## 4 Research Methodology
### FDH, CCR and BCC Models  
FDH model is a non-parametric method to measure the efficiency of production units or DMUs. FDH model reduces the convexity assumption of basic DEA models. The computational technique to solve the FDH program reflects the mixed integer programming problem associated with the DEA model with a linear programming problem [39]. If definite inputs can produce particular outputs, the pairs of these inputs and outputs are producible, so the braces are called the production possibility set or PPS. In Figures 1 and 2, if any activity like 4 (x\({}_{1}\), y\({}_{1}\)) in Figure 1, belongs to the production possibility set (P), then the activity (\(t_{x}\), \(t_{y}\)) belongs to P for any positive scalar t. This property is a CCR model or CRS. This hypothesis can be adapted to allow the production possibility set with diverse hypothesizes. The BCC model is illustrative using by variable returns-to-scale (VRS). It is characterized by increasing returns-to-scale (IRS), decreasing returns-to-scale (DRS), and constant returns to scale \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Type** & **Stage** & **Variable (Stat)** & **Unit** \\ \hline Input & I & Energy Consumption (X) & 10 TCE (Ton of coal equivalent) \\ Intermediate elements & I\&II & Cement Production (M) & 10 Ton \\ Intermediate elements & & Pollution Control & 1000 USD (United States Dollar) \\ (desirable output) & & Investment (N) & \\ Desirable Output & II & Wastewater Removed (Y) & 10 ton \\ Desirable Output & II & Waste Gas Removed (E) & 10 ton \\ Desirable Output & II & Solid Waste Removed (F) & 10 ton \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The inputs, intermediate elements, and outputs for the 1 ST DMUs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stat** & **Max** & **Min** & **Ave** & **SD** \\ \hline X & 35351.31 & 1723.96 & 14241.86 & 8865.93 \\ M & 36781.76 & 2359.98 & 14543 & 8809.97 \\ N & 891.91 & 27.24 & 291.92 & 193.29 \\ Y & 766248.45 & 6531.01 & 164131.92 & 153176.91 \\ E & 29.98 & 0.82 & 9.35 & 6.91 \\ F & 23431.70 & 7.98 & 2761.76 & 4723.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Expressive analysis of the data. (CRS). The production possibility set of the FDH model is attained by defining it inversely with CCR and BCC models. Based on the Figure 2, if 4 (\(x_{1}\), \(y_{1}\)) and 1 (\(x_{2}\), \(y_{2}\)) belong to the production possibility set of CCR and BCC models respectively, then the 5 (a (\(x_{1}+x_{2}\)), b (\(y_{1}+y_{2}\))) with any positive scalar a, b is also measured to be in the same production possibility set. The axiom is called convexity. Free impossibility means if a specific pair of input and output is producible, any pairs of more input and less output for the specific one is also producible. FDH model allows the free impossibility to construct the production possibility set. Accordingly, the frontier line for the FDH model is developed from the observed inputs and outputs, enabling free failure. In Figure 3, the form of the production possibility set in FDH is stepwise. The frontiers determined for the FDH model are presented considering two inputs and one output for six production units  
Figure 1: Production possibility sets. Figure 2: Production possibility sets in increasing returns-to-scale (IRS), decreasing returns-to-scale (DRS), and constant returns to scale (CRS). labeled A through F. In the BCC model, DMUs A, B, and C are efficient, but A, B, C, and F are efficient in FDH model. The efficiency of observation E in BCC model is defined as:  
\[\theta_{E.BCC.input-oriented}=OE_{2}/OE \tag{1}\]  
But the efficiency of observation E in FDH model is defined as:  
\[\theta_{E.FDH.input-oriented}=OE_{1}/OE \tag{2}\]  
One version of the FDH model aims to minimize inputs while satisfying at least the given output levels. This is called the input-oriented model. The other one is called the output-oriented model that attempts to maximize outputs without requiring more inputs. The scores of efficiencies in the FDH model are between 0 and 1.