## 3 Method Description and Integration with KAI
### Framework and Alignment Evaluation Criteria  
The methodology proposed by [PERSON] et al. [22] aims to provide a comprehensive framework to design and assess the consistency of the overall alignment of a hierarchy of indicators in an unstable and changing environment. To do this, an evaluation structure is introduced: the trends and priorities of the internal and external environment are assessed to estimate how the company should position itself in the market and, if there is a change in these factors, a re-evaluation of the feedback of these parameters is triggered. The performance measurement and management system will then be designed according to the procedure described above; at this stage, the tree of indicators, their hierarchy, CSFs, KPIs and KRIs will be formalized. As a final step, the consistency of the tree alignment will be assessed. It should be noted that after analysing the internal and external environment, the organization must identify its CSFs and review them whenever there is a change in perspective due to internal or external changes. This identification is necessary to design the hierarchy of indicators, which will then be evaluated through the proposed criteria. From the critical success factors must derive, following the 10/80/10 rule, about 10 KPIs and 10 KPIs, followed by about 80 PIs and RIs below them, which departments or teams will use to make more specific measurements. The calculations of the three criteria will start from each combination of the latter indicators. There will be an assessment of the consistency between KPIs and KRIs (same for KRIs and RIs), an assessment of the calculability of PIs and RIs and, finally, the identification of guidelines to improve the calculability of PIs/RIs together with the general consistency between PIs and KRIs. The same is performed for the KPIs and KRIs; only the latter will be compared with the corporate CSFs to check the consistency. There are three criteria introduced in the method to assess consistency and alignment:  
1. _The 'Mathematical Connection' (MC)_--This criterion measures the degree to which two items are mathematically connected or related within the industrial PMM; it is common in the literature to assess the relationship between sets of indicators. The measurement scale ranges from 0, no mathematical connection, to 4, maximum mathematical connection. 2. _The 'Alignment/Polarization' (AP)_--This measures the positive/negative influence between two elements, i.e., whether there is an alignment or misalignment between indicators. The score assigned in this criterion can be equal to -1 if the indicators are _strategically misaligned_, 0 if there is _no alignment/misalignment_, and +1 if they are _strategically aligned_. 3. _The criterion of \"Generic Computability\" (C)_--This measures the availability of data needed to calculate a specific element or indicator. This can take values between 0, no data availability, to 4, easily available data and simple calculation. Let us now see how the method is mathematically encoded:  
\(OCA_{strategic level}=\frac{\sum_{i=1}^{n}\sum_{i=1}^{n}MC_{ij}*AP_{ij}}{4*n_{Adj =1}*n_{Adj=1}}\left[\%\right]\)  
\(GC_{KPI}=\frac{\sum_{i=1}^{n}C_{i}}{4*n}\left[\%\right]\)  
\(GC_{PI}=\frac{\sum_{i=1}^{n}C_{i}}{4*n}\left[\%\right]\)  
where:  
* \(MC_{ij}\) is the connection from a mathematical standpoint between \(CSF_{i}\) and \(KPI_{j}\), whereas \(MC_{jh}\) represents the mathematical connection between \(KPI_{j}\) and \(PI_{h}\);
* \(AP_{ij}\) defines the alignment/polarization between \(CSF_{i}\) and \(KPI_{j}\), whereas \(AP_{ih}\) represents the alignment/polarization between \(KPI_{j}\) and \(PI_{h}\);
* \(C_{j}\) represents the calculability of \(KPI_{j}\), whereas \(C_{h}\) represents the calculability of \(PI_{h}\). The OCA (Overall Consistency Alignment) indicator shows how closely the indicators are aligned with both the strategic level (CSF and KPIs) and the operational level (KPIs and Pis) by weighing the values of the mathematical connection of the indicators.