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de
ning the activities and processes of the distributors and the average cost of each
activity performed.
The third cycle introduced the use of benchmark models. To take into account
the evolution of ef
ciencies along time, in the benchmark models is common the
use of indices such as Malmquist index. Jasmab and e Pollit (
2001
), reporting such
increasing importance of benchmarking methods in the context of incentive regu-
lation of distribution and transmission of electricity and analyzing OECD and some
other countries, observe a preference for non-parametric methods.
ANEEL (
2011
) proposed to estimate the evolution of productivity using, as
inputs, operational cost and, as outputs, size of the market served, in MWh/year
offered, number of consumers in the region attended, and extent of the network, in
Km.
Operational costs have considerable variation with a magnitude of millions of
dollars per year for the larger Brazilian companies. The extension of the operated
network reaches hundreds of thousands of miles and is a proxy for the costs of
maintaining services. The number of consumers exceeds the total of a million in
many cases, capturing the business costs incurred by the distributors. Finally,
annual MWh captures the size of the distributor on issues where the voltage level
affects the costs.
In ANEEL (
2011
) was used a decreasing returns to scale and input orientation
DEA model. It is shown in the work of Andrade and Sant
Anna (
2013
), resumed
here, how to replace DEA by the probabilistic composition, using the probabilistic
Malmquist index to assess each company
'
s effort on improving through time,
'
taking into account its speci
c operating conditions.
The same variables in ANEEL (
2011
) study are employed here but the proba-
bilities of maximization and minimization are calculated for the vectors of quotients
of each output by the unique input. Thus, the variables employed are the
ν
aj
below,
ν
aj
denoting the ratio between the j-th output O
aj
of the a-th company and its
operational cost I
aj
. It is assumed that, to measure ef
ν
aj
are modes of
independent random variables with triangular distribution of constant range.
ciency, the
m
a1
¼
O
a1
=
I
a1
;
m
a2
¼
O
a2
=
I
a2
;
m
a3
¼
O
a3
=
I
a3
:
Table
7.1
shows the values of the three ratios for each of the 61 distributors in
the Brazilian electric system in the years 2008 and 2009. Table
7.2
presents
rst the
2009 ef
ciency scores for each company by the progressive and optimistic
approach, followed by the scores for the pessimistic and conservative approach.
Then comes the Malmquist index based on the frontier of excellence (the lower the
score, the higher the evolution in ef
ciency) followed by the Malmquist index
based on the lower frontier and
finally the compound index.
Initially the probabilistic composition used is progressive and optimistic, which
is conceptually similar to the benevolent approach of traditional DEA. The measure
used for this approach is a measure of inef
ciency, the probability of not being the
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