Bounded and discrete data and Likert scales in data envelopment analysis: application to regional energy efficiency in China

Abstract

In data envelopment analysis (DEA), it is usually assumed that all data are continuous and not restricted by upper and/or lower bounds. However, there are situations where data are discrete and/or bounded, and where projections arising from DEA models are required to fall within those bounds. Such situations can be found, for example, in cases where percentage data are present and where projected percentages must not exceed the requisite 100 % limit. Other examples include Likert scale data. Using existing integer DEA approaches as a backdrop, the current paper presents models for dealing with bounded and discrete data. Our proposed models address the issue of constraining DEA projections to fall within imposed bounds. It is shown that Likert scale data can be modeled using the proposed approach. The proposed DEA models are used to evaluate the energy efficiency of 29 provinces in China.

Appendices

Appendix 1: Dual model

The dual to model (1) may reveal returns to scale (RTS) nature with bounded data. Note that Fig. 1 indicates that a DMU is being projected onto the DRS frontier. The dual to model (1) can be written as

$$\begin{aligned}&\alpha _o^{1*} =\min \sum \limits _{i=1}^m {v_i x_{io} } +\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {w_{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } -\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\hat{{w}}_{r_{\textit{Bnd}} } L_{r_{\textit{Bnd}} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {v_i x_{ij} } -\sum \limits _{r=1}^s {u_r y_{\textit{rj}} } \ge 0, \quad j=1,\ldots ,n \\ &{}\sum \limits _{r=1}^s {u_r y_{ro} } +\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } { \left( { w_{r_{\textit{Bnd}} } -\hat{{w}}_{r_{\textit{Bnd}} } } \right) y_{r_{\textit{Bnd}} o} } =1 \\ &{}v_i ,u_r ,w_{r_{\textit{Bnd}} } ,\hat{{w}}_{r_{\textit{Bnd}} } \ge 0\, \quad i=1,\ldots ,m, \quad r=1,\ldots ,s, r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ \end{array} \end{aligned}$$

(6)

Model (6) can be converted into the following ratio model

$$\begin{aligned}&\alpha _o^{1*} =\min \frac{\sum \nolimits _{i=1}^m {\upsilon _i x_{io} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } -\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\hat{{\omega }}_{r_{\textit{Bnd}} } L_{r_{\textit{Bnd}} } } }{\sum \nolimits _{r=1}^s {\mu _r y_{ro} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } { \left( { \omega _{r_{\textit{Bnd}} } -\hat{{\omega }}_{r_{\textit{Bnd}} } } \right) y_{r_{\textit{Bnd}} o} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {\upsilon _i x_{ij} } -\sum \limits _{r=1}^s {\mu _r y_{rj} } \ge 0, \quad j=1,\ldots ,n \\ &{}\sum \limits _{r=1}^s {\mu _r y_{ro} } +\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } { \left( { \omega _{r_{\textit{Bnd}} } -\hat{{\omega }}_{r_{\textit{Bnd}} } } \right) y_{r_{\textit{Bnd}} o} } \ge 0 \\ &{}\upsilon _i ,\mu _r ,\omega _{r_{\textit{Bnd}} } ,\hat{{\omega }}_{r_{\textit{Bnd}} } \ge 0,\quad i=1,\ldots ,m, \quad r=1,\ldots ,s, r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ \end{array} \end{aligned}$$

(7)

To avoid possible zero denominator in the objective function, we use a non-Archimedean positive infinitesimal $\varepsilon$ to restrict it to be positive.

$$\begin{aligned}&\alpha _o^{1*} =\min \frac{\sum \nolimits _{i=1}^m {\upsilon _i x_{io} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } -\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\hat{{\omega }}_{r_{\textit{Bnd}} } L_{r_{\textit{Bnd}} } } }{\sum \nolimits _{r=1}^s {\mu _r y_{ro} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } { \left( { \omega _{r_{\textit{Bnd}} } -\hat{{\omega }}_{r_{\textit{Bnd}} } } \right) y_{r_{\textit{Bnd}} o} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {\upsilon _i x_{ij} } -\sum \limits _{r=1}^s {\mu _r y_{rj} } \ge 0, \quad j=1,\ldots ,n \\ &{}\sum \limits _{r=1}^s {\mu _r y_{ro} } +\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } { \left( { \omega _{r_{\textit{Bnd}} } -\hat{{\omega }}_{r_{\textit{Bnd}} } } \right) y_{r_{\textit{Bnd}} o} } \ge \varepsilon \\ &{}\upsilon _i ,\mu _r ,\omega _{r_{\textit{Bnd}} } ,\hat{{\omega }}_{r_{\textit{Bnd}} } \ge 0,\quad i=1,\ldots ,m, \quad \textit{Bnd}r=1,\ldots ,s, r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ \end{array} \end{aligned}$$

(8)

If we let $\hat{{\mu }}_{r_{Unb} } =\mu _{r_{Unb} } ,r_{Unb} \in O_{Unb}$ and $\hat{{\mu }}_{r_{\textit{Bnd}} } =\mu _{r_{\textit{Bnd}} } +\omega _{r_{\textit{Bnd}} } -\hat{{\omega }}_{r_{\textit{Bnd}} } ,r_{\textit{Bnd}} \in O_{\textit{Bnd}}$, model (8) becomes

$$\begin{aligned}&\alpha _o^{1*} =\min \frac{\sum \nolimits _{i=1}^m {\upsilon _i x_{io} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } -\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\hat{{\omega }}_{r_{\textit{Bnd}} } L_{r_{\textit{Bnd}} } } }{\sum \nolimits _{r=1}^s {\hat{{\mu }}_r y_{ro} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {\upsilon _i x_{ij} } -\sum \limits _{r=1}^s {\hat{{\mu }}_r y_{rj} } +\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } { \left( { \omega _{r_{\textit{Bnd}} } -\hat{{\omega }}_{r_{\textit{Bnd}} } } \right) y_{r_{\textit{Bnd}} j} } \ge 0, \quad j=1,\ldots ,n \\ &{}\sum \limits _{r=1}^s {\hat{{\mu }}_r y_{ro} } \ge \varepsilon \\ &{}\hat{{\mu }}_{r_{\textit{Bnd}} } -\omega _{r_{\textit{Bnd}} } +\hat{{\omega }}_{r_{\textit{Bnd}} } \ge 0,r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ &{}\upsilon _i ,\hat{{\mu }}_{r_{Unb} } ,\omega _{r_{\textit{Bnd}} } ,\hat{{\omega }}_{r_{\textit{Bnd}} } \ge 0\quad \,i=1,\ldots ,m, r_{Unb} \in O_{Unb} , r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ \end{array} \end{aligned}$$

(9)

Note that model (9) is very similar to the ratio form of variable returns to scale (VRS) model (Banker et al. 1984).

Note that under VRS, the bounded constraint of $L_{r_{\textit{Bnd}} } \le \alpha y_{r_{\textit{Bnd}} o} \le U_{r_{\textit{Bnd}} } (r_{\textit{Bnd}} \in O_{\textit{Bnd}} )$ is not required. This is because the convexity constraint of $\sum \limits _{j=1}^n {\lambda _j } =1$ guarantees the projection $\hat{{y}}_{r_{\textit{Bnd}} j}$ to be within the range of current output levels. Although under CRS (constant returns to scale), model (9) indicates that model (1) behaves like a VRS model. Note that $\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } -\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\hat{{\omega }}_{r_{\textit{Bnd}} } L_{r_{\textit{Bnd}} } }$ in model (9) can be regarded as a value for the “free” variable in the VRS ratio model. Note also that, for example, the non-decreasing RTS (NDRS) or non-increasing RTS (NIRS) model restricts the VRS “free” variable to be either positive or negative.

Furthermore, if we only impose the upper bounds, $\alpha y_{r_{\textit{Bnd}} o} \le U_{r_{\textit{Bnd}} } ,r_{\textit{Bnd}} \in O_{\textit{Bnd}}$ in the CRS model (1), the ratio form of its dual model can be expressed as

$$\begin{aligned}&\min \frac{\sum \nolimits _{i=1}^m {\upsilon _i x_{io} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } }{\sum \nolimits _{r=1}^s {\mu _r y_{ro} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } y_{r_{\textit{Bnd}} o} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {\upsilon _i x_{ij} } -\sum \limits _{r=1}^s {\mu _r y_{rj} } \ge 0, \quad j=1,\ldots ,n \\ &{}\upsilon _i ,\mu _r, \omega _{r_{\textit{Bnd}} } \ge 0,\, \quad i=1,\ldots ,m, \quad r=1,\ldots ,s, r_{\textit{Bnd}} \in O_{\textit{Bnd}} \end{array} \end{aligned}$$

(10)

With non-Archimedean positive infinitesimal $\varepsilon$, model (10) is converted to

$$\begin{aligned}&\min \frac{\sum \nolimits _{i=1}^m {\upsilon _i x_{io} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } }{\sum \limits _{r=1}^s {\mu _r y_{ro} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } y_{r_{\textit{Bnd}} o} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {\upsilon _i x_{ij} } -\sum \limits _{r=1}^s {\mu _r y_{rj} } \ge 0, \quad j=1,\ldots ,n \\ &{}\upsilon _i ,\mu _r ,\omega _{r_{\textit{Bnd}} } \ge \varepsilon \, \quad i=1,\ldots ,m, \quad r=1,\ldots ,s, r_{\textit{Bnd}} \in O_{\textit{Bnd}} \end{array} \end{aligned}$$

(11)

If we let $\hat{{\mu }}_{r_{Unb} } =\mu _{r_{Unb} } ,r_{Unb} \in O_{Unb}$ and $\hat{{\mu }}_{r_{\textit{Bnd}} } =\mu _{r_{\textit{Bnd}} } +\omega _{r_{\textit{Bnd}} } ,r_{\textit{Bnd}} \in O_{\textit{Bnd}}$, model (11) becomes

$$\begin{aligned}&\min \frac{\sum \nolimits _{i=1}^m {\upsilon _i x_{io} } +\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } } }{\sum \nolimits _{r=1}^s {\hat{{\mu }}_r y_{ro} } } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{i=1}^m {\upsilon _i x_{ij} } -\sum \limits _{r=1}^s {\hat{{\mu }}_r y_{rj} } +\sum \limits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } y_{r_{\textit{Bnd}} j} } \ge 0, \quad j=1,\ldots ,n \\ &{}\hat{{\mu }}_{r_{\textit{Bnd}} } -\omega _{r_{\textit{Bnd}} } \ge \varepsilon ,r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ &{}\upsilon _i ,\hat{{\mu }}_{r_{Unb} } ,\omega _{r_{\textit{Bnd}} } \ge \varepsilon ,\quad i=1,\ldots ,m, r_{Unb} \in O_{Unb} , r_{\textit{Bnd}} \in O_{\textit{Bnd}} \\ \end{array} \end{aligned}$$

(12)

It can be seen that $\sum \nolimits _{r_{\textit{Bnd}} \in O_{\textit{Bnd}} } {\omega _{r_{\textit{Bnd}} } U_{r_{\textit{Bnd}} } }$ is always non-negative, indicating that model (12) may exhibit DRS.

Appendix 2: Input orientation

Assume some inputs are in a form of non-continuous data, and denote this type of inputs by $I_{\textit{Int}} \subseteq \left\{ { 1,2,\ldots ,m} \right\}$, and its complement by $I_{Cont}$, where $I_{Cont} \cup I_{\textit{Int}} =\left\{ { 1,2,\ldots ,m} \right\}$. Denote the sets of inputs with bounded data and Likert scales as $I_{\textit{Bnd}} \subseteq \left\{ { 1,2,\ldots ,m} \right\}$ and $I_{\textit{Lik}} \subseteq \left\{ { 1,2,\ldots ,m} \right\}$, respectively. We use subscripts $i_{\textit{Bnd}}$ (in $x_{i_{\textit{Bnd}} j} )$ and $i_{\textit{Int}}$ (in $x_{i_{\textit{Int}} j}$) to denote the inputs in subsets $I_{\textit{Bnd}}$ and $I_{\textit{Int}}$, respectively. The input-oriented model for bounded and discrete data and Likert scales is developed as

$$\begin{aligned}&\min \frac{1}{m} \sum \nolimits _{i=1}^m {\alpha _i } \nonumber \\&\begin{array}{ll} s. t.&{}\sum \limits _{j=1}^n {\lambda _j \,x_{ij} } \le \tilde{x}_{io}, \quad i=1,\ldots ,m \\ &{}\tilde{x}_{io} \le \alpha _i x_{io}, \quad i=1,\ldots ,m \\ &{}L_{i_{\textit{Bnd}} } \le \tilde{x}_{{i_{\textit{Bnd}}}^{o}} \le U_{i_{\textit{Bnd}} } , i_{\textit{Bnd}} \in I_{\textit{Bnd}} \\ &{}\tilde{x}_{i_{\textit{Int}} o} integer, i_{\textit{Int}} \in I_{\textit{Int}} \\ &{}1\le \tilde{x}_{{i_{\textit{Lik}}}^{o}} \le L, i_{\textit{Lik}} \in I_{Lik} \\ &{}\tilde{x}_{{i_{\textit{Lik}}}^{o}} integer, i_{\textit{Lik}} \in I_{Lik} \\ &{}\sum \limits _{j=1}^n {\lambda _j \,y_{rj} } \ge y_{ro}, \quad r=1,\ldots ,s \\ &{}\lambda _j \ge 0, \quad j=1,\ldots ,n \end{array} \end{aligned}$$

(13)

where $\tilde{x}_{i_{\textit{Int}} o} , i_{\textit{Int}} \in I_{\textit{Int}}$ are integer variables. Note that $I_{\textit{Int}}$ and $I_{\textit{Bnd}}$ are permitted to share common elements.