Chapter 6: Statistical Quality Control

Operations Management: Chapter 6 - Statistical Quality Control

Learning Objectives

  • LO 1-5:

    • Describe categories of statistical quality control (SQC).

    • Identify and describe causes of variation.

    • Explain the use of descriptive statistics in measuring quality characteristics.

    • Describe the use of control charts.

    • Identify the differences between x-bar and R-charts.

  • LO 6-11:

    • Identify the differences between p- and c-charts.

    • Explain the meaning of process capability and the process capability index.

    • Explain the concept of Six Sigma.

    • Explain the process of acceptance sampling and describe the use of operating characteristic (OC) curves.

    • Identify decisions that managers must make when implementing SPC.

    • Describe the challenges inherent in measuring quality in service organizations.

Describing Categories of Statistical Quality Control (SQC)

What Is SQC?

  • Statistical Quality Control (SQC): A term used to describe tools utilized by quality professionals to evaluate organizational quality.

Three Categories of SQC

  • Descriptive Statistics: Used to describe quality characteristics and relationships, including:

    • Mean

    • Standard Deviation

    • Range

    • Measures of distribution

  • Statistical Process Control (SPC): Involves inspecting random samples of output from a process to determine if they fall within acceptable limits, hence assessing if the process is functioning properly.

  • Acceptance Sampling: Involves randomly inspecting a batch of goods to determine whether to accept or reject the lot.

Identifying and Describing Causes of Variation

Sources of Variation

  • Common Causes:

    • Also known as random causes.

    • They are random and unavoidable, resulting from slight differences in processing.

  • Assignable Causes:

    • Causes that can be identified and corrected.

    • Examples include:

    • Poor employee training

    • Worn tools

    • Machines needing repair

Explaining the Use of Descriptive Statistics in Measuring Quality Characteristics

Descriptive Statistics Defined

  • Mean: Measure of central tendency.

  • Range: The difference between the largest and smallest observations in a dataset.

  • Standard Deviation: Measures the dispersion of data around the mean.

  • Distribution of Data Shape: Can be characterized as normal (bell-shaped) or skewed.

Detailed Definitions

  • Mean Formula:
    xˉ=∑<em>i=1nx</em>in\bar{x} = \frac{\sum<em>{i=1}^{n} x</em>i}{n}

    • Where:

    • xˉ\bar{x} = mean

    • xix_i = observation i

    • n = number of observations

  • Standard Deviation Formula:
    σ=∑<em>i=1n(x</em>i−xˉ)2n\sigma = \sqrt{\frac{\sum<em>{i=1}^{n} (x</em>i - \bar{x})^2}{n}}

    • Where:

    • σ\sigma = standard deviation

    • xix_i = observation i

    • xˉ\bar{x} = sample mean

    • n = number of observations in the sample

  • Distribution Types:

    • Normal Distribution: FIGURE 6.1 illustrates variations in standard deviation.

    • Skewed Distribution: FIGURE 6.2 shows differences between symmetric and skewed distributions.

Describing the Use of Control Charts

Overview of Control Charts

  • Control Charts (Process Charts): Graphs that show sample data plotted with a center line (CL), upper control limit (UCL), and lower control limit (LCL).

    • Control Chart for Variables: Used to monitor measurable characteristics (e.g., length, weight).

    • Control Chart for Attributes: Used for characteristics with discrete values (e.g., percent defective).

    • Example: Quality control chart for Cocoa Fizz FIGURE 6.3.

Setting Control Limits

  • Type I error is the chance of concluding the process is out of control when it is not, set at ±3σ\pm 3\sigma.

  • Distribution of Values: Different ranges of standard deviation impacts coverage (see FIGURE 6.5).

Types of Control Charts

  • Variable Control Charts:

    • x-bar charts

    • R-charts

  • Attribute Control Charts:

    • p-charts (for measuring proportions)

    • c-charts (for measuring counts of defects)

Identifying the Differences Between X-Bar and R-Charts

Uses of X-Bar and R-Charts

  • X-Bar Charts:

    • Monitor changes in mean value of a process.

  • R-Charts:

    • Monitor dispersion or variability of a process.

  • System Performance: It’s possible to display acceptable central tendencies but risky variability and vice-versa.

  • Together Usage: Utilizing both charts show different problems present in the processes.

Constructing X-Bar and R-Charts

  • Example: A quality control inspector uses historical data of 25 samples with 4 observations each to develop control limits based on standard deviation calculations.

  • Center Line Formula: The average of the sample means derived from historical data.

Differences Between P-Charts and C-Charts

Control Charts for Attributes

  • P-Charts:

    • Used for quality characteristics that are binary (yes/no).

    • Measure the proportion of items in a sample that are defective (e.g., broken cookies).

  • C-Charts:

    • Used for actual counts of defects, accommodating the possibility of multiple defects per unit (e.g., number of complaints).

Example Analysis

  • P-Chart Example: Utilized in tire production to assess defective tires in random samples.

  • C-Chart Example: Monitored weekly customer complaints in a hotel to compute control limits based on complaints recorded.

Explaining Process Capability and the Process Capability Index

Process Capability

  • Definition: The ability of a production process to meet or exceed preset specifications.

  • Product Specifications: Also known as tolerances, indicating acceptable quality ranges (e.g., bottle fill specifications).

Measuring Process Capability

  • Cp and Cpk:

    • Cp: Assumes process centering relative to specifications.

    • Cpk: Adapts to evaluate centering deviations.

  • Cp Interpretation:

    • Cp=1Cp = 1 indicates process variability meets specifications.

    • Cp<1Cp < 1 indicates process cannot meet specifications.

    • Cp>1Cp > 1 indicates process exceeds minimum specifications.

Computing Examples

  • Example at Cocoa Fizz evaluating bottling machines. Summary statistics indicate variability near specifications and capability indices to assess machine suitability.

Explaining the Concept of Six Sigma

Six Sigma Overview

  • Definition: Concept developed by Motorola in the 1980s emphasizing high quality, roughly equating to 3.4 defective parts per million.

  • Implementation Across Functions: Engages design, operations, marketing, finance to improve product characteristics and process control uniformly.

  • Figure Representation: FIGURE 6.10 shows PPM defective relationships for quality.

Six Sigma Approach

  • DMAIC Framework: Describes the structured approach (Define, Measure, Analyze, Improve, Control).

Explaining the Process of Acceptance Sampling and Operating Characteristic (OC) Curves

Acceptance Sampling

  • Definition: Random inspection of a selected number of items from a lot to accept or reject an entire batch.

  • Differences with SPC: Acceptance sampling performed before/after processes; it can be more resource-efficient in certain contexts.

Acceptance Sampling Plans

  • Criteria for Acceptance/ Rejection:

    • Lot size (N).

    • Sample size (n).

    • Defect acceptance threshold (c).

  • Sampling Types: Includes single, double, multiple-sampling plans based on costs and defect tolerances.

Operating Characteristic (OC) Curves

  • OC Curves Explanation: Graphs showing the probability to accept a batch based on defective proportions.

    • The x-axis reflects defective percentage; the y-axis reflects acceptance probability.

  • Example Scenario: Utilizing OC curves to statistically reason acceptance probabilities given defect thresholds.

Summary of AQL, LTPD, Consumer’s and Producer’s Risk

  • Acceptable Quality Level (AQL): The small percentage of defects consumers are willing to accept.

  • Lot Tolerance Percent Defective (LTPD): The maximum defect percentage consumers tolerate.

  • Consumer's Risk (β): Probability of accepting a lot above LTPD - related to Type II error.

  • Producer's Risk (α): Probability of rejecting an acceptable lot - related to Type I error.

Developing OC Curves

  • Graphical Depiction: OC curves represent a sampling plan's discriminating power.

  • Example for Curve Construction: Establishing rules for accepting or rejecting lots based on defect counts derived from cumulative probability tables.

Managerial Decisions in Implementing SPC

Implications for Managers

  • Inspection Frequency: Decisions based on product cost, volume, process stability, and lot size considerations.

  • Location of Inspection: Inbound materials, finished products and pre-processing evaluations.

  • Selection of Tools: Control charts recommended for in-process production, while acceptance sampling is better for attributes during inbound/outbound evaluations.

Challenges of Measuring Quality in Service Organizations

Statistical Quality Control in Services

  • Overview: Service organizations lag in statistical quality control application due to intangible product characteristics.

  • Quality Measurement Difficulties: Quality perceptions are subjective; quantifiable measures are key for effective service assessment (e.g., check-in times, complaint counts).

Organizational Influence

  • Organizational Impacts of SQC: SQC involves contributions from marketing, finance (valuing SQC efforts), HR (training and skill requirements), and information systems (data accessibility).

Highlights of the Chapter (LO 1–11)

  • SQC represents a mix of descriptive statistics, acceptance sampling, and SPC. SQC categorizes variations as common versus assignable causes utilizing control charts for monitoring.

  • P-charts and c-charts differentiate between defect proportions and counts. Process capability is quantified by Cp and Cpk indices, guiding the process and product protocols. frameworks like Six Sigma embody a comprehensive quality approach spanning various business functions, echoing the critical importance of effective evaluation and measurement methodologies across manufacturing & service landscapes.