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Procedía Computer Science 55 (2015) 341 - 349

Information Technology and Quantitative Management (ITQM 2015)

Productivity Analysis and Variable Returns of Scale: DEA Efficiency

Frontier Interpretation

Juliana Benicioa, Joäo Carlos Soares de Mellob 1

a Faculdade Cenecista de Rio Bonito - FACERB, Rio Bonito, Rio de Janeiro, Brazil _b Faculdade Federal Flumeinense, Niterói, Rio de Janeiro, Brazil_

bstract

The main objective of this paper is to analyze DMUs efficiency from the perspective of variable returns to scale. Thus, a case study is proposed, where the efficiencies of DMUs suffer variation according to the methods used in the analysis. The classic models of DEA, CCR and BCC, and a new model proposed by the authors, will have their results compared to classical foundations of the economy. The case study will examine the efficiency of administrative units selected of Undergraduate Higher Education.

© 2015 Publishedby ElsevierB.V. This isanopen access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ITQM 2015 "Keywords: Variable scale returns; DEA; Higher Education"

1. Introduction

The efficiency analysis in economic science has its starting point at the concept of Pareto-Koopmans [14-15] efficiency that says that a production unit is fully efficient, if and only if, you cannot improve any input and output without reduce any other input or output [7]. Thus, one unit is considered inefficient, in Koopmans Pareto definition, if it can produce the same output reducing at least one of the inputs; or if you can use the same inputs to produce more outputs.

In this sense, the construction of a production line, also referred to as efficient frontier, aims to define a limit where the production more efficient will be located on this boundary, and the less efficient will be situated in the below border area, known as a set of possible production [22]. The shape of the efficient frontier defines the technology used in production analyzed. [9] points out that the efficiency measure must incorporate a set of production possibilities to maximize the ratio of output / input.

In the original paper [8], Charnes, Cooper and Rhodes used Farrell efficiency concept in linear programming model known as DEA CCR. The DEA (Data Envelopment Analysis) is able to evaluate the level of efficiency of production units (DMUs -Decision Making Units) that perform the same activity, as the use of its resources. The measure of efficiency is obtained because of the weighted sum of outputs by the weighted sum of inputs. This model allows to analyze performance DMU to produce multiple outputs from multiple inputs through [10] compared to the other DMU observed.

The result of the original model CCR is to build an efficient production frontier, so that the DMUs that they have the best ratio "product / input" are considered more efficient and will be located on this border, and the less efficient will be situated in below the border region, known as envelope [16].

* Corresponding author. Tel.: 55 21 99772 9119. E-mail address: juliana.benicio@hotmail.com.

1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of ITQM 2015

doi: 10.1016/j.procs.2015.07.059

As analytical progress, efficiency studies began to incorporate the concept of scale in their analysis. That is, by increasing the volume of production inputs, DMUs considered efficient may have gain, loss or constancy of productivity. As argued by [9] is a mistake to consider only the measure of increased productivity (better relationship between inputs and outputs) as a measure of efficiency. This is because, although they have different scale production, some DMUs may have the same efficiency. That is, the efficient frontier begins to recognize the possibility of improvement or deterioration of the DMU productivity by altering the amount produced [21].

[2] BCC DEA developed a model that incorporates the notion returns to scale [3]. The main objective of the model is to calculate efficiencies that took into account the efficiency of scale of observed DMUs. That is, a DMU which is in scale production where the return is increasing or decreasing, need not necessarily present a virtual output interface/ virtual input as the DMU more efficient. Thus, the condition of proportionality of the CCR is not guaranteed in this model.

This article aims to analyze different models in the definition of efficient DMUs based in a case study, from the concept of variable returns to scale. From real data, the results of DEA CCR and BCC models will be contrasted with the DMUs productivity analysis and with a new model concave frontier model. The proposed case study will be based on analysis of the efficiency of the administrative sector of different higher education units in managerial their enrolled students.

2. A revision of variable returns of scale: an economic perspective

The average productivity of a factor (PMe) is calculated as the quotient of the output (y) and the amount used of the input in question (x). Algebraically:

PMe = f (1)

The marginal productivity of a factor (PMg) is calculated as the quotient between the variation in the amount of output (y) and the variation in the amount used of the input in question (x), namely:

^ = £ (2)

As such, it defines that:

If the variation of inputs is the same as output variation, the production has constant returns to scale. That is, the marginal productivity is equal to 1; or even average productivity remained constant with increasing production scale.

If the variation of inputs is smallest as the output variation, the production has increasing returns to scale. That is, the marginal productivity is greater than 1; or the average productivity increases with increasing production scale.

If the variation of inputs is the largest of the outputs, the production has decreasing returns to scale. That is, the marginal productivity is less than 1; or the average productivity decreases with increasing production scale.

Figure 1 shows the scale returns from possible change in the amount of inputs used, where the different behaviors of the efficient frontier curve production can be verified [20].

Fig. 1. (a) Constant returns of scale; (b) Increasing returns of scale; (c) Decreasing returns of sca[4] highlight the following reasons that contribute to the

technology shows increasing returns to scale:

Existence of indivisibilities techniques or fixed costs, which are diluted with increasing production scale (eg costs of mobile telephone network, product design, music or films).

The division of labor and specialization can enable efficiency gains (eg production line). Inventory needs usually increase less than the scale (eg hypermarkets).

Geometric relationships: for example, duplicating the walls of a warehouse, quadruples the available area. [4] described also reasons that contribute to the existence of decreasing returns to scale:

Excess of work division and loss of overview of the company and its objectives (fruit of great organizational complexity); Supervisory difficulties/management: As the scale of production increases, the supervisors hierarchy tends to increase their efficiency and decreasing (also the result of major organizational complexity); Product limitation (extractive industries).

3. Studied Models

3.1. DEA CCR Model

The DEA CCR model introduced by [8] assumes constant returns to scale, meaning any change in inputs should produce a proportional change in output. The model uses the mathematical programming optimization method to determine the efficiency of a DMU (Decision Making Units) dividing the weighted sum of outputs (virtual output) by the weighted sum of inputs (virtual input), generalizing thus definition of [9] presented above.

The classic model CCR [8] with input orientation (ie minimizing the input and maintains the level of production), considers DMUs analysis unit to be compared according to their efficiency based on the following model (3), called model of Multipliers:

Max Eff0 =

lui= 1 ^iXlQ

Subject to

1, k = 1.....n (3)

U:e Vj > 0 V j, i

Such that, uj e vi are the weights of outputs and inputs, respectively, xik, yjk are the inputs ie outputs j of DMUK and xi0, yj0 are the inputs I and outputs j of the observed DMU.

This model can be defined as a fractional programming problem that can be transformed into a linear programming (LPP), where the denominator of the objective function must necessarily be equal to a constant, usually one [6]. The model (3) can be instructed to output, and thus, its objective function seek to maximize output, holding constant the level of inputs.

3.2. DEA BCC Model

The BCC model, introduced by Banker, Charnes and Cooper [3] introduced a change in the formulation of CCR in order to analyze the variable returns to scale in DEA. That is, the BCC model wanted to give account to interpret the fact that, at different scales, the DMUs could have different productivities and still be considered efficient. The objective of this analysis proposed by the BCC model is to take into account the fact that in different situations the conditions that influence the productivity of production are also diverse. As mentioned in the introduction, different production technologies have their productivities influenced by the scale at which the DMUs are operating [2]. When the production frontier exhibits constant returns to scale, efficient DMUs have the same productivity; however, when the production line has variable returns efficient DMU need not have the same productivity [13].

The formulation (4) of the BCC introduces a restriction on the PSS of the original model CCR. The frontier of this convex set is restricted by Yj=i= 1, making the area BCC production possibilities less that the CCR. Consequently, any projection inefficient DMU in the efficient hyperplane, may be represented by an equation of the line segment of the linear combination border, where the sum of the contributions of efficient DMUs (A_j) must result in 1 [3]. Thus, the BCC efficiency is less than or equal to the CCR efficiency.

In formulating the model are introduced the variables v * (scale factor in the output orientation) and u * (scale factor in the input orientation) to the objective function and constraints. These variables, according to [2] indicate the scale return the DMU.

In the model-driven inputs, when positive, indicating increasing returns to scale; when negative, indicate diminishing returns; if they are null, the situation is of constant returns in oriented model outputs, when positive, indicate decreasing returns to scale; when negative, indicating increasing returns to scale; if they are null, the situation is of constant returns to scale. Follows the model of multipliers DEA BCC with input orientation:

JÎ c-f-f %Sj=iujyjo , Max Eff0 = —--u"

lui = 1 ^iXlQ

subject to

%1, k = 1.....n (4)

T'i=1ViXik

Uje Vj > 0 V j, i

Such that, uj e vi are the weights of outputs and inputs, respectively xik, yjk are the inputs ie outputs j of DMUK and xi0, yj0 are the inputs i and outputs j of the observed DMU; u* is the scale factor.

3.3. New Model: Concave Frontier (FCon)

This nonparametric algorithm is designed to ensure that the efficient frontier shows increasing returns. The shape of the efficient frontier generated in this model is concave. The concave curvature ensures the model that increases returns are verified along the border.

To guarantee the increasing returns of scale, the efficient DMUs must present increasing CCR efficiency. Consider the following conventions:

The DMUs are ordered by the numbers of utilized inputs;

EfccR o is the CCR Efficiency of the observed DMU;

EfccR q is the CCR Efficiency of ALL efficient DMU that previous the observed DMU;

The DMU1 (the DMU with the lower quantity of inputs) consider the EfCCR q equal to zero.

Algorithm (5):

Step 1: CALCULATE THE CCR EFFICIENCY of the analysed DMUs.

Step 2: CALCULATE A^, such that A^ = (EfCCR 0 - EfCCR q), for all q. (5)

Step 3: DEFINE INEFFICIENT THE DMUs that: ef

&0q< 0, for any q.

Step 4: DEFINE EFFICIENT THE DMUs that:

Ae0fa> 0, for all q.

Consider de result of Fcon the reason between the EfCCR 0 and the EfCCR k, such that k is the CCR Efficiency of most efficient DMU that previous the observed DMU.

4. Efficiency of the Administrative Sector in the Higher Education: a Study of Case

Educational institutions occupy a central role in the economies of the world. His ability to transform society makes this important activity case study for different areas of knowledge. The study of any educational institution efficiency should be analyzed very carefully. First, it points out that in this activity, assessing the quality of service is of utmost importance, since in most cases, the reason for the existence of these institutions is the ability to train students effectively. Second, one must consider that the complexity of this activity is related to the long time required to consolidate a strategic policy of education; consequently, the judgment of the objectives achieved must be matured over an extended period and still is a subjective analysis. However, the complexity of the analysis of this service may be reduced when proposing the analysis of the efficiency of the administrative sector of capabilities to manage the academic routine. [5]

It lies in the bibliography specialized studies that seek to measure the effectiveness of educational institutions according to two main approaches. The first search the assessment of quality of education, focusing on the evaluation of student

development [11, 17, 18, 19]. In this group, inputs are related to years of study and resources of the student, and outputs are related to the achievements of these students after training, such as number of employees students, number of students entering college (in the case of schools for assessment of high school), number of new students in graduate (in the case of evaluation of universities). The second group seeking an evaluation of the quality of education with a focus on consolidated structure of the different universities [1, 12]. In this case, is taken into consideration as outputs to be maximized: amount of research performed by the institution, publication and titration of teachers, number of students.

In this case will be studied 21 colleges of the same school system in Brazil; the input is the number of employees in the administrative sector of these DMUs in 2014 and, finally, the output will be the total number of students enrolled in 2014.

4.1. CCR Results

Table 1 shows the results of the CRC efficiencies.

Table 1 Observed DMUs and their inputs and outputs, Average Productivity and Efficiency CCR..

DMU Unit Administrative Number Number of Students Pme CCR

1 SALVADOR 1 1 1 0.0202

2 RONDONOPOLIS 1 0 0.00 0.002

3 NOVA PETROPOLIS 9 140 15.56 0.3151

4 VILA VELHA 13 251 19.31 0.3911

5 ITABORAI 19 379 19.95 0.4041

6 ILHA GOVERNADOR 19 268 14.11 0.2857

7 CAPIVARI 19 395 20.79 0.4212

8 ITABORAI 20 536 26.80 0.5429

9 RIO BONITO 22 130 5.91 0.1197

10 RIO DAS OSTRAS 27 928 34.37 0.6963

11 ITAJAI 27 546 20.22 0.4097

12 CAMPO LARGO 28 606 21.64 0.4384

13 BENTO GONCALVES 28 1,382 49.36 1

14 FARROUPILHA 28 611 21.82 0.4421

15 SETE LAGOAS 29 250 8.62 0.1746

16 GRAVATAI 41 1,135 27.68 0.5608

17 JOINVILLE 53 1,371 25.87 0.524

18 SANTO ANGELO 61 2,214 36.30 0.7353

19 UNAI 94 1,158 12.32 0.2495

20 VARGINHA 124 1,790 14.44 0.2924

21 OSORIO 125 3,038 24.30 0.4924

The results shown in Table 1, it is observed that the DMU 13 Bento Gonfalves is the most efficient, and has the highest average productivity.

4.2. BCC Results

Table 2 shows the results of BCC efficiencies with input and output orientation.

Table 2. DMUs analyzed and their inputs and outputs, Average Productivity and Efficiency BCC -oriented input and output._

__________,, . Administrative „„„ . „„„

DMU Unit , , BCC i BCC o

Number Number of Students Pme _ _

1 RONDONOPOLIS 1 - 1 0.1

2 SALVADOR 1 1 1 1 1

10 11 12

NOVA PETROPOLIS VILA VELHA ITABORAI RIO DE JANEIRO CAPIVARI ITABORAI RIO DE JANEIRO RIO DE JANEIRO

ITAJAI CAMPO LARGO

9 13 19 19

20 22 27

140 251 379 268 395 536 130 928 546 606

15.56 19.31 19.95 14.11

26.80 5.91 34.37 20.22 21.64

0.413 0.4529 0.4415 0.3273 0.458 0.5729 0.16 0.7082 0.4316 0.4581

0.3413 0.4082 0.4112 0.2907 0.4285 0.5509 0.1209 0.6972 0.4102 0.4384

13 BENTO GONCALVES 28 1,382 49.36 1 1

FARROUPILHA SETE LAGOAS GRAVATAI JOINVILLE

28 29 41 53

611 250 1,135 1,371

21.82 0.4616 0.4421

8.62 0.2023 0.1776

27.68 0.5651 0.6638

25.87 0.5242 0.6813

18 SANTO ANGELO 61 2,214 36.30 1 1

19 UNAI 94 1,158 12.32 0.2512 0.4388

20 VARGINHA 124 1,790 14.44 0.3563 0.5917

21 OSORIO 125 3,038 24.30 1 1

From Table 2, it is concluded that the DMU 1, 2, 13, 18 and 21 are efficient. By separating analytically, these efficient DMUs of inefficient, it is observed that:

• From 1 to DMU DMU13 average productivity grows; which signals RETURNS SCALE GROWING.

• The DMU 13 to DMU21 average productivity decreases; which signals RETURNS SCALE DECREASING. However, when considering the DMUs 1 and 2 efficient by default, it is concluded that the truly efficient DMUs (13, 18,

21) have only decreasing returns to scale.

4.3. FConc Results

Table 3 presents the results of the efficiencies of the FConc.

Table 3. Observed DMUs and their inputs and outputs, Average Productivity and Efficiency FConc.

DMU Unit Administrative Number Number of Students Pme Fconc

1 SALVADOR 1 1 1 1

2 RONDONOPOLIS 1 - 0.1

3 NOVA PETROPOLIS 9 140 15.56 1

4 VILA VELHA 13 251 19.31 1

5 ITABORAI 19 379 19.95 1

6 RIO DE JANEIRO 19 268 14.11 0.71

7 CAPIVARI 19 395 20.79 1

8 ITABORAI 20 536 26.80 1

9 RIO DE JANEIRO 22 130 5.91 0.22

10 RIO DE JANEIRO 27 928 34.37 1

11 ITAJAI 27 546 20.22 0.59

12 CAMPO LARGO 28 606 21.64 0.63

13 BENTO GONCALVES 28 1,382 49.36 1

14 FARROUPILHA 28 611 21.82 0.44

15 SETE LAGOAS 29 250 8.62 0.17

16 GRAVATAI 41 1,135 27.68 0.56

17 JOINVILLE 53 1,371 25.87 0.52

18 SANTO ANGELO 61 2,214 36.30 0.74

19 UNAI 94 1,158 12.32 0.25

20 VARGINHA 124 1,790 14.44 0.29

21 OSORIO 125 3,038 24.30 0.49

Unlike the results reported by others models with variable scale returns, Table 3 shows that the model FConc presents the efficient DMUs: 1, 3, 4, 5, 7, 8, 10 and 13.

As is presupposed by the model, this border has not only decreasing returns to scale. Observing ONLY efficient DMUs can conclude that, by increasing the volume of inputs, the productivity of these DMUs always grows.

4.4. Comparative Analisys of Results

The frontiers generated by different presented models can be verified in Figure 2.

3000 2500 (c) /(a)

21 (b)

»'Í8

1500 1000 500 o1 •

13, • • •

// f 10 // / • // w •

// X 8 / /5 // • * * // 4 • U-3

2 20 40 60 SO 100 120 140

Fig. 2. Efficient Frontiers of (a) DEA CCR; (b) DEA BCC; (c) Fconc

Following the basic presupposition of proportionality, the DEA CCR model just introduced one efficient DMU because all others have not achieved the productivity of DMU more efficient.

In models that assume variable returns to scale, it can be seen divergence in results.

The model of the concave border ensures no decreasing returns to scale. However, the DEA BCC has, theoretically, the possibility of a boundary with increasing, decreasing and constant returns of scale. Thus, the BCC border should contain the concave boundary, since by definition returns the scale variables are checked.

In the case study, it is observed that the FConc border, in fact, has increasing returns to scale. Efficient DMU 1, 3, 4, 5, 7, 8, 10 and 13 have increased average productivity (1; 15,56; 19,31; 19,95; 20,79; 26,28 and 34,37 respectively). However, except in the case of DMU13, these DMUs were neglected on the border of BCC.

The efficient DMUs with decreasing returns to scale, according to the BCC, were DMUs 18 and 21. These DMUs were neglected by FConc model due to its basic assumption does not consider decreasing returns to scale.

5. Conclusions

It follows from the results of the case of study presented, there are mistakes in the theoretical analysis proposed by the DEA BCC model. Its boundary, when it no longer identify efficient DMUs with increasing returns to scale, is unable to analyze what the model theoretically proposed.

Already, the results of the case of study, the concave border, was able to identify the DMUs with no increasing returns to scale in order to compare their results with the calculated productivities.

As a future study, we suggest expanding the application of FConc, in order to compare with the results of BCC model to consolidate or not this fragility of the BCC model.

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