## 4 Numerical Examples
### Numerical Example 1  
#### 4.1.1 Model Formulation  
Numerical values from various articles have been compiled in order to create a POMDP model to demonstrate the implementation of the proposed approach. The MDP model proposed by [PERSON], [PERSON], and [PERSON] [6] defined eight equally spaced deterioration states with four actions (do-nothing, inhibitor application, pigging, and replacement) to mitigate deterioration. The transition matrix (do-nothing action) was adopted by modifying data from the article. Thus, the probability of a pipeline inspected to be in each state is computed using a Monte Carlo simulation. A pure birth transition intensity was computed using this probability. Then, the intensity was used to find the probability of transitioning to the succeeding state and remaining in the current state (for \(\Delta\)t = 0.0005 years). For the inspection interval of 5 years, a 10,000-step transition matrix was used to define the transition probability. Tables 1 and 2 present these parameters for the model.  
For the observation matrix, the ILI errors were estimated based on the numerical example published by [PERSON] and [PERSON] (2018). Mean values of 0.93 and 0.9 were obtained for the PoD and PoTC (actual depth 40-80% of wall thickness and mean detection 10-25% of wall thickness). Based on this, a reading error of 15% was considered for the numerical analysis. In other words, a 1.2 state deviation was estimated as the error in reading for an eight-state model (i.e., \(1.2=8*0.15\)). The other factor considered in the model was how the observation errors were applied. The first case defined the multiple observed states overlapping with actual states, whereas the second categorized the observed states as only pre-defined multiple actual states (see Table 3).  
\begin{table}
\begin{tabular}{l l l} \hline
**State** & **Probability of the State** & **Transition Intensity (\(\lambda_{i}\)) in Years** \\ \hline
1 & 0.01\% & 1.842068074 \\
2 & 1.96\% & 0.043999123 \\
3 & 58.87\% & 0.221188399 \\
4 & 32.39\% & 0.173228351 \\
5 & 3.62\% & 0.011165733 \\
6 & 1.10\% & 0.00227737 \\
7 & 0.55\% & 0.001113594 \\
8 & 1.49\% & 0 \\ \hline \end{tabular}
\end{table}
Table 1: State probability and transition intensity (modified from the source: [PERSON], [PERSON], and [PERSON] [6]).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & & & & & & & & \\  & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \multirow{9}{*}{**Purchase**} & 1 & 0.000 & 0.822 & 0.111 & 0.051 & 0.016 & 0.000 & 0.000 & 0.000 \\  & 2 & 0.000 & 0.803 & 0.117 & 0.060 & 0.020 & 0.000 & 0.000 & 0.000 \\  & 3 & 0.000 & 0.000 & 0.331 & 0.414 & 0.250 & 0.005 & 0.000 & 0.000 \\  & 4 & 0.000 & 0.000 & 0.000 & 0.421 & 0.  
Finally, the reward function combined the maintenance and failure cost for each state-action pairing. The failure costs were adopted from [PERSON], [PERSON], and [PERSON] (2016), whereas the failure-type probability was generated based on some understanding of the nature of the failures across the states (Table 4). The reward for donothing decreases as the deterioration of the pipeline increases due to the loss of potential savings by prolonged life through maintenance. Inhibitors, pigging, and replacement can only be applied to certain states, and the remaining states were assigned large values. Table 5 presents the reward values applied to the test model.