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for r varying from 1 to m
1
and s varying from 1 to m
2
,m
1
denoting the number of
outputs and m
2
denoting the number of inputs, O
ra
denoting the value of the r-th
output at the a-th alternative, I
si
denoting the value of the s-th input at the a-th
alternative and i
o
denoting the alternative being evaluated.
This problem is equivalent to its dual linear programming problem of mini-
mizing the fraction by which the values of the inputs of the production unit eval-
uated can be reduced in such a way as to keep its output/input ratio lower than that
obtained from any combination of all the alternatives.
The score for the a
o
-th alternative is the fraction
θ
ao
such that, for some set of
coef
cients
of the values of the input in the n alternatives is smaller than the fraction of the
value of that input for the a
o
-th alternative, while the linear combination with these
same coef
cients
ʻ
1
…
ʻ
n
, for every input, the linear combination with these coef
cients of the n values of every output is larger than the value of that
output for that a
o
-th alternative. This dual problem may be formulated in precise
terms, with the same notation, as:
Minimize
h
ao
subject to
X
k
a
I
si
h
ao
I
sao
0
for all s from 1 to m
1
;
;
a
X
O
rao
k
a
O
ra
0
for all r from 1 to m
2
;
;
a
k
a
0
for all a from 1 to n
;
:
To avoid the hypothesis of constant returns to scale, in the BCC model the
constraint
ʣ
a
ʻ
a
= 1 is added to this formulation
.
The CCR model is completely symmetric with respect to
fixing an upper bound
for inputs and maximizing outputs or
fixing a lower bound for outputs and mini-
mizing inputs, in the sense that, by taking this second approach, the score given by
the lower bound obtained is precisely the inverse 1/
θ
ao
of the upper bound
θ
ao
determined by solving the problem presented above.
In the BCC model, there is no such precise relation. Two different vectors of
scores are then obtained according to minimization of inputs or maximization of
outputs.
Thus, two more algorithms are available: the BCC model oriented to the min-
imization of inputs and the BCC model oriented to the maximization of outputs.
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