Encyclopedia of
Decision Making and
Decision Support
Technologies
Frédéric Adam
University College Cork, Ireland
Patrick Humphreys
London School of Economics and Political Science, UK
Volume II
In-Z
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Encyclopedia of decision making and decision support technologies / Frederic Adam and Patrick Humphreys, editors.
p. cm.
Summary: "This book presents a critical mass of research on the most up-to-date research on human and computer support of managerial decision making, including discussion on support of operational, tactical, and strategic decisions, human vs. computer system support structure, individual and group
decision making, and multi-criteria decision making"--Provided by publisher.
ISBN-13: 978-1-59904-843-7
ISBN-13: 978-1-59904-844-4
1. Decision support systems. 2. Decision making--Encyclopedias. 3. Decision making--Data processing. I. Adam, Frédéric. II. Humphreys, Patrick.
HD30.213.E527 2008
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709
Performance Measurement: From DEA to
MOLP
João Carlos Namorado Clímaco
Coimbra University, Portugal
INESC Coimbra, Portugal
João Carlos Soares de Mello
Federal Fluminense University, Brazil
Lidia Angulo Meza
Federal Fluminense University, Brazil
INTRODUCTION
Data envelopment analysis (DEA) is a non-parametric
technique to measure the efficiency of productive units
as they transform inputs into outputs. A productive unit
has, in DEA terms, an all-encompassing definition. It
may as well refer to a factory whose products were
made from raw materials and labor or to a school that,
from prior knowledge and lessons time, produces more
knowledge. All these units are usually named decision
making units (DMU).
So, DEA is a technique enabling the calculation of
a single performance measure to evaluate a system.
Although some DEA techniques that cater for decision makers’ preferences or specialists’ opinions do
exist, they do not allow for interactivity. Inversely,
interactivity is one of the strongest points of many of
the multi-criteria decision aid (MCDA) approaches,
among which those involved with multi-objective
linear programming (MOLP) are found. It has been
found for several years that those methods and DEA
have several points in common. So, many works have
taken advantage of those common points to gain insight
from a point of view as the other is being used. The idea
of using MOLP, in a DEA context, appears with the
Pareto efficiency concept that both approaches share.
However, owing to the limitations of computational
tools, interactivity is not always fully exploited.
In this article we shall show how one, the more
promising model in our opinion that uses both DEA
and MOLP (Li & Reeves, 1999), can be better exploited
with the use of TRIMAP (Climaco & Antunes, 1987,
1989). This computational technique, owing in part to
its graphic interface, will allow the MCDEA method
potentialities to be better used.
MOLP and DEA share several concepts. To avoid
naming confusion, the word weights will be used for
the weighing coefficients of the objective functions in
the multi-objective problem. For the input and output
coefficients the word multiplier shall be used. Still
in this context, the word efficient shall be used only
in a DEA context and, for the MOLP problems, the
optimal Pareto solutions will be called non-dominated
solutions.
BACKGROUND
Ever since DEA appeared (Charnes, Cooper, & Rhodes,
1978) many researchers have drawn attention to the
similar and supplementary characteristics it bears to the
MCDA. As early as 1993, Belton and Vickers (1993)
commented their points of view supplement each other.
This is particularly relevant for MOLP. For instance,
both MOLP and DEA are methodologies that look
for a set of solutions/units that are non-comparable
between them, that is, are efficient/non-dominated.
This contribution is focused on the synergies between
MOLP and DEA.
Taking into consideration the vast literature and to
be able to follow the theme’s evolution articles should
be classified into different categories. The first two
categories are those in which DEA is used for MOLP
problems and vice versa. Although the differences are
often not very clear, these categories can be useful to
introduce the theme.
Copyright © 2008, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.
P
Performance Measurement: From DEA to MOLP
Works in which DEA is used within MOLP are
not the object of this article. Some of these works are
those of Liu, Huang, & Yen (2000), or Yun, Nakayama,
Tanino, and Arakawa (2001).
Within those articles in which MOLP is used in DEA
problems a further disaggregation is possible:
1. Models that use MOLP to determine non-radial
targets in DEA models. Their own nature makes it
imperative that these models use the DEA envelop
formulation.
2. Models that, besides the classic DEA objective,
use other objectives, generally considered of
lesser importance. The majority of the articles
concerning this approach use the multipliers
formulation.
3. Models in which optimization of more than one
DMU is simultaneously attempted.
Papers in Which MOLP is Used in DEA
The first article explicitly written along this line is
Golany’s (1988). He starts from the assumption that not
all DEA efficient solutions are effective, that is, they do
not equally cater to the decision maker’s preferences.
The article assumes that the inputs vector is constant and
an outputs vector should be computed so the DMU is
both efficient and effective. So, an interactive algorithm
(MOLP), based on STEM (Benayoun, Montgolfier,
Tergny, & Larichev, 1971) is proposed. This algorithm
checks all possible DMU benchmarks and eliminates
in succession those that do not conform to the decision
maker’s interests. In fact, the author uses a multi-objective model in which every output is independently
maximized, maintaining all inputs constant.
Joro, Korhonen, and Wallenius (1998) produce a
structural comparison between MOLP and DEA. They
show that the non Archimedean output-oriented CCR
model displays several similarities with the reference
point MOLP model. This property is used in the “Value
Efficiency Analysis” (Halme, Joro, Korhonen, Salo,
& Wallenius, 2002; Joro, Korhonen, & Zionts, 2003)
to assist a decision maker to find the most preferred
point at the frontier.
Tavares and Antunes (2001) based themselves on
a minimizing Chebyshev’s distance method to put
forward a DEA alternative target calculation.
Lins, Angulo–Meza, and Silva (2004) developed
the models MORO and MORO-D. These models are
710
a generalization of Golany’s (1988) model. The multiobjective method allows simultaneously for output
maximization and input minimization. Quariguasi
Frota Neto and Angulo-Meza (2007) have analyzed
the characteristics of the MORO and MORO-D models
and used them to evaluate dentists’ offices in Rio de
Janeiro.
The fuzzy-DEA multidimensional model (Soares
de Mello, Gomes, Angulo-Meza, Biondi Neto, &
Sant’Anna, 2005) used MORO-D as an intermediary
step to find optimist and pessimist targets in the fuzzy
DEA frontier.
Korhonen, Stenfors, & Syrjanen (2003) minimize
the distance to a given reference point to find alternative and non-radial targets. In an empirical way, they
show that radial targets are very restrictive.
DEA Models with Additional Objective
Functions
The first models of this type were not recognized as
multi-objective by their authors and are rarely mentioned as such. They include the two step model (Ali
& Seiford, 1993) and the aggressive and benevolent
cross evaluation (Doyle & Green, 1994; Sexton, Silkman, & Hogan, 1986). These models are not usually
accepted as multi-objective ones. However, as they
optimize in sequence two different objective functions,
they can be considered as a bi-objective model solved
by the lexicographic method (Clímaco, Antunes, &
Alves, 2003).
Kornbluth (1991) remarked that the formulation of
multipliers for DEA can be expressed as multi-objective
fractional programming.
A similar approach by Chiang and Tzeng (2000)
optimizes simultaneously the efficiencies of all DMUs
in the multipliers model. An objective function corresponds to each DMU. The problem is formulated in the
fractional form so as to avoid its becoming unfeasible
owing to the excessive number of equality restrictions.
The authors use fuzzy programming to solve this multiobjective fractional problem. Optimization is carried
out in such a manner as to maximize the efficiency of
the least efficient DMU. The last two models can also
be classified as models in which more than one DMU
are simultaneously optimized.
Owing to its importance, Li and Reeves (1999)
model is detailed hereafter.
Performance Measurement: From DEA to MOLP
The Li and Reeves Model (1999)
Li and Reeves (1999) presented a multi-objective
model with the aim to solve two common problems
in DEA: (1) increasing discrimination among DMUs
and (2) promoting a better multiplier distribution for
the variables. The first problem occurs when we have
a small number of DMUs or a great number of inputs
and outputs (as standard models class too many DMUs
as efficient). The second problem arises as a DMU
becomes efficient with non nil multipliers in just a
few variables. This benefits those that display a good
performance and leaves aside those that do not. These
two problems are intertwined.
To get around them, the authors proposed a multicriteria approach for DEA in which additional objective
functions are included in the classic CCR multipliers
model (Charnes et al., 1978). The additional objective
functions restrict the flexibility of the multipliers.
In DEA, a given DMU O is efficient when hO = 1,
meaning that the constraint relative to that DMU is active
and, thus, its slack is nil. The basic idea is to consider
this slack as an efficiency measurement instead of h.
The slack symbol becomes d. As h equals one minus d,
the CCR model in (1a) can be reformulated as (1b):
s
Max hO = ∑ ur yrj0
i =1
i ij0
=1
r =1
i =1
ur ,vi ≥ 0, ∀r,i
Min Max d j
subject to
r =1
i ij0
(1a)
=1
s
m
r =1
i =1
∑ ur yrj − ∑ vi xij + d j = 0, j=1,..,n
ur , vi , d j ≥ 0, ∀r, i
=1
m
r
rj
i =1
i ij
+ d j = 0, j=1,...,n
ur , vi ≥ 0, ∀ r,i,j
s


Min dO  or Max hO = ∑ ur yrj0 
r =1


subject to
i ij0
j
∑u y − ∑v x
∑ ur yrj − ∑ vi xij ≤ 0, j = 1,.., n
i =1
j =1
s
m
∑v x
n
∑d
Min
i =1
s
m
Min dO
m
subject to
∑v x
– dO. So, the lesser is d0, the more efficient is the DMU.
To restrict multipliers freedom of choice, the MCDEA
model takes into account two other objective functions:
a “general benevolence” objective, the minimization of
the deviations sum, and another, an “equity” one, that
is, the minimization of the maximum deviation.
Thus, the multiobjective programming problem
known as multi-criteria data envelopment analysis
(MCDEA) is formulated as in (2):
∑v x
r =1
m
where vi e ur are respectively the input i, i=l,...,m, and
output r, r = 1,...,s, multipliers; x­ij and yrj DMU j, j
= l,...,n, inputs i and outputs r ; x­io and yro are DMU
O inputs i and outputs r. Besides, d0 is the deviation
variable for DMU O and dj is the deviation variable
for DMU j, that is, how much the DMU is away from
efficiency. In this model, DMU O is efficient if, and
only if dO = 0, this being the same as hO = 1. If DMU
O is not efficient, its measure of efficiency is hO = 1
(1b)
(2)
The first objective function is the classical efficiency
maximization, the second is the equity function and the
third that of “general benevolence.”
The intention to optimize the evaluation of all
DMUs, as a whole, from the view point of the DMU
under analysis, is bred from the “general benevolence”
function.
The relative efficiency of a given DMU corresponding to the second and third objectives can be defined
thus: a DMU O is minimax efficient if, and only if, the
dO value that corresponds to the solution that minimizes
the second objective function of model (2) is zero; likewise, a DMU O is minisum efficient if, and only if, the
dO value that corresponds to the solution that minimizes
the third objective function of model (2) is zero.
711
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Performance Measurement: From DEA to MOLP
In their original work, the authors have only used the
weighted sum as a solution method and then proceed
to draw considerations about the behavior of either
function when the other one is optimized. They have
used the ADBASE software, Steuer (1986).
As their model is a three objective one, TRIMAP
has some advantages in terms of content and user
friendliness. A brief description follows.
The Trimap
The TRIMAP package (Clímaco & Antunes, 1987,
1989) is an interactive environment. It assists the decision makers to perform a progressive and selective
search of the feasible polyhedron non-dominated region.
The aim is helping the decision maker to eliminate the
subsets of the non-dominated solutions which are of
no interest to her/him. It combines three main procedures: weights space decomposition, introduction of
constraints on the objective functions space and on the
weights space. In each step of the human/computer
dialogue the decision maker just has to give indications
about the regions of the weights space where the search
for new non-dominated solutions should be carried out
optimizing weighted sums of the objective functions.
By cognitive reasons, usually this is performed indirectly introducing progressively during the interactive
process bounds on the objective functions values, that
is, reservation values taking into account the present
knowledge on the non-dominated solutions set. Note
that the introduction of constraints on the objective
functions values is automatically translated into the
weights space. The TRIMAP package is intended
for tri-criteria problems. This fact allows for the use
of graphical means and for the particular case of the
MCDEA model is not a limitation. For instance, the
indifference regions on the weights space (which is
a triangle for the tri-objective case) corresponding to
the non-dominated bases previously calculated are
displayed graphically on the triangle.
Other important tools and graphical facilities not
referred to in this summary are available in TRIMAP.
However, the graphic decomposition of the weights
space in indifference regions corresponding to each
of the previously calculated non-dominated bases is
especially interesting to study the MCDEA model.
Besides the graphical aspect, TRIMAP supplies a
text that condenses all the numerical results. Among
other data, the values of the basic variables and those
712
of the objective functions corresponding to the nondominated solutions, the share of the area occupied by
the indifference regions, and so forth.
THE LI AND REEVES (1999) MODEL
AND THE TRIMAP
In the MCDEA model, the knowledge of the weights
space decomposition, as obtained through TRIMAP,
allows researchers to evaluate the stability of the DEA
efficient solutions. Large indifference regions mean
evaluations remained unchanged when moderate
changes in the formulations of the objective functions
occur. The existence of optimal alternatives is a check
as to whether the optimal evaluation of a given DMU
depends on a unique multipliers vector.
It is easy enough to check whether any DMU is
minimax or minisum efficient: it suffices that indifference regions include simultaneously the triangle corners
that correspond to the optimization of the classical
objective function and one of the other two.
As TRIMAP supplies, for each indifference region,
the multiplier values, a solution can be chosen with a
multipliers distribution that is acceptable to the decision
maker: for instance, one without nil multipliers.
Table 1 shows input and output data for the DMUs
used in an example. They are all CCR efficient.
Figures 1 to 5 show weights space decompositions
for the example used.
It is worthy of notice that just by looking at the right
hand bottom corner of the triangles that only DMUs 3
and 5 have non-dominated alternative optima for the
first objective function. Thus, DMUs 3 and 5 are DEA
extreme efficient. The same can not be concluded on
Table 1. Numerical example data
DMU
Input 1
Input 2
Output
1
0,5
5
8
2
2
1
4
3
3
5
20
4
4
2
8
5
1
1
4
Performance Measurement: From DEA to MOLP
Figures 1, 2, and 3. Decomposition of the weights space for DMUs 1, 2 and 3, respectively
(1)
(2)
P
(3)
Figures 4 and 5. Decomposition of the weights space for DMUs 4 and 5, respectively
(4)
this respect about DMUs 1, 2, and 4 as TRIMAP shows
only non-dominated solutions. Eventual alternative
optima concerning those DMUs have no representation
on the triangle, because are dominated, and so are DEA
efficient but worse than the non-dominated one for at
least one of the other two criteria. A classical DEA
analysis of the PPLs reveals that DMU 1 has alternative optimums, which is not the case for DMUs 2 and
4. These DMUs are not Pareto efficient.
The analysis of the bottom left hand side and top
corners shows that DMU 3 is both minisum and minimax efficient. DMU 5 is only minisum efficient. In a
problem to choose one of these five efficient DMUs it
would be acceptable to get rid of DMUs 1, 2, and 4.
(5)
CONCLUSION AND FUTURE
RESEARCH
TRIMAP can help to analyze the MCDEA model with
greater detail than the one provided by Li and Reeves
(1999) themselves because it is a very flexible decision
support tool dedicated to tri-criteria problems. Several
conclusions can reach from mere visual analysis.
The possible uses of TRIMAP for the MCDEA
model do not end here. Two types of development
look particularly promising. The first is to modify the
model’s objective functions. The minimax function can
be replaced by a minimization of a weighted Chebyshev
distance so as to control the restrictive character of the
objective function.
713
Performance Measurement: From DEA to MOLP
The minisum or minimax functions can also be
replaced by the maximization of the smaller multiplier. This replacement partly solves the too common
DEA problem of having multipliers with a very small
value.
TRIMAP capability to render into the weights
space constraints on the objective function values can
be used in MCDEA for future interactivity processes
of the decision maker.
Another development could be to create an aggregate
index of all analyzed properties.
Clímaco, J. C. N., Antunes, C. H., & Alves, M. J. G.
(2003). Programação Linear Multiobjectivo [Multiobjective Linear Programming] Imprensa da Universidade, Coimbra. (Portuguese)
REFERENCES
Halme, M., Joro, T., Korhonen, P., Salo, S., & Wallenius,
J. (1998). A value efficiency approach to incorporating
preference information in data envelopment analysis.
Management Science, 45(1), 103-115.
Ali, A. I., & Seiford, L. M. (1993). The mathematical
programming approach to efficiency analysis. In H.
O. Fried, C. A. K. Lovell, & S. S. Schmidt (Eds.), The
measurement of productive efficiency (pp. 120-159).
New York: Oxford University Press.
Belton, V., & Vickers, T. J. (1993). Demystifying
DEA—A visual interactive approach based on multiple
criteria analysis. Journal of Operational Research
Society, 44(9), 883-896.
Benayoun, R., Montgolfier, J., Tergny, J., & Larichev,
O. (1971). Linear programming and multiple objective functions: STEP method (STEM). Mathematical
Programming, 1(3), 366-375.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978).
Measurering the efficiency of decision-making units.
European Journal of Operational Research, 2, 429444.
Chiang, C. I., & Tzeng, G. H. (2000). A multiple objective programming approach to data envelopment
analysis. New Frontiers of Decision Making for the
Information Tehnology Era, 270-285. Singapore, New
Jersey, London, Hong Kong: World Scientific.
Doyle, J. R., & Green, R. H. (1994). Efficiency and
Ccoss-efficiency in DEA: Derivations, meanings and
uses. Journal of the Operational Research Society,
45(5), 567-578.
Golany, B. (1988). An interactive MOLP procedure for
the extension of DEA to effectiveness analysis. Journal
of the Operational Research Society, 39(8), 725-734.
Joro, T., Korhonen, P., & Wallenius, J. (1998). Structural
comparison of data envelopment analysis and multiple
objective linear programming. Management Science,
44(7), 962-970.
Joro, T., Korhonen, P., & Zionts, S. (2003). An interactive approach to improve estimates of value efficiency
in data envelopment analysis. European Journal of
Operational Research, 149, 688-699.
Korhonen, P., Stenfors, S., & Syrjanen, M. (2003).
Multiple objective approach as an alternative to radial
projection in DEA. Journal of Productivity Analysis,
20(3), 305-321.
Kornbluth, J. S. H. (1991). Analysing policy effectiveness using cone restricted data envelopment analysis.
Journal of the Operational Research Society, 42(12),
1097-1104.
Li, X.-B., & Reeves, G. R., (1999). A multiple criteria
approach to data envelopment analysis. European Journal of Operational Research, 115(3), 507-517.
Clímaco, J. C. N., & Antunes, C. H. (1987). TRIMAP—
An interactive tricriteria linear programming package.
Foundations of Control Engineering, 12, 101-119.
Lins, M. P. E., Angulo-Meza, L., & Silva, A. C. E.
(2004). A multiobjective approach to determine alternative targets in data envelopment analysis. Journal of the
Operational Research Society, 55(10), 1090-1101.
Clímaco, J. C. N., & Antunes, C. H. (1989). Implementation of a user friendly software package—A
guided tour of TRIMAP. Mathematical and Computer
Modelling, 12, 1299-1309.
Liu, F.-H. F., Huang, C.-C., & Yen, Y.-L. (2000). Using
DEA to obtain efficient solutions for multi-objective
0-1 linear programs. European Journal of Operational
Research, 126, 51-68.
714
Performance Measurement: From DEA to MOLP
Quariguasi Frota Neto, J., & Angulo-Meza, L. (2007).
Alternative targets for data envelopment analysis
through multi-objective linear programming: Rio
De Janeiro odontological public health system case
study. Journal of the Operational Research Society,
58, 865-873.
Roy, B. (1987). Meaning and validity of interactive
procedures as tools for decision making. European
Journal of Operational Research, 31, 297-303.
Sexton, T. R., Silkman, R. H., & Hogan, A. (1986).
Data envelopment analysis: Critique and extensions.
In Richard H. Sikman (Ed.), Measuring efficiency: An
assessment of data envelopment analysis (Publication
No. 32 in the series New Directions of Program). San
Francisco.
Soares de Mello, J. C. C. B., Gomes, E. G., AnguloMeza, L., Biondi Neto, L., & Sant’Anna, A. P. (2005).
Fronteiras DEA Difusas [Fuzzy DEA Frontiers]. Investigação Operacional, 25(1), 85-103.
Steuer, R. (1986). Multiple criteria optimization: Theory, computation and application. New York: Wiley.
Tavares, G., & Antunes, C. H. (2001). A Tchebychev
DEA model. Working Paper. Piscataway, NJ: Rutgers
University.
Wei, Q. L., Zhang, J., & Zhang, X. S. (2000). An inverse
DEA model for inputs/outputs estimate. European
Journal of Operational Research, 121(1), 151-163.
KEY TERMS
Benchmark: Benchmark is an efficient DMU with
management practices that are reference for some
inefficient DMUs.
Data Envelopment Analysis (DEA): DEA is a
non-parametric approach to efficiency measurement.
Decision Making Unit (DMU): DMU is a unit
under evaluation in DEA.
Efficient DMU: An efficient DMU is one located
on the efficient frontier.
Multiple Criteria Data Envelopment Analysis
(MCDEA): MCDEA is a tri-objective linear model
proposed by Li and Reeves (1999).
Multiple Objective Linear Programming
(MOLP): MOLP is a linear program with more than
one objective function.
Non-Dominated Solution: A feasible solution is
non-dominated whether does not exist another feasible
solution better than the current one in some objective
function without worsening other objective function.
Target: Target is a point in the efficient frontier that
is used as a goal for an inefficient DMU.
TRIMAP: TRIMAP is an interactive tri-objective
interactive solver package.
Yun, Y. B., Nakayama, H., Tanino, T., & Arakawa, M.
(2001). Generation of efficient frontiers in multiple
objective optimization problems by generalized data
envelopment analysis. European Journal of Operational Research, 129, 586-595.
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