## 2 Methodology
### Definition and Formulation of KPI
#### 2.3.5 KPIs for Sub-Function \"Administration and Supply\"  
This sub-function mainly addresses the following three areas: (1) compliance of all economic practices and procedures of maintenance; (2) accounting activities and suitable procedures for all maintenance, as well as budget and cost control; and (3) supply chain management to purchase and provide all the required technical support.  
Relative strategy cost (\(A\&S_{1}\)) is defined as:  
\[A\&S_{1}=\frac{SC_{x\%}}{SS_{x\%}} \tag{23}\]  
where:  
\[SC=\sum_{\begin{subarray}{c}i\in I\\ j\in I_{i}\end{subarray}}\left(\text{\em cul}_{i}\times\text{\em ul}_{ij} \right)+\sum_{\begin{subarray}{c}i\in I\\ j\in I_{i}\end{subarray}}\left(\text{\em cc}_{i}\times\int_{0}^{T}\lambda_{i, j}(x)\text{\em dx}\right)+uw\times\left(\text{\em MM}_{1(base)}-\text{\em MM}_{1 _{(current)}}\right)\times T \tag{24}\]  
\[SS=\left(\left|\left\{k\in K:\rho_{k}\
eq 0\right\}\right|_{(base)}-\left|\left\{k \in K:\rho_{k}\
eq 0\right\}\right|_{(current)}\right)\times cs+uw\times\left(\text{\em MM }_{1(current)}-\text{\em MM}_{1_{(base)}}\right)\times T \tag{25}\]  
\[\text{\em cul}_{i}\text{: is the cost of the useful life of activity }i.\] \[\text{\em ul}_{ij}\text{: is the amount of useful life wasted due to the advance of the }jth\text{ execution of activity }i.\] \[\text{\em cc}_{i}\text{: is the cost associated with a corrective intervention of activity }i.\] \[\text{\em uw}\text{: is the cost (or saving) of the system to operate without stoppages in one week.}\] \[\text{\em cs: is the setup cost.}\]  
Description: This KPI measures the ratio between strategy costs (\(SC\)) and strategy savings (\(SS\)), showing the proportion of the strategy savings that is imputable to the strategy costs, i.e., how much of the savings due to the strategy are lost due to additional costs incurred due to the implementation of the strategy. \(SC\) is calculated simply the sum of the wasted useful lifetime plus the corrective intervention cost multiplied by the number of failures plus the inefficiency cost as the cost due to the difference between availability in the current scenario and the base scenario. Finally, \(SS\) is calculated as the sum of the setup cost multiplied by the number of stoppages plus the inefficiency savings. As an example, if a current strategy presents \(AS_{1}=15\%\), the cost of the current strategy represents only \(15\%\) of the total savings achieved by the current strategy.  
Timely inefficiency deviation (\(A\&S_{2}\)) is defined as:  
\[A\&S_{2}=\frac{U_{x\%}-U_{0\%}}{U_{0\%}}, \tag{26}\]  
\[U=\frac{T_{u}}{T} \tag{27}\]  
Description: This KPI indicates the percentage deviation of inefficiency measured as the percentage difference of unavailability between strategies. It is assumed that inefficiency cost is proportional to the unavailability of the system.  
Mean Time Between Orders (\(A\&S_{3}\)) is defined as:  
Before presenting a definition for this indicator, it is necessary to define a set containing all moments in which an order is issued. It is assumed that orders are issued whenever a total quantity of \"\(m\)\" spare parts have been employed. Furthermore, it is also assumed that, without losing generality, the use of spare parts is equal to the number of workers required for each activity. Hence, set \(T^{c}\) is defined as follows:  
\[T^{c}=\left\{t_{n}\in[0,T]:\frac{\sum_{k\in K}w_{k}}{m}=n,\text{ \em n},\text{ \em n },\text{\em m}\in\mathbb{N}\right\} \tag{28}\]  
Therefore, the mean time between orders is:  
\[A\&S_{3}=\frac{\sum_{m=1}^{|T^{c}|-1}(t_{n}-t_{n+1})}{|T^{c}|} \tag{29}\]  
Description: This KPI measures the mean time between the moments where a spare-parts purchase order is issued, assuming they are issued at the same time that \"\(m\)\" amount of the resource has been consumed, where \"\(m\)\" is a parameter that can be set by the decision maker. For illustrative purposes only, this parameter is set to 25, which means that only 25 workers are available in the plant.