### Abstract:

The objective of this study was to quantify and determine trends in the uncertainty in the life cycle cost (LCC) associated with the maintenance and renewal (M&R) of the rail of a railway track under a fixed set of input parameters and conditions. Rail maintenance models were identified in the literature which use the mean or expected value of the input distributions to determine a corresponding mean or expected LCC. Although these models display important trends with regard to input parameters such as inspection intervals, they provide no means to quantify the uncertainty related to maintenance and renewal decisions. Thus, a numerical model was developed and programmed using MATLAB which allows the quantification of the uncertainty in the LCC estimated for a given set of conditions.
The model uses Monte Carlo simulation to determine the LCC associated with the installation, maintenance and renewal of the rail. The model incorporates imperfect inspections, a hazard function for rail fatigue defects modelled using the Weibull probability distribution and a P-F interval for rail fatigue defects modelled using an exponential probability distribution. The model also allows the modelling of maintenance as either perfect or minimal maintenance as well as the use of either flash butt or alumino-thermic welds to conduct the maintenance. This allowed the development of a method to assess which weld type to use to minimise the minimum attainable mean LCC.
The developed model was validated against a similar stochastic rail maintenance model from the literature. However, the model from the literature considers only the expected LCC and does not show any uncertainty related thereto. The novelty in this study therefore lies in the fact that the LCC uncertainty can be quantified in the form of a probability distribution at any given renewal tonnage for a given set of conditions. It was found that the distribution of the LCC at a given renewal tonnage followed a lognormal probability distribution. The standard deviation of the lognormal distributions fitted using the method of maximum likelihood was used as a metric to quantify the uncertainty related to the life cycle cost at a given renewal tonnage. The LCC uncertainty was found to increase with an increase in inspection interval length. Furthermore, the uncertainty was also found to increase with a respective increase in renewal tonnage. For varying inspection interval lengths it was found that the uncertainty of combined maintenance costs (planned plus unplanned maintenance costs) tended more strongly towards the uncertainty in the planned maintenance costs for smaller inspection intervals and more strongly towards the uncertainty in unplanned maintenance costs for larger inspection intervals. A critical cost ratio was found of flash butt weld cost to alumino-thermic weld cost at which the minimum attainable mean LCC was equal. It is more economical to use flash butt welding for maintenance if the cost of flash butt welding maintenance produces a cost ratio lower than the critical cost ratio.
The developed model could allow railway operators to assess the risk associated with renewal of the rail at varying renewal tonnages for given conditions such as inspection interval lengths, detectability of rail fatigue defects and the arrival rate of rail fatigue defects.