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,
λ ≥ 0, Where:
θ is a scalar λ is a Nx1 vector of constants.
3.10 This envelopment form covers fewer constraints than the multiplier form
K+M N+1, and therefore generally the preferred form to solve. The value of θ will become the efficiency score for each firm.
It must be less than or equal to 1 θ ≤ 1, with a value of one indicating a point on the frontier. According to the
Farrell, 1957 definition, this condition is known as technically efficient firm.
3.4 Variable Return to Scale VRS DEA Model
CRS DEA model only appropriate used when a firm is in an optimal scale condition. There are a few reason could make a firm to be not operating at an
optimal scale, such as imperfect competition, finance constraint, etc. The alternative model to solving this problem is the return to scale RTS DEA model
Banker, Charnes, Cooper, 1984. The model is also known as a BCC DEA model.
This model permits the calculation of TE without the scale efficiency SE effect. This effect may occur if the CRS model is used when not all firms are at
optimal scale, and leading to the calculation of TE influenced by this effect. The formulation for this model defined on Equation 3.11.
max
θ,λ
θ, st -y
i
+ Yλ ≥ 0, θx
i
– Xλ ≥ 0, N1΄λ = 1
λ ≥ 0 where:
N1 is an Nx1 vector of ones N1΄λ is the convexity constraint
3.11
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
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