## 3 Methodology
### Output-Oriented of VRS-DEA Modeling  
DEA is a type of data-oriented performance-evaluation approach that measures each Decision Making Unit (DMU) with multiple inputs and multiple outputs. DEA can be categorized into two kinds of DEA models: Constant Returns to Scale (CRS), and Variable Returns to Scale (VRS). TE of CRS is the overall comparison between resource-allocation efficiency and resource-utilization efficiency of those DMUs. TE of VRS refers to the production efficiency influenced by the factors like management and technology. The research shows that the increment of Total Factor Productivity (TFP) comes from the improvement of both technology and efficiency. \(SE\) unveils the gap between the real scale and the optimal scale of the production.  
DEA model can be categorized into two types according to different perspectives of input and output for the research. One is input-oriented and the other is output-oriented. Of these two, the former gives the definition of scale-down production frontiers when output is invariable, while the latter gives the definition of scale-up production frontiers when the input is invariable. Actually, both of them present measurement analysis for the same questions, though in two different perspectives of input and output. Therefore, they share the same production frontiers and a very slight difference for the efficiency value after measurement [28]. Considering the one-year data collected for this research, the output-oriented VRS of the DEA model is adopted in the analysis because people dealing with cassava production always put the maximized output as the priority when input is at a fixed level.  
In order to measure cassava output production efficiency clearly, the output-oriented DEA model was chosen. Let N represents the number of DMUs, each DMU has K input factors produce and M outputs, production-possible collection are T\({}_{\text{v}}\) and T\({}_{\text{n}}\), and calculating the \(i\) DMU's relative efficiency value. Output-oriented of VRS of DEA modeling is as follows:  
\[\text{Max}\left[\alpha_{\text{p}}+\varepsilon\left(e_{1}^{T}SA+e_{2}^{T}SB \right)\right],\]  
\[\text{s.t.} \tag{3}\] \[\left\{\begin{array}{l}\sum\limits_{i=1}^{n}\lambda_{i}X_{i}+ SA=X_{0}\\ \sum\limits_{i=1}^{n}\lambda_{i}X_{i}-SB=\alpha_{\text{v}}Y_{0}\ \ \lambda_{i}\geq 0,\ i=1,2,\ldots,\ n.\ SA\geq 0,\ SB\geq 0\\ \sum\limits_{i=1}^{n}\lambda_{i}=1,\end{array}\right.\]  
where \(SA\) and \(SB\) are the slack variables, \(\varepsilon\) is the perturbation, \(\lambda\) is the unit vector of N \(\times\) 1, and \(\beta\) is the relative efficiency of DMUs. \(\beta\)\(\epsilon\)\([0,1]\). When \(\beta\) equals 1, DMU is efficient and DMU is on the production frontiers. At the same time, the DMU cannot create more output when the input is determined. Input quantity cannot be reduced if output is determined. When \(\beta\ <1\), the DMU is inefficient.  
Meanwhile, its change also reflects the influence placed on TFP by increasing input. \(OTE\) represents the technology efficiency when the returns to scale stay unchanged. In contrast, \(PTE\) refers to the Technical Efficiency when the returns to scale fluctuate. The relationship between \(OTE\), \(PTE\), and \(SE\) is \(OTE=PTE\times SE\). When \(OTE=1\), it shows that, if the output and input of DMUs are efficient, \(PTE\) and \(SE\) are efficient as well. When \(PTE=1\), it shows that at the current technological level, the use of its input resources is efficient. The fundamental reason for its failure to achieve comprehensive and effective results is that its scale is invalid. Therefore, focus is needed on how to make better use of its scale efficiency.