# A Multiobjective Decision-Making Model for Risk-Based Maintenance Scheduling of Railway Earthworks
## 3 Multiobjective Decision-Making Model for Maintenance Planning  
### Model Description  
MAUT is a multiciteria decision-making method [37], based on expected utility theory where the concept of a lottery is used to change the weightings of different attributes to determine the indifference point between different outcomes. The indifference point is the point where the expected values from different outcomes are the same. MAUT allows the decision-maker to strongly influence the preference order through the construction of utility functions, which further allows for the direct consideration of uncertainty. In the maintenance planning process, the uncertainty mainly originates from a lack of data and/or a lack of clear preferences from the stakeholders. These functions are real-valued functions that must be determined for each attribute, where the attributes chosen should represent the characteristics that the decision-maker in question deems important. After defining utility functions for each attribute, objectives and constraints need to be established. If the decision-maker has a personal preference regarding maximizing or minimizing the influence of a given attribute, relative weights can be ascribed. The individual utility functions are then combined into one utility function using the established relative weightings and any constraints set by the decision-maker. This single criterion is optimized to determine the best course of action. MAUT's primary use is to choose the strategy in situations where a strategy must be implemented, but there is no clear indication of which path is of greatest value [26]. MAUT is a well-structured procedure designed to elicit utility functions from multiple attributes, implement relative weighting, and establish an objective ranking. There are several methods available to model an attributes utility, such as linear, log, exponential, logarithmic and quadratic [38]. In this paper, the exponential utility function [39] is utilized to capture the uncertainty in stakeholder preference and the risk tolerance of the decision-maker for slope maintenance planning, considering object reliability, economy and availability as performance criteria. The step-by-step application procedure to implement MAUT on multiple objective maintenance planning problems is shown in Figure 4 and discussed in detail below, see Equations (5)-(8):  
1. Assuming that the objectives of the maintenance planning and related attributes have been established, the first step is to compute a _single utility function (SUF)_ for each attribute by applying the exponential utility functions [40,41] as follows: \[U_{i}(x_{i})=A-Be^{(\frac{-\text{MI}}{Rt})}\] (5) where: \[A=\frac{e^{(\frac{-\text{MI}(x_{i})}{Rt})}}{\left[e^{(\frac{-\text{MI}(x_{i}) }{Rt})}-e^{(\frac{-\text{MI}(x_{i})}{Rt})}\right]}\] (6) \[B=\frac{1}{\left[e^{(\frac{-\text{MI}(x_{i})}{Rt})}-e^{(\frac{-\text{MI}(x_{ i})}{Rt})}\right]}\] (7) \[RT_{i}=\frac{-\text{C}E_{i}}{ln\left(\frac{-\text{C}E_{i}}{\left(\frac{- \text{C}E_{i}}{Rt}\right)-\text{0.5U}l_{i}(\text{MI}(x_{i}))+A}\right)}\] (8) where: \[U_{i}(x_{i})=\text{Single utility value for attribute $i$ of a maintenance alternative $x$}\] \[A\], \[B=\text{Scaling constants};\] \[e=\text{The exponential constant, i.e., 2.718;}\] \[Min(x_{i})=\text{The exponential constant, i.e., 2.718;}\] \[Min(x_{i})=\text{Minimum value of attribute $i$ across all alternatives;}\] \[A=\frac{e^{(\frac{-\text{MI}(x_{i})}{Rt})}}{\left[e^{(\frac{-\text{MI}(x_{i}) }{Rt})}-e^{(\frac{-\text{MI}(x_{i})}{Rt})}\right]}\]0\(RT=\) Risk tolerance. When computing SUF, the scaling constants and \(RT\) are cyclically dependent. Equation (9) can be used with the goal seeker function of MS Excel [42] to solve this. \[A=\frac{e^{(\frac{-\text{MI}(x_{i})}{Rt})}}{\left[e^{(\frac{-\text{MI}(x_{i}) }{Rt})}-e^{(\frac{-\text{MI}(x_{i})}{Rt})}\right]}\]1  
To obtain the exponential SUF, the decision-makers are presented with a lottery question of a maximum value (i.e., \(Max(x_{i})\)) and a minimum value (i.e., \(Min(x_{i})\)) of an attribute.