On comparison of different sets of units used for improving the frontier in DEA models

Abstract

Inevitable simplifications of models of real world activities lead to some inadequacies. In the DEA literature several methods were proposed to overcome such difficulties. Some authors proposed to use specific production units in the primal space of inputs and outputs as a starting point in order to improve the frontier of the DEA models. In our previous papers, we introduced the notion of terminal units. It was proved that only terminal units form necessary and sufficient set of units for improving the frontier. In this paper, the relationship between all sets of units proposed for improving the frontier is established. Our theoretical results are confirmed by extensive and instructive graphical examples and also verified by computational experiments using real-life data sets.

Appendices

Appendix 1

Proof of Theorem 4

Take any extreme efficient unit $Z_o =(X_o ,Y_o )\in T_{anc}^3$. Consider the case when this unit is projected on some weakly efficient face $\Gamma _1$ of set $\widetilde{T}_B$, where set $\widetilde{T}_B$ is formed from set $T_B$ by excluding unit $Z_o$. Since face $\Gamma _1$ is infinite, it can be represented as a convex sum of extreme units plus sum nonnegative combination of unbounded rays whose direction vectors are of the form $e_k$, $k\in I_1$, $-e_i$, $i\in I_2$, $e_i ,\;e_k \in E^{m+r}$ are identity vectors with a one in position $i$ and $k$, respectively.

Let the projection of unit $Z_o$ onto the face $\Gamma _1$ be $\widetilde{Z}_o$. Point $\widetilde{Z}_o$ belongs to $\hbox {ri}\;\Gamma _1$, where $\hbox {ri}\;\Gamma _1$ stands for relative interior of face $\Gamma _1$.

Unit $Z_o$ does not belongs to $\hbox {ri}\;\Gamma _1$, hence according to convex analysis (Goldman 1956; Nikaido 1968; Rockafellar 1970) half-segment $[Z_o ,\widetilde{Z}_o$) does not belongs to face $\Gamma _1$.

Let us build hyperplane $H$ in such a way that it separates unit $Z_o$ and set $\widetilde{T}$ and hyperplane $H$ does not contain unit $Z_o$.

Consider the affine set $S$ of the following form:

$$\begin{aligned} S=\left\{ {\left. {Z} \right| Z=Z_o +\sum _{k\in I_1 } {\mu _k e_k } -\sum _{i\in I_2 } {\rho _i e_i } ,\mu _k \ge 0,k\in I_1 ,\rho _i \ge 0,i\in I_2 } \right\} . \end{aligned}$$

(6)

By construction, hyperplane $H$ separates affine set $S$ and set $\widetilde{T}_B$, since direction vectors $e_k$, $k\in I_1$, $e_i$, $i\in I_2$ are parallel to hyperplane $H$.

Hence unit $Z_o$ emanates rays along directions $e_k$, $k\in I_1$ and $-e_i$, $i\in I_2$. However, these rays are infinite edges for set $T_B$ since interior points of every ray cannot be represented as convex combination of points from set $\widetilde{T}$ due to existence of separating hyperplane $H$. So unit $Z_o$ is a terminal unit.

Now, consider the case when model (5) for unit $Z_o$ has infeasible solution. This means that affine set $S$ of the type (6) can be constructed and infinite rays of this set do not intersect the set $\widetilde{T}_B$. Again, this means that some infinite edges are going out from unit $Z_o$ of set $T_B$. Hence unit $Z_o$ is a terminal one in set $T_B$.

This completes the proof. $\square$

Proof of Theorem 6

Take any unit $Z_q =(X_q ,Y_q )\in T_{ext}$. Assume that unit $Z_q$ is not a point from set $T_{anc}^3$. Then there is only one possibility: unit $Z_q$ is projected on some efficient face $\Gamma _1$ of set $\widetilde{T}_B$, where set $\widetilde{T}_B$ is constructed from set $T_B$ by excluding unit $Z_q$.

Let $Z_q^*$ be projection point of unit $Z_q$ onto the efficient face $\Gamma _1$. Then point $Z_q^*$ can be represented in the form

$$\begin{aligned} Z_q^*=\sum _{s\in I_{eff} } {\lambda _s Z_s } , \quad \sum _{s\in I_{eff} } {\lambda _s } =1, \quad \lambda _s \ge 0, \quad s\in I_{eff}, \end{aligned}$$

where $I_{eff}$ is some set of efficient units of set $\widetilde{T}_B$. The set $T_{eff}^*$, formed by convex combination of efficient units from $I_{eff}$, belongs to the set $T_{ext}^1$ that is constructed as a convex combination of all exterior units. Hence $Z_q^*\in T_{ext}^1$. Therefore point $Z_q^*$ will dominate unit $Z_q$ after “reversing the inputs and outputs” according to the procedure of Edvardsen et al. (2008). This means that point $Z_q$ cannot belong to the set of exterior points, contradicting the assumption.

This completes the proof. $\square$

Appendix 2

Tables 2, 3, 4, 5 and 6.

Table 2 An illustrative example regarding an anchor unit in Fig. 1

Table 3 An illustrative example regarding an anchor unit in Fig. 2

Table 4 An illustrative example regarding a terminal unit in Fig. 3

Table 5 An illustrative example regarding an exterior unit in Fig. 4

Table 6 An illustrative example regarding anchor and exterior units in Fig. 5