## 5 Numerical Examples
### Discussion  
CSPOMDP model formulated the corrosion level of the pipeline as a continuous process. The state definitions adopted continuous distributions, such as Weibull and Normal, to fit the corrosion process and forecast the corrosion level. An empirical function can also be substituted for these probability distributions using real-world data. To implement the DS modeling, these state definitions have to be converted to a discrete value, or the deterioration span needs to be divided into equally ranged levels. These discretization processes may cause a loss of pertinent modeling information about the state of the pipelines. Other elements of the modeling process are adjusted accordingly to accommodate these modeling changes. For example, most action, observation, and reward functions were defined to give a discrete reading since this is the standard practice on the ground. In the case of predefined policy implementation, a range of current states can be used to define preferred actions, primarily for CS models.  
The CS POMDP model was tested at different levels of complexity with multiple policy settings. The CSPOMDP results were similar using either solver, even though the state transitions differed. Experiment 1 demonstrated that even though the initial state of the pipelines was the same, the final total reward obtained after multiple steps transition can vary. The stochastic process of the transitional states as well as the ILI reading of the state, contributed to these variations. Through replication of the model run, the uncertainty of the state transition, hence the total reward, can be improved to make better maintenance decisions. Experiment 2 showed this replication process results for both the CS and DS models. In general, increasing the number of replications reduced the uncertainty of total reward at the cost of run time. The impact model formulation variation between the CS and first DS models was minimal, as can be seen in Figures 4 and 5a. Converting the CS model to DS can give improved results when compared to formulating the problem as DS model directly. However, the second DS model differs, especially when using the ARDESPOT solver. However, the solver selection does not clearly provide a distinction with regard to the total reward obtained. Therefore, the exact total reward assessment would have to be performed based on the best result from the two solvers. Finally, the results of experiment 3 showed the variation among different policies, which would aid decision-making. The CS POMDP provided better policy filtering, enhancing the chance of making better maintenance decisions. This advantage can be attributed to the more detailed state formulation in the CS POMDP. Since the solver attains random cost values in each run, considering results from multiple replications using multiple solvers would be advantageous.  
Most model runs were completed within a short run (less than 5 s) using the above-described computing facility. The only exceptions were the model run for POMCP solver policy in experiment 3, which took 25-30 s, and ARDESPOT, which took 50-55 min. The time significantly increased for runs with more than 100 replications, requiring better computing facilities and longer execution times.