Input-Oriented SBM (SBM-I-C)

The input-oriented SBM under a constant-returns-to-scale assumption [76] is described as follows:

$$\rho\_I^\* = \min\_{\beta, d^-, d^+} 1 - \frac{1}{m} \sum\_{i=1}^m \frac{d\_i^-}{x\_{ik}}$$
 S.t

 
$$\begin{aligned} x\_{ic} &= \sum\_{j=1}^m x\_{ic} \beta\_i + d\_i^- \;/\; i = 1, \; 2, \; \dots \; m \\ y\_{rc} &= \sum\_{j=1}^m y\_{rc} \beta\_i - d\_r^+ \;/\; i = 1, \; 2, \dots \; d \\ \beta\_j &\ge 0, \; k \; (\forall j), \; d\_i^- \ge 0 \; (\forall j), d\_r^+ \ge 0 \; (\forall j) \end{aligned} \tag{27}$$

The DMUs in the reference set R of (*xc*, *yc*) are SBM-input-efficient. In addition, the SBM-input-efficiency score must is lower than the CCR efficiency score.
