by Lins et al., (2003). References Amirteimoori, A., Masrouri, S., Yang, F., & Kordrostami, S. (2017). Context-based competition strategy and performance analysis with fixed-sum outputs: An application to banking sector. Journal of the Operational Research Society, 68, 1461–1469. Aparicio, J., Ruíz, J. L., & Sirvent, I. (2007). Closest targets and minimum distance to the Pareto-efficient frontier in DEA. Journal of Productivity Analysis, 28, 209–218. 123 Annals of Operations Research Arana-Jiménez, M., Sánchez-Gil, M. C., & Lozano, S. (2020). Efficiencya Assessment and Target Setting Using a Fully Fuzzy DEA Approach. International Journal of Fuzzy Systems, 22, 1056–1072. Azizi, H., & Wang, Y. M. (2013). Improved DEA models for measuring interval efficiencies of decision-making units. Measurement, 46, 1325–1332. Banker, R. D., & Morey, R. (1986). Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34, 513–521. Bi, G., Feng, C., Ding, J., Liang, L., & Chu, F. (2014). The linear formulation of the ZSG-DEA models with different production technologies. Journal of the Operational Research Society, 65, 1202–1211. Bouzidis, T., & Karagiannis, G. (2022). An alternative ranking of DMUs performance for the ZSG-DEA model. Socio-Economic Planning Sciences, 81, 101179. Calzada-Infante, L., & Lozano, S. (2016). Analysing Olympic Games through dominance networks. Physica a: Statistical Mechanics and Its Applications, 462, 1215–1230. Chiang, C. I., Hwang, M. J., & Liu, Y. H. (2011). Determining a common set of weights in a DEA problem using a separation vector. Mathematical and Computer Modelling, 54, 2464–2470. Churilov, L., & Flitman, A. (2006). Towards fair ranking of Olympics achievements: The case of Sydney 2000. Computers & Operations Research, 33, 2057–2082. Collier, T., Johnson, A. L., & Ruggiero, J. (2011). Measuring Technical Efficiency in Sports. Journal of Sports Economics, 12, 579–598. Cooper, W. W., Pastor, J. T., Aparicio, J., & Borras, F. (2011). Decomposing profit inefficiency in DEA through the weighted additive model. European Journal of Operational Research, 212, 411–416. Despotis, D. K., Sotiros, D., & Koronakos, G. (2016). A network DEA approach for series multi-stage processes. Omega, 61, 35–48. Ebrahimnejad, A., & Tavana, M. (2014). An interactive MOLP method for identifying target units in outputoriented DEA models: The NATO enlargement problem. Measurement, 52, 124–134. Estellita Lins, M. P., Angulo-Meza, L., & Moreira da Silva, A. C. (2004). A multi-objective approach to determine alternative targets in data envelopment analysis. Journal of the Operational Research Society, 55, 1090–1101. Flegl, M., & Andrade, L. A. (2018). Measuring countries’ performance at the Summer Olympic Games in Rio 2016. Opsearch, 55, 823–846. Gutiérrez, E., & Lozano, S. (2016). Efficiency assessment and output maximization possibilities of European small and medium sized airports. Research in Transportation Economics, 56, 3–14. Jablonsky, J. (2018). Ranking of countries in sporting events using two-stage data envelopment analysis models: A case of Summer Olympic Games 2016. Central European Journal of Operations Research, 26, 951–966. Korhonen, P., Stenfors, S., & Syrjänen, M. (2003). Multiple objective approach as an alternative to radial projection in DEA. Journal of Productivity Analysis, 20, 305–321. Lei, X., Li, Y., Xie, Q., & Liang, L. (2015). Measuring Olympics achievements based on a parallel DEA approach. Annals of Operations Research, 226, 379–396. Li, Y., Lei, X., Dai, Q., & Liang, L. (2015). Performance evaluation of participating nations at the 2012 London Summer Olympics by a two-stage data envelopment analysis. European Journal of Operational Research, 243, 964–973. Li, Y., Liang, L., Chen, Y., & Morita, H. (2008). Models for measuring and benchmarking Olympics achievements. Omega, 36, 933–940. Li, Y., Liu, J., Ang, S., & Yang, F. (2021). Performance evaluation of two-stage network structures with fixed-sum outputs: An application to the 2018winter Olympic Games. Omega, 102, 102342. Lins, M. P. E., Gomes, E. G., Soares de Mello, J. C. C. B., & Soares de Mello, A. J. R. (2003). Olympic ranking based on a zero sum gains DEA model. European Journal of Operational Research, 148, 312–322. Lovell, C. A. K., & Pastor, J. T. (1995). Units invariant and translation invariant DEA models. Operations Research Letters, 18, 147–151. Lozano, S. (2023). Bargaining approach for efficiency assessment and target setting with fixed-sum variables. Omega, 114, 102728. Lozano, S., & Adenso-Díaz, B. (2018). Network DEA-based biobjective optimization of product flows in a supply chain. Annals of Operations Research, 264, 307–323. Lozano, S., & Khezri, S. (2021). Network DEA smallest improvement approach. Omega, 98, 102140. Lozano, S., Soltani, N., & Dehnokhalaji, A. (2020). A compromise programming approach for target setting in DEA. Annals of Operations Research, 288, 363–390. Lozano, S., Villa, G., Guerrero, F., & Cortés, P. (2002). Measuring the performance of nations at the Summer Olympics using data envelopment analysis. Journal of the Operational Research Society, 53, 501–511. Malekmohammadi, N., Hosseinzadeh Lotfi, F., & Jaafar, A. B. (2011). Target setting in data envelopment analysis using MOLP. Applied Mathematical Modelling, 35, 328–338. 123 Annals of Operations Research Marler, R. T., & Arora, J. S. (2004). Survey of multi-objective optimization methods for engineering. Structural Multidisciplinary Optimization, 26, 369–395. Ramchandani, G., (2022). Home advantage in the Summer Olympic Games: evidence from Tokyo 2020 and prospects for Paris 2024, https://olympicanalysis.org/section-4/home-advantage-in-the-summer- olympic-games-evidence-from-tokyo-2020-and-prospects-for-paris-2024/ (accessed May 16th 2022) Sekitani, K., & Zhao, Y. (2021). Performance benchmarking of achievements in the Olympics: An application of data envelopment analysis with restricted multipliers. European Journal of Operational Research, 294, 1202–1212. Soares de Mello, J. C. C. B., Angulo-Meza, L., & Branco Da Silva, B. P. (2009). A ranking for the Olympic Games with unitary input DEA models. IMA Journal of Management Mathematics, 20, 201–211. Soares de Mello, J. C. C. B., Angulo-Meza, L., & Lacerda, F. G. (2012). A DEA model with a non discritionary variable for Olympic evaluation. Pesquisa Operacional, 2, 21–29. Soltani, N., & Lozano, S. (2020). Interactive multiobjective DEA target setting using lexicographic DDF. RAIRO-Operations Research, 54, 1703–1722. Wu, J., Liang, L., & Chen, Y. (2009a). DEA game cross-efficiency approach to Olympic rankings. Omega, 37, 909–918. Wu, J., Liang, L., & Yang, F. (2009b). Achievement and benchmarking of countries at the Summer Olympics using cross efficiency evaluation method. European Journal of Operational Research, 197, 722–730. Wu, J., Zhou, Z., & Liang, L. (2010). Measuring the performance of nations at Beijing summer Olympics using integer-valued DEA model. Journal of Sports Economics, 11, 549–566. Yang, F., Wu, D. D., Liang, L., & O’Neill, L. (2011). Competition strategy and efficiency evaluation for decision making units with fixed-sum outputs. European Journal of Operational Research, 212, 560–569. Yang, M., Li, Y. J., & Liang, L. (2015). A generalized equilibrium efficient frontier data envelopment analysis approach for evaluating DMUs with fixed-sum outputs. European Journal of Operational Research, 246, 209–217. Zhang, D., Li, X., Meng, W., & Liu, W. (2009). Measuring the performance of nations at the Olympic Games using DEA models with different preferences. Journal of the Operational Research Society, 60, 983–990. Zhu, Q., Song, M., & Wu, J. (2020). Extended secondary goal approach for common equilibrium efficient frontier selection in DEA with fixed-sum outputs. Computers & Industrial Engineering, 144, 106483. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123