Selective proportionality and integer-valued data in DEA: an application to performance evaluation of high schools

Abstract

Conventional data envelopment analysis (DEA) models are often extended for constant or variable returns to scale assumptions based on the under-investigated technology. It is assumed that all inputs and outputs are real-valued data. However, in many practical applications, proportionality or convexity axioms require to be modified. This study attempts to further expand upon the hybrid returns to scale DEA models in the presence of integer-valued input and output data. We refine the previous axioms to introduce a new minimal extrapolation technology set. Moreover, we formulate a couple of mixed-integer linear programming models for efficiency evaluation and target setting. An empirical application on 30 high schools in Iran is provided to validate the proposed approach. The data analysis, including efficiency evaluations along with providing benchmark units, is also performed.

Appendices

Appendix 1

Proof of Theorem 1

The following items need to verify:

• a. ${T}_{HRS}^{IDEA}$satisfies the mentioned axioms.

• b. If $\hat{T}$ is an arbitrary technology set that satisfies the same axioms, ${\varvec{B}}1 - {\varvec{B}}4$, then it contains $T_{HRS}^{IDEA}$. The first step is straightforward and a trivial verification points that $T_{HRS}^{IDEA}$

  involves the observed DMUs $\left( {{\varvec{B}}1} \right)$, satisfies natural convexity $\left( {{\varvec{B}}2} \right)$, natural disposability $\left( {{\varvec{B}}3} \right)$, and natural selective proportionality $\left( {{\varvec{B}}4} \right)$. Regarding the second step, consider integer-valued data and assume that $\hat{T} \subseteq Z_{ + }^{m + s}$ be any arbitrary technology set that satisfies ${\varvec{B}}1 - {\varvec{B}}4$. Let $T = {\text{conv}}\left( {\hat{T}} \right) \subseteq R_{ + }^{m + s}$ be the convex hull of $\hat{T}$. According to Kuosmanen, Kazemi Matin (2009), $T$ is still contains the observations and satisfies the continues versions of ${\varvec{B}}2 - {\varvec{B}}4$. As it is shown in Podinovski et al. (2014), the production set $T$ is the smallest set that satisfies the mentioned axioms ${\varvec{B}}1 - {\varvec{B}}4$. So, we have $T_{HRS} \subseteq T$. Now, by restricting the vectors to the integer valued points, we obtain

  $$T_{HRS} \cap Z_{ + }^{m + s} \subseteq T \cap Z_{ + }^{m + s}$$

  (14)

  The left and right sides of (6) can be considered as $T_{HRS}^{IDEA}$ and $\widehat{T}$, respectively. This concludes that ${T}_{HRS}^{IDEA}\subseteq \widehat{T}$, which completes the proof.

Appendix 2

Generalized model: In Sect. 4, we assumed that all input and output data can only take integer values. Now, we extend the proposed model to the case in which some inputs and outputs are real-valued. Suppose the input–output vector (x, y) is partitioned as (${x}^{IP}, {x}^{INP}, {x}^{P}, {x}^{NP}, {y}^{IP}, {y}^{INP}, {y}^{P}, {y}^{NP}$) in which:

• $$x_{{}}^{{IP}} {\text{ are integer}} - {\text{valued proportional input m}}_{1} {\text{ }} - {\text{ vector}},$$
• $$x_{{}}^{{INP}} {\text{ are integer }} - {\text{ valued nonproportional input m}}_{2} {\text{ }} - {\text{ vector,}}$$
• $$x_{{}}^{P} {\text{ are real }} - {\text{ valued proportional input m}}_{3} {\text{ }} - {\text{ vector}},$$
• $$x^{{NP}} {\text{ are real }} - {\text{ valued nonproportional input }}m_{4} {\text{ }} - {\text{ vector}},$$
• $$y^{{IP}} {\text{are integer }} - {\text{ valued proportional output s}}_{1} {\text{ }} - {\text{vector}},$$
• $$y_{{}}^{{INP}} {\text{ are integer}} - {\text{ valued nonproportional output s}}_{2} {\text{ }} - {\text{ vecto}}r,$$
• $$y^{P} {\text{are real }} - {\text{ valued proportional output s}}_{3} {\text{ }} - {\text{ vector,}}$$
• $$y_{{}}^{{NP}} {\text{ are real }} - {\text{ valued nonproportional output s}}_{4} - {\text{ vector,}}$$

The following input-oriented MILP model is proposed to analyze the relative efficiency of ${DMU}_{\mathrm{k}}$:

$$\begin{gathered} \mathop {{\text{min}}}\limits_{{\lambda ,\mu ,\nu ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} }} \theta \hfill \\ {\text{s}}.{\text{t}}. \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \left( {\lambda_{j} + \mu_{j} - \nu_{j} } \right)x_{{i_{1} j}}^{IP} \le \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{{i_{1} k}}^{IP} } & {i_{1} = 1,...,m_{1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{{i_{1} k}}^{IP} \le \theta x_{{i_{1} k}}^{IP} } & {i_{1} = 1,...,m_{1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\sum\limits_{j = 1}^{n} {\lambda_{j} x_{{i_{2} j}}^{INP} \le \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{{i_{2} k}}^{INP} } } & {i_{2} = 1,...,m_{2} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{{i_{2} k}}^{INP} \le \theta x_{{i_{2} k}}^{INP} } & {i_{2} = 1,...,m_{2} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \left( {\lambda_{j} + \mu_{j} - \nu_{j} } \right)x_{{i_{1} j}}^{P} \le \theta x_{{i_{1} k}}^{P} } & {i_{3} = 1,...,m_{3} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{{i_{2} j}}^{NP} \le \theta x_{{i_{2} k}}^{NP} } & {i_{4} = 1,...,m_{4} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \left( {\lambda_{j} + \mu_{j} - \nu_{j} } \right)y_{{r_{1} j}}^{IP} \ge y_{{r_{1} k}}^{IP} } & {r_{1} = 1,...,s_{1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \left( {\lambda_{j} - \nu_{j} } \right)y_{{r_{2} j}}^{INP} \ge y_{{r_{2} k}}^{INP} } & {r_{2} = 1,...,s_{2} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \left( {\lambda_{j} + \mu_{j} - \nu_{j} } \right)y_{{r_{1} j}}^{P} \ge y_{{r_{1} k}}^{P} } & {r_{3} = 1,...,s_{3} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mathop \sum \limits_{j = 1}^{n} \left( {\lambda_{j} - \nu_{j} } \right)y_{{r_{2} j}}^{NP} \ge y_{{r_{2} k}}^{NP} } & {r_{4} = 1,...,s_{4} } \\ \end{array} \hfill \\ \sum\limits_{j = 1}^{n} {\lambda_{j} } = 1 \hfill \\ \begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{{i_{1} k}}^{IP} \in Z_{ + } { }} & {i_{1} = 1,...,m_{1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{{i_{2} k}}^{NIP} \in Z_{ + } } & {i_{2} = 1,...,m_{2} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\lambda_{j} - \nu_{j} \ge 0,{ }\mu_{j} \ge 0,{ }\nu_{j} \ge 0} & {j = 1,...,n} \\ \end{array} \hfill \\ \end{gathered}$$

It should be noted that ${m}_{1}+{m}_{2}+{m}_{3}+{m}_{4}=m$ and ${s}_{1}+{s}_{2}+{s}_{3}+{s}_{4}=s$ and m and s are respectively the total number of inputs and outputs.