FREE SOFTWARE FOR DECISION ANALYSIS

A Software Package for Data Envelopment Models

Lidia Angulo Meza

Instituto de Ciências e Tecnologia, Universidade Veiga de Almeida, Rua Ibituruna 108, Maracanã,Rio de Janeiro, Brasil

Luiz Biondi Neto

Depto. de Engenharia Elétrica e de Telecomunicações – Universidade do Estado do Rio de Janeiro, Rua São Francisco

Xavier, 524, Bl. A, Sala 5036, Maracanã, Rio de Janeiro, Brasil

João Carlos Correia Baptista Soares de Mello

Depto. de Engenharia de Produção – Universidade Federal Fluminense, Rua Passo da Pátria, 156, São Domingos, Niterói,

Brasil

Eliane Gonçalves Gomes

Programa de Engenharia de Produção – Universidade Federal do Rio de Janeiro & Embrapa Monitoramento por Satélite,

Av. Dr. Júlio Soares de Arruda, 803, Parque São Quirino, Campinas, SP, Brasil

Pedro Henrique Gouvêa Coelho

Depto. de Engenharia Elétrica e de Telecomunicações – Universidade do Estado do Rio de Janeiro, Rua São Francisco

Xavier, 524, Bl. A, Sala 5036, Maracanã, Rio de Janeiro, Brasil

Keywords: Software, Data Envelopment Analysis.

Abstract: Data Envelopment Analysis is based on linear programming problems (LPP) in order to find the efficiency

of Decision Making Units (DMUs). This process can be computationally intense, as a LPP has to be run

for each unit. Besides, a typical DEA LPP has a large number of redundant constraints concerning the

inefficient DMUs. That results in degenerate LPPs and in some cases, multiple efficient solutions. The

developed work intends to to fill out a gap in current DEA software packages i.e. the lack of a piece of

software capable of producing full results in classic DEA models as well as the capability of using more

advanced DEA models. The software package interface as well as the models and solution algorithms were

implemented in Delphi. Both basic and advanced DEA models are allowed in the package. Besides the main

module that includes the DEA models, there is an additional module containing some models for decision

support such as the multicriteria model called Analytic Hierarchic Process (AHP). The developed piece of

software was coined as FSDA – Free Software for Decision Analysis.

1 INTRODUCTION

Data Envelopment Analysis (DEA) is an approach to

evaluate efficiency that uses Linear Programming

Problems (LPP) whose results assess the

performance of Decision Making Units (DMUs).

In recent years there has been increasi

ng interest

for DEA and its corresponding LPPs. Several real

case applications have led to a need for new

developments in the classic models, the CCR

(Charnes et. al. 1978) and BCC (Banker et al. 1984

models, in order to include new situations. Thus

several researchers have been aware to the results

yielded by the models in terms of efficiency indexes,

benchmarks and targets.

On the other hand, a LPP has to be solved for

each

DMU. So the task for efficiency evaluation can

207

Angulo Meza L., Biondi Neto L., Carlos Correia Baptista Soares de Mello J., Gonçalves Gomes E. and Henrique Gouvêa Coelho P. (2005).

FREE SOFTWARE FOR DECISION ANALYSIS - A Software Package for Data Envelopment Models.

In Proceedings of the Seventh International Conference on Enterprise Information Systems, pages 207-212

DOI: 10.5220/0002548802070212

 SciTePress

be hard and very time consuming without an

adequate specific piece of software, particularly for

a large number of DMUs. Several software packages

were developed so to minimize those problems.

However, in several situations, the results produced

by those packages either are not complete or are not

consistent to the ones produced by other pieces of

software for the same model. That leads to some

questions about the correctness of the computing

implementation of the model in those softwares.

That motivates the need for developing a reliable

software package yielding complete results and

covering new theoretical developments regarding

DEA models. Besides the capability to cover the

classic DEA models and producing complete results,

it also includes advanced models that were not

considered by other DEA software packages

(Angulo-Meza et al., 2003a). The current version of

the software package is presented in Portuguese

language and is called SLAD (Software Livre de

Apoio à Decisão) although a future release in

English is also under development.

The SLAD was written in Delphi 7.0 due mainly

to allow object Pascal coding for writing the

Simplex algorithm to solve LPPs and also to its

capability to deal with graphics.

The developed software package was shown to

be very useful for testing new models and has been

used in many papers written by the authors.

2 DATA ENVELOPMENT

ANALISYS

Data Envelopment Analysis (DEA) was developed

by Charnes et al. (1978) and is a methodology that

uses linear programming for the comparative

evaluation of DMUs efficiencies

The DEA purpose is to compare a certain

number of DMUs performing similar tasks and that

distinguishes themselves in the number of used

inputs and the number of produced outputs. There

are basically two classic DEA models: the Constant

Return Scale (CRS) model also known as CCR

harnes, Cooper and Rhodes, 1978) and the

Variable Return Scale (VRS) model or BCC

anker, Charnes and Cooper, 1984). The first

model considers constant scale returns and the

second one variable scale returns and does not

assume proportionality among inputs and outputs.

Each k

DMU, k = 1, ..., n, is considered to be a

production unity that uses n inputs x

, i =1, …, r, to

produce m outputs y

, j =1, …, s. The CCR model

described by (1) maximizes the ratio between a

linear combination of outputs and a linear

combination of inputs with the constraint that for

each DMU that ratio can not be greater than one. So

for a particular DMU o, h

is its efficiency; x

and

are its inputs and outputs and v

and u

are the

calculated weights for the inputs and outputs.

jivu

iki

jkj

ioi

joj

, 0,

,...,1 ,1

subject to

max

∀≥

=≤

∑

(1)

After some mathematical manipulations, the

model can be rewritten, yielding in a Linear

Programming Problem (LPP) as shown in (2).

(2)

jivu

nkxvyu

yuh

ikijkj

ioi

jojo

, 0,

,...,1 , 0

subject to

max

∀≥

=≤−

∑∑

∑

As a LPP is solved for each DMU, for n DMUs

n LPPs are solved, with r + s decision variables. The

model just presented is the basis for all other DEA

models.

Besides the efficiency index, DEA models yields

for each DMU the variables weights , benchmarks,

and targets for the inefficient DMUs. The last two

elements are determined from the values of the dual

variables i.e. by solving the dual LPPs or by the use

of the complement slack theorem. The set of all

those results defines what we call in this paper

complete results. In recent years, software packages

were developed due to the great interest and the

large number of applications using the DEA

approach. Those programs include mostly basic

models and were developed to avoid the effort of

running separately LPPs without the integration

provided by DEA packages software in order to get

the final evaluation. On the other hand, theoretical

developments have been made and are widely used.

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208

For example, weights restrictions (Allen et al, 1997)

allow the inclusion of a priori information, i.e.

subjective aspects can be incorporated within DEA

models that are essentially very objective. Thus the

use of some advanced developments have been

essential for DEA analysis and some software

packages already include some advanced models.

However, in some cases different softwares

running the same model yielded different results or

results not following the model constraints which

make them unreliable. That also happened even for

the basic models. Most software packages show only

efficiency indexes and targets leaving out the

variable weights which might be useful in a deep

assessment of the DMUs and for aid in later

theoretical developments (Angulo-Meza e Lins,

2000).

Such problems motivated the authors to build a

package software in order to produce complete

results involving the basic models CCR and BCC

(input or output oriented, Cooper et al, 2000), with

open code access to check eventual problems or

discrepancies among results and the capability to

include models for new theoretical advanced

developments.

3 THE FSDA – FREE SOFTWARE

FOR DECISION ANALYSIS

A fundamental step for the development of a DEA

software is the setup and choice of the algorithm to

solve the LPPs associated with the problem.

The Simplex algorithm is very popular for

solving LPPs, so is the interior points algorithm

particularly for large scale LPPs. The FSDA uses

the Simplex algorithm for the solution of linear

programming problems.

The Simplex algorithm was created by Dantzig

and published in 1948. Briefly, according to Dantzig

(1963) a linear programming problem can be

reduced to a combinatory problem in such a way that

the method for searching the optimum solution is

performed by testing a sequence of combinations in

which the value of the objective function is

incremented gradually. An optimum solution is

considered to be reached after a certain number of

iterations not greater than the number of constraints

or the number of variables whatever is the largest.

The FSDA uses Kuenzi et al ‘s (Kuenzi et al,

1971) approach that includes a subroutine to avoid

degenerating problems, a common problem in DEA

models due to the typical structure of the involved

LPPs that present a large number of redundant

constraints for the inefficient DMUs. In many cases

that leads to multiple optimal solutions.

Matrices and vectors are used to separate the

values of the coefficients corresponding to the basic

variables (variables that are part of the solution) and

to the non basic variables ( variables which are not

part of the solution and whose values are set to

zero). Moreover, two lists are kept containing those

two variables which are updated at each algorithm

iteration.

The mentioned procedure has two phases for

evaluating the solution of a LPP. The first one

searches an initial basic solution. The second starts

from the initial basic solution found in part one.

Both phases have three steps. The first step

determines the value of a secondary objective

function that will be used in the other two steps and

also a variable that will enter the base.

The second step locates the pivot element taking

into consideration a possible degeneration in the

corresponding LPP. If the pivot element can not be

found a message is sent and the search ends.

The last step updates the matrices in order to

change the set of basic and non basic variables.

There are three possible outcomes for a given

LPP: an optimal solution is achieved, no solution is

achieved, in those cases the LPP is said to be non

limited or no initial vector is possible which means

the LPP is not feasible.

The algorithm used in the FSDA software is now

presented in detail:

Phase 1: Determine an initial

feasible basic solution and check if

that solution is the origin, otherwise

search one.

Step 1: Search a variable to enter

the base of the initial feasible

solution and evaluate a secondary

objective function.

If the current solution is

feasible go to phase 2, otherwise if

there is a variable to be sent into

the base go to step 2, otherwise the

LPP is non feasible and the program

ends and a message is sent to the

user.

Step 2: Locate pivot element if

input go to step 3, otherwise END,

LPP not feasible.

Phase 2: There is an initial basic

solution go to step 1.

Step 1: Search a variable to be in

the base and evaluate the secundary

objective function.

If the current solution is optimum

then END, otherwise if there is a

variable to be in the base go to step

2, otherwise LPP is non limited.

FREE SOFTWARE FOR DECISION ANALYSIS: A Software Package for Data Envelopment Models

209

Step 2: Locate pivot element.

Step 3: Update the sets of basic

and non basic variables and go to

step 1.

It should be stressed that in basic DEA models (CCR

or BCC, whatever orientation) at least one solution

can always be found with a possibility of an infinite

number of optimum solutions which is quite

frequent. In that case, only the first optimum

solution is shown to the user.

In the case of models with restrictions to the

weights, there is a possibility of appearing non

feasible LPPs once the additional constraints can

turn the LPP originally presenting an optimum

solution to a non feasible LPP. In that cases, a

message is sent and the user has the choice of

changing the restriction and repeat the process.

The method for solving the LPPs is unique, what

changes is the format of the LPPs, once a different

LPP is performed for each DEA model. The data

input have to be put in the proper format in a matrix

structure depending on the used model. The data

ordering process in the referred matrix in the

appropriate format is the most difficult part in the

software implementation.

3.1 Software Description

The FSDA( SLAD in Portuguese version) opening

window is shown in Figure 1 and was developed for

Windows platform with Delphi 7.0. It is capable for

dealing with 150 DMUs and 20 inputs or outputs

variables.

Although for other research areas 150 DMUs

might be insufficient for DEA applications that

value is able to deal with large scale situations since

in the literature there are few applications dealing

with more than a 100 DMUs. In most applications

10 variables are sufficient.

The software package allows the data input to be

entered in two ways: directly inside the program

using a table with the choice of the number of

DMUs and variables, and by the use of a txt file

having the data that are loaded to the mentioned

table shown in figure 2. Figure 2 also shows the

possible choices of basic models (CCR or BCC) and

orientation (input or output) covering the basic and

most used models.

Moreover, advanced options are also included in

the software as one of the objectives of the FSDA

package is to allow new DEA developments that

may be chosen along with the model and its

orientation. The user can also the choice of using

weight restrictions, or considering data uncertainty.

The advanced option of weight restrictions leads

the user to an additional window indicating the

number of restrictions to be included in the chosen

model.

The choice of uncertainty in some or all

variables also leads the user to another window,

showing the data input. A DEA analysis with

uncertainties uses variation intervals for some or all

variables of analysis.

Results for any model, advanced or not, are

presented in an additional window as illustrated in

figure 3. That window shows the efficiency indexes

for all DMUs. Besides, additional options are

presented to show other results: inverted frontier,

that expand the results window to include the

efficiency index in the inverted frontier and the

composed index (standard and inverted efficiencies);

the variables weights; the benchmarks of all DMUs

(efficients and inefficients) and the targets for the

DMUs including the slacks and the levels that the

variables have to reach for turning the DMUs

efficients

Figure 1: FSDA (SLAD) Opening Window

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Figure 2: Options window of FSDA (SLAD).

Figure 3: Results window of FSDA (SLAD).

It should be noted that the results window just

shows the efficiencies of the DMUs whereas the

other results are shown only when asked for by

clicking the appropriate icons on the same window.

That was done for the sake of easing the

visualization of the results once the presentation of

the complete results in one window might be

difficult to interpret the results particularly for a

large number of variables and DMUs.

In the data input window there is another option,

not yet mentioned, that allows the use of

multicriteria models. By choosing that option the

window changes colour to indicate the user the

substitution of the DEA approach for the

multicriteria approach. As this module is under

development, the only available method is the

Analytic Hierarquic Process (AHP) according to

Clímaco et al. (1996), but other methods will be

include in due time.

4 CONCLUSIONS

The growing interest in recent years of DEA

approach for the evaluation of efficiency and the

theoretical developments due to real world

applications have indicated the need of a reliable

software package that involve jointly a friendly user

interface, yielding reliable, consistent and complete

results and having the capability to work with more

advanced models. There are DEA software

packages but the lack of complete and reliable

results in those softwares led to the development of a

FREE SOFTWARE FOR DECISION ANALYSIS: A Software Package for Data Envelopment Models

211

software package that met the demands of the

authors of this paper.

The FSDA (SLAD) was written in Delphi 7.0

and uses the Simplex algorithm to solve the LPPs

required by DEA. The software includes, besides the

classic models, advanced models such as DEA

models with weight restrictions as well as recent

models developed by the authors such as inverted

frontiers and data uncertainties.

The complete results produced by the software

are suitable for a deep analysis of the DMUs and

could also serve as a basis for the development of

new theoretical models.

The performance of the FSDA in large scale

problems has been satisfactory, e. g. the time

response for problems involving 80 DMUs and 7

variables was less than one second.

A great motivation of the FSDA for the authors

is the possibility of including other models in the

package that has been very useful for the authors in

their new developments and in the testing of new

models.

Finally, new models developed by the authors

are under way to be considered in the FSDA such as

the DEA-GSZ model (Lins et al., 2003), as well as

the expansion of the multicriteria module.

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