## 4 Numerical Examples
### Numerical Example 2  
#### 4.2.1 Model Formulation  
Pipeline maintenance can be performed based on routing and segments, where each consists of multiple pipes. If maintenance decisions need to be made on the routing/segment level, a systemic approach would have to be applied to consider the interaction of components. The individual pipeline model components described above were combined to construct a two-pipeline system model. For identical pipes and ILI readings, the states of the system were formulated as \(s_{1}\), \(s_{2}\) (\(V_{51}\in S_{1}\), \(V_{52}\in S_{2}\)), which combined the states of each pipe [25]. It is assumed that the deterioration of each pipeline is independent and identical. Dependent pipelines and multi-component pipeline systems' [26] deterioration were considered outside of the scope of this study. A total of 64 states were created for an eight-state pipeline system. The actions were considered to be the same since the pipeline system consists of identical pipes. The transition matrix was computed for the individual pipes, which were used to formulate the system's transitions. Therefore, for a one state transition of this two-pipeline system, the transition matrix consisted of both pipes remaining in their current states, one of the pipes moving to the next state, and both pipes transitioning to the next state. A similar procedure was adopted to transform the matrix into an n-state transition matrix (Table 6).  
Figure 2: Input/output interface for reporting results.  
The observations and reward functions have been calculated by multiplying the respective state's values. Table 7 gives the observation probabilities for the overlapping case of the model's formulation and the cost of taking a given action under a specific state.  
The same solver tool, i.e., MDPToolBox [24], was used on the same laptop to run the two-pipeline system POMDP model. Three belief states were defined for each deterioration state in order to meet the computation limit of MATLAB (lack of available memory size). The backward iteration of the Bellman equation method was also selected from the solver toolbox.  
\begin{table}
\begin{tabular}{r r r r r r r r r r r r r r r r r r r r r r r r} \hline \hline  & & & & & & & & & & &  
The two-pipeline system also reported the same maintenance action for the fully observable state under the two setups. The recommended actions for each observable deterioration state are given in Figure 3.  
The result of this model run comprised 45,760 combinations of belief states for the 64 states of pipeline deterioration. The distribution of the recommended actions is given in Table 8, along with a three-belief-state-run of the single pipeline model for comparison.  
The run time is significantly increased for the two-pipeline system, as expected. For the single pipeline, it took 8 milliseconds (fully observable states) and 3 milliseconds (POMDP) for the first observation setup, whereas the second setup needed 0.7 milliseconds (fully observable states) and 0.6 milliseconds (POMDP) to obtain the results. The two-pipeline system required 2 milliseconds (fully observable states) and 1100 milliseconds (POMDP) for the first setup, while 1 millisecond and 300 milliseconds run times were obtained for the second setup.