91Ó°ÊÓ

The following data were considered in quality Engineering ["Parabolic Control Limits for The Exponentially Weighted Moving Average Control Charts in Quality Engineering" (1992, Vol. 4(4)]. In a chemical plant, the data for one of the quality characteristics (viscosity) were obtained for each 12 -hour batch's at the batch completion. The results of 15 consecutive measurements are shown in the following table. $$\begin{array}{cccc}\hline \text { Batch } & \text { Viscosity } & \text { Batch } & \text { Viscosity } \\\\\hline 1 & 13.3 & 9 & 14.6 \\\2 & 14.5 & 10 & 14.1 \\\3 & 15.3 & 11 & 14.3 \\\4 & 15.3 & 12 & 16.1 \\\5 & 14.3 & 13 & 13.1 \\\6 & 14.8 & 14 & 15.5 \\\7 & 15.2 & 15 & 12.6 \\\8 & 14.9 & & \\\\\hline\end{array}$$ (a) Set up a CUSUM control chart for this process. Assume that the desired process target is \(14.1 .\) Does the process appear to be in control? (b) Suppose that the next five observations are 14.6,15.3,15.7 , \(16.1,\) and \(16.8 .\) Apply the CUSUM in part (a) to these new observations. Is there any evidence that the process has shifted out of control?

Step by step solution

01

Understand the CUSUM Chart

A CUSUM (Cumulative Sum) chart is used to monitor small shifts in the process mean. For a CUSUM chart, we calculate cumulative sums of the deviations of sample values from a target value, which is 14.1 in this case. The chart consists of a Cumulative Positive Sums (C+) and Cumulative Negative Sums (C-) to track deviations.

02

Calculate Initial CUSUM Values

Given the target value of 14.1, we will calculate initial CUSUM values using the given data. The formulas for CUSUM are:\[ C^+_i = \max(0, x_i - (\text{Target} + k) + C^+_{i-1}) \]\[ C^-_i = \max(0, (\text{Target} - k) - x_i + C^-_{i-1}) \]Here, \(x_i\) is the observed viscosity, and \(k\) is a constant representing the allowable deviation (commonly set to half the target shift we detect). Assuming \(k = 0.5\), we begin with \(C^+_0 = 0\) and \(C^-_0 = 0\).

03

Apply CUSUM to Initial Measurements

Using the batch data, calculate \(C^+_i\) and \(C^-_i\) for each of the 15 batches:- Batch 1: \(x_1 = 13.3\)- \(C^+_1 = \max(0, 13.3 - 14.6 + 0) = 0\)- \(C^-_1 = \max(0, 13.6 - 13.3 + 0) = 0.3\)Continue computing these values for all 15 batches.

04

Evaluate In-Control Status

For a process to be considered in control based on CUSUM values, neither cumulative sum should exceed a threshold \(h\), commonly set according to the desired control level (such as 5 or 10). After calculating all batch CUSUMs, check if any values exceed \(h\). If no values exceed, the process is currently in control.

05

Analyze New Observations

Add the new observations: 14.6, 15.3, 15.7, 16.1, and 16.8 to the analysis. Calculate the updated \(C^+_i\) and \(C^-_i\) using the last cumulative values from the initial observations:- For each new observation, continue the same calculation method.- Monitor \(C^+_i\) and \(C^-_i\) for each new observation to see if the threshold \(h\) is exceeded.

06

Determine Shift in Process Control

If during Step 5 any CUSUM values exceed the control limit \(h\), then there is evidence that the process has shifted out of control. If they remain below \(h\), the process continues to be in control even with new observations.

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Key Concepts

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

Quality Control

Quality Control plays a vital role in ensuring that products and processes meet specific standards and requirements. It involves a systematic method of checking whether products and processes are aligned with the desired level of quality. This is crucial in industries like chemical manufacturing, where precision in quality characteristics such as viscosity can greatly affect the final product's performance.

Quality Control uses a variety of tools to monitor and control the quality of products and processes. Some common tools include control charts, which help in visualizing whether a process is stable or if corrective actions are needed.

• Ensures that the product meets customer expectations.
• Reduces waste and rework by catching defects early.
• Involves continuous monitoring and refining of processes.

Implementing a robust Quality Control system leads to efficiency and customer satisfaction by delivering high-quality products consistently.

Process Monitoring

Process Monitoring is a continuous activity to check and improve the quality characteristics of manufacturing processes. It involves collecting and analyzing data to detect any deviation from the expected performance. This helps in identifying and addressing issues before they escalate and impact the final product.

Monitoring is critical in environments where even minor changes can significantly affect product quality, such as in chemical plants monitoring characteristics like viscosity.

• Involves using control charts, like CUSUM, to detect changes.
• Aims to maintain process consistency and reliability.
• Provides data feedback loops for real-time decision-making.

By regularly monitoring processes, manufacturers can ensure smooth operations and maintain the required quality standards over time.

Statistical Process Control

Statistical Process Control (SPC) is a scientific and data-driven method used to monitor and control a process. It involves using statistical methods, such as control charts, to analyze the process behavior and variability. This approach helps in determining whether a process is stable and capable of producing quality outputs.

SPC tools are essential for detecting trends and shifts that could lead to process inefficiencies or defects. Employing SPC in quality management helps provide insights into variations and their sources.

• Utilizes mathematical and statistical methods for analysis.
• Focuses on maintaining process stability and improving processes.
• Helps in predicting process performance over time.

By integrating SPC with other quality management practices, organizations can achieve and sustain high levels of process quality and efficiency.

Cumulative Sum Methodology

The Cumulative Sum (CUSUM) Methodology is a type of control chart used for monitoring process deviations by calculating cumulative sums of the deviations from a specified target. It's especially effective in detecting small shifts in a process mean, proving advantageous in situations where subtle changes can indicate significant issues.

The CUSUM chart differs from conventional control charts by focusing on cumulatively adding the differences between the observed data and the target value at each point in the process.

Key features of the CUSUM chart include:

• Tracks cumulative positive and negative deviations separately.
• More sensitive to small shifts in the process mean compared to simple average charts.
• Utilizes h-parameter for decision-making about process control status.

This methodology allows for quicker detection of a drift in the process mean, enabling faster interventions and maintaining control over the process quality.