# A Multiobjective Decision-Making Model for Risk-Based Maintenance Scheduling of Railway Earthworks
## 3 Multiobjective Decision-Making Model for Maintenance Planning
### Model Description
The decision-maker must choose an indifference point between the best and the worst possible values, which is referred to as the certainty equivalent (\(CE\)). The average value between the best possible value or the worst possible value is referred to as the expected value (EV). The indifference point chosen by the decision-maker represents his/her attitude towards risk and risk tolerance. An indifference value where \(CE\) is equivalent to EV represents risk-neutral behavior. If the value of \(CE\) is lower than EV, the decision-maker has a risk-avoiding attitude, whereas the \(CE\) value greater than EV shows a risk-taking attitude. Consequently, a risk-avoiding attitude has a positive risk tolerance, while a risk-taking attitude has a negative risk tolerance value [43] ([PERSON] and [PERSON], 2008). With the consideration of decision-makers' uncertainty in eliciting indifference points, \(CE\), under risk consideration, the final computed utilities of an attribute must lie between \(U\) (\(Max(x_{i})\)) = 0, \(U\) (\(Min(x_{i})\)) = 1. 2. The next step is to perform tradeoffs among the attributes in order to find a solution that either maximizes or minimizes the stated performance goals or objectives. These tradeoffs characterize the relative importance of attributes for their defined objectives/performance goals. A direct rating method is applied to determine the weighting factors of the attributes, which is represented in Equation (10). \[A=\frac{e^{(\frac{-\text{MI}(x_{i})}{Rt})}}{\left[e^{(\frac{-\text{MI}(x_{i}) }{Rt})}-e^{(\frac{-\text{MI}(x_{i})}{Rt})}\right]}\]2 (10) where: \(k(x_{i})\) = Weighting factor of each attribute \(i\) across all alternatives; \(w(x_{i})\) = weighting assigned by an expert for attribute \(i\) for maintenance activity \(x\); \(w(x_{i})\) = weighting assigned by an expert for attribute \(i\) for maintenance activity \(x\). 3. The next step of the MAUT application is the computation of the aggregated utility of each alternative based on both the computed SUF and the relative weighting factors. For the final aggregation, the multiplicative or addictive form can be used. The additive form requires the attributes to be mutually and preferentially independent. Preferentially independent means that the preferences of one attribute are not dependent on the preferences of another. When attributes are not mutually and preferentially independent, the multiplicative form is used [44]. Here, as the attributes (presented in Section 4.1.2) are found to be mutually and preferentially independent, the additive form is used to compute the total aggregated score of each alternative, see Equation (11):  
\[A=\frac{e^{(\frac{-\text{MI}(x_{i})}{Rt})}}{\left[e^{(\frac{-\text{MI}(x_{i}) }{Rt})}-e^{(\frac{-\text{MI}(x_{i})}{Rt})}\right]}\]3 Flowchart for multi-attribute utility theory (MAUT) application. \[A=\frac{e^{(\frac{-\text{MI}(x_{i})}{Rt})}}{\left[e^{(\frac{-\text{MI}(x_{i}) }{Rt})}-e^{(\frac{-\text{MI}(x_{i})}{Rt})}\right]}\]4 (11) where: \(U(x)=\) multi-attribute utility of alternative \(x_{i}\) \(k=\) weighting factor of each attribute \(i\); \(U_{i}(x_{i})=\) single-attribute utility of each attribute \(i\) for an alternative \(x\). 4. Finally, maintenance alternatives are ranked based on the magnitude of their aggregated score. The maintenance alternative that contributes most to the realization of the defined objectives is ranked the highest. There are several successful examples of MAUT in the area of transport infrastructure management. [PERSON] et al. (2008) (2008) used MAUT as a means of quality control for bridge construction, [PERSON] and [PERSON] (2010) (2010) utilized it as the basis for a bridge management system, while [PERSON] et al. (2019) (2019) demonstrated the viability of MAUT for network-level bridge maintenance. MAUT also has been used for the reliability assessment of infrastructure design and for inspection and maintenance planning ([PERSON] et al., 2019).