## 5 Descriptive Statistics of Failure andRepair Database  
Descriptive statistics of the failure database (TBFs and TTRs) for each machine and for the complete ice cream manufacturing system were illustrated in Table 1, in order to measure the central tendency and the dispersion of the data. It was pointed out that: (a) the average TBF of the system is 334.2 min or approximately 5.5 h, hence about 1.5 failures per shift on the ice cream system is shown, while the average TTR a failure is 25.12 min. (b) The maximum value of TTR is 156 min (replacement of  
Figure 3: Diagram of the failure data for the ice cream manufacturing system. Figure 2: Summary Report for time between failure-TBFand time to repaire-TTR (both recorded in minutes) for the ice cream manufacturing system, where (**a**) is the summery report for TBF, and (**b**) is the summery report for TTR. motor inverter) at the aging machine (M2). There are also time-consuming repairs at an ice cream machine (M3, failure of screw pump) with 146 min, and at the pasteurization/homogenization machine (M1, failure of heat exchanger) with 137 min. (c) The highest values of SD are observed at the machine with the least number of failures. For instance, the exogenous (M6, 7 failures), the ice cream machine (M3, 70 failures), and the aging machine (M2, 82 failures) have SDs 34553, 2242, and 1532 min respectively. Whereas, the lowest values of SD are shown at the workstation with the most number of failures, i.e., the pasteurization/homogenization machine (M1, 123 failures) has 1376 min and the packaging machine (M5, 102 failures) stand for 1443 min. (d) The CVs of TBF and TTR for the manufacturing system are slighter than one (with 0.7704 and 0.8648 respectively) and are considered low-variance, i.e.,Erlang distribution. (e) All the breakdowns for all machines and the complete manufacturing system for TBF and TTR (except TTR M6) are positively skewed, then it is valid: mode \(<\) median \(<\) mean. (f) The lowest mean TBFs are at the pasteurization/homogenization machine (M1), packaging machine (M5), and aging machine (M2) with 1343, 1643, and 1914 min respectively, and (g) the highest mean TTRs are at exogenous (M6), ice cream machine (M3), and pasteurization/homogenization machine (M1) with 59.50, 31.71 and 27.22 min respectively. Figure 4 shows the box plots of the failure and repair database for all machines and the complete ice cream system. The box plot shows the median, the quartiles (Q1, Q2, and Q3), and the minimum and maximum data points to reveal the spread and the symmetry of distribution of database values. Moreover, the graph depicts the central tendency, variability, and shape of the failure database. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline
**Variable** & **Total Count** & **Mean** & **SD** & **CV** & **Minimum** & **Maximum** & **Skewness** & **Kurtosis** \\ \hline TBF M1 & 123 & 1343 & 1376 & 1.0246 & 10 & 8332 & 2.29 & 7.03 \\ TBF M2 & 87 & 1914 & 1532 & 0.8006 & 57 & 8948 & 1.75 & 4.66 \\ TBF M3 & 70 & 2376 & 2242 & 0.9437 & 2 & 9460 & 1.49 & 1.65 \\ TBF M4 & 85 & 1958 & 1531 & 0.7818 & 70 & 8150 & 1.70 & 3.28 \\ TBF M5 & 102 & 1634 & 1443 & 0.8831 & 1 & 7410 & 1.57 & 3.37 \\ TBF M6 & 7 & 24,018 & 34,553 & 1.4386 & 2360 & 98,480 & 2.18 & 4.86 \\ TBF Line & 469 & 334.2 & 257.5 & 0.7704 & 1.0 & 1350.0 & 0.88 & 0.54 \\ \hline TTR M1 & 122 & 27.22 & 23.62 & 0.8678 & 4.00 & 137.00 & 2.43 & 7.81 \\ TTR M2 & 86 & 22.95 & 20.16 & 0.8783 & 5.00 & 156.00 & 3.59 & 21.41 \\ TTR M3 & 69 & 31.71 & 27.46 & 0.8658 & 7.00 & 146.00 & 2.03 & 4.61 \\ TTR M4 & 84 & 24.57 & 16.00 & 0.6512 & 6.00 & 80.00 & 1.19 & 1.34 \\ TTR M5 & 101 & 18.35 & 16.49 & 0.8988 & 3.00 & 87.00 & 2.58 & 7.47 \\ TTR M6 & 6 & 59.50 & 24.31 & 0.4086 & 11.00 & 76.00 & \(-\)2.22 & 5.06 \\ TTR Line & 468 & 25.12 & 21.73 & 0.8648 & 3.00 & 156.00 & 2.44 & 8.45 \\ \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics for the machines and the entire line level of the ice cream manufacturing system.