Composition by DEA Distance to the Frontier - Probabilistic Composition of Preferences, Theory and Applications - page 51

outputs belongs to this excellence frontier. Otherwise, it is said to be inef cient and

receives a score of relative ef ciency determined by the quotient between such

productivity ratios.
There are two classic models in DEA. The first is called CCR, in reference to

Charnes et al. ( 1978 ). The second, called BCC, was proposed by Banker et al.
( 1984 ). Later successive developments led to the construction of a series of DEA
models, considering, for instance, restrictions to the vector of weights, as well as
randomness. However, when establishing a setup to combine probabilistic prefer-
ences, it is convenient to not deviate from a fixed pattern. For this reason, only the

two general optimization procedures of the CCR and BCC models will be
employed here in the probabilistic composition of criteria.
In both these procedures, the distance from the vector of values of the inputs and
outputs in the alternative being evaluated to a piecewise linear frontier generated by
linearly combining observed inputs of other alternatives is minimized. In the BCC
approach, the coef cients of the linear combinations that form the frontier are

forced to sum to 1. In the CCR approach, they can present a sum smaller than 1.
This cients of the CCR model corresponds to

allowing for introducing in the composition of the frontier an alternative of null
inputs and null outputs. This is equivalent to allow for reducing the use of inputs
and production of outputs in any unit in the frontier on a proportional basis, as if
returns to scale were constant.
In the BCC model, the frontier of excellence is formed only by points inter-
mediary between those representing observed alternatives. That means that the
scores of ef

flexibility in the sum of coef

ciency of the alternatives cannot be reduced by comparison to fictitious

production units of smaller dimensions.
The ef

ciency of each alternative is calculated by solving an optimization
problem proper for that alternative. The CCR optimization problem has as objective
function to maximize the quotient between the ef ciency ratio of that alternative

and the best ef ciency ratio with the same multipliers in the entire set of alterna-

tives. By fixing the value 1 for the linear combination of the inputs that constitutes

the denominator of the ratio, this fractionary optimization problem can be formu-
lated in the following linear form:

X

Maximize h ao ¼ l r O rao

r

subject to

X

m s Is ao ¼ 1

;

s

X X

l r O ra m s I sa 0 for every alternative a from 1 to n

; ;

r a

l r [ 0 and m s [ 0 for all r and s

;