An extension on super slacks-based measure DEA approach

Abstract

In order to break the tie of efficient decision-making units, super-efficiency data envelopment analysis is proposed to fully discriminate them. Recently, a slacks-based version of the super slacks-based measure (S-SBM) is developed and a novel two-stage approach is proposed to calculate both super-efficiency score by the S-SBM model and efficiency score by the slacks-based measure model. In this paper, we extend the approach to consider continuity of efficiency scores. We illustrate the discontinuity of efficiency measure, and define a continuous slacks-based measure which is proved continuous and directly calculated. An interesting efficiency zone category is also provided. In addition, this paper investigates the relationship among the super-efficiency measures of the proposed approach and some existing approaches under variable returns to scale.

Appendices

Appendix 1: Super-efficiency models under VRS

The standard input-oriented VRS super-efficiency DEA model for $DMU_o$ can be expressed as:

$$\begin{aligned}&{\begin{array}{ll} {\min } \theta \\ \end{array} } \nonumber \\&{\begin{array}{ll} {s.t.} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le \theta x_{io} ,\quad \forall i} \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} } \ge y_{ro} ,\quad \forall r} \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \\&{\begin{array}{ll} \quad \quad {\lambda _j \ge 0} \\ \end{array} },\quad \forall j,\quad j\ne o, \nonumber \\&{\begin{array}{ll} \quad \quad {\theta \,\, \hbox {free}} \\ \end{array} }. \nonumber \end{aligned}$$

(11)

Model (11) can be infeasible considering its VRS assumption. For example, model (11) is infeasible if an output component of $DMU_o$ is the unique maximum among all DMUs (Seiford and Zhu 1999). Similarly, the standard output-oriented VRS super-efficiency DEA model for $DMU_o$ can be expressed as:

$$\begin{aligned}&{\begin{array}{ll} {\max } \phi \\ \end{array} } \nonumber \\&{\begin{array}{ll} {s.t.} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le x_{io} ,\forall i} \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} } \ge \phi y_{ro} ,\forall r} \\ \end{array} }, \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\lambda _j \ge 0} \\ \end{array} },\forall j,j\ne o. \nonumber \end{aligned}$$

(12)

The optimal value of model (12) can be less than unity. Similarly, model (12) is infeasible if an input component of $DMU_o$ is the unique minimum among all DMUs (Seiford and Zhu 1999).

To solve the problem of infeasibility, Cook et al. (2009) developed the modified input-oriented VRS super-efficiency DEA model as follows:

$$\begin{aligned}&{\begin{array}{ll} {\min } {\tau +M\times \beta } \\ \end{array} } \nonumber \\&{\begin{array}{ll} {s.t.} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le (1+\tau )x_{io} ,\quad \forall i,} \\ \end{array} } \nonumber \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} \ge } (1-\beta )y_{ro} ,\quad \forall r} \\ \end{array} }, \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\beta \ge 0,\lambda _j \ge 0,\quad \forall j,\quad j\ne o} \\ \end{array} }. \nonumber \end{aligned}$$

(13)

And the output-oriented VRS super-efficiency DEA model is:

$$\begin{aligned}&{\begin{array}{ll} {\min } {\gamma +M\times \delta } \\ \end{array} } \nonumber \\&{\begin{array}{ll} {s.t.} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le (1+\delta )x_{io} ,\quad \forall i,} \\ \end{array} } \nonumber \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} \ge } (1-\gamma )y_{ro} ,\quad \forall r} \\ \end{array} }, \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{l} j=1 \\ j\ne o \\ \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\delta \ge 0,\lambda _j \ge 0,\quad \forall j,\quad j\ne o} \\ \end{array} }. \nonumber \end{aligned}$$

(14)

Note that models (13) and (14) are always feasible. If models (11) and (12) are infeasible, the super-efficiency scores of models (13) and (14) are defined as $1+\tau +1/{(1-\beta )}$ and $1+\delta +1/{(1-\gamma )}$, respectively.

Following Cook et al. (2009), a two-step approach was proposed to derive the super-efficiency score (See Lee et al. (2011). Chen and Liang (2011) transformed it into a single DEA-based model. The input-oriented VRS super-efficiency DEA model is shown as follows:

$$\begin{aligned} \begin{array}{l} {\begin{array}{ll} {\min }&{} {\tau +M\times \sum _{r=1}^s {\beta _r } } \\ \end{array} } \\ {\begin{array}{ll} {s.t.}&{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le (1+\tau )x_{io} ,\quad \forall i,} \\ \end{array} } \\ {\begin{array}{ll} \quad \quad &{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} \ge } (1-\beta _r )y_{ro} ,\quad \forall r} \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\beta _r \ge 0,\lambda _j \ge 0,\quad \forall j,\quad j\ne o} \\ \end{array} }. \\ \end{array} \end{aligned}$$

(15)

Corresponding output-oriented VRS super-efficiency DEA model is:

$$\begin{aligned} \begin{array}{l} {\begin{array}{ll} {\min }&{} {\gamma +M\times \sum _{i=1}^m {\delta _i } } \\ \end{array} } \\ {\begin{array}{ll} {s.t.}&{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le (1+\delta _i )x_{io} ,\quad \forall i,} \\ \end{array} } \\ {\begin{array}{ll} \quad \quad &{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} \ge } (1-\gamma )y_{ro} ,\quad \forall r} \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o } \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\delta _i \ge 0,\lambda _j \ge 0,\quad \forall j,\quad j\ne o} \\ \end{array} }. \\ \end{array} \end{aligned}$$

(16)

The super-efficiency scores of models (15) and (16) are $1+\tau +\frac{1}{R}\sum _{r\in R} {1/{(1-\beta _r )}}$, R is the set of $\beta _r >0$, and $\frac{1}{I}\sum _{i\in I} {(1+\delta _i )} +1/{(1-\gamma )}$, I is the set of $\delta _i >0$, respectively.

Different from Cook et al. (2009) and Lee et al. (2011) and Chen et al. (2011)’s VRS super-efficiency DEA model considered simultaneous input–output projection. Its super-efficiency score is defined as the ratio of input and output efficiencies. For a DMU that is not extreme-efficient (see the details below), the super-efficiency score can be measured through the following model:

$$\begin{aligned}&{\begin{array}{ll} {\min } {{\theta _o^{sr} }/{\varphi _o^{sr} }} \\ \end{array} } \nonumber \\&{\begin{array}{ll} {s.t.} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o } \end{array}}^n {\lambda _j x_{ij} } \le \theta _o^{sr} x_{io} ,\quad \forall i,} \\ \end{array} } \nonumber \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} \ge } \varphi _o^{sr} y_{ro} ,\quad \forall r,} \\ \end{array} } \\&{\begin{array}{ll} \quad \quad {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array} }^n {\lambda _j =1} } \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {\lambda _j \ge 0,\quad \forall j,\quad j\ne o} \\ \end{array} }, \nonumber \\&{\begin{array}{ll} \quad \quad {0<\theta _o^{sr} \le 1,\varphi _o^{sr} \ge 1} \\ \end{array} }. \nonumber \end{aligned}$$

(17)

For an extreme-efficient DMU, the super-efficiency score can be measured through the following model:

$$\begin{aligned} \begin{array}{l} {\begin{array}{ll} {\min }&{} {{\theta _o^{sr} }/{\varphi _o^{sr} }} \\ \end{array} } \\ {\begin{array}{ll} {s.t.}&{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j x_{ij} } \le \theta _o^{sr} x_{io} ,\quad \forall i} \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j y_{rj} \ge } \varphi _o^{sr} y_{ro} ,\quad \forall r,} \\ \end{array} } \\ {\begin{array}{ll} \quad \quad &{} {\sum \limits _{\begin{array}{c} {j=1} \\ {j\ne o} \end{array}}^n {\lambda _j =1} } \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\lambda _j \ge 0,\quad \forall j,\quad j\ne o} \\ \end{array} }, \\ {\begin{array}{ll} \quad \quad &{} {\theta _o^{sr} \ge 1} \\ \end{array} },0<\varphi _o^{sr} \le 1. \\ \end{array} \end{aligned}$$

(18)

The efficiency ${\theta _o^{sr} }/{\varphi _o^{sr} }$ can be interpreted as a product of the input- and output-oriented standard VRS super-efficiency scores.

Appendix 2: Proofs of Theorems

Theorem 1

The efficiency measure in Eq. (5), i.e. $CSBM_o$, is continuous.

Proof

Note that model (4) is equivalent to SBM model if $DMU_o$ is not SBM-efficient as shown in Fang et al. (2013). According to Chen (2013), we only need to examine the boundary conditions in order to prove continuity of Eq. (5), i.e. the discontinuous gap between the SBM score of a weakly efficient DMU A⁵ and S-SBM score of an SBM-efficient DMU B (DMU A is the projected DMU of DMU B). Without loss of generality, we only consider the case of data perturbation for $DMU_o$ under evaluation. Assume there exists a data perturbation $\delta >0$ for $DMU_o$ such that $\left\| {({X}'_o ,{Y}'_o )-(X_o ,Y_o )} \right\| \le \delta$. Assume that $({X}'_o ,{Y}'_o )$ and $(X_o ,Y_o )$ are input–output vectors of DMUs A and B, respectively. Then, ${x}'_{io} =x_{io} +w_i^{-*} ,{y}'_{ro} =y_{ro} -w_r^{+*}$. According to $\left\| {({X}'_o ,{Y}'_o )-(X_o ,Y_o )} \right\| \le \delta$, we have $\left| {{x}'_{io} -x_{io} } \right| =w_i^{-*} \le \delta$ and $0\le \mathop {\lim }\limits _{(X_o ,Y_o )\rightarrow ({X}'_o ,{Y}'_o )} (CSBM_o^B -CSBM_o^A )\le \mathop {\lim }\limits _{\delta \rightarrow 0} (\frac{1-\frac{1}{m}\sum _{i=1}^m {s_i^{-*} /x_{io} } +\delta /x_{io} }{1+\frac{1}{s}\sum _{r=1}^s {s_r^{+*} /y_{io} } -\delta /y_{io} }-\frac{1-\frac{1}{m}\sum _{i=1}^m {s_i^{-*} /x_{io} } }{1+\frac{1}{s}\sum _{r=1}^s {s_r^{+*} /y_{io} } })=0$. The efficiency score of DMU A is $CSBM_o^A =\frac{1-\frac{1}{m}\sum _{i=1}^m {s_i^{-*} /x_{io} } }{1+\frac{1}{s}\sum _{r=1}^s {s_r^{+*} /y_{io} } }$, and the efficiency score of DMU B is as follows:

$$\begin{aligned} {\textit{CSBM}}_o^B= & {} \frac{1-\frac{1}{m}\sum _{i=1}^m {(s_i^{-*} -w_i^{-*} )/x_{io} } }{1+\frac{1}{s}\sum _{r=1}^s {(s_r^{+*} -w_r^{+*} )/y_{io} } }\le \frac{1-\frac{1}{m}\sum _{i=1}^m {(s_i^{-*} -\delta )/x_{io} } }{1+\frac{1}{s}\sum _{r=1}^s {(s_r^{+*} -\delta )/y_{io} } }\\= & {} \frac{1-\frac{1}{m}\sum _{i=1}^m {s_i^{-*} /x_{io} } +\delta /x_{io} }{1+\frac{1}{s}\sum _{r=1}^s {s_r^{+*} /y_{io} } -\delta /y_{io} } \end{aligned}$$

Note that $CSBM_o^B \ge CSBM_o^A$. Then, we have:

$$\begin{aligned} 0\le & {} \mathop {\lim }\limits _{(X_o ,Y_o )\rightarrow ({X}'_o ,{Y}'_o )} (CSBM_o^B -CSBM_o^A )\le \mathop {\lim }\limits _{\delta \rightarrow 0} \left( \frac{1-\frac{1}{m}\sum _{i=1}^m {s_i^{-*} /x_{io} } +\delta /x_{io} }{1+\frac{1}{s}\sum _{r=1}^s {s_r^{+*} /y_{io} } -\delta /y_{io} }\right. \\&\left. -\frac{1-\frac{1}{m}\sum _{i=1}^m {s_i^{-*} /x_{io} } }{1+\frac{1}{s}\sum _{r=1}^s {s_r^{+*} /y_{io} }}\right) =0 \end{aligned}$$

Then, we have $\mathop {\lim }\limits _{(X_o ,Y_o )\rightarrow ({X}'_o ,{Y}'_o )} (CSBM_o^B -CSBM_o^A )=0$. So the efficiency measure $CSBM_k$ in Eq. (5) is continuous. It completes the proof of Theorem 1. $\square$

Theorem 2

If model (8) is feasible, then $\theta ^{*}\ge \eta ^{*}$ and $\phi ^{*}\ge \phi _0^*\ge 1$.

Proof

If model (8) is feasible, then $0<\theta _0^*\le 1,\phi _0^*\ge 1$. This leads to a feasible solution of model (11) subject to $\theta =\theta _0^*$. So the optimal value of model (11) is subject to $\theta ^{*}\le \theta _0^*\le 1$. Thus leads to a feasible solution of model (8) subject to $\theta _i =\theta ^{*},\phi _r =1,\forall i,r;\eta =\theta ^{*}$. So $\theta ^{*}\ge \eta ^{*}$. Analogously, we have $\phi ^{*}\ge \phi _0^*\ge 1$. It completes the proof of Theorem 2. $\square$

Theorem 3

The following is true for model (10):

• (i) model (10) is always feasible and $\hat{{\eta }}^{*}\ge 1$.

• (ii) $DMU_o$ is extreme-efficient if and only if $\hat{{\eta }}^{*}>1$.

• (iii) If model (11) is feasible and $\theta ^{*}>1$, then $\theta ^{*}\ge \hat{{\eta }}^{*}>1$.

• (iv) If model (11) is feasible and $0<\theta ^{*}\le 1$, then $\theta ^{*}\le \hat{{\eta }}^{*}=1$.

• (v) If model (12) is feasible and $0<\phi ^{*}<1$, then $1/\phi ^{*}\ge \hat{{\eta }}^{*}>1$.

• (vi) If model (12) is feasible and $\phi ^{*}\ge 1$, then $1/\phi ^{*}\le \hat{{\eta }}^{*}=1$.

Proof

Let $\hat{{\theta }}_0 =\max \{\hat{{\theta }}_i \}={\mathop {\max }\limits _{i,j} \{x_i^j \}}/{\mathop {\min }\limits _i \{x_i^k \}},\hat{{\phi }}_0 =\min \{\hat{{\phi }}_r \}={\mathop {\min }\limits _{r,j} \{y_r^j \}}/{\mathop {\max }\limits _r \{y_r^k \}}$, then there must exist a feasible solution $(\hat{{\lambda }}_j ,\hat{{\theta }}_0 ,\hat{{\phi }}_0 )$ for model (10) and $\hat{{\eta }}^{*}\ge 1$. (ii) Assume that $DMU_o$ is extreme-efficient and $\hat{{\eta }}^{*}=1$, then $\hat{{\theta }}_i^*=1,\hat{{\phi }}_r^*=1,\forall i,r$. This leads to feasible solution of models (11) and (12), i.e., $\theta ^{*}=1,\phi ^{*}=1$. Thus, $DMU_o$ is not extreme-efficient, a contradiction. So $\hat{{\eta }}^{*}>1$. On the other hand, assume that $\hat{{\eta }}^{*}>1$ and $DMU_o$ is not extreme-efficient. Then, models (11) and (12) are feasible and $0<\theta ^{*}\le 1,\phi ^{*}\ge 1$. Thus, $\hat{{\theta }}_i^*=1,\hat{{\phi }}_r^*=1,\forall i,r$ is an optimal solution of model (10). So $\hat{{\eta }}^{*}=1$, a contradiction. Thus, $DMU_o$ is extreme-efficient. (iii) If model (11) is feasible and $\theta ^{*}>1$, then it leads to a feasible solution for model (10) subject to $\hat{{\theta }}_0 =\theta ^{*},\hat{{\phi }}_r =1,\forall r$. Then, $\theta ^{*}\ge \hat{{\eta }}^{*}>1$. (iv) If $0<\theta ^{*}\le 1$, then $\hat{{\theta }}_i^*=1,\hat{{\phi }}_r^*=1,\forall i,r$ is an optimal solution for model (10). Thus, $\theta ^{*}\le \hat{{\eta }}^{*}=1$. Analogous to the proofs for (iii) and (iv), It is easy to prove (v) and (vi). It completes the proof of Theorem 3. $\square$

Theorem 4

The optimal values of models (8) and (10) are not greater than those of models (17) and (18), respectively, i.e., $\eta ^{*}\le \rho _1^*$ and $\hat{{\eta }}^{*}\le \rho _2^*$.

Proof

Let $\theta _0 =\max \{\theta _i \}=\theta _0^{s1*} ,\phi _0 =\min \{\phi _r \}=\varphi _0^{s1*}$, then there must exist a feasible solution $(\lambda _j ,\theta _i ,\phi _r )$ of model (8) with an objective function value $\eta$ subject to $\eta \le \rho _1^*$. Thus, the optimal value of model (8) is not greater than that of model (17), i.e. $\eta ^{*}\le \rho _1^*$. Similarly, we can have $\hat{{\eta }}^{*}\le \rho _2^*$. It completes the proof of Theorem 4. $\square$

Theorem 5

For a DMU, the super-efficiency score of Eq. (5) is minimal among all the efficiencies in models (11)–(18) if models (11) and (12) are feasible.

Proof

First, it is clear that a DMU’s efficiency score obtained by Eq. (5) is not larger than that calculated by model (8) or (10). For a DMU, if models (11) and (12) are feasible, then the input- and output-oriented super-efficiency score under models (13) and (14) or models (15) and (16) are equal to models (11) and (12), respectively. For an extreme-efficient DMU, we have $\hat{{\eta }}^{*}>1$ according to (ii) in Theorem 3. As models (11) and (12) are feasible, we have $\theta ^{*}\ge \hat{{\eta }}^{*}$ and $1/\phi ^{*}\ge \hat{{\eta }}^{*}$ according to (iii) and (v) in Theorem 3, respectively. It indicates that the super-efficiency score of model (10) is not greater than that of model (11) or (12). For a DMU that is not extreme-efficient, model (8) is feasible. According to Theorem 2, the super-efficiency score under model (8) is not greater than that under models (11) and (12). Hence, the super-efficiency score of Eq. (5) is not greater than that of models (11) or (12). According to Theorem 4, the super-efficiency of models (8) and (10) is not greater than that of models (17) and (18), respectively. Then the super-efficiency score of Eq. (5) is minimal if a DMU is feasible under models (11) and (12). It completes the proof of Theorem 5. $\square$