## 3 Methodology
### Fuzzy Deterioration Framework  
The fuzzy set theory was first established by [30], where it was found to overcome the shortcomings of binary logic and was mainly utilized to accommodate the uncertainties within decision-making methodologies. In the following years, various scholars have explored the concept of fuzzy sets and how they can be applied to facilitate the decision-making process of diverse types of problems. In a traditional crisp set \"A\", an element \"x\" is evaluated based on whether they belong inside the set or not. A typical fuzzy set extends this concept in which it illustrates the inherent relationship between an element \"x\" and a given set \"A\" by means of a fuzzy membership function \"\(\mu\)\", ranging from 0 to 1. The mathematical formulation representing the fuzzy set \"A\" in a universe \"X\" is illustrated in Equation (1). \[\mathrm{A}\ =\left\{\ (\mathrm{x}\,\ \mu_{\mathrm{A}}(\mathrm{x}))\ |\ \mathrm{x}\ \in\ \mathrm{X}\ \right\} \tag{1}\]  
where \(\mu_{\mathrm{A}}\) (\(\cdot\)): \(\mathrm{X}\rightarrow[0,1]\) is the membership function of elements present in \"A\", all belonging to universe X, and \(\mu_{\mathrm{A}}\) (\(\mathrm{x}\)) denotes the membership degree of element x within set A. A membership function can take any form, but the chosen form needs to be justified by given information [31]. Membership functions are most commonly illustrated into triangular fuzzy numbers (TFN) or trapezoidal fuzzy numbers (ZFN) that are both most commonly used to demonstrate the linguistic variable used by the experts interviewed or subjective data available. A fuzzy set is typically normal, convex or of a bounded form in order to qualify as a fuzzy number. As part of the current study, TFNs are chosen to demonstrate the relative membership of the various parameters involved in the model for the sake of their relevant simplicity and applicability without imposing a declination in the generality. As a first step in the model design phase of this study, three parameters were involved as a foundation for the prediction process. The process of defining the inputs and outputs of a fuzzy-based framework is often referred to as the process of knowledge base creation. In the present study, the fuzzy model developed takes two inputs to derive a single output parameter as explained below. The first input into the fuzzy model was the normalized physical condition of the assets as measured in the last preceding visual inspection of system components and is a representation of the level of defects and damage within each system. The physical condition is evaluated by specialized engineers on a six-point scale ranging from excellent to critical. A system evaluated as having an excellent condition is probably new and very recently installed and is working with near 100% performance, while the evaluation resulting in a critical condition suggests that the system is in a significant state of damage, almost working with 10% of its performance, and this in turn suggests that its replacement is due and inevitable. The second parameter input to describe the system was its relative age since it was initially installed, compared with its life expectancy. This parameter includes five possibilities represented by a percentage value for the age divided by the design life, ranging from recently installed to very old. In the case of oxygen gas systems, the average design life is 28 years as per [32], and this was the value used in the developed model to compare the age against. The final output parameter was the expected deterioration rate as a result of the previous two indicators, which was evaluated using a variable r that was initially assumed and then calibrated using the proposed optimization model.