On classifying decision making units in DEA: a unified dominance-based model

Abstract

In data envelopment analysis (DEA), the concept of efficiency is examined in either Farrell (DEA) or Pareto senses. In either of these senses, the efficiency status of a decision making unit (DMU) is classified as either weak or strong. It is well established that the strong DEA efficiency is both necessary and sufficient for achieving the Pareto efficiency. For the weak Pareto efficiency, however, the weak DEA efficiency is only sufficient, but not necessary in general. Therefore, a DEA-inefficient DMU can be either weakly Pareto efficient or Pareto inefficient. Motivated by this fact, we propose a new classification of DMUs in terms of both DEA and Pareto efficiencies. To make this classification, we first demonstrate that the Farrell efficiency is based on the notion of FGL dominance. Based on the concept of dominance, we then propose and substantiate an alternative single-stage method. Our method is computationally efficient since (1) it involves solving a unique single-stage model for each DMU, and (2) it accomplishes the classification of DMUs in both input and output orientations simultaneously. Finally, we present a numerical example to illustrate our proposed method.

Appendix

Proof of Lemma 3.1

Since part (ii) is an immediate consequence of part (i), we prove only part (i).

Let $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in NE_I$. By Definition  2.1, there exists $\hat{{\theta }}<1$ such that $\left( {\hat{{\theta }}\mathbf{x}_o ,\mathbf{y}_o } \right) \in T^{DEA}$. Since $\hat{{\theta }}<1$, we have $\mathbf{x}_o >^{+}\hat{{\theta }}\mathbf{x}_o$. Hence, by Definition 3.1, it follows that $\left( {\hat{{\theta }}\mathbf{x}_o ,\mathbf{y}_o } \right) {\succeq }_I \left( {\mathbf{x}_o ,\mathbf{y}_o } \right)$.

Conversely, let $\left( {{{\hat{\mathbf{x}}}},{{\hat{\mathbf{y}}}}} \right) \in T^{DEA}$ dominates $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right)$ in the FGL sense. By Definition 3.1, we have $\mathbf{x}_o >^{+}{{\hat{\mathbf{x}}}}$, and ${{\hat{\mathbf{y}}}}{\ge }\mathbf{y}$. We define $\hat{{\theta }}:=\max \left\{ {\frac{\hat{{x}}_i }{x_{io} }\left| {\hbox { }x_{io} >0} \right. } \right\}$. Then, $\hat{{\theta }}<1$ and $\hat{{\theta }}\mathbf{x}_o \,{\geqq }\,{{\hat{\mathbf{x}}}}$. Since $\left( {{{\hat{\mathbf{x}}}},{{\hat{\mathbf{y}}}}} \right) \in T^{DEA}$, as per the strong disposability assumption, $\left( {\hat{{\theta }}\mathbf{x}_o ,\mathbf{y}_o } \right) \in T^{DEA}$. Hence, the optimal value of $\theta ^{*}$ in model (5) is less than one, indicating $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in NE_I$.

Proof of Proposition 3.1

Note that parts (iii), (v) and (vii) are immediate consequences of parts (iv), (vi) and (viii), respectively. Moreover, the proof of part (vi) is similar to that of part (iv). Therefore, we are left with in proving only parts (i), (ii), (iv) and (viii).

Part (i) Let $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in E$ and assume by contradiction that $\left( {\begin{array}{l} \mathbf{x}_o -{{\hat{\mathbf{s}}}}^{-} \\ \mathbf{y}_o +{{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) \in \Omega _o$. Let ${{\hat{{\varvec{\upmu }}}}}$ be the intensity variable corresponding to this element in (11). For any direction vector $\mathbf{g}$, $\left( {{\beta }^{\prime },{\varvec{\upmu }^{\prime }},\mathbf{s}^{{-}^{\prime }},\mathbf{s}^{{+}^{\prime } }} \right) :=\left( {0,{{\hat{{\varvec{\upmu }} }}},{{\hat{\mathbf{s}}}}^{-},{{\hat{\mathbf{s}}}}^{+}} \right)$ is then a feasible solution to model (8). Since the first phase of model (8) is a minimization problem, the optimal value of $\beta$ is non-positive, which is a contradiction by Proposition 2.4.

In a similar way, the converse can be proved by the way of contradiction and with the help of Proposition 2.4.

Part (ii) Let $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in {E}^{\prime }$. Then, part (ii) of Proposition 2.4 follows that $\left\{ {\left( {\begin{array}{l} \mathbf{x}_o \\ \mathbf{y}_o \\ \end{array}} \right) } \right\} \subseteq \Omega _o$. To prove the equality, assume by contradiction that $\left( {\begin{array}{l} \mathbf{x}_o -{{\hat{\mathbf{s}}}}^{-} \\ \mathbf{y}_o +{{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) \in \Omega _o$ such that $\left( {\begin{array}{l} {{\hat{\mathbf{s}}}}^{-} \\ {{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) \ge \mathbf{0}_{m+s}$. Let ${{\hat{{\varvec{\upmu }} }}}$ be the intensity variable corresponding to this element in (11). For any direction vector $\mathbf{g}$, $\left( {{\beta }^{\prime },{\varvec{\upmu }^{\prime }},\mathbf{s}^{{-}^{\prime }},\mathbf{s}^{{+}^{\prime } }} \right) :=\left( {0,{{\hat{{\varvec{\upmu }} }}},{{\hat{\mathbf{s}}}}^{-},{{\hat{\mathbf{s}}}}^{+}} \right)$ is then a feasible solution to model (8). Since the second phase of model (8) is a maximization problem, the optimal sum of slacks is positive, which is a contradiction by Proposition 2.4.

In a similar way, the converse can be proved by the way of contradiction and with the help of Proposition 2.4.

Part (iv) By Lemma 3.1, $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in NE_I$ if and only if there exists $\left( {{{\hat{\mathbf{x}}}},{{\hat{\mathbf{y}}}}} \right) \in T^{DEA}$ such that $\mathbf{x}_o >^{+}{{\hat{\mathbf{x}}}}$ and $\mathbf{{y}^{\prime }}{\ge }\mathbf{y}_o$. We define ${{\hat{\mathbf{s}}}}^{-}:=\mathbf{x}_o -{{\hat{\mathbf{x}}}}$ and ${{\hat{\mathbf{s}}}}^{+}:={{\hat{\mathbf{y}}}}-\mathbf{y}_o$. Then, $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in NE_I$ if and only if (i) ${{\hat{\mathbf{s}}}}^{-}\,{\geqq }\,\mathbf{0}_m$ and ${{\hat{\mathbf{s}}}}^{+}\,{\geqq }\,\mathbf{0}_s$, and (ii) $\hat{{s}}_i^- >0\Leftrightarrow x_{io} >0$; or, if and only if $\left( {\begin{array}{l} \mathbf{x}_o -{{\hat{\mathbf{s}}}}^{-} \\ \mathbf{y}_o +{{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) \in \bar{{\Omega }}_o$ and $n^{+}\left( {{{\hat{\mathbf{s}}}}^{-}} \right) =n^{+}\left( {\mathbf{x}_o } \right)$.

Part (viii) By Definition 2.3, $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in NE_P$ if and only if there exists $\left( {{{\hat{\mathbf{x}}}},{{\hat{\mathbf{y}}}}} \right) \in T^{DEA}$ that strongly dominates $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right)$. We define ${{\hat{\mathbf{s}}}}^{-}:=\mathbf{x}_o -{{\hat{\mathbf{x}}}}$ and ${{\hat{\mathbf{s}}}}^{+}:={{\hat{\mathbf{y}}}}-\mathbf{y}_o$. Then, $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right) \in NE_P$ if and only if $\left( {\begin{array}{l} {{\hat{\mathbf{s}}}}^{-} \\ {{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) >\mathbf{0}_{m+s}$; or, if and only if $\left( {\begin{array}{l} \mathbf{x}_o -{{\hat{\mathbf{s}}}}^{-} \\ \mathbf{y}_o +{{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) \in \bar{{\Omega }}_o$ and $n^{+}\left( {{{\hat{\mathbf{s}}}}^{-}} \right) +n^{+}\left( {{{\hat{\mathbf{s}}}}^{+}} \right) =m+s$. $\square$

Proof of Lemma 3.2

First, assume that model (15) is infeasible. Then, system (14) is infeasible and, by Corollary 3.1, $\left( {\mathbf{x}_o ,\mathbf{y}_o } \right)$ is an extreme-efficient DMU.

Now, let model (15) be feasible. Then, this model has an optimal solution since its objective function is upper bounded by $m+s$. Hence, let $\left( {{{\varvec{\updelta }} }^{*},\mathbf{t}^{-*},\mathbf{t}^{+*}} \right)$ be an optimal solution to model (15). We claim that the positive components of the vectors $\mathbf{t}^{-*}$ and $\mathbf{t}^{+*}$ are all equal to one. Since the proofs are similar for these vectors, we prove the assertion only for $\mathbf{t}^{-*}$.

By way of contradiction, assume that $0<t_g^{-*} <1$ for some $g\in \left\{ {1,\ldots ,m} \right\}$. Dividing both sides of the constraints of (15) at optimality by $t_g^{-*}$ yields

$$\begin{aligned}&\mathbf{X}_o \left( {\frac{1}{t_g^{-*} }{{\varvec{\updelta }} }^{*}} \right) +\frac{1}{t_g^{-*} }\mathbf{t}^{-*}{\leqq }\left( {\frac{\mathbf{1}_{n-1}^T {{\varvec{\updelta }} }^{*}}{t_g^{-*} }} \right) \mathbf{x}_o ,\nonumber \\&\mathbf{Y}_o \left( {\frac{1}{t_g^{-*} }{{\varvec{\updelta }} }^{*}} \right) -\frac{1}{t_g^{-*} }\mathbf{t}^{+*}\,{\geqq }\,\left( {\frac{\mathbf{1}_{n-1}^T {{\varvec{\updelta }} }^{*}}{t_g^{-*} }} \right) \mathbf{y}_o , \nonumber \\&\frac{\mathbf{1}_{n-1}^T {{\varvec{\updelta }} }^{*}}{t_g^{-*} }\ge 1, \end{aligned}$$

(16)

Then, according to (16), the vector $\left( {{\varvec{\updelta }^{\prime }},\mathbf{t}^{{-}^{\prime }},\mathbf{t}^{{+}^{\prime }}} \right)$ defined by

$$\begin{aligned}&{\varvec{\updelta }^{\prime }}:=\frac{1}{t_g^{-*}}{{\varvec{\updelta }} }^{*}, \nonumber \\&t_i^{- ^{\prime }}:=\min \left\{ {1,\frac{t_i^{-*} }{t_g^{-*} }} \right\} ,\hbox { for }i=1,\ldots ,m,\nonumber \\&t_r^{+ ^{\prime }}:=\min \left\{ {1,\frac{t_r^{+*} }{t_g^{-*} }} \right\} ,\hbox { for }r=1,\ldots ,s, \end{aligned}$$

(17)

is a feasible solution to model (15) whose objective function value is strictly greater than $\mathbf{1}_m^T \mathbf{t}^{-*}+\mathbf{1}_s^T \mathbf{t}^{+*}$. This contradicts the optimality of $\left( {{{\varvec{\updelta }} }^{*},\mathbf{t}^{-*},\mathbf{t}^{+*}} \right)$ and proves our claim. $\square$

Proof of Proposition 3.4

According Lemma 3.2, it is obvious that $\mathbf{1}_m^T \mathbf{t}^{-*}=n^{+}\left( {\mathbf{t}^{-*}} \right)$ and $\mathbf{1}_s^T \mathbf{t}^{+*}=n^{+}\left( {\mathbf{t}^{+*}} \right)$. Moreover, as in the paragraph below proposition 3.2, it is straightforward to show that $\left( {\begin{array}{l} \mathbf{x}_o -\frac{1}{\sigma ^{*}}\mathbf{t}^{-*} \\ \mathbf{y}_o +\frac{1}{\sigma ^{*}}\mathbf{t}^{+*} \\ \end{array}} \right) \in \Omega _o$.

Let $\left( {\begin{array}{l} \mathbf{x}_o -{{\hat{\mathbf{s}}}}^{-} \\ \mathbf{y}_o +{{\hat{\mathbf{s}}}}^{+} \\ \end{array}} \right) \in \Omega _o$ for which $n^{+}\left( {{{\hat{\mathbf{s}}}}^{-}} \right) +n^{+}\left( {{{\hat{\mathbf{s}}}}^{+}} \right)$ is maximum. Then, we have $\mathbf{1}_m^T \mathbf{t}^{-*}+\mathbf{1}_s^T \mathbf{t}^{+*}=n^{+}\left( {\mathbf{t}^{-*}} \right) +n^{+}\left( {\mathbf{t}^{+*}} \right) \le n^{+}\left( {{{\hat{\mathbf{s}}}}^{-}} \right) +n^{+}\left( {{{\hat{\mathbf{s}}}}^{+}} \right)$. From Proposition 3.2, we also know that there exists a feasible solution $\left( {{{\hat{{\varvec{\updelta }} }}},{{\hat{\mathbf{t}}}}^{-},{{\hat{\mathbf{t}}}}^{+}} \right)$ for model (15) such that $\mathbf{1}_m^T {{\hat{\mathbf{t}}}}^{-}+\mathbf{1}_s^T {{\hat{\mathbf{t}}}}^{+}=n^{+}\left( {{{\hat{\mathbf{s}}}}^{-}} \right) +n^{+}\left( {{{\hat{\mathbf{s}}}}^{+}} \right)$. Therefore, the proof is complete by the fact that model (15) is a maximization LP problem. $\square$