Cross Malmquist Productivity Index in Data Envelopment Analysis

Abstract

In Data Envelopment Analysis (DEA), the Malmquist Productivity Index (MPI) is an important instrument used to assess dynamic performance. In this paper, to deal with overestimation, to increase rationality, and to reduce dependence on self-assessment results, we equip MPI with the cross-evaluation strategy. The MPI values measured from self and peer points of view are not the same necessarily, even significantly different or strongly inconsistent, and none of them can be ignored. Therefore, to match the results for the two common up-down and bottom-up strategies in building MPI indices, and to retain the multiplicative structure at the aggregate level, we use the geometric mean to aggregate the self and peer MPIs. Because the MPI indices of all units are calculated with several common bundle weights, the resulting values are consistent and therefore comparable. We will show the usability of the proposed method using a real example of 20 branches of an Iranian bank in 2017–2018.

Appendix

As indicated by Pastor and Lovell (2005), the global MPI is circular and different from the super efficiency case, where under some conditions the associated LP could be infeasible (Zhu 1996; Seiford and Zhu 1999), the LP in the global MPI is always feasible. In this appendix, we present a global version of the CRMPI proposed in this paper. Following Pastor and Lovell (2005), we assume a global technology that is formed from data of all DMUs in two time periods (all periods in the case which we may have more periods). First, we get two weight vectors for each DMUj, so that ${\text{(u}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{,v}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{)}}$ and ${\text{(u}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{,v}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{)}}$ are the optimal values obtained by evaluating the unit j at time t and t + 1 relative to the global frontier. Then, we calculate ${\text{E}}_{{_{{{\text{j,}}\,{\text{o}}}} }}^{{{\text{t,G}}}} =\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}$, ${\text{E}}_{{_{{{\text{j,}}\,{\text{o}}}} }}^{{{\text{t + 1,G}}}} =\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}$, in which ${\text{E}}_{{_{{{\text{j}}\,{\text{,o}}}} }}^{{{\text{k,G}}}} =\frac{{{\text{u}}_{{\text{j}}}^{{{\text{k,G}}}} {\text{y}}_{{\text{o}}}^{{\text{k}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{k,G}}}} {\text{x}}_{{\text{o}}}^{{\text{k}}} }}$ is the cross global efficiency of DMUo in time k = t, t + 1 using optimal weights of DMUj in time k relative to the global technology frontier. By changing k, we obtain two Cross Global Efficiency Matrices (CRGEMs) (Table

Table 16 Two CRGEMs over time (k = t, t + 1)

16).

Using ${\text{GMPI}}_{{{\text{jo}}}}^{{}} =\frac{{{\text{E}}_{{_{{{\text{j,.o}}}} }}^{{{\text{t + 1,G}}}} }}{{{\text{E}}_{{_{{{\text{j,.o}}}} }}^{{{\text{t,G}}}} }}$ we calculate the cross global MPI matrix (CRGMPIM) (Table

Table 17 CRGMPIM

17).

We calculate the CRGMPI score as the average of cross global MPIs.

$$ {\text{CRGMPI}}_{{\text{o}}} = \left( {\mathop {\prod \,}\limits_{{{\text{j = 1}}}}^{{\text{n}}} {\text{GMPI}}_{{{\text{j}}\,{\text{o}}}} } \right)^{{{\text{1/n}}}} =\left( {\frac{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}}\, \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}} \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}}} \right)^{{\frac{{\text{1}}}{{\text{n}}}}} $$

Since the usual DEA models and global technology are used to calculate the efficiency values used in CGMPI, this index is clearly safe from the impossibility. It is also a circular index, because,

$$ \begin{aligned} & {\text{CRGMPI}}_{{_{{\text{O}}} }}^{{{\text{(t,t + 1)}}}} \times{\text{ CRGMPI}}_{{_{{\text{O}}} }}^{{{\text{(t + 1,t + 2)}}}}\\&\quad=\left( {\frac{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}}\, \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}} \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}}} \right)^{{\frac{{\text{1}}}{{\text{n}}}}} \times \left( {\frac{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t + 2,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 2}}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t + 2,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 2}}}} }}}}{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}\, \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t + 2,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 2}}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t + 2,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 2}}}} }}}}{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}} \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t + 2,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 2}}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t + 2,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 2}}}} }}}}{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t + 1,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 1}}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t + 1,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 1}}}} }}}}} \right)^{{\frac{{\text{1}}}{{\text{n}}}}} \\ & \quad =\left( {\frac{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t + 2,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 2}}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t + 2,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 2}}}} }}}}{{\frac{{{\text{u}}_{{\text{1}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{1}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}}\, \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t + 2,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 2}}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t + 2,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 2}}}} }}}}{{\frac{{{\text{u}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{j}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}} \times \cdots \times \frac{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t + 2,G}}}} {\text{y}}_{{\text{o}}}^{{{\text{t + 2}}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t + 2,G}}}} {\text{x}}_{{\text{o}}}^{{{\text{t + 2}}}} }}}}{{\frac{{{\text{u}}_{{\text{n}}}^{{{\text{t,G}}}} {\text{y}}_{{\text{o}}}^{{\text{t}}} }}{{{\text{v}}_{{\text{n}}}^{{{\text{t,G}}}} {\text{x}}_{{\text{o}}}^{{\text{t}}} }}}}} \right)^{{\frac{{\text{1}}}{{\text{n}}}}} {\text{ = CRGMPI}}_{{_{{\text{O}}} }}^{{{\text{(t,t + 2)}}}} \\ \end{aligned} $$