Input Orientated Efficiency Measurement Concepts
The convexity constraint ensures that the VRS model take into accounts the variation of the efficiency with respect to the scale size of firm. The measurement
of the efficiency of each firm only benchmarked against firms that have similar size. Hence, the inefficient firm is a result of measurement with firm that has a
similar size. The convexity constraint is not applied in the CRS model. Therefore, a firm might be compared to the firms, which are larger or smaller size than it.
In TE calculation using the VRS model, the value of scale efficiencies for each firm will be obtained. It is derived from the ratio of the TE value both the
CRS DEA and VRS DEA. Based on this calculation, it is known that the TE value in the CRS includes two components, one due to scale inefficiency and one due to
pure technical inefficiency. The firm is indicated that scale is inefficiencies if there is a difference in the
CRS and VRS TE value. Otherwise, the firm only has pure technical inefficiency if there is the same in the CRS and VRS value, and surely has the TE value
smaller than one. This concept is illustrated on the Figure 8.
Source: Coelli, Rao, Battese, 1998 Figure 6 Calculation of Scale Efficiency in DEA
CRS Frontier
VRS Frontier J
J
c
M J
v
K L
y
x NIRS Frontier
In Figure 8 illustrated scale inefficiency using one input and producing one output, under input orientated assumption. There are CRS and VRS frontier, and
non-increasing return to scale NIRS. Under CRS condition, the technical inefficiency of the point J is the distance between J and J
c
JJ
c
. While, the technical inefficiency of VRS condition would only be the distance between J and
J
v
JJ
v
. This difference is due to scale inefficiency. This is defined on Equation 3.12.
TE
CRS
= MJ
c
MJ TE
VRS
= MJ
v
MJ SE = MJ
c
MJ
v
3.12 Shortcoming arising from scale efficiency is the value does not indicate
whether the firm is in a condition increasing return to scale IRS or decreasing return to scale DRS. An additional DEA problem with non-increasing return to
scale NIR S can be determines this condition by changing the N1΄λ = 1 into N1΄λ
≤ 1 as on Equation 3.13. min
θ,λ
θ, st -y
i
+ Yλ ≥ 0, θx
i
– Xλ ≥ 0, N1΄λ ≤ 1
λ ≥ 0 3.13
Based on Figure 3.7, the firm is under IRS condition when the NIRS value is not equal to VRS value. This condition is illustrated by point J. While, if both
of values are equal then the DRS exist for that firm. This condition is shown by point K.