## 3 Materials and Methods
### The Proposed FMEA-DEA Method
#### 3.3.1 Stage 1: Modified FMEA  
The suggested modified FMEA in this paper can be conducted according to the following steps:  
Step 1.1: Form the FMEA team; let \(T\) be the number of experts participating in the study. Step 1.2: Perform a detailed study of the system, the process, or the product being analyzed. Step 1.3: Identify and predict all possible failure modes (FMs) and their causes and effects, and treat the causes as sub-failure modes (SFMs). Let the total number of failure modes be \(M\), and the total number of sub-failure modes be \(N\); it follows that each failure mode \(FM_{i}\) has a number \(n_{i}\) of sub-failure modes \(SFM_{ij}\), where \(i=1,2,3,\ldots,M\) and \(j=1,\ldots,n_{i}\). It can be clearly seen that  
\[N=\sum_{1}^{M}n_{i} \tag{2}\]  
Step 1.4: Set the rating scale for each risk factor. Step 1.5: Prepare the FMEA tableau or sheet; there should be a field for the components, a second for the failures of those components, a third for the causes, and then a fourth for the effects. It must also include columns for risk factors ratings and two additional columns containing the expenses and time necessary to fix each cause of failure (or sub-failure mode). Step 1.6: Collect the data from the FMEA tables filled by the \(T\) experts. Let \(O_{ije}\), \(S_{ije}\), \(D_{ije}\), \(Cost_{ije}\), and \(Time_{ije}\) be the occurrence, the severity, the detection, the cost, and the time rates of the sub-failure mode \(SFM_{ij}\) assigned by the \(e^{th}\) expert, respectively, where \(i=1,\ldots,M\) ; \(j=1,\ldots,n_{i}\) and \(e=1,\ldots T\). Step 1.7: Assign a weight \(\lambda_{e}\) to the \(e^{th}\) expert reflecting the diversity of the members' background and experience. \[\sum_{e=1}^{T}\lambda_{e}=1 \tag{3}\]  
Step 1.8: Aggregate the assessments and construct the \((N\times 5)\) judgment matrix as follows: for each sub-failure mode \(SFM_{ij}\), the aggregated rates of the occurrence, severity, detection, cost, and time are \(O_{ij}\), \(S_{ij}\), \(D_{ij}\), \(Cost_{ij}\), and \(Time_{ij}\), respectively, and are calculated according to the following equations:  
\[ \begin{split} O_{ij}=\sum_{e=1}^{T}\lambda_{e}O_{ije};S_{ij}= \sum_{e=1}^{T}\lambda_{e}S_{ije};\ D_{ij}=\sum_{e=1}^{T}\lambda_{e}D_{ije};\\ Cost_{ij}=\sum_{e=1}^{T}\lambda_{e}Cost_{ije}\ ;\ Time_{ij}= \sum_{e=1}^{T}\lambda_{e}Time_{ije}\end{split}  \tag{4}\]  
Step 1.9: Compute the RPNs for each sub-failure mode \(SFM_{ij}\). \[RPN_{ij}=O_{ij}\times S_{ij}\times D_{ij}\ ;where\ i=1,\ldots,M\ and\ j=1,\ldots,n_{i} \tag{5}\]  
Step 1.10: Assign a rank \(\text{a}_{ij}\) (\(\text{a}_{ij}=1,2,\ldots N\)) for each \(SFM_{ij}\) related to the \(FM_{i}\) based on the descending order of the \(RPN_{ij}\) obtained in step 8. Step 1.11: Synthetize the procedure for the \(n_{i}\) sub-failure modes \(SFM_{ij}\) that belong to the same parent failure mode \(FM_{i}\) in order to eliminate the rank gaps. Thus, the common weighted RPN for the SFMs belonging to the same parent FM will attempt to give more weight to the SFM with the highest \(RPN_{ij}\) (or the smallest \(a_{ij}\)). On the other hand, solving an FM can be achieved only if all the related SFMs are solved; the total cost and time (aggregated cost and aggregated time) of each FM will be equal to the sum of the costs and times of its related SFMs. As a result, the aggregated occurrence \((AGGO_{i})\), the aggregated severity \((AGGS_{i})\), the aggregated detection \((AGGD_{i})\), the aggregated cost \((AGGOST_{i})\), and the aggregated time \((AGGTIME_{i})\) will be calculated as follows:  
\[ \begin{split} AGGO_{i}=\frac{\sum_{j=1}^{n_{i}}a_{ij}\times O_{ ij}}{\sum_{j=1}^{n_{i}}a_{ij}};\ AGGS_{i}=\frac{\sum_{j=1}^{n_{i}}a_{ij}\times S_{ij}}{ \sum_{j=1}^{n_{i}}a_{ij}};\ AGGD_{i}=\frac{\sum_{j=1}^{n_{i}}a_{ij}\times D_{ ij}}{\sum_{j=1}^{n_{i}}a_{ij}};\\ AGGCOST_{i}=\sum_{j=1}^{n_{i}}Cost_{ij}\ ;\ AGGTIME_{i}=\sum_{j=1}^{n_{i}} Time_{ij}\end{split}