Appendix A
Derivation of the simplified version of the unit-specific estimate of the production possibility set
Consider the general estimate \( \hat{\wp }_{o} ({\varvec{\Gamma}}_{{\varvec{\uplambda}}} ) \) of the production possibility set \( \wp_{o} ({\varvec{\Gamma}}_{{\varvec{\uplambda}}} ) \) formulated in (2), in which only two input and two output trade-offs are applied in order to translate the weight restrictions (1) into envelopment space. Whilst the input trade-offs are represented by vectors \( {\mathbf{p}}_{o1} = ((1 - w_{oU}^{{\mathbf{x}}} ){\kern 1pt} ,\; - w_{oU}^{{\mathbf{x}}} {\mathbf{x}}_{o}^{obs \prime} )^{\prime} \) and \( {\mathbf{p}}_{o2} = - {\mathbf{p}}_{o1} \), the output trade-offs are captured similarly by vectors \( {\mathbf{q}}_{o1} = ((1 - w_{oU}^{{\mathbf{y}}} ){\kern 1pt} ,\; - w_{oU}^{{\mathbf{y}}} {\mathbf{y}}_{o}^{obs \prime} )^{\prime} \) and \( {\mathbf{q}}_{o2} = - {\mathbf{q}}_{o1} \). It is easy to see that the inequalities in (2) may be rewritten in the fashion of Podinovski (2004, p. 1314; 2015, p. 122) by means of complementary non-negative vectors \( {\mathbf{e}}^{{\mathbf{x}}} = (e_{U}^{{\mathbf{x}}} ,{\mathbf{e}}_{{}}^{{{\mathbf{x}},obs \prime}} )^ \prime \in \Re_{ \ge 0}^{m + 1} \) and \( {\mathbf{e}}^{{\mathbf{y}}} = (e_{U}^{{\mathbf{y}}} ,{\mathbf{e}}_{{}}^{{{\mathbf{y}},obs \prime}} )^ \prime \in \Re_{ \ge 0}^{s + 1} \) as
$$ {\tilde{\mathbf{x}}} = \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{x}}_{i}} + \sum\limits_{k = 1}^{k = 2} {\pi_{k} {\mathbf{p}}_{ok}} + {\mathbf{e}}^{{\mathbf{x}}},\;\;\;\;\;{\tilde{\mathbf{y}}} = \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{y}}_{i}} + \sum\limits_{l = 1}^{l = 2} {\phi_{l} {\mathbf{q}}_{ol}} - {\mathbf{e}}^{{\mathbf{y}}}, $$
(11)
or
$$ {\tilde{\mathbf{x}}} = \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{x}}_{i}} + (\pi_{1} - \pi_{2}){\mathbf{p}}_{o1} + {\mathbf{e}}^{{\mathbf{x}}},\;\;\;\;\;{\tilde{\mathbf{y}}} = \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{y}}_{i}} + (\phi_{1} - \phi_{2}){\mathbf{q}}_{o1} - {\mathbf{e}}^{{\mathbf{y}}}. $$
(12)
Splitting the vectors into unobservable and observable parts and allowing then for \( x_{1U} = \cdots = x_{nU} = \tilde{x}_{U} = 1 \) and \( y_{1U} = \cdots = y_{nU} = \tilde{y}_{U} = 1 \), it is obtained that
$$ \begin{aligned} \left({\begin{array}{*{20}c} 1 \\ {{\tilde{\mathbf{x}}}^{obs}} \\ \end{array}} \right) = \left({\begin{array}{*{20}c} {\sum\limits_{i = 1}^{i = n} {\lambda_{i}}} \\ {\sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{x}}_{i}^{obs}}} \\ \end{array}} \right) + (\pi_{1} - \pi_{2})\left({\begin{array}{*{20}c} {1 - w_{oU}^{{\mathbf{x}}}} \\ {- w_{oU}^{{\mathbf{x}}} {\mathbf{x}}_{o}^{obs}} \\ \end{array}} \right) + \left({\begin{array}{*{20}c} {e_{U}^{{\mathbf{x}}}} \\ {{\mathbf{e}}^{{{\mathbf{x}},obs}}} \\ \end{array}} \right), \\ \left({\begin{array}{*{20}c} 1 \\ {{\tilde{\mathbf{y}}}^{obs}} \\ \end{array}} \right) = \left({\begin{array}{*{20}c} {\sum\limits_{i = 1}^{i = n} {\lambda_{i}}} \\ {\sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{y}}_{i}^{obs}}} \\ \end{array}} \right) + (\phi_{1} - \phi_{2})\left({\begin{array}{*{20}c} {1 - w_{oU}^{{\mathbf{y}}}} \\ {- w_{oU}^{{\mathbf{y}}} {\mathbf{y}}_{o}^{obs}} \\ \end{array}} \right) - \left({\begin{array}{*{20}c} {e_{U}^{{\mathbf{y}}}} \\ {{\mathbf{e}}^{{{\mathbf{y}},obs}}} \\ \end{array}} \right). \\ \end{aligned} $$
(13)
The unobservable parts of these equalities then yield the expressions \( \pi_{1} - \pi_{2} = (1 - w_{oU}^{{\mathbf{x}}} )^{ - 1} \left(1 - \sum_{i} \lambda_{{{\kern 1pt} i}} - e_{U}^{{\mathbf{x}}} \right) \) and \( \phi_{1} - \phi_{2} = (1 - w_{oU}^{{\mathbf{y}}} )^{ - 1} \left(1 - \sum_{i} \lambda_{{{\kern 1pt} i}} + e_{U}^{{\mathbf{y}}} \right) \), in consequence of which the observable part of the equalities simplifies into
$$ \begin{aligned} {\tilde{\mathbf{x}}}^{obs} = \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{x}}_{i}^{obs}} + (1 - w_{oU}^{{\mathbf{x}}})^{- 1} (\sum_{i} \lambda_{{{\kern 1pt} i}} - 1 + e_{U}^{{\mathbf{x}}})w_{oU}^{{\mathbf{x}}} {\mathbf{x}}_{o}^{obs} + {\mathbf{e}}^{{{\mathbf{x}},obs}}, \\ {\tilde{\mathbf{y}}}^{obs} = \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{y}}_{i}^{obs}} + (1 - w_{oU}^{{\mathbf{y}}})^{- 1} (\sum_{i} \lambda_{{{\kern 1pt} i}} - 1 - e_{U}^{{\mathbf{y}}})w_{oU}^{{\mathbf{y}}}{\mathbf{y}}_{o}^{obs} - {\mathbf{e}}^{{{\mathbf{y}},obs}}. \\ \end{aligned} $$
(14)
The non-negativity of \( {\mathbf{e}}^{{\mathbf{x}}} \) and \( {\mathbf{e}}^{{\mathbf{y}}} \) implies that
$$\begin{aligned} {\tilde{\mathbf{x}}}^{obs} &\le \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{x}}_{i}^{obs}} +(1 - w_{oU}^{{\mathbf{x}}})^{- 1} (\sum_{i} \lambda_{{{\kern 1pt} i}} - 1)w_{oU}^{{\mathbf{x}}} {\mathbf{x}}_{o}^{obs}, \;\; \\ {\tilde{\mathbf{y}}}^{obs} &\ge \sum\limits_{i = 1}^{i = n} {\lambda_{i} {\mathbf{y}}_{i}^{obs}} + (1 - w_{oU}^{{\mathbf{y}}})^{- 1} (\sum_{i} \lambda_{{{\kern 1pt} i}} - 1){w_{oU}^{{\mathbf{y}}} }{\mathbf{y}}_{o}^{obs},\end{aligned} $$
(15)
which ends the derivation and testifies the validity of (3) since the remaining constraint \( {\varvec{\uplambda}} \in {\varvec{\Gamma}}_{{\varvec{\uplambda}}} \) was preserved intact.
Appendix B
Proof of the property that explicit consideration of slacks on the unobservable variable increases SBM efficiency scores
Property 5
Consider a weighted SBM model in the form described by (4) with conditions (4a) and (4g). Consider an analogical weighted SBM model given by the optimization problem
$$ \mathop {\min}\limits_{{{\varvec{\uplambda}},\;{\mathbf{s}}_{o}^{{\mathbf{x}}},\;{\mathbf{s}}_{o}^{{\mathbf{y}}},\;{\varvec{\uppi}},\;{\varvec{\upphi}},\;{\mathbf{e}}^{{\mathbf{x}}} \;,{\mathbf{e}}^{{\mathbf{y}}},\;\theta,\;\eta}} \quad \rho_{o}^{U} = \frac{{1 - (1 - w_{oU}^{{\mathbf{x}}})\sum\limits_{i = 1}^{i = m} {w_{oi}^{{\mathbf{x}}} s_{oi}^{{\mathbf{x}}}/x_{oi}}}}{{1 + (1 - w_{oU}^{{\mathbf{y}}})\sum\limits_{r = 1}^{r = s} {w_{or}^{{\mathbf{y}}} s_{or}^{{\mathbf{y}}}/y_{or}}}}, $$
(16)
subject to the same set of conditions (4a) and (4g). The optimal values of these two programs satisfy \( \rho^{*}_{o} \le \rho^{U^*}_{o} \).
Proof
First note that the minimization of the slacks-based ratio in both (4) and (16) depends on the weighted average input contractions and output expansions that appear in the numerator and denominator, respectively. Regardless of a particular solution for \( {\varvec{\uplambda}} \), \( {\mathbf{s}}_{o}^{{\mathbf{x}}} \), \( {\mathbf{s}}_{o}^{{\mathbf{y}}} \), \( {\varvec{\uppi}} \), \( {\varvec{\upphi}} \), \( {\mathbf{e}}^{{\mathbf{x}}} \), \( {\mathbf{e}}^{{\mathbf{y}}} \), θ and η, the SBM is a function of average input contraction \( \bar{s}_{ \, o}^{{\mathbf{x}}} = \sum_{i} w_{oi}^{{\mathbf{x}}} s_{ \, oi}^{{\mathbf{x}}} /x_{oi} \) and average output expansion \( \bar{s}_{ \, o}^{{\mathbf{y}}} = \sum_{r} w_{or}^{{\mathbf{y}}} s_{ \, or}^{{\mathbf{y}}} /y_{or} \). A useful inequality turns out to be the relationship
$$\begin{aligned} \frac{{1 - (1 - w_{oU}^{{\mathbf{x}}})\bar{s}_{o}^{{\mathbf{x}}}}}{{1 + (1 - w_{oU}^{{\mathbf{y}}})\bar{s}_{o}^{{\mathbf{y}}}}} &= \frac{{1 - \bar{s}_{o}^{{\mathbf{x}}} + w_{oU}^{{\mathbf{x}}} \bar{s}_{o}^{{\mathbf{x}}}}}{{1 + \bar{s}_{o}^{{\mathbf{y}}} - w_{oU}^{{\mathbf{y}}} \bar{s}_{o}^{{\mathbf{y}}}}}\\ &= \frac{{1 - \bar{s}_{o}^{{\mathbf{x}}}}}{{1 + \bar{s}_{o}^{{\mathbf{y}}}}} \cdot \frac{{1 + w_{oU}^{{\mathbf{x}}} \bar{s}_{o}^{{\mathbf{x}}}/(1 - \bar{s}_{o}^{{\mathbf{x}}})}}{{1 - w_{oU}^{{\mathbf{y}}} \bar{s}_{o}^{{\mathbf{y}}}/(1 + \bar{s}_{o}^{{\mathbf{y}}})}} \\ & \ge \frac{{1 - \bar{s}_{o}^{{\mathbf{x}}}}}{{1 + \bar{s}_{o}^{{\mathbf{y}}}}} \cdot 1 = \frac{{1 - \bar{s}_{o}^{{\mathbf{x}}}}}{{1 + \bar{s}_{o}^{{\mathbf{y}}}}} \end{aligned} $$
(17)
that holds obviously for any \( w_{oU}^{{\mathbf{x}}} ,w_{oU}^{{\mathbf{y}}} \in [0,1) \) and for any \( \bar{s}_{ \, o}^{{\mathbf{x}}} \in [0,1) \) and \( {\bar{s}_{ \, o}^{{\mathbf{y}}} \in [1,\infty )} \). Because objective functions (4) and (16) are minimized subject to the same set of constraints (4a) and (4g), their optimal values must satisfy a similar relationship
$$ \rho_{o}^{U^*} = \frac{{1 - (1 - w_{oU}^{{\mathbf{x}}})\bar{s}_{o}^{{{\mathbf{x}}U^*}}}}{{1 + (1 - w_{oU}^{{\mathbf{y}}})\bar{s}_{o}^{{{\mathbf{y}}U^*}}}} \ge \frac{{1 - \bar{s}_{o}^{{{\mathbf{x}}O^*}}}}{{1 + \bar{s}_{o}^{{{\mathbf{y}}O^*}}}} = \rho_{o^*}, $$
(18)
in which U or O in superscripts indicate whether optimized average input contractions or output expansions arise from solving program (4) or (16) alongside constraints (4a) and (4g). If this were not true, and the equality would be reverse so that \( \rho_{o}^{U^*}\;<\;\rho_{o}^* \), the inequality in (17) would then imply
$$ \rho_{o}^{*} = \frac{{1 - \bar{s}_{o}^{{{\mathbf{x}}O^*}}}}{{1 + \bar{s}_{o}^{{{\mathbf{y}}O^*}}}} > \frac{{1 - (1 - w_{oU}^{{\mathbf{x}}})\bar{s}_{o}^{{{\mathbf{x}}U^*}}}}{{1 + (1 - w_{oU}^{{\mathbf{y}}})\bar{s}_{o}^{{{\mathbf{y}}U^*}}}} \ge \frac{{1 - \bar{s}_{o}^{{{\mathbf{x}}U^*}}}}{{1 + \bar{s}_{o}^{{{\mathbf{y}}U^*}}}}. $$
(19)
This, however, means that \( \rho_{o}^{U^*}\) would not be optimal, which is a contradiction.