A variation of two-stage SBM with leader–follower structure: an application to Chinese commercial banks

Abstract

The two-stage slack-based measure (SBM) model has many applications in the real world. Due to the limitations of the SBM model on which it is based, the two-stage SBM model unfortunately gives unrealistically low efficiencies and rather far projections (Tone in Eur J Oper Res 197(1):243–252, 2010) for inefficient decision-making unit. Based on the novel idea in Tone (2010), this paper proposes a variation of the two-stage SBM model by incorporating a leader–follower structure and applies the proposed approach to Chinese commercial banks. The results show that our proposed approach can increase efficiencies of inefficient banks and halve the projection distance of some inefficient banks.

Appendix

Models (10) (the second stage) and (11) (the first stage) are suitable for the situation in which, contrary to the example above, the first stage is follower stage and the second stage is leader stage. In such situations, the calculation should be started from the second stage. After obtaining the optimal solution $\left( {\rho_{2}^{*} ,\varvec{\lambda}^{{2\varvec{*}}} ,\varvec{c}^{ - *} ,\varvec{s}^{ + *} } \right)$ from model (10), substitute $\varvec{c}^{ - *}$ into model (11) to get the first-stage optimal solution $\left( {\rho_{1}^{*} ,\varvec{\lambda}^{{1\varvec{*}}} ,\varvec{s}^{ - *} } \right)$. Then calculate $\rho_{1}^{*} *\rho_{2}^{*}$ as the two-stage process efficiency.

$$ \rho_{2} = \hbox{min} \frac{{1 - \frac{1}{D}\varSigma_{d = 1}^{D} \frac{{c_{d}^{ - } }}{{z_{0d} }}}}{{1 + \frac{1}{s}\varSigma_{r = 1}^{s} \frac{{s_{d}^{ + } }}{{y_{{0{\text{d}}}} }}}} $$

subject to

$$ \begin{aligned} \begin{array}{*{20}l} {\varSigma_{j = 1}^{n} z_{jd} \lambda_{j}^{2} = z_{0d} - c_{d}^{ - } ,} \hfill & {\left( {\forall d} \right)} \hfill \\ {\varSigma_{j = 1}^{n} y_{jr} \lambda_{j}^{2} = y_{0r} + s_{r}^{ + } ,} \hfill & {\left( {\forall r} \right)} \hfill \\ {\lambda_{j}^{1} \ge 0,} \hfill & {\left( {\forall j} \right)} \hfill \\ {c_{d}^{ - } ,s_{r}^{ + } \ge 0,} \hfill & {\left( {\forall d,\forall r} \right).} \hfill \\ \end{array} \hfill \\ \hfill \\ \end{aligned} $$

(10)

$$ \rho_{2} = \hbox{min} \left( {1 - \frac{1}{m}\varSigma_{i = 1}^{m} \frac{{s_{i}^{ - } }}{{d_{0i} }}} \right) $$

subject to

$$ \begin{aligned} \begin{array}{*{20}l} {\varSigma_{j = 1}^{n + 1} x_{ji} \lambda_{j}^{1} = x_{0i} - s_{i}^{ - } ,} \hfill & {\left( {\forall i} \right)} \hfill \\ {\varSigma_{j = 1}^{n + 1} z_{jd} \lambda_{j}^{1} = z_{0d} - c_{d}^{ - *} ,} \hfill & {\left( {\forall d} \right)} \hfill \\ {\lambda_{j}^{1} \ge 0,} \hfill & {\left( {\forall j} \right)} \hfill \\ {s_{i}^{i} , \ge 0,} \hfill & {(\forall i).} \hfill \\ \end{array} \hfill \\ \hfill \\ \end{aligned} $$

(11)