Extra Resource Allocation: A DEA Approach in the View of Efficiencies

Abstract

Data envelopment analysis has been successfully used in resource allocation problems. However, to the best of our knowledge, there are no allocation models proposed in the literature that simultaneously take both the global efficiency and growing potential into account. Hence, this research aims at developing an allocation model for extra input resources, which maximizes the global technical efficiency and scale efficiency of a decision-making unit (DMU) set while maintaining the pure technical efficiency (i.e., growing potential) of each DMU. To this purpose, we first discuss the optimal resources required by each DMU. We prove that the optimal inputs for the DMU are actually the inputs of some most productive scale size (MPSS). We then propose the allocation model based on the discussion on the case of one DMU. The allocation model is illustrated using two numerical examples.

Appendix: Technical Proofs

1.1 Proof of Lemma 2.1

Proof

Since model $(\mathrm LP_0)$ is the dual model of model $(\mathrm DLP_0)$ (see [31]), only one of them needs to be discussed. Here we only discuss model $(\mathrm DLP_0)$.

According to the conventional DEA theory, the calculation of the efficiency score of $\mathrm {DMU_{o}^{'}}$ follows the following model:

$$\begin{aligned}&(\mathrm DLP_0') \quad \max \quad \phi _\mathrm{o}' \nonumber \\&\mathrm{s.t.} \quad \varLambda X^\mathrm{T} + \lambda _{n+1}(X_\mathrm{o}+{\Delta X}_\mathrm{o}) \leqslant X_\mathrm{o}+{\Delta X}_\mathrm{o}, \end{aligned}$$

(5.1)

$$\begin{aligned}&\qquad \,\, \varLambda Y^\mathrm{T} + \lambda _{n+1}(Y_\mathrm{o}+{\Delta Y}_\mathrm{o}) \geqslant \phi _\mathrm{o}' (Y_\mathrm{o}+{\Delta Y}_\mathrm{o}), \nonumber \\&\qquad \,\, \delta _1(\varLambda \textit{e}+\delta _2(-1)^{\delta _3}w)=\delta _1, \nonumber \\&\qquad \,\, \varLambda \geqslant 0,\quad w \geqslant 0, \quad \lambda _{n+1} \geqslant 0, \nonumber \\&\qquad \,\, \phi _\mathrm{o}' \quad \mathrm {unconstrained}. \end{aligned}$$

(5.2)

Suppose that the optimal solution of model (DLP$_0)$ is $(\phi ^*,\varLambda ^*,w^*)$ and that of model (DLP$_0' )$ is $(\overline{\phi }, \overline{\varLambda }, \overline{\lambda }_{n+1}, \overline{w})$. Then, $(\phi ^*,\varLambda ^*,\lambda _{n+1}=0,w^*)$ is a feasible solution of (DLP$_0')$, which indicates that $\phi ^* \leqslant \overline{\phi }$. Considering constraints (2.2), we know that $\phi _\mathrm{o}'=1$ is a feasible solution of (DLP$_0)$. Thus, we have

$$\begin{aligned} 1 \leqslant \phi ^* \leqslant \overline{\phi }. \end{aligned}$$

(5.3)

Considering constraint (5.1) in model (DLP$_0')$, we have $\lambda _{n+1} \leqslant 1$. Otherwise, we will have $\varLambda X^\mathrm{T} < 0$ which contradicts the constraint $\varLambda \geqslant 0$. Next, we discuss in terms of $\overline{\lambda }_{n+1}$.

If $\overline{\lambda }_{n+1}=1$, we have $\varLambda X^\mathrm{T}=0$ and $\varLambda =0$. Considering constraint (5.2), $\overline{\phi }= 1$ holds. Therefore, $\phi ^* = \overline{\phi }= 1$.

If $0< \overline{\lambda }_{n+1} < 1$, the following transformation is performed:

$$\begin{aligned}&\widetilde{\varLambda } = \frac{{\overline{\varLambda }}}{{1 - \overline{\lambda }_{n + 1} }}, \quad \widetilde{w} = \frac{{\overline{w} }}{{1 - \overline{\lambda }_{n + 1} }}, \quad \widetilde{\phi }= \frac{{\overline{\phi }- \overline{\lambda }_{n+1} }}{{1 - \overline{\lambda }_{n + 1} }}. \end{aligned}$$

We can conclude that $(\widetilde{\phi }, \widetilde{\varLambda }, \widetilde{w})$ is a feasible solution of model (DLP$_0)$. Therefore, $\widetilde{\phi }\leqslant \phi ^*$ holds.

If $\overline{\phi }= 1$, formula (5.3) indicates $\phi ^* = \overline{\phi }= 1$.

If $\overline{\phi }> 1$, then we have

$$\begin{aligned} \overline{\phi }- \widetilde{\phi }= \overline{\phi }-\frac{{\overline{\phi }- \overline{\lambda }_{n+1} }}{{1 - \overline{\lambda }_{n + 1} }}=\frac{{\overline{\lambda }_{n+1}(1-\overline{\phi })}}{{1 - \overline{\lambda }_{n + 1} }}<0 \end{aligned}$$

and $\overline{\phi }< \widetilde{\phi }\leqslant \phi ^*$, which contradicts formula (5.3). Therefore, $\overline{\phi }>1$ is impossible.

If $\overline{\lambda }_{n+1}=0$, $(\overline{\phi },\overline{\varLambda },{\overline{\varLambda }_{n+1}},\overline{w})$ is a feasible solution of model (DLP$_0)$. Thus, we find that $\phi ^* \geqslant \overline{\phi }$. Therefore, $\phi ^*=\overline{\phi }$ holds due to formula (5.3).

To summarize, model (DLP$_0)$ has the same optimal solution as model (DLP$_0')$. Therefore, when $\mathrm DMU_{o}^{'}$ satisfies constraints (2.2), models (DLP$_0)$ and (DLP$_0')$ are equivalent, and both of them can be used to obtain the efficiency score of $\mathrm DMU_{o}^{'}$.

1.2 Proof of Theorem 2.1.1

Proof

According to Lemma 2.1, use the CCR version of model (LP$_0)$ to solve the CCR efficiency of $\mathrm DMU_{o}^{'}$. The optimal value of model (LP$_0)$ is the CCR efficiency of $\mathrm DMU_{o}^{'}$ and we assume that the optimal solution is $(\widetilde{v}, \widetilde{u})$. $(v^*, u^*)$, which is part of the optimal solution of model (OPTSE), is a feasible solution of model (LP$_0)$ . We have

$$\begin{aligned} {\widetilde{v}}^\mathrm{T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) \leqslant v^\mathrm{*T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*). \end{aligned}$$

And considering that $\widetilde{S}=(\widetilde{v}, \widetilde{u}, \Delta X_\mathrm{o}^*, \Delta Y_\mathrm{o}^*, \varLambda ^*)$ is a feasible solution of model (OPTSE),

$$\begin{aligned} {\widetilde{v}}^\mathrm{T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) \geqslant v^\mathrm{*T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) \end{aligned}$$

holds.

Therefore, we have proved ${\widetilde{v}}^\mathrm{T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) = v^{v*\mathrm{T}}(X_\mathrm{o}+\Delta X_\mathrm{o}^*)$ indicating that the CCR efficiency score of $\mathrm DMU_{o}^{'}$ is the optimal value of model (OPTSE).

1.3 Proof of Lemma 2.2

Proof

Proof of sufficiency:

The Pareto solution of model (InvBCC) is $S^*=(\Delta X_\mathrm{o}^*,\varLambda ^*)$. Considering the case of multi-solutions, $\varLambda ^*$ can have many proper values that meet condition (2.6). Applying the BCC version of model (DLP$_0)$ to $\mathrm DMU_{o}^{'}$, we can find that $(\phi _\mathrm{o}^\mathrm{BCC},\varLambda ^*)$ is one of its feasible solutions. Suppose that its optimal solution is $(\overline{\phi }, \overline{\varLambda })$, we have $\overline{\phi }\geqslant \phi _\mathrm{o}^\mathrm{BCC}$ and

$$\begin{aligned} \overline{\varLambda }{Y^\mathrm{T}} \geqslant \overline{\phi }\left( {{Y_\mathrm{o}} + \Delta Y_\mathrm{o}^0} \right) \geqslant \phi _\mathrm{o}^\mathrm{BCC}\left( {{Y_\mathrm{o}} + \Delta Y_\mathrm{o}^0} \right) . \end{aligned}$$

Therefore, $(\Delta X_\mathrm{o}^*,\overline{\varLambda })$ is a feasible solution of model (InvBCC), and its corresponding objective value is $\Delta X_\mathrm{o}^*$, indicating that $(\Delta X_\mathrm{o}^*, \overline{\varLambda })$ is a Pareto solution of model (InvBCC).

If we assume that $\overline{\phi }> \phi _\mathrm{o}^\mathrm{BCC}$, then the inequality $\overline{\varLambda }Y^\mathrm{T} > \phi _\mathrm{o}^\mathrm{BCC} (Y_\mathrm{o}+\varLambda Y_\mathrm{o}^0 )$ holds, which contradicts condition (2.6). Thus, the assumption is invalid, and $\overline{\phi }=\phi _\mathrm{o}^\mathrm{BCC}$ holds.

Proof of necessity:

Suppose that the optimal solution generated by solving the BCC efficiency of $\mathrm DMU_{o}^{'}$ is $(\overline{\phi }, \overline{\varLambda })$ and assume that $\overline{\phi }=\phi _\mathrm{o}^\mathrm{BCC}$. Considering model (InvBCC), if there exists a Pareto solution, e.g., $(\Delta X_\mathrm{o}^*, {\widetilde{\varLambda }}^*)$, that corresponds to the Pareto optimal value $\Delta X_\mathrm{o}^*$ and satisfies

$$\begin{aligned} {\widetilde{\varLambda }}^* Y^\mathrm{T} > \phi _\mathrm{o}^\mathrm{BCC}(Y_\mathrm{o}+\Delta Y_\mathrm{o}^0), \end{aligned}$$

which means that condition (2.6) is not met, then there exists $\widetilde{\phi }$ that satisfies $\widetilde{\phi }> \phi _\mathrm{o}^\mathrm{BCC}$ and ${\widetilde{\varLambda }}^* Y^\mathrm{T} \geqslant \widetilde{\phi }(Y_\mathrm{o}+\Delta Y_\mathrm{o}^0)$. Therefore, $({\widetilde{\varLambda }}^*, \widetilde{\phi })$ is a feasible solution of model (DLP$_0)$ indicating that the BCC efficiency score of $\mathrm DMU_{o}^{'}$ is no less than $\widetilde{\phi }$. This contradicts the assumption that $\overline{\phi }=\phi _\mathrm{o}^\mathrm{BCC}$.

1.4 Proof of Theorem 2.2.1

Proof

Consider the notations defined earlier: the optimal solution of model (OPTSE) is $S=(v^*, u^*, \Delta X_\mathrm{o}^*, \Delta Y_\mathrm{o}^*, \varLambda ^*)$ and $\mathrm{DMU_{o}^{'}}=(X_\mathrm{o}+\Delta X_\mathrm{o}^*, Y_\mathrm{o}+\Delta Y_\mathrm{o}^*)$. Let $\Delta Y_\mathrm{o}^0$ in model (InvBCC) equal to $\Delta Y_\mathrm{o}^*$. We need to prove that $(\Delta X_\mathrm{o}^*, \varLambda ^*)$ is the Pareto solution of model (InvBCC) and also meets condition (2.6).

It is obvious that $(\Delta X_\mathrm{o}^*,\varLambda ^*)$ is a feasible solution of model (InvBCC). If it is not a Pareto solution, then there exists a Pareto solution $({\Delta } \overline{X}_\mathrm{o},\overline{\varLambda })$ of model (InvBCC) that satisfies ${\Delta } \overline{X}_\mathrm{o} < \Delta X_\mathrm{o}^*$.

According to Lemma 2.1, we can use model (LP$_0)$ to solve the CCR efficiency score of $\mathrm{DMU_{o}^{''}}=(X_\mathrm{o}+{\Delta } \overline{X}_\mathrm{o},Y_\mathrm{o}+\Delta Y_\mathrm{o}^*)$. Suppose its optimal solution is $(v'', u'')$, $\overline{S}=(v'', u'', {\Delta } \overline{X}_\mathrm{o}, \Delta Y_\mathrm{o}^*, \overline{\varLambda })$ is a feasible solution of model (OPTSE) and

$$\begin{aligned} {v''}^\mathrm{T}(X_\mathrm{o}+{\Delta } \overline{X}_\mathrm{o}) \geqslant v^\mathrm{*T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) \end{aligned}$$

(5.4)

holds.

Solve the CCR efficiency score of $\mathrm DMU_{o}^{'}$ using model (LP$_0)$, and suppose its optimal solution is $(v', u')$. $(v^*, u^*)$ is a feasible solution of model (LP$_0)$ about $\mathrm DMU_{o}^{'}$. We have

$$\begin{aligned} {v'}^\mathrm{T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) \leqslant v^\mathrm{*T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*). \end{aligned}$$

(5.5)

Regarding $\mathrm DMU_{o}^{''}$ and $\mathrm DMU_{o}^{'}$, they share the same outputs. As a result, they also share the same feasible solutions of model (LP$_0)$ about them. Considering that ${\Delta } \overline{X}_\mathrm{o} < {\Delta X_\mathrm{o}^*}$, we have the following inequality:

$$\begin{aligned} {v'}^\mathrm{T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) > {v'}^\mathrm{T}(X_\mathrm{o}+{\Delta } \overline{X}_\mathrm{o}) \geqslant {v''}^\mathrm{T} (X_\mathrm{o}+{\Delta } \overline{X}_\mathrm{o}). \end{aligned}$$

(5.6)

According to inequalities (5.5) and (5.6), we can deduce that

$$\begin{aligned} v^\mathrm{*T}(X_\mathrm{o}+\Delta X_\mathrm{o}^*) > v''^\mathrm{T} (X_\mathrm{o}+{\Delta } \overline{X}_\mathrm{o}), \end{aligned}$$

which contradicts inequality (5.4). Therefore, the assumption that $(\Delta X_\mathrm{o}^*, \varLambda ^*)$ is not a Pareto solution of model (InvBCC) is invalid.

If condition (2.6) is not met, which means that there exists $\varLambda ^*$ that satisfies $\varLambda ^* Y^\mathrm{T}>\phi _\mathrm{o}^\mathrm{BCC} (Y_\mathrm{o}+\Delta Y_\mathrm{o}^*)$, then we can increase $\Delta Y_\mathrm{o}^*$ to ${\Delta } \overline{Y}_\mathrm{o}^*$ such that the inequality ${\Delta } \overline{Y}_\mathrm{o}^* > \Delta Y_\mathrm{o}^*$ and equality $\varLambda ^* Y^\mathrm{T}=\phi _\mathrm{o}^\mathrm{BCC} (Y_\mathrm{o}+{\Delta } \overline{Y}_\mathrm{o}^*)$ hold. Hence, $(v^*,u^*,\Delta X_\mathrm{o}^*,{\Delta } \overline{Y}_\mathrm{o}^*,\varLambda ^*)$ is a feasible solution of model (OPTSE).

By applying model (DLP$_0)$ to and $\mathrm{DMU_o}'$ and considering the fact that has the same inputs and larger outputs compared to $\mathrm DMU_{o}^{'}$, we can deduce that the CCR efficiency of is less than that of $\mathrm{DMU_o}'$ whose CCR efficiency is $v^\mathrm{*T} (X_\mathrm{o}+ \Delta X_\mathrm{o}^* )$. According to Theorem 2.1.1 and considering that $(v^*,u^*,\Delta X_\mathrm{o}^*,{\Delta } \overline{Y}_\mathrm{o}^*,\varLambda ^*)$ is a feasible solution of model (OPTSE), we can deduce that there must exist an optimal solution which is different from S and corresponds to an objective value that is less than $v^\mathrm{*T} (X_\mathrm{o}+ \Delta X_\mathrm{o}^* )$. This is a contradiction indicating that condition (2.6) is met.

To summarize, $(\Delta X_\mathrm{o}^*,\varLambda ^*)$ is a Pareto solution of model (InvBCC) where $\Delta Y_\mathrm{o}^0=\Delta Y_\mathrm{o}^*$, which meets condition (2.6). According to Lemma 2.2, the BCC efficiency score of $\mathrm DMU_{o}^{'}$ is $\phi _\mathrm{o}^\mathrm{BCC}$.

1.5 Proof of Theorem 2.2.2

Proof

Denote the CCR efficiency score of $\mathrm DMU_{o}^{'}$ as $\phi _{{o}^{'}}^\mathrm{CCR}$. According to Theorem 2.2.1 and the fact that the output-oriented CCR efficiency score of a DMU is always no less than its BCC efficiency score, we have $\phi _{{o}^{'}}^\mathrm{CCR} \geqslant \phi _\mathrm{o}^\mathrm{BCC}$. As a result, to prove this theorem, we need to construct a new DMU from $\mathrm DMU_o$. Its CCR and BCC efficiency scores are both $\phi _\mathrm{o}^\mathrm{BCC}$, and its input and output variations compared with $\mathrm DMU_o$ compose a feasible solution of model (OPTSE).

Among the original DMU set with n DMUs, find the DMU whose CCR efficiency score is 1. Let us assume that the DMU we found is $\mathrm{DMU_s}=(X_s,Y_s)$ whose CCR and BCC efficiency scores are both 1. After applying model (BCCDLP) and model (CCRLP) to $\mathrm DMU_s$, we obtain the optimal solutions denoted as $(\phi _s^\mathrm{BCC}=1,\varLambda ^*)$ and $(v^*,u^*)$, respectively.

$$\begin{aligned}&(\mathrm {CCRLP}) \quad \min \quad v^\mathrm{T} X_\mathrm{o}\\&\mathrm {s.t.}\quad u^\mathrm{T} Y_\mathrm{o}=1,\\&\qquad \,\, v^\mathrm{T} X_j-u^\mathrm{T} Y_j \geqslant 0,\quad j=1,\cdots ,n,\\&\qquad \,\, v \geqslant 0, \quad u \geqslant 0.\\ \end{aligned}$$

Construct the new DMU (denoted as $\mathrm{DMU_{o}^{'}}=(X_\mathrm{o}',Y_\mathrm{o}')$) as follows:

$$\begin{aligned}&X_\mathrm{o}'=X_\mathrm{o}+\Delta X_\mathrm{o}=X_s, \\&Y_\mathrm{o}'=Y_\mathrm{o}+\Delta Y_\mathrm{o}=Y_\mathrm{o}/\phi _\mathrm{o}^\mathrm{BCC}. \end{aligned}$$

According to Lemma 2.1, we can apply model (DLP$_0)$ to solve the BCC efficiency score of $\mathrm DMU_{o}^{'}$. In this case, model (DLP$_1)$ is applied.

Let $\phi '=\phi /\phi _\mathrm{o}^\mathrm{BCC}$, model (DLP$_1)$ can be transformed to (DLP$_2)$. Considering the optimal solution of the BCC model about $\mathrm DMU_s$, we have that the optimal solution and optimal value of model (DLP$_2)$ are $(\phi '=\phi _s^\mathrm{BCC}=1, \varLambda ^*)$ and $\phi _\mathrm{o}^\mathrm{BCC}$, respectively. Therefore, the BCC efficiency score of $\mathrm DMU_{o}^{'}$ is $\phi _\mathrm{o}^\mathrm{BCC}$.

Apply model (LP$_0)$ to solve the CCR efficiency of $\mathrm DMU_{o}^{'}$:

Let $v'=v/\phi _\mathrm{o}^\mathrm{BCC}$, $u'=u/\phi _\mathrm{o}^\mathrm{BCC}$ and transform model (LP$_1)$ to model (LP$_2)$:

$$\begin{aligned}&(\mathrm {LP}_2) \quad \min \quad v^\mathrm{T} X_\mathrm{o}'=\phi _\mathrm{o}^\mathrm{BCC} v'^\mathrm{T} X_\mathrm{o}'=\phi _\mathrm{o}^\mathrm{BCC} v'^\mathrm{T} X_s\\&\mathrm{s.t.} \quad u'^\mathrm{T} Y_s=1, \\&\qquad \, v'^\mathrm{T} X_j-u'^\mathrm{T} Y_j \geqslant 0,\quad j=1,\cdots ,n,\\&\qquad \, v' \geqslant 0, \quad u' \geqslant 0. \end{aligned}$$

Considering the optimal solution of the CCR model about $\mathrm DMU_s$, we have that the optimal solution of model (LP$_2)$ is $(v^*,u^*)$. Its optimal value is $\phi _\mathrm{o}^\mathrm{BCC} v^\mathrm{*T} X_s=\phi _\mathrm{o}^\mathrm{BCC}$.

To summarize, the CCR and BCC efficiency scores of $\mathrm{DMU_{o}^{'}}=(X_\mathrm{o}',Y_\mathrm{o}')$ are both $\phi _\mathrm{o}^\mathrm{BCC}$. And according to models (LP$_1)$ and (DLP$_1)$, $S=(\phi _\mathrm{o}^\mathrm{BCC} v^*, \phi _\mathrm{o}^\mathrm{BCC} u^*, X_s-X_\mathrm{o}, Y_s/\phi _\mathrm{o}^\mathrm{BCC}-Y_\mathrm{o}, \varLambda ^*)$, which corresponds to the variation of $\mathrm DMU_{o}^{'}$ compared with $\mathrm DMU_o$, is a feasible solution of model (OPTSE). Hence, we find the required DMU and accomplish the proof of Theorem 2.2.2.

1.6 Proof of Theorem 2.2.3

Proof

Proof of necessity:

For $\mathrm{DMU_o}=(X_\mathrm{o},Y_\mathrm{o})$ whose BCC efficiency is $\phi _\mathrm{o}^\mathrm{BCC}$, suppose it is not in the MPSS region. As a result, neither $\mathrm DMU_o$ nor $\mathrm{DMU_{o}^{'}}=(X_\mathrm{o},\phi _\mathrm{o}^\mathrm{BCC} Y_\mathrm{o}) \in T$ is an MPSS. There must exist $\alpha$ and $\beta$ that satisfy $\mathrm{DMU_{o}^{''}}=(\beta X_\mathrm{o},\alpha \phi _\mathrm{o}^\mathrm{BCC} Y_\mathrm{o})$ in T and $\alpha / \beta > 1$.

Apply model (CCRDLP) to solve for the CCR efficiency of $\mathrm DMU_o$, and suppose the optimal solution is $(\phi _\mathrm{o}^\mathrm{CCR},\varLambda ^*)$. Since $\mathrm{DMU_{o}^{''}} \in T$, we can deduce that there exists $\varLambda '$ which makes $(\alpha \phi _\mathrm{o}^\mathrm{BCC}/\beta ,\varLambda '/\beta )$ a feasible solution of model (CCRDLP), and $\alpha \phi _\mathrm{o}^\mathrm{BCC}/\beta \leqslant \phi _\mathrm{o}^\mathrm{CCR}$. This indicates that

$$\begin{aligned} \mathrm{SE}=\frac{\phi _\mathrm{o}^\mathrm{BCC}}{\phi _\mathrm{o}^\mathrm{CCR}} \leqslant \frac{\beta }{\alpha } < 1. \end{aligned}$$

Therefore, the efficiency of $\mathrm DMU_o$ is less than 1, which proves that the DMU whose SE is 1 must be in the MPSS region.

Proof of sufficiency:

Suppose $\mathrm{DMU_o}=(X_\mathrm{o},Y_\mathrm{o})$ is in the MPSS region, there must exist a production possibility $\mathrm{DMU_o^*}=(X_\mathrm{o},Y_\mathrm{o}^*) \in T$ that is an MPSS. And the CCR and BCC efficiency scores of $\mathrm DMU_o^*$ are both 1.

We claim that the BCC and CCR efficiency scores of $\mathrm DMU_o$ have the following relationship:

$$\begin{aligned} \phi _\mathrm{o}^\mathrm{BCC}=\phi _\mathrm{o}^\mathrm{CCR}=\mathop {\min }\limits _{r \in \left\{ {1, \cdots ,p} \right\} } \frac{{y_{ro}^*}}{{{y_{ro}}}}. \end{aligned}$$

We first prove $\phi _\mathrm{o}^\mathrm{BCC} = {\min _{r \in \{ 1, \cdot ,p\} }} y_{ro}^* / y_{ro}$. We solve the BCC efficiency score of $\mathrm DMU_o^*$ by applying model (DLP$_0)$ and suppose the optimal solution is $(\phi _{o^*}^\mathrm{BCC}=1,\varLambda ^*)$. Hence $(\phi _\mathrm{o}^\mathrm{BCC},\varLambda ^*)$ is a feasible solution of model (BCCDLP) about $\mathrm DMU_o$. Suppose the optimal solution of model (BCCDLP) about $\mathrm DMU_o$ is $(\overline{\phi }_\mathrm{o}^\mathrm{BCC}, \overline{\varLambda }^*)$, we have $\overline{\phi }_\mathrm{o}^\mathrm{BCC} \geqslant \phi _\mathrm{o}^\mathrm{BCC}$.

Let us assume that $\overline{\phi }_\mathrm{o}^\mathrm{BCC} > \phi _\mathrm{o}^\mathrm{BCC}$. It is obvious that $(\overline{\phi }_\mathrm{o}^\mathrm{BCC} / \phi _\mathrm{o}^\mathrm{BCC}, \overline{\varLambda }^*)$ is a feasible solution of model ($\hbox {DLP}_0$) about $\mathrm DMU_o^*$. However , $\overline{\phi }_\mathrm{o}^\mathrm{BCC}/\phi _\mathrm{o}^\mathrm{BCC}>1$ and this contradicts the assumption that the BCC efficiency score of $\mathrm{DMU_o^*}$ is 1. Therefore, the assumption that $\overline{\phi }_\mathrm{o}^\mathrm{BCC} > \phi _\mathrm{o}^\mathrm{BCC}$ is invalid, and $\overline{\phi }_\mathrm{o}^\mathrm{BCC} = \phi _\mathrm{o}^\mathrm{BCC}$ is proved true. The BCC efficiency score of $\mathrm DMU_o$ is ${\min _{r \in \{ 1, \cdot ,p\} }} y_{ro}^* / y_{ro}$.

For the case of the CCR efficiency, the proof is the same as the case of the BCC efficiency.

Hence, we have confirmed the claim above and proved SE = 1 and that any DMU in the MPSS region is scale efficient.

1.7 Proof of Theorem 3.0.4

Proof

Considering model (OPTSE), it is obvious that model (ALLO) gives the allocation mechanism to maximize the overall GTE and SE of the DMU set.

Model (ALLO) without constraints (3.1) and (3.2) is equivalent to applying model (OPTSE) to every DMU and adding all the optimal values together. According to Theorems 2.1.1, 2.2.1 and 2.2.2, the optimal value of model (OPTSE) about each DMU is the DMU’s BCC efficiency score, and their sum is $\sum _{j=1}^n \phi _j^\mathrm{BCC}$.