Inputoriented constant returns to scale DEA (CRS): min_{θ,λ}θ, subject to

Inputoriented variable returns to scale DEA (VRS): min_{θ,λ}θ, subject to

Inputoriented nonincreasing returns to scale DEA (NIRS): min_{θ,λ}θ, subject to


− q_{i} + Qλ ≥ 0

− q_{i} + Qλ ≥ 0

− q_{i} + Qλ ≥ 0

θ x_{i} − Xλ ≥ 0

θ x_{i} − Xλ ≥ 0

θ x_{i} − Xλ ≥ 0

λ ≥ 0

N1 ’ λ = 1

N1 ’ λ ≤ 1


λ ≥ 0

λ ≥ 0

 Where θ is the total technical efficiency score of the ith firm and λ is a Nx1 vector of constants. The value of θ must satisfy the restriction: 0 ≤ θ ≤ 1. If θ = 1; it indicates that the firm is on the production frontier and is technically efficient. When θ ≤ 1, the firm is technically inefficient.
 In the VRS DEA model, N1’λ = 1 is added to show a convexity constraint which ensures that an inefficient firm is only benchmarked against firms of a similar size. On the other hand, in the NIRS DEA model, N1’λ = 1 is replaced by N1’λ ≤ 1 to ensure that the ith firm is not “benchmarked” against firms that are substantially larger than it, but may be compared with firms smaller than it.
 Source: Summarized from Coelli et al. (1998).