25 Statistical Process Control Methods: Control Chart for Variables

Kajal Kiran

 

25.1 OBJECTIVES

 

This chapter will enable the students to understand:

  • The concept of control chart
  • The situations under which control chart for variables can be used
  • The working of Charts, R Charts and σ charts
  • Limitations associated to control charts for variables

 

25.2 INTRODUCTION

 

A control chart is a statistical technique to control the quality of a product being manufactured. A control chart also known as process chart or quality control chart is the graphic presentation which depicts whether sample quality falls within the normal range of variation. The normal range of variation is described by the use of control chart limits. It was first devised by Dr. Walter A. Shewart in 1924 after whose name these charts are also called Shewart charts.

 

A control chart has upper and lower control limits within which deviations are allowed from the mean value. The process is said to be out of control if the data plotted reveals that one or more sample fall beyond the control limits. Thus the two lines on the control chart indicate the tolerance limits within which the variation of quality will be allowed.

 

The upper and the lower control limits separate common cause of variation from assignable causes of variation. If the dots plotted fall outside the tolerance limits (UCL and LCL), the process is considered to be out of control and the sample is rejected. It warrants the determination of assignable causes and immediate actions are initiated to improve the quality. The various reasons responsible for making production process out of control may be poor quality of material, negligence of machine operator or defective machine etc.

 

There are mainly two types of control charts

  • Control charts for Variables and
  • Control charts for Attributes

 

25.3 CONTROL CHARTS FOR VARIABLES

 

The quality characteristic of a product which can be measured and expressed in specific units is called a variable e.g. height, weight, density and diameter. Thus the control charts which are based on measureable quality characteristics are called control charts for variables.

 

Control charts for variables are of three types:

  • Mean or  Chart
  • Range or R chart
  • Standard deviation or σ chart.

 

CONTROL CHART

 

Control charts for Range – R chart

 

R charts are used to control the variability in process. R charts are normally presented along with chart. Control chart for range requires determination of control limits for the value of Range. If the sample data falls beyond the control limits, the process variability can be considered out of control.

 

The control limits for R-chart are

 

Upper Control Limit for Range = D₄R

 

Lower Control Limit for Range  = D₃R

 

Here, D₃ and D₄ are constants which provide standard deviation limits for a given sample size. The standard statistical and quality control tables can be used to obtain the value of D₃ and D₄ corresponding to a sample size.

 

Example 3

 

Draw R chart from the given information

Sample No (Sample Size 7) 1 2 3 4 5 6 7 8 9 10
R 2 3 4 3 2 2 3 4 2 3

 

Sol:R-   = ∑R / N     = 28/10  = 2.8

 

UCL for Range =D₄R

 

Value of D₄ for sample size 7 (as per statistical and quality control chart) = 1.92

 

= 1.92 (2.8) = 5.376

 

LCL for Range = D₃ R

 

Value of D₃ for sample size 7 (as per statistical and quality control chart) = 0.08

 

= 0.08 (2.8) = 0.224

 

There control chart can be mapped out as under.

 

CONTROL CHART

 

 

σ charts or Control charts for the standard deviation

 

Standard deviation is considered as an ideal measure of dispersion. So a combination of control chart for mean and control chart for standard deviation is more appropriate than control chart for mean and control chart for range.

 

Two limits for σ chart are :

 

Upper Control Limit for Range = B₄ S

Lower Control Limit for Range = B₃ S

Here S = Sum of Sample Standard Deviation

Number of Samples

 

B₃ and B₄ = Constants for different sample sizes whose values can be obtained from standard statistical and quality control tables

 

Example 4

 

If sum of sample standard deviation of 18 samples of 10 items each is 8.28. Determine the control limits for standard deviation if B₃ and B₄ for sample size 10 is 0.28 and 1.72 respectively.

 

Sol:-  S   = ∑σ/ N   =  8.28 / 18  = 0.46

UCL   = B₄ S

= (1.72) (0.46)

= 0.7912     or   0.79 (approx)

LCL  = B₃ S

= (0.28) (0.46)

= 0.1288     or   0.13 (approx)

 

25.4 CASE STUDY OF AN INFORMATION TECHNOLOGY COMPANY

 

IT services is a competitive field populated with companies that all deliver important online and call centre support to a variety of customers. Most IT services businesses come to realize that their clients have choices and, within the same pricing range, they gravitate to the support organization where the service is best. In this case study of an IT services business, benchmarking helped quantify what the business already knew – its competitive position was not totally secure. There are a number of ways the company might have responded to the challenge. While the company had built up a reasonable capability in Six Sigma, its management realized improvement was not as simple as forming a project team and turning them loose on the problem. Senior managers had learned that an important part of their responsibility as leaders is to find the issues that are well- enough defined and of a scope to be suitable for a Six Sigma DMAIC project team to take on.

 

The company could see a strong indication that a DMAIC (Define, Measure, Analyze, Improve, Control) project to reduce support costs should be quite doable – and should return significant dollars to the bottom line.

 

Management also could see that the DMAIC team should look for the improved customer experience connected with reduced wait times and service times to improve new account growth – bringing dollars to the top line.

 

The company assigned a Champion from its leadership team to take responsibility for the new project and identify a team leader and key team members. The team was given its top level goals and scope – to reduce support costs while improving new account growth. The work with the benchmark data was helpful in orienting the team to the project rationale. The team began working on their project charter.

 

The Define Phase

 

The senior leadership of the IT services company completed the important pre-project work and found an area of the business worthy of attention by a DMAIC (Define, Measure, Analyze, Improve, Control) project team. The team then began work on understanding and articulating the project goals, scope and business case.

 

The DMAIC roadmap called for work in these areas during the Define phase:

 

D1. Project Charter: Spelling out the project’s goal statement.

 

D2. Customer Requirements: Identifying all the internal and external customers who depend on the outputs of the process under study, the deliverables and measures connected with those outputs, and the process steps, process inputs and (as appropriate) the suppliers of those inputs.

 

D3. High Level Process Map: Showing the flow of information, materials and resources, from key process inputs, through process steps and decision points, to create the process outputs. The map describes the flow of what happens within the scope of the target process and it defines the boundaries of that scope.

 

The Measure Phase

 

Having developed a good understanding of the project’s business case and customer requirements (identifying the Y‘s), and the as-is process, the Six Sigma project team of the IT services business began to focus on the Measure phase. The team identified the measures and data collection plan for gathering the right amount of the right data to impel their learning about root causes and drivers that impact the project Y‘s. The DMAIC (Define, Measure, Analyze, Improve, Control) roadmap called for work in these areas during the Measure phase:

 

M1. Refine the Project Y‘s: Getting even clearer about how the project’s key outcome measure(s) will be defined, measured and reported.

 

M2. Define Performance Standards for the Y‘s: Identifying how performance will be measured – usually somewhere on the continuum from capability measures like Cp and Cpk for “variables” data that is normally distributed to percentile or other capability metrics for “attribute” and other data that may be skewed in distribution.

 

M3. Identify Segmentation Factors for Data Collection Plan: Starting with the natural segmentation of project Y‘s and moving though consideration of prospective driving factors (X‘s), segmentation suggests the packets of data that should be collected in order compare and contrast segments to shed light on Y root causes and drivers.

 

M4. Apply Measurement Systems Analysis (MSA): In any project, raw data is gathered and then converted into measures. That process comprises a “measurement system” that should be characterized and strengthened in terms of its accuracy and repeatability.

 

M5. Collect the Data: Gathering data, preserving its meaning and noting any departures from the discipline put in place under MSA.

 

M6. Describe and Display Variation in Current Performance: Taking an initial look at the data for its distribution, extreme values and patterns that suggest special variation.

 

The Analyze Phase

 

Having refined the project’s key outcome measures, defined performance standards for project Y‘s, identified segmentation factors and defined measurement systems, the Six Sigma project team of the IT services business began to focus on the Analyze phase. The DMAIC (Define, Measure, Analyze, Improve, Control) roadmap called for work in these areas during the Analyze phase:

 

A1. Measure Process Capability: Before segmenting the data and “peeling the onion” to look for root causes and drivers, the current performance is compared to standards (established in step M2 of the Measure phase).

 

A2. Refine Improvement Goals: If the capability assessment shows a significant departure from expectations, some adjustment to the project goals may need to be considered. Any such changes will, of course, be made cautiously, supported with further data, and under full review with the project Champion and sponsors.

 

A3. Identify Significant Data Segments and Patterns: By segmenting the Y data based on the factors (X‘s) identified during the Measure phase – the team looks for patterns that shed light on what may be causing or driving the observed Y variation.

 

A4. Identify Possible X’s: Asking why the patterns seen in A3 are as observed highlights some factors as likely drivers.

 

A5. Identify and Verify the Critical X’s: To sort out the real drivers from the “likely suspects” list built in A4, there is generally a shift from graphical analysis to statistical analysis.

 

A6. Refine the Financial Benefit Forecast: Given the “short list” of the real driving x’s, the financial model forecasting “how much improvement?” may need to be adjusted.

 

The Improve Phase

 

Having identified and verified ways that support cost is driven by staffing ratios, process factors like transfers and call backs, and the proportion of phone and web traffic, the Six Sigma project team of the IT services business began identifying and selecting among solutions. It had entered the Improve phase. The DMAIC (Define, Measure, Analyze, Improve, Control) roadmap called for work in these areas during the Improve phase:

 

I1. Identify  Solution  Alternatives  to  Address  Critical  X’s: Consider  solution  alternatives  from the  possibilities identified earlier and decide which ones are worth pursuing further.

 

I2. Verify the Relationships Between X’s and Y‘s: What are the dynamics connecting the process X‘s (inputs, KPIVs) with the critical outputs (CTQs, KPOVs)?

 

I3. Select and Tune the Solution: Using predicted performance and net value, decide what is the best solution alternative.

 

I4. Pilot / Implement Solution: If possible, pilot the solution to demonstrate results and to verify no unintended side effects.

 

The Control Phase

 

The Six Sigma project team reached the final step in making significant improvements to the operation and profitability of the call center of the IT services business. After the company’s senior leadership did the pre- project work, the team followed the first four steps of the DMAIC (Define, Measure, Analyze, Improve, Control) methodology and entered the final phase. The DMAIC roadmap called for work in these areas during the Control phase:

 

C1. Develop Control Plan: Include both management control dashboards that focus on Y(s) and operational control indicators that monitor the most significant process variables, focusing on the x’s. The Control plans addressed two views – one concerned with management control and the other with operational control. Management control includes a focus on the Y‘s or outcomes of the process and often some of the X‘s as well. The level of detail was decided upon based on the interests of the specific managers concerned – some want a lot of detail, some much less. Hence, the management control plan needed to consider individual preferences so as to deliver enough – but not too much – information.

 

The operational control plan was more concerned with the X‘s that are predictive of outcome Y’s. Operational control information included both controllable and “noise” variables. Operational control information was provided more frequently than management control information.

 

C2. Determine Improved Process Capability: Use the same measures from Define and Measure in order to provide comparability and monitor impact in a consistent way. The team linked the capability of the improved process to the baselines and targets identified during Define and Measure. It was important to use the same measures. (If it was necessary to change the measures, then baselines and targets would have had to been restated in those terms to enable comparison.) Many different statements of capability were considered, including mean/median, variance, Cp, Cpk, DPMO, sigma level, percentile rank, etc. The team knew that to a great extent these alternate characterizations are equivalent and the choice is largely one of preference. However, the team made its choices so that all concerned could have a common understanding of the meaning of the measure.

 

C3. Implement Process Control: Create, modify and use data collection systems and output reports or dashboards consistent with the control plan. The team began by planning the data collection process to be used, including preparing operational definitions for each data element and automated tools whenever possible to minimize expense and effort. Heeding W. Edward Deming’s message to “drive out fear,” the team was careful to prepare a well-thought-out communication plan to ensure the staff knew how the data was to be used and to address any concerns about punitive uses of the data. The team recognized that if members of the staff thought the data would be misused, they might be tempted to distort the data.

 

The team also verified that the process was under procedural control (i.e., standards and documentation were up-to-date and the staff understood and followed the intended process). In preparation for implementing control charts on some of the process variables, the team recognized the segmented some of the data, such as “issue type.” Significant variations were expected across, but not within, issue types (e.g.,  “problem” versus “question”).

 

The team selected the appropriate form of control chart to suit each situation to be monitored.

 

C4. Close Project: The team’s final effort was aimed at wrapping up the project and transferring control to the call centre group. This last step included:

  • Developing and executing a plan to implement the improved process, including any necessary training.
  • Developing and executing a communication plan that informed all those affected by the change.
  • Conducting a transition review with key managers and staff, making adjustments and improvements they suggested.
  • Establishing the timeline and responsibilities for the transfer, and executing the transition process.
  • After an agreed interval, validating the financial benefits in conjunction with a representative of the finance department.
  • Conducting a project post-mortem from multiple perspectives – the team, the Champion/sponsor, and the financial results. (Emphasis on process improvement, not critiques of individual performance.)

 

25.5  LIMITATIONS OF CONTROL CHARTS FOR VARIABLES

  • Control charts for variables can be used only for those quality characteristics which can be expressed numerically.
  • These charts use only one quality characteristic at a time.
  • It is a costly affair to use  and R chart.
  • These charts are used for critical characteristics as the cost involved is quite high.

 

25.6 SUMMARY

 

The control charts for variables are an important tool to monitor the quality characteristic of goods and services which can be expressed in numerical terms. Control charts for mean help to determine whether sample mean falls within the tolerance limits of mean  or not. Similarly control chart for range  and  control chart  for standard deviation enable fixation of tolerance limits for range and standard deviation respectively. If the sample data plotted is within the tolerance limits, the process is said to be under control.

 

25.7 GLOSSARY

  • Variable: Quality Characteristic of goods or services which can be measured in quantitative terms like height, weight, diameter etc.
  • Sample Mean:- The average of values of items in a given sample.
  • Sample Range :- The difference between the highest and lowest value in a sample.
  • Sample Std Deviation :- A measure of extent of variation among different values of a sample.

 

25.8 REFERENCES/ SUGGESTED READINGS

  • Chary, Production and Operations management, McGraw-Hill
  • Selvedy G. , Handbook of Industrial Engineering, Wiley Inter Science, N.Y., 1982
  • Nair N G, Production and Operation Management, Tata McGraw- Hill Publishing Company Limited, New Delhi
  • Robert Fetter B. , Quality Control Systems, Richard D. Irwin, Illinois, USA