## 3 Methodology
### Fuzzy Deterioration Framework
The five-point scale associated with this parameter was represented using multiples of r which is a variable representing the deterioration speed expected for this specific system, starting from 0 which represents a nondeteriorated system, moving on to r, 2r, 3r, and 4r which indicates a \"very fast\" pace of deterioration. The variable r is initially assumed to be a 10% deterioration loss from the system's initial condition state to the following lower state. This is further calibrated as explained in Section 3.3. After the formation of the fuzzy knowledge base, a fuzzy set of rules was developed in order to govern the relationship and dependence between all the parameters to further calculate the expected deterioration rate of the system based on its fuzzified condition rating in the last inspection, as well as its fuzzified age. The distribution of the membership functions for the three previously presented parameters as well as an overview of the rule-based relationship between them obtained from the model execution on MATLAB(r) using the Fuzzy Logic Toolbox(tm) are illustrated in Figure 1.  
Furthermore, the fuzzification process of the condition, age and rate of deterioration parameters of the selected system was done using the following generic formula that can be adjusted to fit any number of subsets included in a triangular fuzzy set to derive their respective membership functions. \[\mu_{x}=\left\{ \begin{array}{r@{\quad\quad}l}0\,,\;x_{1}=\;V_{i}\quad\quad \\ \frac{x_{2}-V_{i}}{V_{i}-V_{i}}\;\;V_{i}<\;x_{2}<V_{j}\quad\quad\\ 1\,,\;x_{3}=V_{j}\quad\quad\\ \frac{V_{k}-x_{k}}{V_{k}-V_{j}}\;\;V_{j}<\;x_{4}<V_{k}\quad\quad\\ 0\,,\;x_{5}=V_{k}\quad\quad\end{array} \right. \tag{2}\]An illustration of the previous equations is presented in Figure 2 and provides a guide to calculate the membership function \(\upmu\) for different possible positions of x falling within the range of a single subset. The succeeding step after establishing the parameters, their respective membership functions as well as the governing rules, was the design of the Markov model that was used in the deterioration prediction of the future states of the system being studied. A fuzzy deterioration rate for a given system was arrived at using the previously demonstrated procedure, and was consequently defuzzied to be used in further stages of the developed model. The defuzzification methodology implemented in this study was the Mamdani inference approach where the y coordinate of the center of gravity  
Figure 1: Parameters and rules of the fuzzy deterioration framework. (COG) of the area under a fuzzy set B' was calculated using Equation (3) to obtain a de-fuzzified crisp output d [33]. \[\text{d }=\text{ COG }(\text{ B'})=\frac{\sum\limits_{i=1}^{N}\text{ }\mu_{B}\text{, } \left(y_{i}\right)y_{i}}{\sum\limits_{i=1}^{N}\text{ }\mu_{B}\text{, }\left(y_{i}\right)} \tag{3}\]  
where N is the total number of elements y_i in the continuous domain Y; Y needs to be discretized to allow for the calculation of the COG. The output derived from the previously illustrated framework was fed as an initiation input to the Markovian model that was further calibrated and modified as per the procedure explained in Section 3.2.