# A New Hybrid Dynamic FMECA with Decision-Making Methodology: A Case Study in an Agri-Food Company
## 2 Methodology
### Criteria's Weights' Calculation through Entropy Method and Best Worst Method (BWM)  
Once obtained, the starting matrix in which the possible alternatives are on the rows, the criteria are on the columns, and the evaluations constitute the heart of the matrix; moreover, each criterion's weight must be identified.  
In the traditional FMEA, a severe weakness, much discussed over the years, is the lack of different weights for the criteria; one factor may be predominant compared to the others.  
This problem has been solved using a combination of the Entropy method and BWM to obtain the weights. A choice was made to not use methods that use language variables because these are difficult to manage and require adept experience to be used most effectively. There is also a risk of incorrectly considering some subjective values, as it is related to individual skills [31].  
Although experts' subjective opinions are intrinsically present in the data structure, the combination with BWM has been made to take more account of business ideas [28]. Moreover, using the two methods makes the method replicable in any company, even where no objective maintenance data is available.  
After calculating the weights with the two methods, a simple average or a weighted average of the values can be done; this depends mainly on how much the business ideas influence the maintenance aspects.  
The Entropy method, according to [32], will be applied as follows:  
* The data of the matrix will be normalised to ensure a homogeneous and direct comparison between the criteria: \[Z_{ij}=\frac{x_{ij}}{\sum_{i=1}^{n}x_{ij}}\] (6)
* Entropy is calculated for each criterion: \[E_{j}=\left(-\frac{1}{\ln(n)}\right)*\sum_{i=1}^{n}\left(Z_{ij}*\ln z_{ij}\right)\] (7)
* The values \(d_{j}\) are calculated: \[d_{j}=1-E_{j}\] (8)Finally, weights \(W_{j}\) are calculated:  
\[W_{j}=\frac{d_{j}}{\sum_{j=1}^{m}d_{j}} \tag{9}\]  
The greater the weight of criterion \(j\), the more critical the criterion \(j\). If the values of criterion \(j\) are almost equal, then it will be assigned a small weight, because the entropy method is an objective method based solely and exclusively on the data structure and is in no way influenced by managerial policies [32].  
According to the aforementioned studies, the BWM will be applied as follows:  
* The most important criterion and the least important criterion will be identified;
* Preferences of the most important criterion are expressed over the others by giving a number from one to nine, which obtains a line vector;
* Preferences of the least important criterion are expressed by giving a number from one to nine. A column vector is obtained;
* Finally, a problem of optimisation of the type is solved: \[\begin{array}{ll}\{\min_{\xi_{L}}&\\ &\begin{array  
Calculate the mean negative distance \(\rm{NDA_{ij}}\) for the benefit and disadvantage criteria: Benef \[\rm{NDA_{j}}=\frac{\left(\max\{0,\;AV_{j}-x_{ij}\}\right)}{AV_{j}}\] (15) Disadvantage \[\rm{NDA_{j}}=\frac{\left(\max\{0,\;x_{ij}-AV_{j}\}\right)}{AV_{j}}\] (16)
* Using the weights of the previously calculated criteria, the weighted sums are calculated, \(\rm{SP_{i}}=\sum_{j=1}^{m}w_{j}\times PDA_{j}\) (17)
* Using the weights of the previously calculated criteria, the weighted sums are calculated, \(\rm{SN_{i}}=\sum_{j=1}^{m}w_{j}\times NDA_{j}\) (18)
* The weighted sums are normalised \[\rm{NSP_{i}}=\frac{SP_{i}}{max\{SP_{i}\}}\] (19) \[\rm{NSN_{i}}=1-\frac{SN_{i}}{max\{SN_{i}\}}\] (20)
* The priority index, \(\rm{AS_{i}}\), is calculated \[\rm{AS_{i}}=\frac{1}{2}(\rm{NSP_{i}}+\rm{NSN_{i}})\] (21)  
The ranking of the possible causes of failure is evaluated following the descending order of the AS index.