Basic Concepts of Statistical Process Control
In general, a production process has many sources or causes of variation. These can be further subdivided as process inputs and process operational characteristics including equipment, procedures and environmental conditions. Environmental conditions consist of factors such as temperature and humidity or work-tools. Visual guides for instance, might not allow operators to precisely position parts on fixtures. The complex interactions between material, tools, machine, work methods, operators, and the environment combine to create variability in the process. Factors that are permanent, as a natural part of the process, are causing chronic problems and are called common causes of variation. The combined effect of common causes can be described using probability distributions. It is important to recognize that recurring causes of variability affect every work process and that even under a stable process there are differences in performance over time. Failure to recognize variation leads to wasteful actions and detrimental overcontrol. The only way to reduce the negative effects of chronic, common causes of variability is to modify the process. This modification can occur at the level of the process inputs, the process technology, the process controls or the process design. Some of these changes are technical (e.g. different process settings), some are strategic (e.g. different product specifications) and some are related to human resources management (e.g. training of operators). Special causes, assignable causes, or sporadic spikes arise from external temporary sources that are not inherent to the process. These terms are used here interchangeably. For example, an increase in temperature can potentially affect the piston’s performance. The impact can be both in terms of changes in the average cycle times and/or the variability in cycle times.
In order to signal the occurrence of special causes we need a control mechanism. Specifically in the case of the piston such a mechanism can consist of taking samples or subgroups of 5 consecutive piston cycle times. Within each subgroup we compute the subgroup average and standard deviation.
 Fig. 2.5: $X$-bar Chart of Cycle Times under Stable Operating Conditions |
 Fig. 2.6: $S$ Chart of Cycle Times under Stable Operating Conditions |
Figures 2.5 and 2.6 display charts of the average and standard deviations of 20 samples of 5 time measurements. To generate these charts with Python we use:
simulator = mistat.PistonSimulator(n_simulation=20, n_replicate=5, seed=1)
Ps = simulator.simulate()
Ps['seconds'].groupby(Ps['group']).apply(np.mean)
Output:
group
1 0.044902
2 0.042374
3 0.043812
4 0.048865
5 0.047265
6 0.043910
7 0.048345
8 0.041833
9 0.041135
10 0.045080
11 0.044307
12 0.047490
13 0.045008
14 0.045684
15 0.046281
16 0.044656
17 0.044445
18 0.044227
19 0.041077
20 0.044947
Name: seconds, dtype: float64
The chart of averages is called an X-bar chart, the chart of standard deviations is called an S chart. All 100 measurements were taken under fixed operating conditions of the piston (all factors set at the maximum levels). We note that the average of cycle time averages is 0.045 seconds and that the average of the standard deviations of the 20 subgroups is 0.0048 seconds. All these numbers were generated by the piston computer simulation model that allows us to change the factors affecting the operating conditions of the piston. Again we know that no changes were made to the control factors. The observed variability is due to common causes only such as variability in atmospheric pressure or filling gas temperature.