2 - Cross-efficiency evaluation approach to Olympic.pdf

76 Int. J. Applied Management Science, Vol. 2, No. 1, 2010

Copyright © 2010 Inderscience Enterprises Ltd.

Cross-efficiency evaluation approach to Olympic ranking and benchmarking: the case of Beijing 2008

Jie Wu* and Liang Liang School of Management, University of Science and Technology of China, Hefei 230026, Anhui Province, P.R. China Fax: +86-551-3600025 E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: A number of studies have used data envelopment analysis (DEA) to evaluate the performance of the countries in the Olympic Games. In this paper, an extension tool of DEA, namely cross-efficiency evaluation method, is used to measure the performance of the nations participating in the Summer Olympic Games of Beijing 2008. The model in the paper considers two inputs (GDP per capita and population) and three outputs (number of gold, silver and bronze medals won) and the weight restrictions are included to guarantee a higher valuation for gold medals than for silver medal and higher for the latter than for bronze medals. The results for the Beijing 2008 Olympic Games are analysed and a unique ordering of the participants based on average cross-efficiency is provided, also cluster analysis technique is used to select the more appropriate targets for poorly performing countries to use as benchmarks.

Keywords: management science; data envelopment analysis; DEA; cross-efficiency; Beijing 2008 Olympic ranking.

Reference to this paper should be made as follows: Wu, J. and Liang, L. (2010) ‘Cross-efficiency evaluation approach to Olympic ranking and benchmarking: the case of Beijing 2008’, Int. J. Applied Management Science, Vol. 2, No. 1, pp.76–92.

Biographical notes: Jie Wu is a Lecturer at the School of Management, University of Science and Technology of China, Hefei, Anhui Province, P.R. China. His research interests focus on performance evaluation and benchmarking. His published work has appeared in OR, EJOR, OMEGA, IJPE, JORSJ, ESWA and other top journals.

Liang Liang is a Professor of Operations Management and the Executive Dean of the School of Management at the University of Science and Technology of China, Hefei, Anhui Province. He holds an MA and a PhD, both in Engineering, from Hefei University of Technology and Southeastern University of China, respectively. His research interests focus on decision analysis, performance evaluation and benchmarking and supply chain management.

Cross-efficiency evaluation approach to Olympic ranking and benchmarking 77

1 Introduction

As the largest-scale sport event globally, Olympic Games, especially the Summer Olympic Games, have gained increasingly more attention and have exerted significant influence on hosting countries. By participating this event, the comprehensive strengthen, nation culture and ideology of each country can be extended. In each time, nations, without exception, struggle for the Olympic medals to demonstrate their strength and to enhance honour of nationality. Meanwhile, hosting countries seize this golden opportunity to develop commerce and to improve the image of the hosting cities even that of the countries. The 29th Summer Olympic Games of Beijing held in August 2008 has the largest number of countries and participants, which is also the largest-scale beyond the memory of men. The successful host of the 2008 Olympic Games exerts great influence on Beijing, as well as the economy and social development of China. Therefore, it is of great necessity to carry out scientific research on the Olympic Games, especially for the medal list, which cannot only make a fair appraise, but provide appropriate improvement approach for the participants in the Olympic Games.

In general, the competition is among athletes and not among countries, but the countries from where the winners originated would grant them numerous prerogatives clearly showing that the country felt it had won as well. The Olympic Committee has never published an official ranking that would allow a country to be classed as the overall Olympic Games winner, however, more and more researches have been done in an effort to evaluate the performance of the nations with respect to the number of gold, silver and bronze medals won.

As a non-parametric method to provide a relative efficiency measure for peer decision-making units (DMUs) with multiple inputs and outputs, data envelopment analysis (DEA), proposed by Charnes et al. (1978), has been widely used in many cases such as supply chains (Dharmapala, 2008), mobile phone selection (Mahdavi et al., 2008), banking industry (Pramodh et al., 2008), financial services industry (Malhotra et al., 2009) and so on. DEA has also been used in the Olympic rankings to evaluate the achievements of those Olympic participating countries, such as Lozano et al. (2002), Lins et al. (2003), Churilov and Flitman (2006) and so on. A major advantage of DEA is that it does not require any assumptions about the functional form of the model that underpins the relationships between the input and output variables. However, despite the increase in research on the Olympic ranking, very few studies have investigated the benchmarking of the participants, which is important to ensure their continuing competitiveness.

In this paper, we propose an effective method for evaluating the performance of the nations participating at the Summer Olympic Games of Beijing 2008 and identifying useful benchmarks for efficiency improvement in less efficient countries. The cross-efficiency DEA model is used to examine the efficiency performance of the participants and the cross-efficiency scores of the participants are calculated. Cluster analysis is then used to select the performance targets that will serve as benchmarks for poorly performing countries. Countries with inherently similar inputs and outputs are classified into the same cluster and the country with the highest score in a given cluster is considered to be the primary benchmark for efficiency improvement for the other countries in the cluster. The rest of this paper is organised as follows: Section 2 provides a literature review of the Olympic ranking based on DEA method. The methods used in

78 J. Wu and L. Liang

this paper are introduced in Section 3. Section 4 shows the results of the Beijing 2008 Olympic Games. The conclusions are made in Section 5.

2 Literature review

As a non-parametric method to provide a relative efficiency measure for peer DMUs with multiple inputs and outputs, DEA has been used in the Olympics. The DEA applications in the Olympics can be classified into two groups: the first is to assess the benefit or harm of the home country (for example, Balmer et al., 2001, 2003; Bernard and Busse, 2004) and the second is to evaluate the achievements of those Olympic participating countries. We stress on the second group since this paper focus on the achievement evaluation.

Lozano et al. (2002) presented a simple DEA-based tool to assess the performance of participants at the Olympics, taking into consideration two inputs (GNP and population) and three outputs (number of gold, silver and bronze medals won), on which weight restrictions are imposed in order to increase the validity of the results. Taking into consideration the limited number of medals to be won, Lins et al. (2003) proposed a zero sum gains DEA model, in which two inputs (GDP and population) are considered and weights are constrained also. Apparently, Lozano et al. (2002) and Lins et al. (2003) used different approaches, taking into consideration different input measures and resulting in different ranking orders for the same nations, but the same technique, namely reference-set count, was used to rank efficient countries, that is, the efficient units are ranked by simply counting the number of times they appear in the reference sets of inefficient units, an idea first developed in Charnes et al. (1985). However, a complete ranking cannot be assured since many DMUs may receive the same ranked score, like Germany and Australia (both 23 times) in Lozano et al. (2002). For the improvement of inefficient countries, Lins et al. (2003) did not consider this topic, while Lozano et al. (2002) identified reference sets for each of the inefficient countries, a linear combination of DMUs in reference set is the target that an inefficient DMU uses to become efficient. However, according to Doyle and Green (1994), a difficulty with these conventional reference sets is that an inefficient DMU and its reference set may not be inherently similar in their practices. Therefore, it is possible that these reference targets are unattainable goals for the inefficient DMUs.

Churilov and Flitman (2006) combined self-organising maps (SOMs) and DEA to analyse the real achievements of the various Olympics participants. Based on four input measures [GDP per capita in US dollars, population, disability adjusted life expectancy (DALE) and index of equality of child survival (IECS)], they used SOMs to cluster participating countries into groups, each of which contains members of similar profile, but the combination of the data mining and DEA-based ranking systems suggested in the case study is not perfect. From the results, we can observe that certain clusters of countries use their own members as peers for benchmarking purposes, while there are clusters which do not utilise any of the ‘peers’ in clustering sense as a ‘peer’ for benchmarking and DEA comparison.

Hai (2007) integrates the vote-ranking technique into the cross-evaluation methods to assess the performance of the nations at the Olympics. The main idea of cross-evaluation is to use DEA in a peer evaluation instead of a self-evaluation mode. The weight bundle in the cross-efficiency case is more acceptable than those from the CCR models since the former is a common weight originated from averaging all DMUs’ favourite weights. So,


the cross-efficiency technique is accepted by many researchers because it is implemented under the common-weight logic. In the Olympics application, the weights for gold, silver and bronze medals for one country should be as same as that for another country, so Hai (2007) follows the common-weight logic and employ the cross-efficiency technique in the Olympics evaluation. The technique proposed by Hai (2007) can work effectively in ranking, however, not in target setting, since it also ignores the inherent similarity and difference between one inefficient DMU and its frontier target.

Recently, Li et al. (2008) used context-dependent assurance region DEA (CAR-DEA) to incorporate multiple sets of ARs in an effort to better characterise the Olympic performance of each nation. Besides, because of the competitions between participating countries in the Olympic Games, DEA game cross-efficiency model of Liang et al. (2008) was modified and used to measure the performance of the nations participating in the last six Summer Olympic Games (1984–2004) by Wu et al. (2009a). They extended the approach of Liang et al. (2008) to the variable returns to scale (VRS) DEA model which could generate negative cross-efficiency scores. They presented a modification to the VRS DEA model so that the cross-efficiency scores were non-negative. Besides, Wu et al. (2009b) used an extension tool of DEA, namely cross-efficiency evaluation method, to measure the performance of the nations participating in the last six Summer Olympic Games (1984–2004).

These studies indicate that DEA has become an important analytical tool in assessing the performance of the nations participating at the Olympic Games. This paper extends this line of research by examining the benchmarking of the participants at the Summer Olympic Games of Beijing 2008, using a new variant of DEA known as the cross-efficiency approach. An advantage of cross-efficiency over simple efficiency is that the former can be used to rank all DMUs and Doyle and Green (1994) argued that cross-efficiency evaluation method is more general and more powerful than the reference-set count and other DEA-based ranking system. Meanwhile, cluster analysis is used for identifying more practical and feasible benchmarks for poorly performing DMUs.

3 Methodology

The main idea of cross-evaluation is to use DEA in a peer evaluation instead of a self-evaluation mode. Cross-efficiency evaluation has been used in various applications, e.g., efficiency evaluations of nursing homes (Sexton et al., 1986), R&D project selection (Oral et al., 1991), preference voting (Green et al., 1996) and others. In this paper, we will perform a DEA analysis of nations in the Beijing 2008 Olympic Games. The DMUs considered correspond to the participating nations which won at least one medal and there covers 86 different countries in the Beijing 2008 Olympic Games. Based upon the previous DEA studies, we use two inputs, namely GDP per capita (in US dollars) and population of the country. As noted in Lozano et al. (2002), these two input attributes have captured the most important factors expressing the economic and demographic power of nations. The outputs in the evaluation include the total number of gold, silver and bronze medals which DMUs won in the Olympics.

In order to formulate the specific mathematical model used, we need to introduce some notations as follows:


n number of DMUs (countries)

j index of countries

d index of country being assessed

x1j GDP per capita of country j

x2j population of country j

y1j number of gold medals won by country j

y2j number of silver medals won by country j

y3j number of bronze medals won by country j

ω1d multiplier assigned by DMUd to input x1d

ω2d multiplier assigned by DMUd to input x2d

μ1d multiplier assigned by DMUd to output y1d



α number of silver medals that are equivalent to one gold medal

β number of bronze medals that are equivalent to one silver medal.

The multiplier form of linear programming model for assessing DMUd can thus be stated as:

1 1 2 2 3 3

1 1 2 2 1 1 2 2 3 3

1 1 2 2

1 2

2 3

max . . 0,

1, 2,..., .

1

0

0

0, 1, 2.

0, 1,2,3.

d d d d d d

d j d j d j d j d j

d d d d

d d

d d

id

rd

y y ys t x x y y y

j n

x x

i

r

μ μ μω ω μ μ μ

ω ω

μ αμ

μ βμ

ω

μ

+ +

+ − − − ≥

=

+ =

− ≥

− ≥

≥ =

≥ =

(1)

The main meaning of above model (1) is to maximise the efficiency of DMUd that keep the efficiency scores of all DMUs at or below unity and this model has five non-negative and continuous variables and the two weight constraints define an assurance region for the output multipliers.

3.1 Cross-efficiency

For each DMUd (d = 1,…, n) under evaluation in model (1), after solving the corresponding linear programming model (1) by a mathematical tool like MATLAB and Lingo, we can obtain a set of optimal weights (multipliers) * * * * *

1 2 1 2 3( , , , , ).d d d d dω ω μ μ μ Using this set, the d-cross-efficiency for any DMUj (j = 1,…, n), is then calculated as:


* * *1 1 2 2 3 3

* *1 1 2 2

, , 1, 2,...,d j d j d jdj

d j d j

y y yE d j n

x x

μ μ μ

ω ω

+ += =

+ (2)

For DMUj (j = 1,…, n), the average of all Edj (d = 1,…, n), namely:

1

1 n

j djd

E En =

= ∑ (3)

can be used as an efficiency measure for DMUj and will be referred to as the cross-efficiency score for DMUj.

In general, the weights of input and output measure in model (1) may not be unique. To resolve choice amongst multiple solutions, Sexton et al. (1986) and Doyle and Green (1994) suggested the use of aggressive and benevolent cross-evaluation. A cross-evaluation is aggressive/benevolent in the sense that it selects a set of weights which not only maximise the efficiency of a particular DMU under evaluation, but also minimise/maximise the efficiencies of all other DMUs in some sense.

In situations where relative dominance among the DMUs is to be evaluated, the aggressive formulation is more appropriate than the benevolent formulation. The aggressive formulation aims to maximise the own efficiency and minimise that of other DMUs, so it is suitable to the scenario where one DMU prefers to discriminate against other DMUs. The corresponding aggressive formulation of (1) developed by Doyle and Green (1994) is shown as (4):

1 1 2 2 3 3

1 1 2 2 1 1 2 2 3 3

1 1 2 2

1 1 2 2 3 3 1 1 2 2

1 2

2 3

min

. . 0,

.

1

( ) 0

0

0

0

d l d l d ll d l d l d

d j d j d j d j d j

d l d ll d l d

d d d d d d dd d d d d

d d

d d

id

y y y

s t x x y y y

j d

x x

y y y E x x

μ μ μ

ω ω μ μ μ

ω ω

μ μ μ ω ω

μ αμ

μ βμ

ω

≠ ≠ ≠

≠ ≠

+ +

+ − − − ≥

≠

+ =

+ + − × + =

− ≥

− ≥

≥

∑ ∑ ∑

∑ ∑

, 1, 2.

0, 1,2,3.rd

i

rμ

=

≥ =

(4)

where Edd is the efficiency of DMUd obtained from (1). In model (4), an aggressive cross-evaluation is selected in that DMUd selects a set of weights which not only maximise its own efficiency (i.e., Edd), but also minimise the efficiencies of all other DMUs [the objective of model (4)]. The benevolent formulation has the same set of constraints except that the objective function is maximised.

3.2 Identification of benchmarks

In traditional DEA, a linear combination of DMUs in the reference set is the target that an inefficient DMU uses to become efficient. According to Doyle and Green (1994), a


difficulty with these conventional reference sets is that an inefficient DMU and its reference sets may not be inherently similar in their practices. Therefore, it is possible that these reference targets are unattainable goals for the inefficient DMUs.

In this paper, cluster analysis is used in this methodology for identifying benchmarks for poorly performing DMUs. Computing the correlation coefficient between pairs of columns in a CEM tells us how similarly those DMUs are appraised by their peers. A high positive correlation coefficient indicates that the two DMUs are inherently similar with respect to their inputs and outputs. This means that these DMUs are performing similarly when evaluated with the optimal weights of other DMUs. Thus, using these correlation coefficients as the elements in a resemblance matrix and executing a clustering method yields clusters with inherently similar DMUs. The DMU with the highest column mean in a given cluster can be used as the primary benchmark for improvement by other DMUs in that cluster.

4 Results and discussions

In Section 2, we have presented cross-efficiency evaluation method used to analyse the performance of the Olympics participants. In this section, we apply cross-efficiency evaluation method to analyse the performance of nations participating in the last six Summer Olympic Games. The performance evaluation of the participants is assessed by the aggressive formulation (4) proposed by Doyle and Green (1994). The aggressive formulation is chosen for the analysis because a relative dominance among the countries is desired.

4.1 Rankings

Table 1 shows for each country that won at least one medal in the Beijing 2008 its efficiency in model (1) and average cross-efficiency. Only two countries, namely Jamaica and Mongolia are considered to be efficient in the Beijing 2008. Since model (1) fails to differentiate among efficient countries, we utilise the reference-set count to rank the efficient countries and Jamaica (75 times) appears more frequently as benchmarks and is ranked in the first place, followed by Mongolia (25 times).

The average cross-efficiency is shown in column 3 of Table 1, which indicates that Jamaica (efficient country) has the highest score of 0.9922 and is rated as the country with the best performance by the other countries. Mongolia, which is shown to be efficient by the analysis of model (1), achieves a score of 0.6098 and is ranked second in the entire set. Although Bahamas which has an efficiency score of 0.7582 and is ranked fourth in model (1), it achieves an average cross-efficiency of 0.4312, so it is ranked third in the entire set and has better overall performance than Zimbabwe which is ranked third from the evaluation of model (1). Finally, we compare the results of cross-evaluation method with those of model (1) and the ranking differences between the two methods are listed in the last column of Table 1. The comparison result reveals that 66 of the 86 DMUs occupy different rank positions in one than in the other, which illustrates that there are significant differences in the rankings of the two methods.


Table 1 Beijing 2008 results (α = β = 2)


Table 1 Beijing 2008 results (α = β = 2) (continued)


Table 2 Sensitivity of cross-efficiency and rankings for the Beijing 2008


Table 2 Sensitivity of cross-efficiency and rankings for the Beijing 2008 (continued)


The results in Table 1 have been computed using α = β = 2, i.e., one gold medal is worth at least two silver medals and one silver medal at least two bronze medals. Sensitivity of the results to the values of parameters α and β are listed in Table 2, in which the average cross-efficiencies and the corresponding ranking results obtained from different values of the parameters are shown. As the parameters increase, the size of the output multipliers feasible region in model (1) gets smaller and, therefore, the efficiency in model (1) generally decreases. Note, however, that the average cross-efficiency has changed irrespective of the values of parameters used, for it not only relates to its own weights, but also to the weights of other countries.

To test the consistency of those rankings results based on different α and β values shown in Table 2, we employ Spearman test by SPSS computation and the test results are shown in Table 3. It can be found obviously that the results show a high consistency. For example, the results under α = β = 4 and under α = β = 5 are the highest with the consistency coefficient 0.997, and the lowest, the results under α = β = 1 and under α = β = 5 are also consistent enough with the consistency coefficient 0.876. Besides, the testing value under the hypothesis that all ranking results are totally independent is 0.0000, that is to say, the independent hypothesis should be rejected. So we can conclude that, the ranking results under different α and β values are distinctly correlated. In other words, the choice of different α and β values affects the rankings little.

Table 3 Spearman rank-order correlations among the results by different α and β values

α = β = 1 α = β = 2 α = β = 3 α = β = 4 α = β = 5

α = β = 1 1 0.973 (0.0000)

0.935 (0.0000)

0.901 (0.0000)

0.876 (0.0000)

α = β = 2 1 0.988 (0.0000)

0.968 (0.0000)

0.953 (0.0000)

α = β = 3 1 0.994 (0.0000)

0.985 (0.0000)

α = β = 4 1 0.997 (0.0000)

α = β = 5 1

Note: The p-values are reported in parentheses for the non-trivial correlations.

4.2 Benchmarking

As discussed in the introduction, although traditional CCR model or model (1) provides benchmarks for inefficient countries, it has certain limitations. The main issue is that an inefficient country and its benchmarks may not be inherently similar in their practices (Doyle and Green, 1994).

To identify proper benchmarks that can be used by poorly performing countries as a basis for improvement, we firstly compute the correlation coefficient between a pair of columns in cross-efficiency matrix, which can tell us how similarly those two DMUs were appraised by their peers. Then, we use these correlation coefficients as the elements in a resemblance matrix and the clustering technique based on an average linkage method is executed. The tree is truncated to obtain a reasonable level of clustering among the countries.


The cluster analysis used in this paper is a hierarchical clustering technique and the average linkage method is utilised in deriving the clusters1. Figure 1 illustrates the clusters that are identified from the above technique to the Olympics Game of Beijing 2008.

Figure 1 Cluster result of the Beijing 2008 based on cross-efficiency

Figure 1 illustrates a total of seven clusters, A, B, C, D, E, F and G which are identified in this analysis. The best performers in each of the clusters are utilised by other countries in those clusters as the primary benchmarks for improvement. For example, in cluster G, DMU 42 (Jamaica) with an average cross-efficiency score of 0.9922 should be utilised by DMU 24 (Dominica), DMU 38 (Iran) and DMU 75 (Thailand) as a benchmark for improvement. These countries should identify the operating practices of Jamaica and implement those practices in order to become more competitive.

DMU 55 = 0.6098 31, 36, 46, 9, 53, 45, 28, 18, 86, 82

DMU 27 = 0.267722, 63, 11, 57, 66, 13, 32, 47, 43, 81, 58, 73, 64, 60, 70, 5, 23, 41, 29, 49, 30, 67, 16, 65, 34, 62, 15

DMU 8 = 0.4312 50, 44, 12, 20, 33, 6, 39, 79, 35,

69, 72, 77, 17, 3

DMU 10 = 0.2639 68, 78, 14, 40, 51, 85, 83

DMU 48 = 0.0992 74, 84, 37, 61, 71, 54, 80, 1, 76, 59

DMU 21 = 0.2889 56, 26, 2, 19, 52, 4, 7, 25

DMU 42 = 0.9922 75, 38, 24

A

B

C

D

E

F

G


It is evident that in some of these clusters, such as cluster E, the benchmark country itself is not a very good performer. Although this is the case, it is still important for the countries in this cluster to utilise it as a potential benchmark because improvements have to be performed in an incremental manner by first achieving an attainable target before making radical improvement strides. It so happens that the countries in this cluster are relatively poor performers and the most appropriate benchmark for improving their current practices is DMU 48 (Kyrgyzstan). The clustering results for other clusters can be interpreted in the same manner as discussed in this section.

5 Conclusions

In this paper, cross-efficiency evaluation method has been used to assess the performance of the participants in the Beijing 2008 Summer Olympic Games. Two inputs (GDP per capita and population) and three outputs (number of gold, silver and bronze medals won) are considered. Weight restrictions on the output multipliers are also imposed in order to ensure the validity of the results. The average cross-efficiency is calculated to provide a unique ordering of the participating nations and the cluster analysis technique is also utilised to select the more appropriate and inherently similar targets for poorly performing countries to use as a benchmark.

We point out that while we have concentrated herein on the CCR model, further research is to extend the ideas to other DEA models, such as the VRS DEA model. On the other hand, the model proposed in the paper assumes weight restrictions apply uniformly across all nations. In fact, different (sets of) nations may impose different weight restrictions on the output multipliers, for example, different nations may value the gold, silver and bronze medals differently, so one can easily extend the model proposed by incorporating multiple sets of nation-specific weight restrictions.

Acknowledgements

The authors are grateful to two anonymous referees for their insightful comments and helpful suggestions which helped to improve the previous version of this paper significantly. The research is supported by the National Natural Science Funds of China for Innovative Research Groups (No. 70821001).

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Notes 1 Clustering execution was completed on a PC version of SPSS. The clusters are obtained

by truncating the dendrogram at a rescaled distance measure of 5 on a scale of 0–25, where 0 represents a correlation coefficient of 1.0000.

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