1 Introduction

The Rio de Janeiro Carnival, a main festival in Brazil, has international repercussions, attracts thousands of tourists and is an important jobs and income source (Costa, Silva & Ramalho, 2010). The “folia climax” is the special group samba schools parade, considered the largest outdoor audiovisual event in the world (Pompeu & Perez, 2008). The growing competition for the Carnival champion title has made samba schools adopt modern administrative practices (Rego & Lopes, 2008; Tureta & Araújo, 2013), so that, the efficiency is one of the most desired attributes (Lopes, Malaia & Vinhais, 2009).

Recent papers utilized the Data Envelopment Analysis (DEA) technique to mensurate the banking efficiency (Paradi & Zhu, 2013), European soccer teams (Pyatunin et al., 2016), NBA basketballs teams

In addition to this introduction, this article contains four more sessions. The next one emphasizes the DEA technique as a measurement tool of technical and scale efficiency. In the third one, the research methods are shown and in the fourth the results founded are discussed. Finally, the conclusions were presented.

2 Measuring the technical and scale efficiency through the DEA model

The Data Envelopment Analysis it is considered a non parametric measuring efficiency technique, especially widespread by the Charnes, Cooper & Rhodes (1978) and Banker, Charnes & Cooper (1984) seminal studies. It is admitted that N Decision Making Units (DMU) uses a common technologyto transform an input vectorinto an output vectorTherefore, it is posible do define a set of production possibilities, formed by the feasible planes (x, y), such that:

Though it is not observable, the DEA technique, by the use of linear programming, calculates the estimative, given by (1), when setting the smallest space subset that contains the pair (X_n, Y_n), y and satisfies: the free inputs and outputs discard; the convexity production possibilities set as well as its scale returns, that can be considered constants (CRS) or variables (VRS) (Bogetoft & Otto, 2010):

(1)

(2)

(3)

The DEA models can be oriented by inputs, when the purpose is reducing inputs, keeping the outputsconstants, or oriented by outputs, when the goal is to increasing outputs, fixing the inputs. The second one model will be adopted in the present paper and the efficiency score,, is giving by:

The DMU0 efficiency score obtained under the assumption of constant scale returns is designed by the global technical efficiency while the another one, obtained under the assumption of variable scale returns it is characterized by the technical efficiency pure. The ratio between these two measures provides the scale efficiency (Ferreira & Gomes, 2009). In other words, the global technical inefficiency is composed by the pure technical inefficiency and the scale technical inefficiency.

The Figure 1 illustrates geometrically the presented concepts. For this, it is considered an input and an output and a model based on an outputs orientation. In this way, the efficient limits calculated by the DEA model are presented under the constant return to scale (CRS) assumptions, to which DMU C belongs and the variable scale returns (VRS), consisting of the DMUs A, B, C, D and E. The last one contains a part with non-decreasing scale returns (A to C) and one with non-increasing scale returns (C to E). Realize that the DMU P does not belong to those borders, therefore, it is inefficient. Under the assumption of constant scale returns, the global technical inefficiency of P is characterized by the distance PPc. By the oder hand, under the assumption of variable returns to scale, the pure technical inefficiency of P is designed by the distance PPv. The difference between these two techniques, given by the distance PvPc, it is called by the scale inefficiency of P.

Figure 1
Technical efficiency and scale efficiency
Source: Ferreira & Gomes (2009, p. 192, adapted).

If the scale efficiency it is equal to one, then the DMU will be operating with constant returns to scale, but if it is smaller than one, there may be increasing or decreasing scale returns. That is, the scale efficiency does not identify the type of scale return that a DMU is operating (Banker, Charnes & Cooper, 1984). To do so, for all optimal solutions (4), one of them must check the following conditions (Banker et al., 2004):

1. i. The scale returns will be increasing if, and only if,

2. ii. The scale returns will be decreasing if, and only if,

3. iii. The scale returns will be constantly if, and only if,,

3 Methods

The present research adopted a cross section approach when analyzing the Carnival data of Rio de Janeiro for the year 2017, which were collected on the site of the Independent League of Samba Schools (LIESA). The selected DMUs were the 12 samba schools that paraded the aforementioned year in the special group: Paraíso do Tuiuti, Grande Rio, Imperatriz, Vila-Isabel, Salgueiro, Beija-Flor, União da Ilha, São Clemente, Mocidade, Unidos da Tijuca, Portela e Mangueira.

The input utilized was the components number (COMP) that each school decided to led to the parade, which, as a rule, should be between 2,500 and 3,500 (LIESA, 2017b). This indicator it is important because it reflects the amount of labor employed, an obvious input from any production process (Cook, Tone & Zhu, 2014).

Outputs are called process results (Charnes et al., 2013). Thus, it is considered as outputs the sums of the valid notes obtained by the samba schools in each one of the questions judged during the parade (Chart 1). In 2017 nine items were scored, each one by four jurors, totaling 36 judgments. Each jury awarded a score ranging from nine to ten points, which could be fractioned in tenths. In the result calculation, for each item, only the three highest grades were accepted as valid, that is, the lowest grade was discarded (LIESA, 2017b). Mocidade and Portela, the champion ones, obtained exactly 269.9 points in the sum of the valid scores, according to all the judgment criteria.

Chart 1
Criteria and Judged Factors

Source: LIESA (2017a)

Therefore, for each of the 12 DMUs, an input and nine outputs were considered, whose statistics are shown in Table 1. However, it is recommended that the number of DMUs it is equal to at least three times the number of inputs and outputs (Banker et al., 1989), since the large number of inputs and outputs compared to the DMUs number, decreases the DEA discriminatory power (Cook, Tone & Zhu, 2014). Since it was not possible to increase the DMUs number, once that only 12 samba schools belong to the Rio de Janeiro Carnival special group, and in order to meet that recommendation, it was decided to reduce the number of outputs to three.

Table 1
Variables descriptive statistics

In order to select the three most relevant outputs it was used the Principal Component Analysis (PCA) technique, which aims to explain a random vector: variance and covariance structure, composed of random p-variables, by constructing some linear combinations of k (k < p) original variables - the main components - not correlated with each other (Hongyu, Sandanielo & Oliveira Júnior, 2016). Thus, from the original variables set, initially correlated, it is possible obtain a substantially smaller set of uncorrelated variables that contain most of the information.

Table 2 presents the applying PCA results to the covariance matrix of the valid scores questions sum. The total variance, given by the eigenvectors sum of the covariance matrix, it is equals to 9.001. The first major component (PC1), which contains the most relevant information from the original data, explains 62.72% of the total variance.

Table 2
Principal component analysis

The PC1, given by (5), can be understood as a vector of items weights linked to the samba schools overall performance. Because they have the highest coefficients, the valid scores sum of the EV, HAR and A&A are the three most important variables (Mingoti, 2004) of PC1. The eigenvectors, which are composed of coefficients that correspond to each variable, are used to calculate the scores of the major components. The coefficients indicate the relative weight of each variable in the component (Mingoti, 2004). Therefore, these were considered as outputs.

(5)

In the data analysis, output-oriented DEA models were adopted. Initially, under the assumption of constant returns to scale, the global technical efficiency scores were calculated. With this assumption relaxed, the pure technical efficiency scores were computed. Of the reason between these measures, the scale efficiency scale was ascertained. Finally, the returns to scale were classified as constant, increasing or decreasing.

4 Results and Discussion

Table 3 presents the efficiency scores and scale returns to the 12 special group samba schools of Rio de Janeiro Carnival in 2017. It is possible to perceive that, under the assumption of constant scale returns, only the Mocidade obtained the maximum global technical efficiency. In this criterion the average level of inefficiency was 0.08 (1-0,920), which means that schools could, on average, increase valid grades in Evolution, Harmony and Allegories and Adornments by up to 8%, without increasing the number of components.

The global technical inefficiency can be caused by pure technical inefficiency and / or scale inefficiency. In order to evaluate the scale influence, the assumption of constant returns was relaxed. Once this was done, pure technical efficiency scores were obtained, with a mean of 0.996. Therefore, it can be stated that, on average, the overall technical inefficiency of 8% is due less to the pure technical inefficiency, which was 0.4% (1 - 0.996), and more to the scale inefficiency, whose average was 7.6% (1- 0.924). It is also verified that in addition to Mocidade, the Grande Rio, Beija-Flor, Portela, Salgueiro and Mangueira schools presented pure technical efficiency.

It is still shown in Table 3 that Mocidade, Imperatriz and São Clemente had constant scale returns, which indicates that these schools operated at an optimal scale. However, only the first one was technically efficient; the other two, to be efficient must, using the same components number, increase the valid notes in the criteria: Evolution, Harmony and Allegories & Adornments. The other schools presented decreasing scale returns, that is, they operated above the optimal scale. In the case of Grande Rio, Beija-Flor, Portela, Salgueiro and Mangueira, which presented pure technical efficiency, the current valid notes should be maintained in the requirements of Evolution, Harmony and Allegories & Adornments and reduced the number of components. In order to correct the problems of the Paraíso do Tuiuti, Vila Isabel, União da Ilha and Unidos da Tijuca schools, which operated above the optimal scale and were still technically inefficient, respectively, the components number should be reduced and the grades, should be increased in the following criteria: Evolution, Harmony and Allegories & Adornments. In order to become efficient, inefficient schools should target the input and outputs of the Mocidade, the efficiency benchmark school: 3,000 components and 30.0 points sum of valid grades in the Evolution, Harmony and Allegories & Adornments requirements.

Table 3
Escores de eficiência e retornos de escala das escolas de samba do Rio de Janeiro

5 Conclusion

The present research showed that the DEA technique can be useful to measure the technical and scale efficiencies of samba schools. It was verified that only one samba school inserted on the special group of Rio de Janeiro Carnival in 2017, Mocidade, had the global technical efficiency. It was also found that, although Grande Rio, Beija-Flor, Portela, Salgueiro and Mangueira, as well as Mocidade, presented pure technical efficiency, the average global technical inefficiency was 8%.

This result was strongly influenced by the low scale efficiency of the samba schools, which mostly, presented decreasing scale returns, which shows that they operated above the optimal scale. Therefore, in order to become efficient, samba schools must reduce the components number and / or increase its performance in the evaluating criteria, with reference to Mocidade, the samba school benchmark.

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Author notes