# PAL-VMEA: A Novel Method for Enhancing Decision-Making Consistency in Maintenance Management
## 4 Best-Worst Method (BWM)  
According to [PERSON] (2016), the Best-Worst Method (BWM) is a multicriteria decision-making technique where the best and worst choices, among the alternatives, are defined by means of a peer comparison between the analyzed alternatives (with a weight scale of 1-9 used for comparison). This method also stands out for its simple application and the reduction in the time for carrying out the peer comparison between the variables, in addition to the good performance in maintaining consistency between the judgments (2016, 2016). In this work, the BWM is used as an auxiliary tool for PAL2v, evaluating the ability (in percentage) of decision-makers (engineering experts) to assign weights to critical variables that affect (directly and indirectly) the performance of KPCs analyzed in the VMEA method.  
In general applications, the algorithm to structure the Best-Worst Method is presented as follows (2016, 2016):  
* Step I: list the variables (decision criteria) involved in the decision process.
* Step II: decision-makers must determine which criterion is the most important (BEST criterion) and which has the least impact on the decision (WORST criterion).
* Step III: carry out the parity comparison (using [PERSON]'s scale: 1-9 (2016)) between the preference of BEST criteria over other criteria. The result of the Best-to-Others vector is AB = (\(a_{B1}\), \(a_{B2}\),..., \(a_{Bn}\)), where \(a_{Bj}\) indicates the degree of preference of the best criterion, \(B\), over criterion \(j\), and \(1\leq j\leq n\).
* Step IV: make the parity comparison (using [PERSON]'s scale: 1-9) between the preferences of the criteria involved in the decision-making process over the determined WORST criteria. The result of the Others-to-Worst vector is \(A_{W}\) = (\(a_{1W}\), \(a_{2W}\),..., \(a_{nW}\)), where \(a_{jW}\) indicates the degree of preference of the criterion \(j\) over the worst criterion, \(W\).
* Step V: obtain the optimal weights (percentages), (\(w_{1}\)*, \(w_{2}\)*,..., \(w_{n}\)*), of each criterion from the formulation of a Maximization and Minimization problem (MAX-MIN), given by Equation (1), in order to satisfy the conditions that for each (\(w_{B}\)/\(w_{j}\)) and (\(w_{j}\)/\(w_{W}\)), (\(w_{B}\)/\(w_{j}\)) = \(a_{Bj}\) and (\(w_{j}\)/\(w_{W}\)) = \(a_{jW}\), being \(w_{B}\) the importance weight of the best criterion and \(w_{W}\) the importance weight of the worst criterion. \[\  
* Step VI: Perform the consistency verification of the assigned values (weights) by the decision-makers for the criteria of the decision-making process. This verification can be performed by the Consistency Ratio calculated by Equations (3) and (4): \[\xi^{2}-(1+2a_{BW})\xi+\left(a_{BW}^{2}-a_{BW}\right)=0\] (3) Consistency Ratio \[=\xi*/\max(\xi).\] (4)  
Being the maximum possible \(\xi\)--max (\(\xi\))--called Consistency Index (CI), found as function of \(a_{BW}\), according to Table 1[51].  
Once the consistency verification is done, the result of the percentage weight of each criterion involved in the decision-making process is obtained. In conclusion, BWM is an easy-to-apply method for determining the most appropriate weights for decision criteria, with guaranteed reliability of results, as it performs a consistency check of the judgments (even comparison between the criteria) made by the decision-makers.