In the realm of Lean Six Sigma, a methodology focused on improving process efficiency and effectiveness, Control Plans play a pivotal role in sustaining the gains achieved during the improvement phases. Among the various tools utilized in these plans, Control Charts stand out for their ability to monitor process performance and variability over time. A specific type of control chart, the U Chart, or Defects per Unit Chart, is instrumental in situations where the monitoring of defects per unit across varying sample sizes is required. This article delves into the theory behind U Charts, guiding through their construction and application.

Understanding U Charts

U Charts fall under the category of attribute control charts, which are used for data that can be counted for occurrences of defects (nonconformities) rather than measured on a continuous scale. The 'U' in U Chart stands for "defects per unit," and it is specifically designed to track the number of defects in each of a series of samples, with each sample possibly having a different number of units. This makes the U Chart particularly useful in situations where the sample size varies but the interest lies in understanding the average number of defects per unit.

Theory Behind U Charts

The theoretical foundation of U Charts is grounded in the Poisson distribution, a probability distribution that estimates the likelihood of a given number of events happening in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event. In the context of U Charts, each defect is considered an event, and the chart assumes that the occurrence of defects in a unit is random and independent of the occurrence of defects in another unit.

Construction of U Charts

To construct a U Chart, follow these steps:

1. Collect Data: Gather data on the number of defects and the corresponding number of units for each sample. It's crucial to have enough data points to establish a reliable baseline for process performance.

2. Calculate Defects per Unit (u): For each sample, calculate the defects per unit by dividing the total number of defects by the number of units in the sample. This yields the 'u' value for each sample.

3. Determine the Average Defects per Unit (uˉ): Calculate the overall average of the 'u' values across all samples to establish a central line for the chart.

4. Calculate Control Limits: The control limits for a U Chart are determined using the uˉ and the actual size of each sample. The upper control limit (UCL) and lower control limit (LCL) for each sample are calculated using the formulae:

where n is the sample size. For smaller sample sizes, the LCL can be negative; in such cases, it is set to zero, as the number of defects cannot be negative.

5. Plot the Chart: On the chart, plot the 'u' value for each sample against the sample number or time period. Draw the central line (uˉ) and the UCL and LCL for each sample.

6. Interpret the Chart: Regularly update and review the chart to monitor process performance. Investigate any points outside the control limits or patterns within the limits indicating potential process shifts or trends.

Application of U Charts

U Charts are versatile tools applicable in various industries, including manufacturing, healthcare, and service sectors. They are particularly beneficial when dealing with high-volume production processes or services where the sample size varies significantly. By identifying trends and signals of process instability, organizations can take preemptive actions to address issues, thereby reducing defects, improving quality, and enhancing customer satisfaction.

In conclusion, U Charts offer a robust framework for monitoring defects per unit, accommodating varying sample sizes. Through their construction and analysis, businesses can gain valuable insights into process performance, facilitating continuous improvement in line with Lean Six Sigma principles.

Scenario:

For our example, let's consider a scenario in a paint and coating company involved in aircraft manufacturing. The company is focused on improving the quality of its paint application process to minimize defects, such as uneven coating, air bubbles, and impurities. Lean Six Sigma projects often utilize U Charts to monitor defects per unit over time, aiming for process improvement and consistency. The U Chart helps in identifying variations in the process and signals when a process may be out of control. The quality control department has decided to track the number of defects in the paint and coating process over a period of 8 days. Each aircraft section painted is considered a unit. The number of sections inspected each day varies due to production schedules.

Data Collection:

Day: The day of the inspection

Units Inspected (n): The number of aircraft sections inspected each day

Defects Found (c): The total number of defects found in those sections

Here's the collected data:

Calculations for U Chart:

Calculate the Defects per Unit (U): This is the average number of defects per unit over the observed period.

U = 35 / 4800 = 0.007291667

Calculate the Upper Control Limits:

DAY 1 UCL = 0.007291667 + 3 * SQRT ( 0.007291667 / 500 ) = 0.018748106 DAY 2 UCL = 0.007291667 + 3 * SQRT ( 0.007291667 / 700 ) = 0.016974125 DAY 3 UCL = 0.007291667 + 3 * SQRT ( 0.007291667 / 600 ) = 0.017749917 DAY 4 UCL = 0.007291667 + 3 * SQRT ( 0.007291667 / 800 ) = 0.016348777 ....

Calculate the Lower Control Limits:

DAY 1 LCL = 0.007291667 - 3 * SQRT ( 0.007291667 / 500 ) = -0.004164773

DAY 2 LCL = 0.007291667 - 3 * SQRT ( 0.007291667 / 700 ) = -0.002390792

DAY 3 LCL = 0.007291667 - 3 * SQRT ( 0.007291667 / 600 ) = -0.003166584

DAY 4 LCL = 0.007291667 - 3 * SQRT ( 0.007291667 / 800 ) = -0.001765444 .... Since all LCL calculations are below 0, we consider LCL = 0, as defects per unit cannot be negative.

Complete the table with UCL & LCL

Plot the U chart

Chart Analysis

The chart displays the defects per unit for each day, with the average number of defects per unit (Uˉ) shown as a dashed green line. The Upper Control Limit (UCL) and Lower Control Limit (LCL) are represented by dashed red and purple lines, respectively. The area between UCL and LCL is highlighted, indicating the range within which the process is considered to be in control.

As you can see, all the observed defects per unit fall within the control limits, suggesting that the process is in control regarding the defects per unit over the days observed.