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Input allocation with the Ellipsoidal Frontier Model

Abstract

This work aims at complementing the development of the EFM (Ellipsoidal Frontier Model) proposed by Milioni et al. (2011a). EFM is a parametric input allocation model of constant sum that uses DEA (Data Envelopment Analysis) concepts and ensures a solution such that all DMUs (Decision Making Units) are strongly CCR (Constant Returns to Scale) efficient. The degrees of freedom obtained with the possibility of assigning different values to the ellipsoidal eccentricities bring flexibility to the model and raises the interest in evaluating the best distribution among the many that can be generated. We propose two analyses named as local and global. In the first one, we aim at finding a solution that assigns the smallest possible input value to a specified DMU. In the second, we look for a solution that assures the lowest data variability.

Date Envelopment Analysis; DEA-Parametric Models; Ellipsoidal Frontier Model


Input allocation with the Ellipsoidal Frontier Model

Luciene Bianca Alves; Armando Zeferino Milioni* * Corresponding author ; Nei Yoshihiro Soma

ITA – Instituto Tecnológico de Aeronáutica, DCTA – ITA, 12228-200 São José dos Campos, SP, Brasil. E-mails: bianca@ita.br; milioni@ita.br; soma@ita.br

ABSTRACT

This work aims at complementing the development of the EFM (Ellipsoidal Frontier Model) proposed by Milioni et al. (2011a). EFM is a parametric input allocation model of constant sum that uses DEA (Data Envelopment Analysis) concepts and ensures a solution such that all DMUs (Decision Making Units) are strongly CCR (Constant Returns to Scale) efficient. The degrees of freedom obtained with the possibility of assigning different values to the ellipsoidal eccentricities bring flexibility to the model and raises the interest in evaluating the best distribution among the many that can be generated. We propose two analyses named as local and global. In the first one, we aim at finding a solution that assigns the smallest possible input value to a specified DMU. In the second, we look for a solution that assures the lowest data variability.

Keywords: Date Envelopment Analysis, DEA-Parametric Models, Ellipsoidal Frontier Model.

1 INTRODUCTION

DEA (Data Envelopment Models) Models of Constant Sum refer to problems in which a new (or already existing) input or output variable has to be assigned (or reassigned) to a group of DMUs (Decision Making Units) such that the total sum of this new (or existing) variable across all DMUs has to remain constant.

Such models may be parametric or nonparametric. Examples of nonparametric DEA Models of Constant Sum are Cook & Kress (1999), Wei at al. (2010), Beasley (2003), Lins et al. (2003) and Gomes & Lins (2008). Parametric DEA models were first proposed by Kozyreff & Milioni (2004). Further publications on Parametric DEA Models are Avellar (2004, 2010), Avellar et al. (2005, 2007 and 2010), Milioni et al. (2011a and 2011b), Silva & Milioni (2012), Guedes (2007) and Guedes et al. (2012).

Parametric DEA models are characterized by the assumption of the geometrical shape or locus of points of the production frontier. This may be considered a strong assumption, but parametric DEA models are also the only ones for which it is possible to prove a desirable property known as coherent property, within the context of sensitivity analysis (see, for instance, Milioni et al., 2011a and Guedes et al., 2012).

This work aims to complement the development of an input allocation model of constant sum, parametric, adapted to the characteristics of Data Envelopment Analysis (DEA) and that guarantees a strongly efficient solution to all the Decision Making Units (DMUs), in models with constant return of scale. This is the Ellipsoidal Frontier Model (EFM), based on an efficiency frontier with ellipsoidal shape that is capable of distributing inputs taking into account the problem.

The EFM provides several possibilities for different solutions due to its flexibility derived from the degrees of freedom (eccentricities) of the model. For this reason, it became interesting to guide the decision maker to select the best solution, which is the purpose of this work. Therefore, two different types of analyzes are proposed, classified as: Local (LA) and Global (GA). The first one (LA) has the objective of finding the lowest possible input value associated to a specific DMU. The second (GA) searches for a solution with the lowest total variability of the data set: input and output values.

2 EFM MODEL

According to Avellar (2010) and Milioni et al. (2011a), the Ellipsoidal Frontier Model (EFM) is a parametric model of constant sum such that the efficiency frontier has an ellipsoidal shape. It assures several solutions that are CCR strongly efficient for all DMUs, by distributing (redistributing) a new (already existing) input variable among all DMUs, taking into account all other input and output variables involved in the problem.

Model construction is presented in three different cases:

(i) two outputs and a single input,

(ii) s Outputs and a single input and

(iii) s Outputs and m + 1 Inputs.

According to Avellar (2010): "Consider yr j (> 0) the measured value of output r (r = 1, ... , s) for DMU j ( j = 1, ... , n); F (> 0) the total fixed input (or cost) to be distributed to all DMUs,i.e., F = fj , where fj is the input value to be allocated to each DMU j."

Thus, the coordinate values and the values of the new Input (fj) to be distributed for each case will be:

Case (i):

The value assigned to e refers to the eccentricity of the ellipse. Thus, the model allows a different solution (frontier) for each different value of e. It is noteworthy that an ellipse with zero eccentricity is a sphere (thus, spherical frontier is a particular case of this model).

Case (ii):

Case (iii):

In order that we assure that fj > 0, we must have:

As it is shown in Avellar (2010), EFM solution can be obtained with the use of a Linear Programming Problem (LPP) presented in the following set of equations:

According to the author, their properties and characteristics make EFM a Cooperative, Competitive and Flexible model. Namely:

• Frontier Homogeneity property: replaces the original piece-wise linear DEA frontier by a smooth frontier.

• DEA control weights Effective solutions generating property (flexibility model): gives the decider the possibility of obtaining a weight distribution for each combination of eccentricity, strongly efficient solutions CCR (characteristic competitive);

• Coherent Distribution Ownership (cooperation characteristics): Inputs to distribute special consistently in the presence of errors;

• Input distribution characteristic considering input and output values existing in the current problem.

• No DMU has to increase input value to become efficient.

Further details of the model can be found in Avellar (2010) e Milioni et al. (2011a).

3 METHOD

EFM model is in essence flexible due to its many degrees of freedom. By using different eccentricities values one can generate many different solutions. Moreover, in the illustrative example presented in Milioni et al. (2011a) the authors show that by choosing different values to the eccentricities one can gain control on the weights assigned to each input and output variable in the DEA solution.

In their example, they propose two solutions, analyze the characteristics of each one of them and then claim:

"So, for both strongly efficient solutions, the decision maker can choose which weight distribution is more adequate to his or her reality (...)".

Further ahead they point out that:

"The kind of procedure could provide a guideline for how one chooses specific parameters based on the ellipsoidal shape of the frontier, an issue so important and dense that we intend to address it in another paper".

This is precisely our goal in this paper. We propose two different analyses denominated as Local

(LA) and Global (GA):

(i) in the first one (LA), by varying the values assigned to the eccentricities (0 < e < 1), we investigate the solution that achieves the smallest possible input value to a specific and previously chosen DMU;

(ii) in the second (GA), we suppose that there is a prior solution (i.e., the input variable with constant sum is already somehow distributed among all DMUs) and seek the values of eccentricities for which one has the smallest total variability, considering the current existing distribution.

For this variability, it is used the Euclidean Distance as a metric. We mean the total sum of squares of the differences between prior (currently existing) and new (provided by the solution) input variable for each DMU.

4 EXAMPLE

We use the same real data presented in Gomes & Lins (2008). In their case, DMUs are countries and the problem is to fairly distribute a single input, which is emission of CO2 (carbon equivalent ton3) considering three outputs: population (in million), energy (million BTU) and Gross Domestic Product (GDP, in billions of dollars).

Since this is a three outputs problem, the model formulation has two degrees of freedom, i.e., there are two eccentricities to be chosen. The 64 countries regarded as DMUs are:

(01) Argentina (02) Australia (03) Austria

(04) Belgium

(05) Bolivia (06) Brazil (07) Bulgaria

(08) Canada

(09) Chile (10) China (11) Costa Rica

(12) Croatia

(13) Czech Republic (14) Denmark (15) Egypt (16) El Salvador (17) Estonia (18) Finland (19) France (20) Germany (21) Greece (22) Guatemala (23) Honduras (24) Indonesia (25) Ireland (26) Israel (27) Italy (28) Japan (29) Kazakhstan (30) Latvia (31) Lithuania (32) Luxembourg (33) Malaysia (34) Maldives (35) Malta (36) Mexico (37) Netherlands (38) New Zealand (39) Nicaragua (40) Norway (41) Panama (42) Paraguay (43) Peru (44) Philippines (45) Poland (46) Portugal (47) Republic of Korea (48) Romania (49) Russian Federation (50) Seychelles (51) Slovakia (52) Slovenia (53) Spain (54) Sweden (55) Switzerland (56) Thailand (57) Turkmenistan (58) Ukraine (59) United Kingdom (60) United States (61) Uruguay (62) Uzbekistan (63) Vietnam (64) Zambia

Input and output real data values for each country are presented in Table 1.

In the Local Analysis (LA) we seek the values of eccentricities that assign the lowest possible input value for a single chosen DMU, still assuring the existence of a solution in which all DMUs are strongly CCR efficient.

Table 2 shows the results obtained for the analysis carried out for each DMU.

If we look at the result for DMU 1 presented in Tables 1 and 2 we conclude that Argentina's current CO2 emission, which 34.85 carbon equivalent ton3 has to climb to a minimum of 41.81 such that there will still be a solution for which global total CO2 emission remains constant and all DMUs (countries) are strong CCR efficient.

On the global analysis we investigated in all distributions (varying eccentricity values), the less variability on the data conjunct. The results are presented in Table 3:

As we can see from Table 3, the pair of eccentricities that minimizes total variability is the pair (0.8, 0.0).

For this study, all Local Analysis (Table 2) of the EFM model (Equation 2) were conducted using MSExcel. The Global Analysis (Table 3) was implemented on Matlab.

5 RESULTS AND CONCLUSION

In the LA, the largest differences between current and minimum possible values were observed for DMUs 10 (China), 28 (Japan), 49 (Russian Federation) and 60 (United States). For these Countries, the differences are, 503.57, 203.64, 351.60 and 602.23 ton3 CO2 found in eccentricities (e1, e2 ) = (0.99, 0.99), (0.00, 0.00), (0.90, 0.99) and (0.50, 0.99), respectively. Among these countries, only DMU 28 (Japan) has a minimum value that is greater than its current value of emission.

Regarding Global Analysis (GA), it was possible to realize once more the merits of the EFM model to provide various distributions in which the decision maker can choose how to redistribute their data with strongly efficient solutions. The values in Table 3 illustrate these possibilities ranging from 1520 ton3 CO2 (minimum) – found in the eccentricities values (e1, e2 ) = (0.80, 0.00) – which represents 28% of the total F value, to 5346 ton3 CO2 (maximum). The average was 1990 ton3 CO2.

While observing both analysis results (LA and GA) the minimum value found on GA distribution do not reveal any of the cases of LA distribution. However, the merit of this study is attained by indication of distributions with eccentricities values that occurs the solution of analyses LA and GA, according to each problem. The guidance while choosing one solution from many possibilities is now feasible, since it requires minimum data treatment. This is a relevant fact mainly due to the nature of input associated with many resources quantity.

Received March 4, 2013

Accepted October 21, 2013

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  • *
    Corresponding author
  • Publication Dates

    • Publication in this collection
      08 Apr 2014
    • Date of issue
      Apr 2014

    History

    • Accepted
      21 Oct 2013
    • Received
      04 Mar 2013
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