4.2 The concept of Cox regression-based root-
4.3.2 Results of the Cox regression analysis
First, the distribution of the time demand of the whole changeover process is analyzed. The impact of the studied variables has been checked by the sensitivity analysis shown in Figure 4.7. As this gure shows, the curve related to the cross-section change (CSAChange) deviates the most from the baseline, which indicates that this variable has the most signicant inuence on the time demand of the changeover.
The rst row (Overall process) of Table 4.5 summarizes the parameters of the identied Cox model. When a parameter is smaller than the related variable, the changeover time increases. The analysis highlights that OperatorSeniority does not aect the time demand of the changeover process, while CSAChangeis
0 100 200 300 400 500 600 700 800 900 1000
Figure 4.7: Survival functions of the whole process and the inuences of the variables.
the most signicant feature. The N o.of T ools and N o.of W ires variables have a considerable inuence on the times. Note that these variables are not independent, and both N o.of T oolsand CSAChange change when the tools are changed.
The estimated uncertainty parameters are summarized in Table 4.6, while the p values are shown in the rst row in Table 4.7, also conrming thatOperatorSeniority does not signicantly inuence the time demand of the process.
A changeover is decomposed into dierent states to further explore the root causes of the losses. Figure 4.8 shows the four studied steps of the process: the setup-changeover, setup-sample, setup-short fault, and setup learning steps.
Figure 4.8 a) presents the time demand of the Setup-Changeover step. Based on the Cox parameters (shown in the second row of Table 4.5), the N oOf W ireshas the most signicant inuence on the time, and the curves showing the sensitivity of the CSAChange and N oOf T ools variables also signicantly deviate from the baseline.
The results of the Cox regression of the setup-sample process step can be seen in Figure 4.8 b). The parameters are given in the third row of Table 4.5. Note that N oOf W ires decreases the activity time because, during the setup-changeover step, the wire spool is replaced, and there are more preparation activities than there would be otherwise.
Figure 4.8: a.) Setup-Changeover b.) Setup-Sample c.) Setup-Short Fault d.) Setup-Learning. This gure shows the survival functions of individual machine
states.
Figure 4.8 c) shows the result of the setup-short fault step. The Cox parameters are not statistically signicant in this case (see the fourth row in Table 4.5), and the parameters are close to zero, which correctly reects that this process step occurs randomly.
The Cox parameters of the setup-learning step are shown in Figure 4.8 d. The small parameter values shown in the last row in Table 4.5 reect a well-controlled process. The change in cross-section increases the activity time, which is entirely in line with the experience of the process engineers.
The application of the method assumes that the proportional hazard assumption (PHA) is satised for each predictor. As presented in the previous section, when the Schoenfeld residuals are correlated and the rank order of the survival times are not correlated, then the PHA is satised for the studied predictor. As the cor-relation values in Table 4.8 show, the PHA assumption is not satised in the case of the total activity time for theN oOf W ires,N oOf T erminals and N oOf T ools variables; in the case of the setup-changeover, theN oOf W iresvariable should be modelled by stratied Cox regression.
A graphical validation of the PHA has also been performed. There are two graph-ical methods. One method is the log-log method. The Kaplan-Meier curves for
0 100 200 300 400 500 600
Figure 4.9: a.) Kaplan-Meier distribution of marked events b.) Graphical checking of the PHA being satised in log-log scale c.) Graphical checking of the PHA being satised in log scale d.) Predicted vs expected examination of
satisfying the PHA.
each value of the predictors must be twice logarithmized, and the proportional hazard analysis is satised if these curves are parallel [218]. The functions of N oOf W ires of the data are plotted in Figure 4.9 b),c). Figure 4.9 b) has the abscissa label also logarithmized. The other method is the predicted vs expected method. The Kaplan-Meier curve and the baseline curve of the Cox regression need to be compared. If the curves are close to each other, the PHA assumption is satised [218]. These functions are displayed in Figure 4.9 d). The application of both methods leads to the previous conclusion, as the PHA is not satised for the N oOf W ires variable; thus, the stratied Cox model should be applied for modeling the overall process and the setup-changeover process step.
As Figure 4.10 and Table 4.10 show, this model describes the Setup-changeover process step well, while the overall process should be modelled when bothN oOf W ires and N oOf T ools are used to form separate groups in the survival analysis. As Figure 4.10 and Figure 4.11 illustrate, these models show realistic results, and the eects of CSAChange and N oOf T erminals are in line with expectations.
Figure 4.10: Stratied Cox model for setup-changeover. a) No. of Wires is 0;b) No. of Wires is 1.
Figure 4.11: Stratied Cox model for the whole process. a) No. of Wires is 0, No. of Tools is 0; b) No. of Wires is 0, No. of Tools is 1; c) No. of Wires is 0, No. of Tools is 2; d) No. of Wires is 1, No. of Tools is 0; e) No. of Wires is 1,
No. of Tools is 1; f) No. of Wires is 1, No. of Tools is 2.
Table 4.1: Integrated information database and the key variables
Database Variable Range Description
Machine
log-data Time stamp
[datetime]
[−] The start time stamp
of the status
Duration[s] [1−] The duration of
ac-tual status
Number of
Wires [pcs]
[0,1] If the wire is
changed, then it is 1and 0 otherwise.
Number of Ter-minals[pcs]
[0,1,2] If the terminal is changed, then it is 1 or2 and 0otherwise.
Number of Tools
[0,1] If the cross section of wire is changed, then is changed, then it is 1 and 0otherwise.
Order
de-scriptions Order ID [−] [−] The identication of
produced order CSA[mm2] [0.13−35] Wire cross section Length[mm] [30−8514] Length of the wire Type of terminal
[−]
[−] Type of the terminal
Type of wire[−] [−] Type of the wire Operator
data Operator ID [−] [−] Operator IDs
Seniority [weeks]
[0−] Seniority of the
op-erator (number of weeks working at the company)
Table 4.2: Description of branches
B Production If no need for more learning
Setup-Learning - -
-Table 4.3: Input variables of the Cox regression model
CaseID Duration Censored Features representing the changeover
1 t1 s1 x1,1 x1,2 . . x1,m
Table 4.4: Logged machine states
Status Description
Production Logged when the actual production is started Production end Logged at the end of the
order (all batches are done)
Production-Short
fault Micro-stoppages
between two batches or during a fault
Setup-Changeover Logged when the
changeover is started Setup-Learning Logged at the beginning
of the learning (meas-urement, tool/machine setup)
Setup-Sample Logged when the ac-tual sample production is started
Setup-Short Fault Micro-stoppages during changeovers
No Cause Starts when the down-time is longer than 30 second and the operator did not log any cause.
Targeting model of operators
0 2 4 6 8 10 12 14 16 18 20
0 5 10 15 20 25 30 35
Figure 4.12: Scatter plot of the prediction of the targeting model (t,ˆ p= 50%) and the measured (t) activity times. The red diamonds represent the average
measured activity times of identical changeovers.