# VRS Model Based on DEA Method

The VRS model is mainly used to study the efficiency evaluation in the case of variable scale income. This paper is dealing with the input-oriented VRS model. The model assumes that there are K DMUs and each DMU can obtain Y output with the use of X input. The input vector of j DMU is Xj, and the output vector is Yj. OS is the output relaxation of DMU and is the input relaxation of DMU. (Constant vector. Through the K linear programming solver, the relative technical efficiency value of each DMU can be obtained v, the value range of v is from 0 to 1. If v is 1, the DMU is just on the frontier, indicating that the technique is effective. Namely, at the current input level, the output of DMU is optimal; if v < 1, it indicates that the technology has not reached the effective level. That is, there is a gap between the actual output and the optimal output of DMU). VRS’s linear programming model is as follows(cf. Huang et al. 2018, p.6 )

The technical efficiency is then decomposed in two parts. The first is pure technical efficiency  induced by management level, the second is scale efficieny induced by enterprise size. The closer the efficiency value is to 1, the more ineffective DMU is. (cf. Huang et al. 2018, p.6).

## Malmquist Index Model

Malmquist index is the geometric average of total factor productivity index in t + 1 relative to t period. The model is as follows (cf. Huang et al. 2018, p.7 ).

(xt, yt) is the input and output vector in the period of t, while (xt+1, yt+1) is the vector in t + 1 period; DtO. (xt, yt) is the output distance function of the input-output vector in t period with the technology in the period of t as a parameter, and DtO (xt+1, yt+1) is the output distance function in t + 1 period. The Malmquist index can be transformed into (cf. Huang et al. 2018, p.7).

The left part of formula (3) represents the change of technical efficiency, and the left part represents technological progress. the Malmquist index (M) can be decomposed into technical progress and technical efficiency change index (TECH). The latter can be decomposed into pure technical efficiency change index (PTECH) and scale efficiency change index (SECH). The following formula is generated (cf. Huang et al. 2018, p.7).

M> 1 means that the total factor productivity of DMU shows an increasing trend from the t period  to t + 1 period, while M < 1 means that the total factor productivity of DMU shows a declining trend from the t period to t + 1 period (cf. Huang et al. 2018, p.8)

## Reference

Huang, Yanni ; Luo, Sumei ; Xu, Guohu ; Zhou, Guanyou (2018): Quantitative Analysis and Evaluation of Enterprise. Group Financial Company Efficiency in China. University Shanghai et al.

https://www.mdpi.com/2071-1050/10/9/3210/pdf