Performance prediction of DMUs using integrated DEA-SVR approach with imprecise data: application on Indian banks

Abstract

Data Envelopment Analysis (DEA) is a widely used performance analysis tool, but for every newly added unit in a data set, it requires complete re-processing of the DEA model. In real applications, data sets are usually imprecise and enormously increasing, which demands a large computer processing time. A predictive capability of DEA, typically in uncertain environment, can solve this problem. This paper aims to establish an integrated DEA and support vector machine for regression (SVR) approach with imprecise data for efficiency estimation and prediction in optimistic and pessimistic environments. To illustrate the potential of the proposed approach, it is applied on Indian banks for evaluation and prediction of different efficiency measures (technical, pure technical, and scale) during the period 2011–2020. To study the sensitivity analysis and productivity of Indian banks’ efficiency with time, DEA window analysis (window width five years) and Malmquist productivity index (MPI) in optimistic and pessimistic environments have been improvised with interval data. The new approach results in very reliable and accurate prediction (mean square error: $2\times 10^{-3}$ to $5\times 10^{-3}$) of different efficiency measures that benefits bank experts for planning and decision making on bank addition/merger. Other findings conclude that (i) technical efficiency is more influenced by managerial (pure technical) inefficiencies rather than bank size (scale efficiency) in all windows, and (ii) on the contrary to all other banks, Yes Bank Ltd. examined decreased productivity in both the environments, (iii) technical efficiency change has higher impact on MPI in both environments as compared to technological change.

Appendices

Appendix A Kernel function

Kernel function is a map, say $\phi$ that transform a m-dimensional vector $X\in {\mathbb {R}}^m$ into another higher n-dimensional vector $X\in {\mathbb {R}}^n$ in which they become separable (Williamson et al. 1999). Thus, the map can be expressed as:

$$\begin{aligned}&\phi :{\mathbb {R}}^m\rightarrow {\mathbb {R}}^n&\\&\text {i.e.}, X =(x_1,x_2,\ldots ,x_m)^T \rightarrow (x_1,x_2,\ldots ,x_n)^T=\phi (X)&\end{aligned}$$

Some common inner-product kernel functions (Amari and Wu 1999) available in the literature are :

• Polynomial kernel: The equation is $\kappa (x_i,x_j)=(x_i \cdot x_j + 1)^d$, where d is degree of polynomial.

• Gaussian radial basis function (RBF) : $\kappa (x,y)=\exp \left( -\frac{\parallel x-y \parallel ^2}{2\sigma ^2}\right)$, where $\sigma ^2$ is the width specified prior by the user.

• Hyperbolic Tangent Kernel Function: $\kappa (x_i,x_j)\!=\!\tanh (x_i \cdot x_j+c)$

Appendix B Interval MPI

The following radial models are presented for calculating the lower and upper bounds of both single period (within-time period) and mixed period efficiency measures (adjacent-time period) for $\text {DMU}_k$ with interval data in both optimistic and pessimistic environments under CRS mode, respectively, where ‘k’ is the index for unit under evaluation:

Model B.1

(Optimistic environment for within-time period)

$$\begin{aligned}&\text {min}~~ \Theta _k^{U,f}(\hat{\textrm{X}}_k^f, \hat{\textrm{Y}}_k^{g,f}, \hat{\textrm{Y}}_k^{b,f}\,|\,f=t,t+1)\\&\quad =\theta _k - \epsilon \left( \sum _{i=1}^{m} s_{ik}^- + \sum _{r=1}^{s_1} s_{rk}^{+g}+\sum _{p=1}^{s_2} s_{pk}^{+b}\right) \\&\text {subject to}~~ \sum _{j=1}^{n} \lambda _{jk} y_{rj}^{gU,f} - \sum _{j=1}^{n} \alpha _{jk} y_{rj}^{gU,f} - s_{rk}^{+g} = y_{rk}^{gU,f},\\&\quad \forall \,r=1,2,\ldots ,s_1;\\&\sum _{j=1}^{n} \lambda _{jk} y_{pj}^{bL,f} - \sum _{j=1}^{n} \alpha _{jk}y_{pj}^{bL,f} + s_{pk}^{+b} = y_{pk}^{bL,f},\quad \forall \,p=1,2,\ldots ,s_2;\\&\theta x_{ik}^{L,f} - \sum _{j=1}^{n} \lambda _{jk} x_{ij}^{L,f} - s_{ik}^- = 0,\\&\quad \forall \,i=1,2,\ldots ,m;\\&\lambda _{jk}, \alpha _{jk}, s_{ik}^-, s_{rk}^{+g}, s_{pk}^{+b} \ge 0,\quad \forall \,j,i,r,p; \,\epsilon >0,\quad \theta _k \,\text {unrestricted}. \end{aligned}$$

Model B.2

(Pessimistic environment for within-time period)

$$\begin{aligned}&\text {min}~~\Theta _k^{L,f}(\hat{\textrm{X}}_k^f, \hat{\textrm{Y}}_k^{g,f}, \hat{\textrm{Y}}_k^{b,f}\,|\,f=t,t+1)\\&\quad =\theta _k - \epsilon \left( \sum _{i=1}^{m} s_{ik}^- + \sum _{r=1}^{s_1} s_{rk}^{+g}+\sum _{p=1}^{s_2} s_{pk}^{+b}\right) \\&\text {subject to}~~ \sum _{j=1}^{n} \lambda _{jk} y_{rj}^{gU,f} - \sum _{j=1}^{n} \alpha _{jk} y_{rj}^{gU,f} - s_{rk}^{+g} = y_{rk}^{gL,f},\\&\quad \forall \,r=1,2,\ldots ,s_1;\\&\sum _{j=1}^{n} \lambda _{jk} y_{pj}^{bL,f} - \sum _{j=1}^{n} \alpha _{jk}y_{pj}^{bL,f} + s_{pk}^{+b} = y_{pk}^{bU,f},~\forall \,f=1,2,\ldots ,s_2;\\&\theta x_{ik}^{U,f} - \sum _{j=1}^{n} \lambda _{jk} x_{ij}^{L,f} - s_{ik}^- = 0,~\forall \,i=1,2,\ldots ,m;\\&\lambda _{jk}, \alpha _{jk}, s_{ik}^-, s_{rk}^{+g}, s_{pk}^{+b} \ge 0,\quad \forall \,j,i,r,p; \,\epsilon >0,\quad \theta _k \,\text {unrestricted}. \end{aligned}$$

Model B.3

(Optimistic environment for adjacent-time period)

$$\begin{aligned}&\text {min}~~ \Theta _k^{U,f}(\hat{\textrm{X}}_k^h, \hat{\textrm{Y}}_k^{g,h}, \hat{\textrm{Y}}_k^{b,h}\,|\,f,h=t,t+1,f\ne q)\\&\quad = \theta _k - \epsilon \left( \sum _{i=1}^{m} s_{ik}^- + \sum _{r=1}^{s_1} s_{rk}^{+g}+\sum _{p=1}^{s_2} s_{pk}^{+b}\right) \\&\text {subject to}~~ \sum _{j=1}^{n} \lambda _{jk} y_{rj}^{gU,f} - \sum _{j=1}^{n} \alpha _{jk} y_{rj}^{gU,f} - s_{rk}^{+g} = y_{rk}^{gU,h},\\&\quad \forall \,r=1,2,\ldots ,s_1;\\&\sum _{j=1}^{n} \lambda _{jk} y_{pj}^{bL,f} - \sum _{j=1}^{n} \alpha _{jk}y_{pj}^{bL,f} + s_{pk}^{+b} = y_{pk}^{bL,h},~\forall \,f=1,2,\ldots ,s_2;\\&\theta x_{ik}^{L,h} - \sum _{j=1}^{n} \lambda _{jk} x_{ij}^{L,f} - s_{ik}^- = 0,~\forall \,i=1,2,\ldots ,m;\\&\lambda _{jk}, \alpha _{jk}, s_{ik}^-, s_{rk}^{+g}, s_{pk}^{+b} \ge 0,\quad \forall \,j,i,r,p; \,\epsilon >0,\quad \theta _k \,\text {unrestricted}. \end{aligned}$$

Table 6 Descriptive statistics of input–output data for all windows

Model B.4

(Pessimistic environment for adjacent-time period)

$$\begin{aligned}&\text {min}~~\Theta _k^{L,f}(\hat{\textrm{X}}_k^h, \hat{\textrm{Y}}_k^{g,h}, \hat{\textrm{Y}}_k^{b,h}\,|\,f=t,t+1)\\&\quad =\theta _k - \epsilon \left( \sum _{i=1}^{m} s_{ik}^- + \sum _{r=1}^{s_1} s_{rk}^{+g}+\sum _{p=1}^{s_2} s_{pk}^{+b}\right) \\&\text {subject to}~~ \sum _{j=1}^{n} \lambda _{jk} y_{rj}^{gU,f} - \sum _{j=1}^{n} \alpha _{jk} y_{rj}^{gU,f} - s_{rk}^{+g} = y_{rk}^{gL,h},\\&\quad \forall \,r=1,2,\ldots ,s_1;\\&\sum _{j=1}^{n} \lambda _{jk} y_{pj}^{bL,f} - \sum _{j=1}^{n} \alpha _{jk}y_{pj}^{bL,f} + s_{pk}^{+b} = y_{pk}^{bU,h},~\forall \,f=1,2,\ldots ,s_2;\\&\theta x_{ik}^{U,h} - \sum _{j=1}^{n} \lambda _{jk} x_{ij}^{L,f} - s_{ik}^- = 0,~\forall \,i=1,2,\ldots ,m;\\&\lambda _{jk}, \alpha _{jk}, s_{ik}^-, s_{rk}^{+g}, s_{pk}^{+b} \ge 0,\quad \forall \,j,i,r,p; \,\epsilon >0,\quad \theta _k \,\text {unrestricted}. \end{aligned}$$

where real variable $\theta _k$ and non-negative variables $\lambda _{jk}$ and $\alpha _{jk},~\forall \,(j=1,2,\ldots ,n)$ are the dual variables, $s_{ik}^-$ are input excess for ith input, and $s_{rk}^{+g}$, $s_{pk}^{+b}$ are the output shortfalls for rth desirable and pth undesirable output, respectively, associated with $\text {DMU}_k$, and $\epsilon$ is a non-Archimedean infinitesimal.

Appendix C List of Banks and descriptive statistics of input–output data variables

See Appendix Tables 6 and 7.

Table 7 List of Indian Banks with bank code involved in study