Unobservable or omitted production variables in data envelopment analysis through unit-specific production trade-offs

Abstract

The paper focuses upon situations in which decision-making units carry out their production activities with some inputs or outputs unobservable (or possibly omitted), and when there are a priori known constraints on the relative significance of otherwise observable (or explicitly considered) inputs and outputs. For such settings, the paper proposes a modification that alters traditional construction of the production possibility set and isolates the role of the unobservable (or omitted) variables in production by means of restrictions on virtual inputs and outputs being converted into production trade-offs. In effect, the proposed procedure induces unit-specific production possibility sets that derive from production trade-offs framed for units assessed separately to reflect their specific observed production conditions. The modification is implemented within a weighted slacks-based measure with restricted direction of slacks in order to make technical efficiency measurement more informative and consistent with the operating conditions under which production activities are accomplished. These ideas are illustrated and models implemented in a case study of bank branch performance measurement.

Appendices

Appendix A

1.1 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

2.1 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.