Anyone who has faced a production problem with a need to solve it by using production data can relate to the notion of a brain teaser. The brain teasers presented here are based on real-world situations encountered by workers in manufacturing environments. The brain teasers have three parts: (1) the situation, (2) available data or other supporting information and (3) questions that various workers need answered for continual improvement. Recommended solutions follow in the next issue and on the Web at Quality Online (www.qualitymag.com).

Situation

Lidia, a process engineer, has been assigned to set up short-run capability studies to test critical characteristics of new product designs. Current practice is to make a production run of nine, 18 or 30 units to test. Because time and cost are concerns, most of the process engineers conduct the short-run capability studies with nine units. However, Lidia is interested in finding out how many units she really needs to test in order to detect a shift in the average of each characteristic of a new design. For her current assignment, she has been asked to determine the capability of the diameter of a shaft. This critical dimension must meet specifications of 1.100 inch ±0.008 inch to perform correctly in final assembly. Lidia wants to know how many production units to make so that her capability study will be reliable.

Available data

First, Lidia decides to check data for the diameter of shafts used in previous designs. She discovers that the standard deviation of shaft diameter has been predictable for three months. The most recent estimate from a process behavior chart using one month of data is 0.002 inch. In addition to the capability calculations, she wants to make certain that the diameters will still meet specifications if the average shifts 0.0015 larger or smaller than the nominal of 1.100 inch.

Questions

1. Based on a standard deviation of 0.002 inch, how large a sample does Lidia need to detect a shift in the average diameter of 0.0015? Hint: Check your reference texts for sample size formulas. These will ask you to select an alpha value to use. Use alpha values of 0.05 and 0.01 to compare the amount of data required. Alpha is the chance of getting an unusual value in the data when there is no underlying change.

2. Lidia met with the product design team and found that they had completed a run of nine prototype shafts. She decided to use the data for a preliminary capability study and to check the sample size she needs to use in the next study. The nine data values are 1.100, 1.099, 1.097, 1.100, 1.101, 1.101, 1.099, 1.098 and 1.092. What can she learn about the capability of the diameter of the shafts using these data?

3. How much data would Lidia need if she wanted to detect a shift of 0.0015 in the average shaft diameter with a power of 0.8 or 0.9? Use available software packages to do this calculation.

Answers to August Brain Teaser

Colin, a measurement study expert for a company that makes electrical components, is concerned that his company’s measurement devices have too much variation to be useful under tighter specifications from a major customer. Specifically, specifications for resistance and watts for a particular electrical component have been cut in half, from ±5% of the nominal to ±2.5% of nominal. He has used two reference standard components to conduct measurement studies for resistance and watts.

Q: What is the behavior of the two measurement standards for resistance and watts? What is the behavior of the two production models for resistance and watts?
A: Both measurement standards have a predictable behavior for resistance and watts. Also, the behavior of the two production models is predictable for resistance and watts. This predictability is established using individuals and moving range process behavior charts.

Q: Based on the production data provided, what is the capability of production models 4 and 15 for resistance and watts?
A: Capability indexes, based on the current specifications of ±5% of the nominal for resistance and watts, are summarized in the table, “Capability Analyses for Production Models 4 and 15.”

Under the current specifications, the Cpk values for Model 4 indicate that it is marginally meeting the specification on watts, but not on resistance. Model 15 is not meeting the specifications for either characteristic.

Q: Using the variation from the measurement study and the variation from the production process, determine the variation of the production process for the two production models.
A: The production data for resistance and watts for both models vary because of the production process itself plus the measurement process. Calculate the total standard deviation from the production data and the measurement process standard deviation from the measurement study data using the average moving ranges from the respective process behavior charts. The standard deviations are not additive, but the variances (square of the standard deviations) are. This leads to the expression for the total variance.

^σ2 Total = ^σ2Production + ^σ2Measurement

First, calculate the standard deviations from the moving ranges charts using the formula:

mR
d2

Then, square these to get the variances and solve for the production variance. Take the square root of the production variance to get the standard deviation for production. For Model 4 resistance, the average range is 1.668 from the production data and 0.452 from the measurement study. The variances are 2.187 from the production data and 0.161 from the measurement study. Using the equation for variances above, the variance for the production process only is 2.026 and the standard deviation is 1.423. The other production process standard deviations can be found by applying the formulas to the remaining data.

Q: If the new specifications were applied to the current production models, what is the capability of resistance and watts?
A: With the current production processes and measurement processes, neither model is capable for resistance or watts under the new specifications. Even if the variation of the measurement process was reduced to zero and there were no bias, neither resistance nor watts for either model would be capable based on the production process variation alone.