# A Multiobjective Decision-Making Model for Risk-Based Maintenance Scheduling of Railway Earthworks
## 4 Case Study
### Application of Multi-Attribute Utility Theory (MAUT)
#### 4.1.1 Assessment of Single Utility Function  
Utility functions are inspired by lottery problems wherewith equal probability of obtaining the best or worst possible value; a gambler needs to take a certain risk under uncertainty. We have adopted the convention from [52] to present the lottery question along with the risk preferences of the decision-maker. It is worth mentioning that the utility of an attribute is relative to the decision-maker's choices, which can change over time. To determine a single utility function, the utility function of all attributes must be computed, i.e., reliability benefit, annual maintenance cost and user cost. The reliability benefit is determined using Equation (12), where the initial reliability is taken from the Irish Rail database, and the final reliability is taken from the row relating to the maintenance activity in question in Table 1 or Table 2. The greater the change in the reliability index, the safer the slope will be. Therefore, the objective is to maximize the reliability benefit while keeping the annual maintenance cost and user cost minimal. Figure 6 presents the lottery set up to elicit the \(CE\) (certainty equivalent)/indifference point of increasing the reliability index. Figure 5: Irish Rail network (www.irishrail.ie). The expected value (EV) of increasing the reliability index is determined by setting up a lottery question where there is an equal chance of getting the best possible increase in reliability index and the worst possible increase in reliability index. The expected value is then easily obtained by getting the mean value. In this case, the expected increase in the reliability index is equal to 2.89. Due to the maximization objective, the indifference point (CE) was chosen to be 2. The risk tolerance (\(RT\)) value was calculated by satisfying optimizing Equation (9). The exponential single utility function of all the maintenance alternatives is calculated by solving Equations (6)-(8) iteratively, which resulted in Equation (15). \[U_{ri}(x_{ri})=-0.18+3.23\mathrm{e}^{(\frac{-x_{ri}}{15})} \tag{15}\]  
Figure 7 describes the change in utility as reliability increases. Due to the conflicting nature of the attributes, the utility scores of the benefit from the reliability index were reversed. Therefore, the slopes which gain the most from increasing reliability will have the lowest utility score. The rationale behind this is to keep better alignment between the attributes as minimum values are preferred in the other two attributes. The value reversal will reduce the maximal values (the largest increase in reliability) into smaller utility scores, as shown in Figure 7 so as to rank them higher in the final minimization function. Figure 8 shows the procedure for determining the \(CE\) (certainty equivalent)/indifference point for annual maintenance cost (AMC). As with the reliability attribute, the expected value must first be obtained by setting up a lottery problem with the best and worst case. Figure 6: Lottery setup to discern utility values for the benefit of reliability. Figure 7: Utility plot of reliability benefit. In this case, the expected value was equal to \(\Theta 69672\). Assuming that the decision-maker is willing to spend \(\upxi 7000\) each year (\(CE\)) on maintaining a critical asset. The Equations (5)-(8) solved iteratively yield the Risk Threshold and AMC scaling constants resulting in Equation (16). \[U_{amc}(x_{amc})=1.02-1.04e^{(\frac{-x_{i}}{500})} \tag{16}\]  
By substituting the AMC value for each slope into Equation (16), the utility score of all the slopes are computed. Figure 9 presents a plot of annual maintenance cost versus utility where the utility values increase steadily with increasing expenditure. Since the objective is to have minimal AMC, smaller utility sores are preferred. The lottery setup to discern the utility score of user delay cost (UDC) is shown in Figure 10. To evaluate the UDC metrics were needed on passenger numbers, Value of time and line speed.