# 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)  
The DNSBM model was proposed by [18] and the non-radial approach is chosen because (i) it assumes non-proportional reduction/increase in inputs/outputs. that is, by  
Figure 1: Map of Piped Gas Distributors in Brazil. Source: ABEGAS [3]. relaxing the radial proportionality, the slacks-based approach allows for capturing non-radial slacks that cause inefficiencies [31], enabling individual handling of inputs/outputs and allowing for non-proportional changes; (ii) it combines the global dynamic efficiency analysis of a given DMU, considering the connection of various time periods (carry-over variables), with the assessment of the dynamic efficiency of each interconnected division (link variables) that represents the internal process structure of the DMU [32]; and (iii) the non-radial approach, in terms of networks, provides greater discriminative power in identifying efficiency and inefficiency [33], compared to the radial model proposed by [34]. The DNSBM model formulation proposed by [PERSON] [18] is as follows:  
The efficiency calculations in DNSBM are performed for n DMUs (j = 1,..., n) consisting of \(k\) divisions (k = 1,..., K) over \(T\) time periods (t = 1,..., T). In each term, the gas distribution companies have m\({}_{k}\) inputs and r\({}_{k}\), outputs for division k. We denote the link from division k to division h as (k, h)\({}_{i}\) and the set of links as L\({}_{kh}\) The classification of the observed data is categorized into inputs, outputs, links, and carry-overs. Links connect the divisions, and [PERSON] and [PERSON] [18] classify them into four categories: the free link (LF), in which linking activities are determined discretionarily, maintaining continuity between inputs and outputs; the fixed link (LN), in which activities remain unchanged, typically when intermediate products are beyond the control of DMUs or management; the as-input link (LB), in which linking activities are treated as inputs for the next division, with excesses contributing to input inefficiency; and the as-output link (LG), in which linking activities are treated as outputs from the preceding division, with shortages contributing to output inefficiency. Carryovers are responsible for the intertemporal transition of efficiency within the division and, according to [PERSON] and [PERSON], they are classified into four categories. Desirable (good) (CG): Desirable transfers are treated as outputs, and their values are restricted to not being less than those observed. The comparative scarcity of transitions in this category is considered inefficiency; Undesirable (bad) (CB): Undesirable transitions are treated as inputs, and their values are restricted to not being greater than those observed. Comparative excess of transitions in this category is accounted for as inefficiency; Discretionary (free) (CF): Its value can be increased or decreased from the observed one. The deviation from the current value is not directly reflected in the efficiency evaluation, but the continuity condition between the two explained periods exerts an indirect effect on the efficiency score; and Non-Discretionary (fixed) (CN): Its value is fixed at the observed level. Similarly to the free carry-over, the fixed carry-over affects the efficiency score directly through the continuity condition between the two periods. Based on the objectives of this study, we adopted the non-oriented model approach, which defines overall efficiency as well as efficiency scores by period, by division, and divisionally by period. The choice of the non-oriented model is due to the fact that the piped gas segment falls within the scope of public concessions, implying that the tariff paid for the product (output) should be modest, facilitating access to the service by everyone, thus fulfilling the principle of universalization [35]. Furthermore, with the application of the non-oriented model, as highlighted by [36], it is possible to assess the deficiencies of inputs, outputs, and links for inefficient DMUs.