A new family of models with generalized orientation in data envelopment analysis

TL;DR

Problem: classical radial/ directional DEA cannot handle simultaneous improvements in inputs and outputs under CRS. Approach: introduce a generalized oriented framework with LO and QO formulations and an extended Farrell efficiency measure; they also develop monotonicity results and zero-data handling, plus case-based illustrations. Key findings: QO targets are CRS-balanced and β^*_Q ≤ β^*_L, with isotropic cases reducing to CCR Farrell; in special cases LO can be solved via linearization of QO. Significance: the framework yields flexible, interpretable targets under CRS and opens paths to stochastic and non-convex extensions.

Abstract

In the framework of data envelopment analysis, we review directional models \citep{Chambers1996, Chambers1998, Briec1997} and show that they are inadequate when inputs and outputs are improved simultaneously under constant returns to scale. Conversely, we introduce a new family of quadratically constrained models with generalized orientation and demonstrate that these models overcome this limitation. Furthermore, we extend the Farrell measure of technical efficiency using these new models. Additionally, we prove that the family of generalized oriented models satisfies some desired monotonicity properties. Finally, we show that the new models, although being quadratically constrained, can be solved through linear programs in a fundamental particular case.

Figures (5)

• Figure 1: Scheme of a set $\mathcal{D}=\left\{ A,B,C,D,E\right\}$ of 5 DMUs with 2 inputs and 1 output. The production possibility set $P$ is a closed set represented in grey (regardless of returns to scale), containing all the feasible activities defined by $\mathcal{D}$. The efficient frontier $\partial ^{\text{S}}P$ is represented by the blue line, and the weakly efficient frontier $\partial ^{\text{W}}P$ is obtained by the addition of the red lines. Efficient activities are those in $\partial ^{\text{S}}P$ but also those outside $P$. Efficient DMUs are $A,B,C$, while $E,D$ are not efficient. However, $D$ is weakly efficient.
• Figure 2: Input target contractions and output target dilations, made by (a) LO and (b) QO models. Considering DMUs with $m=4$ inputs and $s=4$ outputs, and orientation vectors $\mathbf{d}^{ \hbox{$-$}}=\mathbf{d}^{ \hbox{$+$}}=(1,0.75,0.5,0.25)$, we have represented on the left axis input target contractions, $\theta ^*_i$ for $i=1,\ldots ,4$, and output target dilations, $\phi ^*_r$ for $r=1,\ldots ,4$, corresponding to optimal solutions with (a) $\beta ^*_{\text{L}}=0.5$, and (b) $\beta ^*_{\text{Q}}=0.5$, according to \ref{['eq:thetalin']} and \ref{['eq:thetaquad']}, respectively. Moreover, according to \ref{['eq:thetagen']}, the length of a bar is the value of the corresponding relative target slack, represented on the right axis.
• Figure 3: The same plots represented in Figure \ref{['fig1']}, but the scale of dilations is inverse with respect to the scale of contractions, which continues being linear. It is noted that QO models are according to CRS because the target dilation of an output with orientation coefficient $d$ is the inverse of the target contraction of an input with the same orientation coefficient $d$. This property can be seen in (b), where the dilation bars are an exact reflection of the contraction bars.
• Figure 4: Plots of $\beta ^*_{\text{L}}-\beta ^*_{\text{Q}}$ in the particular cases $\mathbf{d}^{ \hbox{$-$}}=\mathbf{1}$ and $d^{ \hbox{$+$}}_r=d^{ \hbox{$+$}}$ for $r=1,\ldots ,s$ under CRS, with different values $d^+=1,0.75,0.5,0.25$, according to \ref{['eq:alphapc']}.
• Figure 5: Plots of $\beta ^*_{\text{Q}}$ in the particular cases $d^{ \hbox{$-$}}_i=d^{ \hbox{$-$}}$ for $i=1,\ldots ,m$ and $\mathbf{d}^{ \hbox{$+$}}=\mathbf{1}$ under CRS, with different values $d^{ \hbox{$-$}}=1,0.75,0.5,0.25$, according to \ref{['eq:alphapc']}. Note that $\beta ^*_{\text{L}}$ ranges from $0$ to $1/d^{ \hbox{$-$}}$, while $\beta ^*_{\text{Q}}$ ranges from $0$ to $1$.

Theorems & Definitions (21)

• Remark 3.1: Input and output oriented radial models
• Remark 3.2: Non-oriented radial models. Balance
• Remark 3.3: Balanced according to CRS
• Remark 3.4: Efficient projection and second stage
• Proposition 3.1
• Proposition 3.2
• Definition 4.1
• Proposition 5.1
• Proposition 5.2
• Lemma 7.1