EE
i
= 0B0A Where: EE
i
is the input orientated economic efficiency 3.3
The distance BA could also be interpreted in terms of cost reduction. The EE is also could be calculated by Equation 3.4.
TE
i
x AE
i
= 0D0A x 0B0D 3.4
It could be done because the product of the TE and AE measures provides the measures of overall economic efficiency. All of three measures bounded by
zero and one.
3.2.2 Output Orientated
The alternative question about eff iciency is “How much quantity of
outputs can be proportionally increased without changing the input quantity used?” The output oriented measures is opposed to the input oriented measures.
The difference between the input and output oriented measures can be seen in Figure 4, which is using a simple example involving one input x and one output
y.
Source: Coelli, Rao, Battese, 1998 Figure 4 Input and Output Orientated TE Measures and Return to Scale
a b
fx fx
P P
R R
Q Q
S S
y y
x x
A A
Figure 4a illustrated a decreasing return to scale in technology, represented by fx, and an inefficient firm operating at the point A. The input orientated
measure of TE would be equal to the ratio PQPA, while the output-orientated measure of TE would be RARS. Whereas only when constant return to scale
exist, the input orientated measures would equal to the output orientated measures Fare Lovell, 1978. This case is illustrated in Figure 4b, where PQPA =
RARS, for the inefficient firm operating at point A. The TE and AE of an output orientation are illustrated in Figure 5. This
illustrates the output-orientated measures by considering the where production involves two outputs y
1
and y
2
and a single input x. The input quantity is set as fixed at particular level, and then the technology represented by a production
possibility curve in two dimensions. The production possibility curve is illustrated in Figure 5
as line JJ΄, and the point K shown inefficient firm. The firm, which is operating at point K, is an inefficient because it lies below the curve. The curve
JJ΄ represents the upper bound of production possibilities.
Source: Coelli, Rao, Battese, 1998 Figure 5 Technical and Allocative Efficiencies of Output Orientation
y
2
x
y
1
x J΄
J L
΄ K
M N
Q
Referring to Figure 5, the distance KM represents technical inefficiency. It means the number of outputs could be increased without requiring extra input.
Therefore a measure of output orientated TE is ratio as on Equation 3.5. TE
o
= 0K0M Where: TE
o
is the output orientated technical efficiency 3.5
If there is input information then w e can draw the isorevenue line, LL΄, and
can be calculated the AE as on Equation 3.6. AE
o
= 0M0N Where: AE
o
is the output orientated allocative efficiency 3.6
Like input orientation, which the cost is reducing interpretation of allocative efficiency, the output orientation has a revenue increasing interpretation. The EE
define as Equation 3.7. EE
o
= 0K0N = 0K0M x 0M0N = TE
o
x AE
o
Where: EE
o
is the output orientated economic efficiency
3.7
Similar as in input orientated, all of these three measures are bounded by zero and one. The three important points about the efficiency measures Coelli,
Rao, Battese, 1998, namely: 1.
TE has been measured along a ray from the origin to the observed production point. Therefore, these measures hold a relative set of inputs outputs
constant. The advantage of the radial efficiency measure is changing the units of measurement does not change the value of the efficiency measures.
2. AE could be calculated from a cost minimizing and from the revenue
maximizing, but not from a profit maximizing perspective. It is able to accommodate in a number of ways. The principal difficulty is related to the
selection of the orientation in which to measure TE input, output, or both. 3.
The Farrell input and output orientated TE measures are equivalent to the input and output distance function.
3.3 Constant Returns to Scale CRS DEA Model
In 1978, the traditional DEA model was introduced by Charnes, Cooper, Rhodes, known as CCR model, had an input orientation and assumed CRS. This
model was the first to be widely applied. The optimal weights are acquired by solving the mathematical problem Equation 3.8:
max
u.y
u΄y
i
v΄x
i
st u΄y
j
v΄x
j
≤ 1, j=1, 2, ..., N,
u, v ≥ 0 3.8
An intuitive of DEA is via ratio form. The aims are to obtain a measure of the ratio of all outputs overall all inputs for each firm. For example, as in Equation
3.8 u΄y
i
v΄x
i
where u is an Mx1 vector of output weights and v is a Kx1 vector of input weights.
Values for u and v will also find, so that the efficiency measures for i-th firm is maximized. Subject to st the constraints that all of measures must be less
than or equal to one. There is one problem for this particular ratio formulations, it has an infinite number of solutions. To avoid this problem, st the constraint that
all efficiency must be equal to one v΄x
i
= 1, define as Equation 3.9. max
µ,v
µ΄y
i
, st
v΄x
i
= 1, µ΄y
j
- v΄x
j
≤ 0, j=1, 2, ..., N, µ, v ≥ 0,
3.9 The changing of notation u and v to µ and v is used to stress that this is a
different linear programming problem. Equation 3.9 is known as the multiplier form of the DEA linear programming problem.
The above formulation is the primal formulation, while the alternative is called the dual formulation. It can be derived an equivalent envelopment form of
this problem as Equation 3.10. min
θ,λ
θ, st -y
i
+ Yλ ≥ 0, θx
i
– Xλ ≥ 0,
