## 5 Conclusions  
The accurate determination of corrosion in pipelines is difficult in most real-world cases. These inaccuracies emanate from different sources, including equipment errors, human limitations, the environment, and/or the nature of pipeline transportation systems. Maintenance operations are performed with all these limitations, which can lead to additional resource expenditure and maintenance costs. This paper proposed a POMDP maintenance optimization model, which accommodates these inaccuracies (using equipment error as an example) to identify the pipelines' operating condition and propose the optimal maintenance action. Each state represented a uniformly progressive level that covered the entire corrosion deterioration range. The different corrosion mitigation methods were considered as maintenance actions for the model. Transition intensity was used to define the probability of transition between states. The analysis of the inspection errors formulated the observation probability. The cost of maintenance and failure made up the reward function for a given state and action.  
A single-pipeline and two-pipeline system corrosion maintenance optimization problem model were formulated to demonstrate the implementation of the proposed model. The result of the model run showed that the optimal action and the cost of implementing it could be obtained for the different probabilities of deterioration states. This loosens the need for a strictly defined deterioration state, which would provide flexibility in decision-making for maintenance technicians and managers.  
The implementation of the POMDP for optimizing maintenance operations can be further extended in different directions. The problem formulation and size of the model could be enhanced to create a very complex pipeline system. With this approach, the study could address the implementation of the POMDP method for the maintenance of a system consisting of tens and hundreds of pipelines. The model could also include the interaction among different components of the pipeline system. Furthermore, the method can be explored for its suitability in determining maintenance criticality assessment and policy optimization. Maintenance operations criticality can improve the utilization of limited maintenance resources. Different preventive maintenance policies, including multi-policy approaches, could also be adopted in the implementation of maintenance actions in the models.  
Companies in the oil and gas industries carry out pipeline maintenance under uncertain conditions. The proposed model attempts to address these uncertainties to improve the pipeline's overall performance and safety. Maintenance is one of the industry's major functions that affect the success of a company.  
**Author Contributions:** Conceptualization, E.W. and W.Z.; methodology, E.W.; software: E.W.; Validation: E.W., W.Z. and G.L.; formal analysis, E.W.; investigation, E.W.; resources, W.Z.; data curation, E.W.; writing--original draft preparation, E.W.; writing--review and editing, E.W., W.Z. and G.L.; supervision, W.Z. and G.L.; project administration, W.Z.; funding acquisition, W.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by National Science Foundation: 1855147, 1950036, 2141674, 2220683; National Academies of Sciences, Engineering, and Medicine: 200011064; United States Department of Agriculture: 13111855, 13424031; University of Houston: Advanced Manufacturing Institute. The APC was waived by the journal.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.