# An Epistemic Utility-Theoretic Model in Fortifying Oil-and-Gas Production Networks
## 3 Solution Strategy
### Sub-Problem  
In this section, the sub-problem or attacker-operator interactions are simulated to generate the operator's objective of minimizing and the attacker's objective of maximizing flow and damage cost. The operator's dual problem, and the attacker problem representing the model's sub-problem. As explained, the sub-problem consists of two problems, the dual problem that represents the maximization of the operator problem presented in Equations (24)-(27), and the common attacker problem constraints presented in Equations (28)-(30), where the attacker seeks to maximize the demand shortage and the oil/gas delivery cost. The dual problem is sensitive to the cost of transportation, as shown in Equation (26), and it is growing up to its maximum rate, when there is a shortage in a specific production facility or any pipeline, as shown in Equation (27). The sub-problem will import the defender strategies \((\hat{\phi}_{i}^{DF},\hat{\psi}_{ij}^{DF})\) parameters from the master problem as illustrated in Equation (29); the pipelines defending parameter \(\hat{\psi}_{ij}^{DF}\) and the production facilities defending parameter \(\hat{\phi}_{i}^{DF}\), respectively.  
\[\min_{\Gamma^{S}}-\left(\sum_{ip}\left[\lambda_{ip}D_{ip}+\mu_{ip}^{s} \overline{S_{ip}}\left(1-\phi_{i}^{A}\right)+\mu_{ip}^{sh}\ \overline{U_{ip}}-\mu_{ip}^{n}\psi_{ip}\left(1-\phi_{i}^{A}\right)\right]- \sum_{i,j,p}\left[\mu_{ip}^{a}\overline{x_{ij}}\left(1-\psi_{ij}^{A}\right) \right]\right) \tag{24}\]  
Subject to  
\[\lambda_{ip}+\mu_{ip}^{s}=0\ ;\ \forall i,p \tag{25}\]  
\[C_{ip}^{T}-\lambda_{ip}h_{ip}+\lambda_{jp}h_{ip}+\mu_{ip}^{n}+\mu_{ip}^{a}=0\ ; \ \forall ij,p \tag{26}\]  
\[C_{ip}^{Sh}-\lambda_{ip}+\mu_{ip}^{sh}=0;\ \forall i,p \tag{27}\]  
\[\psi_{ij}^{A}\leq 1-\hat{\psi}_{ij}^{DF};\ \forall ij \tag{28}\]  
\[\phi_{i}^{A}\leq 1-\hat{\phi}_{i}^{DF};\ \forall i \tag{29}\]  
\[\sum_{ij}\psi_{ij}^{A}+\sum_{i}\phi_{i}^{A}\leq B^{A} \tag{30}\]  
where  
\[\Gamma^{S}=\left\{\psi_{ij}^{A},\phi_{i}^{A},\mu_{ip}^{s},\mu_{ip}^{n},\mu_{ip }^{a},\mu_{ip}^{sh}\ \right\}\]  
Equation (24) applies the Lagrangian formed from the dual variables (\(\lambda\)) and (\(\mu\)). The method is utilized to maximize the cost of all oil and gas operations within the network in the dual problem, where the first summation is to dual representation on the demand associated with the dual variable \(\lambda_{ip}D_{ip}\), the production level associated with the dual variable and attacker strategy \(\mu_{ip}^{s}\overline{S_{ip}}\left(1-\phi_{i}^{A}\right)\), the maximum lost and dual variable \(\mu_{ip}^{sh}\ \overline{U_{ip}}\), and the upper limit of the production level, dual variable, and attacker strategy of production sources \(\mu_{ip}^{n}\psi_{ip}\left(1-\phi_{i}^{A}\right)\). The second part is the pipeline upper bound, dual variable and attacker strategy on pipelines \(\mu_{ip}^{a}\overline{x_{ij}}\left(1-\psi_{ij}^{A}\right)\). Constraint (25), it is used to calculate the cost of optimal levels of production at the production sources. Constraint (26) is used to calculate the cost of optimal transportation flow in the pipelines. Equation (27), is to calculate the cost of the levels of shortage in the dual problem. Constraint (28), represents the beginning of the attacker problem, which states that if the pipeline is defended (\(\hat{\psi}_{ij}^{DF}\)=1), it is invulnerable, as the attacker variable \(\left(\psi_{ij}^{A}=0\right)\). Constraint (29) is to implement the attack and defend plans for the production sources, if the source defending strategy parameter (\(\hat{\phi}_{i}^{DF}=1\)), means that this sources is safe and the attacker's source variable would be (\(\phi_{i}^{A}=0\)). Equation (30) is used to control the total attacker resources for interruptions.