Control Charts for Attributes

p-chart: A chart used for controlling the proportion of defective services or products generated by the process.

pp = = pp(1 – (1 – pp)/)/nnWheren = sample sizep = central line on the chart, which can be either the historical average population proportion defective or a target value.

z = normal deviate (number of standard deviations from the average)

Control limits are: UCLp = p+zpp and LCLp = p−zp– –

Hometown BankHometown BankExampleExample

The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel.

Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The results for the past 12 weeks are shown in the following table.

Is the booking process out of statistical control? Use three-sigma control limits.

Sample Wrong ProportionNumber Account # Defective

1 15 0.006 2 12 0.0048 3 19 0.0076 4 2 0.0008 5 19 0.0076 6 4 0.0016 7 24 0.0096 8 7 0.0028 9 10 0.00410 17 0.006811 15 0.00612 3 0.0012

Total 147

Hometown BankHometown BankUsing a p-Chart to monitor a processUsing a p-Chart to monitor a process

n = 2500

p =147

12(2500)= 0.0049

pp = = pp(1 – (1 – pp)/)/nn

pp = = 0.00490.0049(1 – (1 – 0.00490.0049)/)/25002500

pp = 0.0014 = 0.0014

UCLp = 0.0049 + 3(0.0014)

= 0.0091 LCLp = 0.0049 – 3(0.0014) = 0.0007

Hometown BankHometown BankUsing a p-Chart to monitor a processUsing a p-Chart to monitor a process

Example

In class Problem

Control ChartsTwo types of error are possible with control

charts• A type I error occurs when a process is

thought to be out of control when in fact it is not

• A type II error occurs when a process is thought to be in control when it is actually out of statistical control

These errors can be controlled by the choice of control limits

Process Capability

• Process capability is the ability of the process to meet the design specifications for a service or product.

• Nominal value is a target for design specifications.

• Tolerance is an allowance above or below the nominal value.

2020 2525 3030

UpperUpperspecification specification

LowerLowerspecificationspecification

NominalNominalvalue value

Process Capability

Process is capable

Process distributionProcess distribution

Process is not capableProcess is not capable

2020 2525 3030

UpperUpperspecification specification

LowerLowerspecificationspecification

NominalNominalvalue value

Process distributionProcess distribution

Process Capability

Process capability ratio, Cp, is the tolerance width divided by 6 standard deviations (process variability).

Process Capability Ratio, Cp

CCpp = =Upper specification - Lower specificationUpper specification - Lower specification

66

Cpk = Minimum of Upper specification – x

3

x – Lower specification

3,

= =

Process Capability Index, Cpk, is an index that measures the potential for a process to generate defective outputs relative to either upper or lower specifications.

Process Capability Index, Cpk

We take the minimum of the two ratios because it gives the worst-case situation.

Intensive Care Lab Example

Upper specification = 30 minutesUpper specification = 30 minutesLower specification = 20 minutesLower specification = 20 minutesAverage service = 26.2 minutes Average service = 26.2 minutes = 1.35 minutes= 1.35 minutes

The intensive care unit lab process has an average turnaround time of 26.2 minutes and a standard deviation of 1.35 minutes.

The nominal value for this service is 25 minutes with an upper specification limit of 30 minutes and a lower specification limit of 20 minutes.

The administrator of the lab wants to have three-sigma performance for her lab. Is the lab process capable of this level of performance?

Cpk = Minimum of Upper specification – x

3x – Lower specification

3,

= =

Upper specification = 30 minutesLower specification = 20 minutesAverage service = 26.2 minutes = 1.35 minutes

Intensive Care Lab Assessing Process Capability

CCpkpk = = Minimum ofMinimum of 26.226.2 – 20.0 – 20.0

3(3(1.351.35)) ,,30.0 – 30.0 – 26.226.2

3(3(1.351.35))

CCpkpk = = Minimum of 1.53, 0.94Minimum of 1.53, 0.94 = 0.94= 0.94Process Capability Index

Example

Cp = =Upper specification - Lower specification

6

Cpp = =30 - 20

6(1.35)= 1.23 Process Capability Ratio

Before Process ModificationUpper specification = 30.0 minutes Lower specification = 20.0 minutes

Average service = 26.2 minutes = 1.35 minutes Cpk = 0.94 CCp p = 1.23

Intensive Care Lab Assessing Process Capability

Does not meet 3 (1.00 Cpkpk) target due to a shift in mean (Note variability is ok since Cp p is over 1.0)

Example

In Class Problem

In Class Problem

LowerLowerspecificationspecification

MeanMean

UpperUpperspecification specification

Nominal valueNominal valueSix sigmaSix sigma

Four sigmaFour sigma

Two sigmaTwo sigma

Effects of Reducing Variability on Process Capability

Range Population in range Expected frequency outside range

Approx. frequency for daily event

μ ± 1σ 0.682689492137 1 in 3 Twice a week

μ ± 2σ 0.954499736104 1 in 22 Every three weeks

μ ± 3σ 0.997300203937 1 in 370 Yearly

μ ± 4σ 0.999936657516 1 in 15,787 Every 43 years (twice in a lifetime)

μ ± 5σ 0.999999426697 1 in 1,744,278 Every 5,000 years (once in history)

μ ± 6σ 0.999999998027 1 in 506,842,372 Every 1.5 million years

What it means to operate at 6-sigma

Six Sigma

• 3.4 defects per million• Cpk = 2• Impact of number of parts or production steps

on yield:– 6 sigma 4 sigma 3 sigma

• 1 100% 99% 99%• 5 100% 97% 71%• 10 100% 94% 50%• 100 99.97% 54% 0%

Designing in 6-sigma

• Reduce the number of parts in a product• Reduce the number of steps in a process

Six Sigma

• Six Sigma is a comprehensive and flexible system for achieving, sustaining, and maximizing business success by minimizing defects and variability in processes.

• It relies heavily on the principles and tools of TQM.

• It is driven by a close understanding of customer needs; the disciplined use of facts, data, and statistical analysis; and diligent attention to managing, improving, and reinventing business processes.

Six Sigma Improvement Model

1. Define Determine the current process characteristics critical to customer satisfaction and identify any gaps.

2. Measure Quantify the work the process does that affects the gap.

3. Analyze Use data on measures to perform process analysis.

4. Improve Modify or redesign existing methods to meet the new performance objectives.

5. Control Monitor the process to make sure high performance levels are maintained.

Six Sigma Implementation

Top Down Commitment from corporate leaders. Measurement Systems to Track Progress Tough Goal Setting through benchmarking best-

in-class companies. Education: Employees must be trained in the

“whys” and “how-tos” of quality. Communication: Successes are as important to

understanding as failures. Customer Priorities: Never lose sight of the

customer’s priorities.

Six Sigma Education

• Green Belt: An employee who achieved the first level of training in a Six Sigma program and spends part of his or her time teaching and helping teams with their projects.

• Black Belt: An employee who reached the highest level of training in a Six Sigma program and spends all of his or her time teaching and leading teams involved in Six Sigma projects.

• Master Black Belt: Full-time teachers and mentors to several black belts.

International Quality International Quality Documentation StandardsDocumentation Standards

ISOISO90009000

A set of standards governing documentation of a quality program.

ISOISO1400014000

Documentation standards that require participating companies to keep track of their raw materials use and their generation, treatment, and disposal of hazardous wastes.

1. Category 1 ─ Leadership

120 points2. Category 2 ─ Strategic Planning

85 points3. Category 3 ─ Customer and Market Focus

85 points4. Category 4 ─ Measurement, Analysis, and

Knowledge Management

90 points5. Category 5 ─ Human Resource Focus

85 points6. Category 6 ─ Process Management

85 points7. Category 7 ─ Business Results

450 points

Malcolm Baldrige National Quality Award

Named after the late secretary of commerce, a strong proponent of enhancing quality as a means of reducing the trade deficit. The award promotes, recognizes, and publicizes quality strategies and achievements.