*3.1. Directional Distance Function, DDF*

This study uses [31] to extend the non-oriented method in the DDF model based on the SBM described by [32]. All models can evaluate the general efficiency value (≤1) at the same time, and its calculation method is as follows:

Non-oriented DD model In this case, we have max *β* s.t. *Xλ* + *βgx* ≤ *xk* (1) *Yλ* − *βgy* ≥ *yk* ∑*λ* = 1 *λ* ≥ 0 (*d*(*I*) , *d*(*IN*) , *d*(*O*) , *d*(*ON*) , *d*(*OBad*) )=( *x* (*I*) *<sup>o</sup>* , 0, *y* (*O*) *<sup>o</sup>* ,0, *<sup>y</sup>* (*OBad*) *<sup>o</sup>* ) (2) [DD-C] *ξ*∗ = *MAXξ* st.*X*(*I*) *λ* + *ξx* (*I*) *<sup>o</sup>* + *s* (*I*) = *x* (*I*) *o X*(*IN*) *λ* + *s* (*IN*) = *x* (*IN*) *<sup>o</sup> Y*(*O*) *λ* − *ξy* (*O*) *<sup>o</sup>* <sup>−</sup> *<sup>s</sup>* (*O*) = *y* (*O*) *<sup>o</sup>* (3) *Y*(*ON*) *λ* − *s* (*ON*) = *y* (*ON*) *<sup>o</sup> Y*(*OBad*) *λ* + *ξy* (*OBad*) *<sup>o</sup>* <sup>+</sup> *<sup>x</sup>*(*OBad*) <sup>=</sup> *<sup>y</sup>* (*OBad*) *<sup>o</sup>* ξ ≥ 0 , *λ* ≥ 0 , *s* (*I*) <sup>≥</sup> 0, *<sup>s</sup>* (*IN*) <sup>≥</sup> 0, *<sup>s</sup>* (*O*) <sup>≥</sup> 0, *<sup>s</sup>* (*ON*) <sup>≥</sup> 0 , *<sup>s</sup>* (*OBad*) <sup>≥</sup> 0 . We define the efficiency value of DMU(*xo*, *yo*) as

$$
\theta^\* = 1 - \xi^{\mu\_\*}.
$$

#### *3.2. Technology Gap Ratio, TGR*

Since the production boundary of g groups is included in the common production boundary, the technical efficiency under the common boundary must be less than that under the group boundary. The ratio of the two is called the technical efficiency gap ratio (TGR), as follows:

$$\text{TGR} = \frac{\text{Technical efficiency under common boundary}}{\text{Techanical efficiency under group boundary}} \tag{4}$$

## **4. Data Analysis and Empirical Results**
