ISE 423 Project

ISE 423 Statistical Quality Control: Process AnalysisAlexander Davis

Abstract:

The purpose of this document is to analyze the data that was reported from the Watfactory simulation. The analysis will cover numerous tools that were taught in the ISE 423 class this semester, including control charts and basic capability ratios. The sigma limits and summary statistics were also determined and noted in this document.

Introduction:

Using the Watfactory Virtual Process simulation, data was created in the form of ten input variables (x1-x10) and three output variables (y100-y300). The final product is part number y300. All of the data was generated randomly and assumed to follow a normal distribution- this was verified later in the document (Watfactory).

This data was analyzed using Microsoft Excel and MiniTab to answer the following questions:

1. Provide descriptive statistics of the output and input variables.

2. Is the process in statistical control?3. Is the process capable?4. What is the sigma level?5. What is the faction nonconforming for this

process?

After analyzing the data, a project charter was filled out to address one of the problems that was evident from the data. Steps to complete the project were also to be provided.

1. Descriptive Statistics

Microsoft Excel was used to calculate the descriptive statistics of each input variable and output variable. The results of this analysis can be seen below, in Exhibits 1-3. Histograms of the output variables can be found in Appendix 1A.

Statistic y100 y200 y300 x1 x2Mean -0.31475 -1.716 -0.9975 2.5124 5.24575Standard Error 0.200126 0.20485 0.209939 0.247305 0.074164Median -0.25 -1.75 -1.125 2.4 5.25Mode 0.5 -2.5 0.5 -0.2 5.5Standard Deviation 6.32854 6.477938 6.638859 7.820468 2.345271Sample Variance 40.05042 41.96368 44.07444 61.15973 5.500295

Exhibit 1. Summary Statistics for outputs, x1 and x2

Statistic x3 x4 x5 x6Mean 2.1588 4.9133 2.5547 1.9021Standard Error 0.047476 0.059701 0.152317 0.2273Median 2.2 4.9 2.5 1.9Mode 2.6 6.9 2.5 -2.8Standard Deviation 1.501332 1.887913 4.816698 7.187856Sample Variance 2.253997 3.564217 23.20058 51.66527

Exhibit 2. Summary Statistics for x3-x6

Statistic x7 x8 x9 x10Mean 5.77445 2.5849 5.0654 2.9413Standard Error 0.042999 0.220844 0.118118 0.046851Median 5.8 2.4 5.2 3Mode 6.05 5.4 6.8 3.1Standard Deviation 1.359759 6.983708 3.735221 1.481546Sample Variance 1.848944 48.77217 13.95187 2.194979

Exhibit 1. Summary Statistics for x7-x10

2. Is the Process in Control?

To answer this question, Minitab was used. Minitab was able to create Xbar and S charts to quickly give a summary of how the process was operating. S charts were used instead of R charts because a subgroup size of 50 was used. 50 is much greater than twelve, so the S chart was the correct choice instead of the R. This created 20 subgroups for the charts.

After the charts were created, it was easily seen that the input variable x3 was the only process to plot out of control. All nine of the other input process plotted in control. There existed one point from x3 that appeared outside the control limits; below the lower control limits on the S chart, to be specific.

The same could not be said for the output variables. Two of the output variables, y200 and y300 both had points plotting out of the control

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limits: two from y200 (one above and one below on the X bar chart) and one from y300 (one above the upper control limit on the X bar chart).

A few samples of these control charts can be found in Appendix 1B, including the out of control processes.

3. Is the process capable?

To determine capability, the standard deviation and the specification limits of the process must be known. Also, the data must be normal in distribution. Before we can find the capability, these requirements must be satisfied.

The Watfactory website lists that the specification limits on the final output variable, y300, is -14 and 14. These are the only specification limits provided, so we can only find the capability of the final output, which is the capability of the entire process.

To ensure normality, Minitab was used. The final output data was plotted on eight different goodness of fit tests to guarantee that it was normal in nature (Montgomery, 2009). The full results can be found in Appendix 1C, but the goodness of fit test for the normal distribution can be seen below, in Exhibit 4.

Exhibit 2. Goodness of Fit Test for y300

Taking this data and plugging it into Minitab, we obtain that the process capability Cp = 0.72 and

its Cpk = 0.67. This can be seen in Exhibit 5 below. The full Minitab output can be seen in Appendix 1C.

Exhibit 3. Capability Chart

Since the output y300, is less than 1 it is not capable. Because y300 is the final output, and is not capable, the entire process is not capable.

4. What is the sigma level?

To find the sigma level of the process, the following equation can be used:

σ=3 (C pk )+1.5 (1)

Subbing in the value of Cpk gives us the following:

σ=3(0.67)+1.5

So the sigma level of the process is:

σ=3.51

5. What is the fraction nonconforming for this process?

To the fraction nonconforming is used to find out how many parts per million can be expected to be nonconforming. The equation is:

P PM defective=p∗1,000,000 (2)

The Minitab output from the capability chart of the final output, y300, provides us with a PPM estimation (see Appendix 1C). Having this PPM (Observed PPM Total = 41000), we can use the equation and back out the fraction nonconforming, p. The equation becomes:

p= PPM defective1000000 (3)

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This calculation returns a p value of .041. Expressed as a percentage, the fraction nonconforming for the entire process is 4.1%.

Conclusion:

Using y300, the final output, it is determined that the entire process is not in statistical control and also not capable:

Y300 has one point that plots above the upper control limit on the Xbar chart, and is thus deemed out of control.

The Cp being less than one showed that the process was not capable.

Using the Cpk, the process is currently operating at a sigma level of 3.51.

Using the PPM, the fraction nonconforming, p, is found to be 4.1%.

References:

Montgomery, Douglas C., Introduction to Statistical Quality Control, sixth edition, Hoboken, NJ: Wiley, 2009."Watfactory Login." Faculty of Mathematics. Web.

<http://www.student.math.uwaterloo.ca/~stat435/login.htm>

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Appendix 1A: Output Variable Histograms

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Appendix 1B: Control Charts for Out of Control Output Variable Processes

Test Results for Xbar Chart of y200

TEST 1. One point more than 3.00 standard deviations from center line.

Test Failed at points: 16, 20

Test Results for Xbar Chart of y300

TEST 1. One point more than 3.00 standard deviations from center line.

Test Failed at points: 20

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Test Results for S Chart of x3

TEST 1. One point more than 3.00 standard deviations from center line.Test Failed at points: 13

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Appendix 1C: Best Fit Plots and Capability Charts

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ISE 423 Project

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