Appendix 1
Table 4 Correlations between output and potential input variables by SPSS linear regression
Table 5 Coefficients between output and potential input variables by SPSS linear regression
Appendix 2
Table 6 Hotel chain/consortia systems, members in Taiwan, and requirements for memberships, green hotel policies
Appendix 3
Table 7 Green certifications and respective conditions to pass certification
Appendix 4. Methodology
To model the technology to produce desired outputs with jointly undesired outputs, some conditions are needed. If we denote desired outputs by \( y\in {R}_{+}^M \), undesired outputs by \( b\in {R}_{+}^I \), and inputs by \( x\in {R}_{+}^N \), we can denote the output sets as:
$$ D(x)=\left\{\left(y,b\right):x\ \mathrm{can}\ \mathrm{produce}\ \left(y,b\right)\right\}. $$
(1.1)
The reduction of undesired outputs is costly and can be modeled as:
$$ \left(y,b\right)\ni D(x)\ \mathrm{and}\ 0\le \theta \le 1\ \mathrm{imply}\left(\theta y,\theta b\right)\ni D(x). $$
(1.2)
The desired outputs jointly produced with the undesired outputs can be modeled by:
$$ \mathrm{if}\ \left(y,b\right)\ni D(x)\ \mathrm{and}\ b=0\ \mathrm{then}\ y=0. $$
(1.3)
Based on conditions 1.1~1.3, Chung et al.’s definition for the ML index of productivity between period t and t + 1 is as:
$$ {\displaystyle \begin{array}{l}{\mathrm{ML}}_t^{t+1}=\frac{{\left[\left(1+D{\prime}_0^{t+1}\left({x}^t,{y}^t,{b}^t;{y}^t,-{b}^t\right)\right)\left(1+D{\prime}_0^t\left({x}^t,{y}^t,{b}^t;{y}^t,-{b}^t\right)\right)\right]}^{1/2}}{{\left(1+D{\prime}_0^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1};{y}^{t+1},-{b}^{t+1}\right)\right)\left(1+D{\prime}_0^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1};{y}^t,-{b}^{t+1}\right)\right)}^{1/2}}\\ {}=\frac{{\left[\left(1+D{\prime}_0^t\left({x}^t,{y}^t,{b}^t;{y}^t,-{b}^t\right)\right)\right]}^{\ast }}{\left(1+D{\prime}_0^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1};{y}^{t+1},-{b}^{t+1}\right)\right)}\\ {}\begin{array}{l}\frac{{\left[\left(1+D{\prime}_0^{t+1}\left({x}^t,{y}^t,{b}^t;{y}^t,-{b}^t\right)\right)\left(1+D{\prime}_0^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1};{y}^{t+1},-{b}^{t+1}\right)\right)\right]}^{1/2}}{{\left(1+D{\prime}_0^t\left({x}^t,{y}^t,{b}^t;{y}^t,-{b}^t\right)\right)\left(1+D{\prime}_0^t\left({x}^{t+1},{y}^{t+1},{b}^{t+1};{y}^{t+1},-{b}^{t+1}\right)\right)}^{1/2}}\\ {}={\mathrm{ML}\mathrm{EFFCH}}_t^{t+1}\ast {\mathrm{ML}\mathrm{TECH}}_t^{t+1}\end{array}\end{array}} $$
where D′0 is a directional distance function and
$$ {D}_0^{\prime}\left(x,y,b;g\right)=\sup \left\{\beta :\left(y,b\right)+\beta g\in D(x)\right\} $$
where g is a vector of “directions” in which outputs are scaled.
ML index uses directional distance function than traditional Malmquist index using Shephard’s output distance function. The distance functions can be illustrated by Fig. 1. The meaning of point A in efficiency frontier in Malmquist index can be modeled as to scale OC/OA by OA/OC which increases both desired and undesired outputs proportionally from point C. The meaning of point B can be modeled as to scale OC by CB in efficiency frontier in ML index by increasing desired outputs and decreasing undesired outputs in direction Og (the same as CB). From the difference, it means that to increase productivity growth in Malmquist index one may have an increasing cost of pollution as a by-product, which is not what we expected in a sustainability effort. Thus, ML index is preferred since it can be expected to increase productivity growth without increasing undesired outputs.
For the efficiency of calculation, ML index can be indirectly derived from Malmquist index. To associate ML index with Malmquist index can be achieved through their respective based Shephard’s output distance function and directional distance function. The relation between directional distance function and Shephard’s output distance function is as:
$$ {\displaystyle \begin{array}{l}{D}_0^{\prime}\left(x,y,b;y,b\right)=\sup \left\{\beta :{D}_0\left(x,\left(y,b\right)+\beta \left(y,b\right)\right)\leqq 1\right\}\\ {}=\left(1/{D}_0\left(x,y,b\right)\right)-1\end{array}} $$
where D0(x, y, b) is Shephard’s output distance function used in Malmquist index.