91Ó°ÊÓ

A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each lot of tablets is measured to control the compression process. The process has been operating in control with mean at the target value \(\mu=12.3\) kilograms \((\mathrm{kg}\) ) and estimated standard deviation \(\sigma=0.2\) kg. Table \(31.2\) gives three sets of data, each representing \(\mathrm{x}^{-} \bar{x}\) for 20 successive samples of \(n=4\) tablets. One set remains in control at the target value. In a second set, the process mean \(\mu\) shifts suddenly to a new value. In a third, the process mean drifts gradually. (a) What are the center line and control limits for an \(\mathrm{x}^{-} \bar{x}\) chart for this process? (b) Draw a separate \(x^{-} \bar{x}\) chart for each of the three data sets. Mark any points that are beyond the control limits. (c) Based on your work in part (b) and the appearance of the control charts, which set of data comes from a process that is in control? In which case does the process mean shift suddenly, and at about which sample do you think that the mean changed? Finally, in which case does the mean drift gradually?

Step by step solution

01

Calculate the Center Line for the Chart

For an \( \bar{x} \) chart, the center line is the target mean \( \mu \). Therefore, the center line is \( \mu = 12.3 \) kg.

02

Calculate the Control Limits

The control limits for an \( \bar{x} \) chart are calculated using the formula: \[ \text{Upper Control Limit (UCL)} = \mu + A_2 \times \sigma \]\[ \text{Lower Control Limit (LCL)} = \mu - A_2 \times \sigma \]where \( A_2 \) is a constant that depends on the sample size \( n \). For \( n = 4 \), \( A_2 \approx 0.729 \). Thus, \[ \text{UCL} = 12.3 + 0.729 \times \frac{0.2}{\sqrt{4}} = 12.3729 \]\[ \text{LCL} = 12.3 - 0.729 \times \frac{0.2}{\sqrt{4}} = 12.2271 \]

03

Plot Each Dataset on an x-bar Chart

For each of the three datasets, plot the \( \bar{x} \) values for the 20 samples on a chart. The y-axis should represent the \( \bar{x} \) values, and the x-axis should represent the sample numbers. Draw the center line at 12.3 and the control limits at 12.3729 (UCL) and 12.2271 (LCL).

04

Analyze the x-bar Charts

Identify any points in any dataset that fall outside of the control limits. A process is in control if all points lie between the control limits. A sudden shift in mean is indicated by a sharp set of points deviating upward or downward outside the control limits. A gradual drift is suggested by points moving progressively away from the center line towards and possibly beyond one control limit.

05

Identify the Control Status of Each Dataset

- For the dataset which remains between the control limits throughout, this indicates the process is in control. - A dataset showing an abrupt deviation from the center line and staying in a new zone (inside or beyond) after few samples indicates a sudden shift. Approximate the sample where it occurs based on the plot. - A dataset that slowly moves away from the center line often toward one limit suggests a gradual drift in the process mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Process Control

In the context of manufacturing, process control is crucial for ensuring product consistency and quality. It involves monitoring and controlling a process to maintain it at a desired state or behavior. By designing a reliable process control system, manufacturers can minimize variability and ensure high quality in their output.

Consider a pharmaceutical company producing tablets. The hardness of each tablet batch is measured. If this property fluctuates outside of acceptable bounds, it could affect the drug's efficacy and safety. By maintaining a process in control, it is ensured that the hardness remains within target limits, thus avoiding costly rework or product recalls.

• Process control involves predefined standards to gauge product attributes.
• It reduces process variation and assures quality consistency.
• Regular monitoring helps in taking timely corrective actions, ensuring continuous control.

Mean Shift

A mean shift in process control occurs when the average level of a process output changes. This shift can be either sudden or gradual and represents a significant source of variability in manufacturing processes.

Understanding mean shifts is essential because they often indicate underlying issues in the process. It could be due to machinery wear and tear, operator errors, or changes in material properties. Identifying and correcting a mean shift quickly is paramount to maintaining process quality.

• Sudden mean shift often appears as an abrupt spike or drop in measured values.
• Gradual mean shift manifests as a slow deviation over time.
• Recognizing shifts early helps maintain product standards and reduces defects.

X-bar Chart

The X-bar chart is a tool used in quality control to track data over time and determine if a process is in control. It specifically monitors the mean of samples from the process.

In an X-bar chart, each point on the graph represents the average of a subset of data (a sample), with sample means plotted on the vertical axis and the sample number on the horizontal axis. The chart includes a center line at the process target mean and control limits, calculated using specific parameters and constants, to provide visual guidance for assessment.

• The center line represents the target mean one wishes to maintain.
• Control limits help in distinguishing between normal variation and signals of process issues.
• Ensures immediate visibility into process variations that need intervention.

Statistical Process Control

Statistical Process Control (SPC) is a method using statistical techniques to monitor and control a process. The main goal is ensuring that a process operates efficiently, producing more specification-conforming products with less waste.

SPC employs tools like control charts to detect deviations from process stability. By looking at patterns and trends in the data, businesses are alerted to potential issues before they become problematic, allowing proactive responses to sustain control.

• SPC helps in identifying the onset of problems due to machine faults, operator errors, or materials issues.
• Encourages timely interventions to adapt processes and uphold quality.
• Enhances prediction of process behavior, aiding in strategic planning.