BridgeRock is a major manufacturer of tires in the U.S.. The company had five manufacturing facilities where tires were made and another 20 facilities for various components and materials used in tires. Each manufacturing facility produced 10,000 tires every hour. Quality had always been emphasized at BridgeRock, but lately quality was a bigger issue because of recent fatal accidents involving tires made by other manufacturers due to tread separation. All tire manufacturers were under pressure to ensure problems did not arise in the future. However, even with multiple visual quality inspections in place, adhesion flaws and other internal problems were not visible, so manufacturers randomly pulled tires from the production line and cut them apart to look for defects. Given the large number of steps in building a tire, errors tended to accumulate, which possibly results in larger process quality variability. Another issue was related to the settings on various machines. Over time, these settings tended to vary because of wear and tear on the machines. In such a situation, a machine would produce defective product even if the machine had the correct setting.
To detect process variations, the company implemented a statistical process control (SPC) program. At the extruder, the rubber for the AX-123 tires had thickness specifications of 400±10 thousandth of an inch (thou) (these are the specification limits). The quality team at BridgeRock analyzed many samples of output from the extruder and determined that if the extruder settings were accurate, the output produced by the machine had a thickness that was normally distributed with a mean of 400 thou and a standard deviation of 4 thou.
Answer the following questions:
a. If the extruder setting is accurate, what proportion of the rubber extruder will be with specifications?
b. The quality team asked operators to take a sample of 10 sheets of rubber each hour from the extruder and measure the thickness of each sheet. Based on the average thickness of this sample, operators will decide if the extrusion process is in control or not. Given that z=3 for constructing control limits, what upper and lower control limits should they specify to the operators?
c. If a bearing is worn out, the extruder produces a mean thickness of 403 thou when the setting is 400 thou. The standard deviation is still 4 thou. Under this condition, what proportion of defective sheets will the extruder produce? Assuming the control limits in question b, what is the probability that a sample taken from the extruder with the worn bearings will be out of control? On average, how many hours are likely to go by before the worn bearing is detected?
d. Now consider the case where extrusion is a six sigma process. In this case, the extruder output should have a mean of 400 thou and a standard deviation of 1.667 thou. Still using the same 400±10 thou quality specifications, what proportion of the rubber extruded will be within specifications in this case?
e. Assuming the operators will continue to collect samples of 10 sheets each hour to check if the process is in control, what control limits should they set for the case when extrusion is a six sigma process (keep in mind Z=3)?
f. Return to the case of the worn bearings in question c where extrusion produces a mean thickness of 403 thou when the setting is 400 thou. Now the process standard deviation is 1.667 thou. Under this condition, what proportion of defective sheets will the extruder produce? Assuming the control limits calculated in question e, what is the probability that a sample taken from the extruder with the worn bearings will be out of control? On average, how many hours are likely to go by before the worn bearing is detected?