# Failure Mode and Effects Analysis Considering Consensus and Preferences Interdependence
## 5 Case Study
### Implement the Proposed Method
\begin{array}{ccccc}([0.8594,0.8928],[0.2048,0.2573])&([0.8781,0.908 8],[0.1784,0.2287])&([0.8441,0.8796],[0.2264,0.2806])\\ ([0.9154,0.9422],[0.1199,0.1669])&([0.9000,0.9281],[0.1441,0.1925])&([0.8904,0.9 195],[0.1608,0.2100])\\ ([0.8912,0.9203],[0.1591,0.2082])&([0.8517,0.8863],[0.2153,0.2687])&([0.9226,0. 9483],[0.1081,0.1547])\\ ([0.8740,0.9052],[0.1835,0.2343])&([0.8441,0.8802],[0.2255,0.2802])&([0.7693,0.8192],[0.3166,0.3836])\\ ([0.8923,0.9213],[0.1571,0.2061])&([0.8740,0.9052],[0.1835,0.2343])&([0.7667,0. 8169],[0.3210,3.3885])\\ ([0.9293,0.9543],[0.9056,0.1418])&([0.9028,0.9305],[0.1421,0.1901])&([0.782,0.8318],[0.2980,0.3650])\\ ([0.9052,0.9327],[0.1356,0.1835])&([0.9337,0.9583],[0.0898,0.1353])&([0.8796, 0.9101],[0.1762,0.2264])\\ ([0.9640,0.9867],[0.0403,0.0801])&([0.8374,0.8740],[0.2343,0.2894])&([0.8011,0. 8441],[0.2806,0.3413])\\ ([0.8629,0.8960],[0.1988,0.2509])&([0.8740,0.9052],[0.1835,0.2343])&([0.8606,0. 8937],[0.2037,0.2560])\end{array} \right).\]  
**Step 4:** Calculate the combination weights of risk factors. **Step 4.1:** Determine the subjective weights of risk factors. The collective evaluation matrix \(H\) of risk factors is constructed by aggregating all individual assessments for risk factors. Subsequently, the normalized subjective weights of risk factors based on Equation (21) are determined to be \(w_{j}^{s}=(0.4021,0.3602,0.2378)^{\text{T}}\). **Step 4.2:** Compute the objective weights of risk factors by the entropy method. Based on the comprehensive evaluation matrix \(D\), the entropy value of each failure mode with regard to each risk factor can be calculated by applying Equation (22). Then, the objective weights for each risk factor can be derived as \(w_{j}^{o}=(0.3568,0.3390,0.3042)^{\text{T}}\) by utilizing Equation (23). **Step 4.3:** Obtain the combination weights of risk factors. Based on the subjective weights and objective weights of risk factors, the combination weights of risk factors are calculated as \(w_{j}=(0.3795,0.3496,0.2709)^{\text{T}}\) according to Equation (24). **Step 5:** Compute the weighted group decision matrix \(Y\). The weighted decision matrix can be calculated by Equation (25) as follows. \[Y=\left( \begin{array}{ccccc}([0.6317,0.6739],[0.5479,0.5974])&([0.6346,0.6762], [0.5474,0.5970])&([0.5353,0.5757],[0.6687,0.7088])\\ ([0.7062,0.7509],[0.4471,0.5069])&([0.6636,0.7063],[0.5080,0.5621])&([0.5892,0. 6301],[0.6095,0.6552])\\ ([0.6717,0.7138],[0.4978,0.5513])&([0.6030,0.6541],[0.5846,0.6317])&([0.6349,0. 6804],[0.5473,0.6032])\\ ([0.6494,0.6912],[0.5255,0.5765])&([0.5944,0.6373],[0.5941,0.6409])&([0.4643, 0.5100],[0.7323,0.7714])\\ ([0.6732,0.7154],[0.4954,0.5492])&([0.6295,0.6711],[0.5528,0.6021])&([0.4620, 0.5078],[0.7350,0.7740])\\ ([0.7284,0.7748],[0.4102,0.4765])&([0.6676,0.7103],[0.5055,0.5597])&([0.4756, 0.5225],[0.7204,0.7611])\\ ([0.6912,0.7340],[0.4684,0.5255])&([0.7157,0.7638],[0.4306,0.4969])&([0.5757, 0.6161],[0.6248,0.6687])\\ ([0.7964,0.8648],[0.2957,0.3837])&([0.5870,0.6295],[0.6021,0.6482])&([0.4927, 0.5353],[0.7088,0.7473])\\ ([0.6359,0.6782],[0.5416,0.5917])&([0.6295,0.6711],[0.5528,0.6021])&([0.5533, 0.5935],[0.6498,0.6914])\end{array} \right)\]  
**Step 6:** Determine the border approximation area vector \(G\). According to Equation (26), the border approximation area vector \(G\) can be obtained as  
\[G=(([0.6881,0.7345],[0.4752,0.5323]),([0.6367,0.6796],[0.5442,0.5951]),([0.5323,0.5757],[0.6696,0.7120])).\]  
**Step7:** Construct the distance matrix \(X\). The distance of failure modes from the border approximation area is computed by Equation (27); the distance matrix is determined as  
\[X=\left( \begin{array}{ccccc}-0.0799&-0.0041&0.0038\\ 0.0261&0.0382&0.0772\\ -0.0261&-0.0456&0.1459\\ -0.0567&-0.0567&-0.0880\\ -0.0240&-0.0105&-0.0920\\ 0.0591&0.0416&-0.0715\\ 0.0071&0.1161&0.0589\\ 0.1846&-0.0662&-0.0528\\ -0.0743&-0.0105&0.0275\end{array} \right)\]  
**Step 8:** Determine the ranking of failure modes. The closeness coefficient \(\theta_{i}\) can be calculated by Equation (28) as follows.