# Governance and Dynamic Efficiency with Network Structure in the Brazilian Natural Gas Utilities
## 2 Materials and Methods
### Dynamic DEA with Network Structure: A Slacks-Based (DNSBM)
In this way, the efficiency scores can be defined from the equations presented below:  
* Overall efficiency: (1) where Wt (vt) is the weight for the period t and wk (vt) is the weight for the division k. These weights satisfy the condition of \(\sum\limits_{t=1}^{T}\) Wt = 1, \(\sum\limits_{k=1}^{K}\) wk = 1, Wt \(\geq\) 0 (vt), wk \(\geq\) 0 (vt) and \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Variable** & **Unit** & \multicolumn{2}{c}{**Tone and Tsutsui [18] Description**} \\ \hline Kpipeline & Km & \(x_{Kpipeline j TO}^{t}\) & Kilometers of pipeline to DMU/\(j\) for technical–operational division at year \(t\). \\  & & & Classified as Input. \\  & & & Quantity of employees to DMU/\(j\) for technical–operational division at year \(t\). \\ Employees & Unit. & \(x_{Employees j TO}^{t}\) & technical–operational division at year \(t\). \\  & & & Classified as Input. \\  & & & Controllable operational expenditure to \\ COpex & BRL 1.000 & \(x_{COpx j EF}^{t}\) & DMU/\(j\) for economic–financial division at year \(t\). \\  & & & \(t\). Classified as Input. \\  & & & Capital expenditure to DMU/\(j\) for economic–financial division at year \(t\). \\  & & & Classified as Input. \\  & & & Accounts Receivable to DMU/\(j\) for economic–financial division at year \(t\). \\  & & & Classified as Input. \\  & & & Municalities served to DMU/\(j\) for technical–operational division at year \(t\). \\  & & & Classified as Output. \\  & & & Delivered volume to DMU/\(j\) for technical–operational division at year \(t\). \\  & & & Quantity of customers to DMU/\(j\) for technical–operational division at year \(t\). \\  & & & Quantity of customers to DMU/\(j\) for technical–operational division at year \(t\). \\  & & & Classified as Output. \\  & & & Operating revenue to DMU/\(j\) for economic–financial division at year \(t\). \\  & & & & Classified as Output. \\  & & & Revenue collection to DMU/\(j\) for economic–financial division at year \(t\). \\  & & & Classified as Output. \\  & & & Total Assets to DMU/\(j\) at year \(t\) (\(l=1\) when there is only one link). Classified as \\  & & & & Underside (bad) link from \\  & & & & technical–operational division to \\  & & & & economic–financial division. \\  & & & & \\  & & & & \\ Area & Km\({}^{2}\) & \(Z_{j TO\ l\ fix}^{(t,t+1)}\) & \(Z_{j TO\ l\ fix}^{(t,t+1)}\) & when there is only carry-over of the type in the division). Classified as Non-Discretionary \\  & & & & (Fixed) Carry-Over at technical–operational division from year \(t\) to year \(t+1\). \\  & & & & Total Liabilities to DMU/\(j\) for economic–financial division at year \(t\) (\(l=1\)). \\  & & & & Classified as Undesirable (Bad) Carry-Over at economic–financial division from year \(t\) to year \(t+1\). \\  & & & & Non-controllable operational expenditure to \\  & & & & DMU/\(j\) for economic–financial division at year \(t\) (\(l=1\)). Classified as Non-Discretionary \\  & & & & (Fixed) Carry-Over at economic–financial division from year \(t\) to year \(t+1\). \\  & & & & Net Equity to DMU/\(j\) for economic–financial division at year \(t\) (\(l=1\)). Classified as \\  & & & & Discretionary (Free) Carry-Over at economic–financial division from year \(t\) to year \(t+1\). \\ \hline \hline \end{tabular} Source: Author’s own elaboration.  
\end{table} 
Table 1: Description of Inputs, Outputs, Link, and Carry-Over Variables. With the application of the DNSBM model, we will have a general efficiency score that encompasses the efficiency results of two divisions applied to this study: the technical division and the economic-financial division. We adopted the variables from the DEA DNSBM model as carry-overs, namely the concession area (Area), liabilities (Liabilities), non-controllable OPEX (NOpex), and net equity (Equity). In the link, the total asset (Assets) is used as a transition between the end of each year and the beginning of the following year.