Closest targets and minimum distance to the Pareto-efficient frontier in DEA

Abstract

In this paper, we propose a general approach to find the closest targets for a given unit according to a previously specified criterion of similarity. The idea behind this approach is that closer targets determine less demanding levels of operation for the inputs and outputs of the inefficient units to perform efficiently. Similarity can be interpreted as closeness between the inputs and outputs of the assessed unit and the proposed targets, and this closeness can be measured by using either different distance functions or different efficiency measures. Depending on how closeness is measured, we develop several mathematical programming problems that can be easily solved and guarantee to reach the closest projection point on the Pareto-efficient frontier. Thus, our approach leads to the closest targets by means of a single-stage procedure, which is easier to handle than those based on algorithms aimed at identifying all the facets of the efficient frontier.

Appendix: proof of the theorem

First, we prove the sufficient condition.

Let ${({\tilde {\hbox{X}}, \tilde {\hbox{Y}}})\in \hbox{D}_0.}$ Since this point is efficient, the optimal value of the following pair of dual LP problems will be zero:

$$ {\begin{array}{lll} {\hbox{Min}}& {-\left\{ {\sum\limits_{{\rm i}=1}^{\rm m} {\hbox{s}_{\rm i}^{-} +\sum\limits_{{\rm r}=1}^{\rm s} {\hbox{s}_{\rm r}^{+} } } } \right\}}& \\ {\hbox{s.t.:}}& {\sum\limits_{{\rm j}\in {\rm E}} {\lambda_{\rm j} \hbox{x}_{\rm ij}} +\hbox{s}_{\rm i}^{-} =\tilde {\hbox{x}}_{\rm i} }& {\hbox{i}=1, \ldots,\hbox{m}} \\ & {\sum\limits_{{\rm j}\in {\rm E}} {\lambda_{\rm j} \hbox{y}_{\rm rj}} -\hbox{s}_{\rm r}^{+} =\tilde {\hbox{y}}_{\rm r} }& {\hbox{r}=1,\ldots,\hbox{s}} \\ & {\lambda_{\rm j} \ge 0}& {\hbox{j}\in \hbox{E}} \\ & {\hbox{s}_{\rm i}^{-} \ge 0}& {\hbox{i}=1,\ldots,\hbox{m}} \\ & {\hbox{s}_{\rm r}^{+} \ge 0}& {\hbox{r}=1,\ldots,\hbox{s}} \\ \end{array} } \quad {\begin{array}{lll} {\hbox{Max}}& {-\sum\limits_{{\rm i}=1}^{\rm m} {\nu_{\rm i} \tilde {\hbox{x}}_{\rm i} +\sum\limits_{{\rm r}=1}^{\rm s} {\mu_{\rm r} \tilde {\hbox{y}}_{\rm r}}}}& \\ {\hbox{s.t.:}}& {-\sum\limits_{{\rm i}=1}^{\rm m} {\nu_{\rm i} \hbox{x}_{\rm ij} +\sum\limits_{{\rm r}=1}^{\rm s} {\mu_{\rm r} \hbox{y}_{\rm rj} \le 0}}}& {\hbox{j}\in \hbox{E}} \\ & {\nu_{\rm i} \ge 1}& {\hbox{i}=1,\ldots,\hbox{m}} \\ & {\mu_{\rm r} \ge 1}& {\hbox{r}=1,\ldots,\hbox{s}} \\ & & \\ & & \\ \end{array}} $$

Since ${({\tilde {\hbox{X}}, \tilde {\hbox{Y}}})\ne 0_{{\rm m+s}}},$ we can find a set of non-negative scalars ${\tilde {\lambda }_{\rm j} ,\;\hbox{j}\in \hbox{E},}$ with at least one of them being strictly positive, such that ${\left({\tilde {\hbox{X}},\tilde {\hbox{Y}}} \right)= ({\sum_{{\rm j}\in {\rm E}} {\tilde {\lambda }_{\rm j} \hbox{X}_{\rm j} }, \sum_{{\rm j}\in {\rm E}} {\tilde {\lambda}_{\rm j} {\rm Y}_{\rm j} } } )}.$ Moreover, for complementary slackness, for each ${\tilde {\lambda }_{\rm j} > 0,}$ the corresponding dual constraint is binding, that is, ${\exists \;\tilde {\nu}_{\rm i} \ge 1,\;\hbox{i}=1,\ldots,\hbox{m},}$ and ${\tilde {\mu}_{\rm r} \ge 1,\;\hbox{r}=1,\ldots,\hbox{s},}$ satisfying ${-\sum_{{\rm i}=1}^{\rm m} {\tilde {\nu}_{\rm i} \hbox{x}_{\rm ij} +\sum_{{\rm r}=1}^{\rm s} {\tilde {\mu}_{\rm r} \hbox{y}_{{\rm rj}} =0}}}.$

Then, let us define ${\tilde {\hbox{d}}_{\rm j} =\tilde {\hbox{b}}_{\rm j} =0,}$ for each ${\tilde {\lambda}_{\rm j} > 0,}$ and ${\tilde {\hbox{b}}_{\rm j} =1}$ and ${\tilde {\hbox{d}}_{\rm j} =\sum_{{\rm i}=1}^{\rm m} {\tilde {\nu}_{\rm i} \hbox{x}_{\rm ij} -\sum_{{\rm r}=1}^{\rm s} {\tilde {\mu}_{\rm r} \hbox{y}_{\rm rj} \ge 0}},}$ for each ${\tilde {\lambda}_{\rm j} =0.}$ Finally, since ${({\tilde {\hbox{X}}, \tilde {\hbox{Y}}})}$ dominates DMU₀, we can also define ${\tilde {\hbox{s}}_{{\rm i}0}^{-} =\hbox{x}_{{\rm i}0} -\tilde {\hbox{x}}_{\rm i} \ge 0,\;\hbox{i}=1, \ldots,{\rm m},}$ and ${\tilde {\hbox{s}}_{{\rm r}0}^{+} =\tilde {\hbox{y}}_{\rm r} -\hbox{y}_{{\rm r}0} \ge 0,\;\hbox{r}=1,\ldots,\hbox{s}.}$

Therefore, ${\tilde {\lambda }_{\rm j},\;\tilde {\hbox{d}}_{\rm j} ,\;\tilde {\hbox{b}}_{\rm j},\;\hbox{j}\in \hbox{E},\;\tilde {\nu}_{\rm i},\;\tilde {\hbox{s}}_{{\rm i}0}^{-} ,\;\hbox{i}=1,\ldots,\hbox{m},}$ and ${\tilde {\mu}_{\rm r},\tilde {\hbox{s}}_{{\rm r}0}^{+},\;\hbox{r}=1,\ldots,\hbox{s},}$ satisfy all the desired conditions.

Conversely, let ${\tilde {\lambda}_{\rm j},\;\tilde {\hbox{d}}_{\rm j} \ge 0,\;\tilde {\hbox{b}}_{\rm j} \in \{{0,1}\},\;\hbox{j}\in \hbox{E},\; \tilde {\nu}_{\rm i} \ge 1,\;\tilde {\hbox{s}}_{{\rm i}0}^{-} \ge 0,\;\hbox{i}=1,\ldots,\hbox{m},}$ and ${\tilde {\mu}_{\rm r} \ge 1, \; \tilde {\hbox{s}}_{{\rm r}0}^{+} \ge 0,\;\hbox{r}=1,\ldots,\hbox{s},}$ be scalars satisfying (t.3)–(t.7) and define ${\left({\tilde {\hbox{X}},\tilde {\hbox{Y}}} \right)=({\sum_{{\rm j}\in {\rm E}} {\tilde {\lambda }_{\rm j} \hbox{X}_{\rm j} }, \sum_{{\rm j}\in {\rm E}} {\tilde {\lambda }_j Y_j } } )}.$ Then we will show that ${\left({\tilde {\hbox{X}},\tilde {\hbox{Y}}} \right)\in \hbox{D}_0 .}$

Obviously, ${({\tilde {\hbox{X}}, \tilde {\hbox{Y}}})}$ dominates (X₀, Y₀) as a consequence of the constraints (t.3) and (t.4). Therefore, we only need to prove that ${\left({\tilde {\hbox{X}}, \tilde {\hbox{Y}}} \right)}$ belongs to the efficient frontier. To do it we will show that this point is a linear combination of extreme efficient DMUs lying on the same efficient facet of the frontier and, consequently, that point will also be an efficient point.

For each ${\tilde {\lambda}_{\rm j} > 0,}$ we have ${\tilde {\hbox{d}}_{\rm j} =\tilde {\hbox{b}}_{\rm j} =0}$ by virtue of (t.6) and (t.7) (note that there always exists such a positive intensity as a result of equalities (t.3)–(t.4) and the fact that at least one input and one output of DMU₀ is non-zero). Hence, all the extreme efficient units with a non-zero coefficient in the linear combinations (t.1) and (t.2) satisfy the equation ${-\sum_{{\rm i}=1}^{\rm m} {\tilde {\nu}_{\rm i} \hbox{x}_{\rm i} +\sum_{{\rm r}=1}^{\rm s} {\tilde {\mu}_{\rm r} \hbox{y}_{\rm r} =0}}}.$ Since, in addition, these multipliers hold (t.5) and they all are strictly positive (in particular, greater than or equal to the unity), we have that such equation defines a facet of the efficient frontier.