## 1 Introduction
### Methodological Steps  
To provide a prioritization process, a monetary evaluation of barrier enhancement was conducted with the help of the machine learning technique. The results of the technique were used to inform policy makers about how much money they would expect to gain after optimizing the underdesigned barriers, and which barriers would be more cost-effective to optimize first. To fulfill these points, several steps were taken in this study, highlighted as follows:  
1. Two barrier types, box beam and W-beam barriers, were aggregated in the dataset on both the state's highways and the interstate highways. The two barrier types namely: box beam and W-beam, were aggregated in the dataset. It was expected that the data aggregations would result in high heterogeneity. However, to account for the heterogeneity, a finite mixture model (FMM) was used to allocate a cluster to each observation based on their distributions
2. After a model was trained across all barriers in the state highway and interstate system, the trained model was implemented only on those barriers that were below/above recommended heights
3. The heights of the barriers were optimized, and the trained model was implemented again to predict costs
4. The barriers' optimal heights were selected based on the recommended heights from a literature review
5. The cost of barrier enhancements were considered in the cost-benefit analysis when barriers were enhanced
6. All the variables applying the trained model were kept constant, except for shoulder width, barrier height, and their interaction term
7. After finding predicted EPDOs for years before and after enhancement, the predicted EPDO values were converted to their monetary values so the barrier enhancement cost could be added to the EPDO costs
8. The barriers were ranked based on their benefits so that WYDOT could first enhance the barriers with higher benefits.  
It should be noted that this work is an extension of previous work conducted by the authors [3]. While the previous work used the Bayesian hierarchical two component model, this method just considers a single component negative binomial. Also, while the previous work could account for zero observations through its first layer, this work, due to its nature, could not optimally account for zero observations so we just consider crash barriers. However, we used another methodology for this paper by accounting for the unseen heterogeneity through a finite mixture, while with a simple negative binomial, the second layer of the previous work, only used a subjective hierarchy assigned to the data. In summary, each method has its strengths and shortcomings as each study took a different approach.  
It should be noted that the maximum likelihood method was implemented for FMM instead of the Bayesian technique, due to the shortcomings of the Bayesian technique in estimating clusters, especially labeling the observations. Label switching is a main issue for the Bayesian technique. The problem is due to symmetry in likelihood of the model parameter. Although the problem might be expected to be resolved by removing symmetry by using artificial identifiable constraints, it has been argued that the technique might not be a reliable method for FMM, especially since we used that technique for labeling the observations. The description of the Bayesian hierarchical finite mixture model is detailed in our previous work and the readers are referred to that study [4]. Thus, maximum likelihood was used for the purpose of the FMM method. The summary of the methodological steps are depicted in Figure 1.