Quality Control Techniques Chandran Udumbasseri, Technical consultant

Introduction:

Quality is an essential requirement for a product. The objective of quality is to provide a

definite quality assurance to a product while maintaining the cost of production at

optimum. Customer expects a product having quality standards and this is also the

objective of the product manufacturing company. The building of the quality starts from

designing the product, purchase of the raw materials, product manufacturing, stages

through which the product reaches the consumer, his reactions to the quality of the

product and use and actions taken by the company to satisfy the customer. So quality

embraces all characteristics at all stages of manufacturing

Meaning of Quality

Two distinct meanings are given to quality and they are quality of design and quality of

conformance.

Quality of design refers to product construction method, processing, materials for

construction, safety, aesthetic factors, etc. So same end use products can be from

different producers with their own specifications.

Quality of conformance ensures that the product has the ability to perform the specified

quality of design. The quality control techniques are used to assure the quality of

conformance.

Quality characteristics

1. Directly measurable: weights, thickness, shear strength, etc.

2. Non-measurable: cracks, dents, breakages, etc.

3. Counting defects: surface defects, packing defects, assembly defects, etc.

Control

When a specification is designed for a product the next step is to produce according to the

specification. This process involves men, machines and materials. So at every stage

Design

Test Require

ments

Manufac

turing Producer

Customer

conformance to specified quality should be strictly followed and the processes monitored

to check conformance. Every repeated manufacturing process is not truly identical and

there is an inherent variability in each process. So it is necessary to determine and include

in the specification the extent to which this variability is tolerable so the fitness for use of

the product is not affected. If the variation is inevitable, then the question is to how much

of the variation should be allowed and how can the variation be determined.

A manufacturing process consists of a number of steps and the pattern of variation is a

complex one. If the variation is caused without any change in the process, then this

variation is the inherent variation. In addition there are variations due to the changes

caused due to factors like, men, machines, material, environment, etc, called external

variability. So the variation observed in the process is due to two types of variability –

inherent and external variability.

The external variability has assignable causes that can be identified and eliminated. When

assignable causes are identified and eliminated then the processes is said to have brought

under control. Conformance to quality characteristics is used as the method to bring a

process under control. These characteristics are given upper and lower limiting values so

that the process can be judged for its status being under control.

Objectives of quality control

1. Evaluation of quality standards of incoming material, product under processing

and outgoing product.

2. Judging the conformity of the process to the established standards and taking

suitable action when deviations are noted.

3. Estimation of optimum quality obtainable under the given conditions.

4. To improve quality and productivity by process control and experimentation.

5. Developing procedures for good vender – vendee relation.

6. Developing quality consciousness, both within and without the organization.

Concepts of quality control

There are three concepts of quality control – inspection, statistical quality control, and

reliability.

Inspection aims at assuring the workmanship demanded by the specification of the

product. The process involves detecting and isolating faulty material. The process of

inspection is carried on incoming material, in process and outgoing product.

Statistical quality control concept is based on understanding the behavior of a process and

exercising control over a process. SQC helps in developing an acceptable process model,

determining all deviations from this model, identifying and eliminating these deviations

so that the process continues to produce acceptable quality.

Reliability is defined as the probability of performing without failure a specified function

under given conditions for a specified period of time. Reliability program has two aspects

– planning and control. The planning establishes the standards of performance and

control measures the performance and evaluates performance against standards and takes

corrective actions if deviates from standards. So reliability is an extension of SQC

concepts.

Principles of quality control

a. Principle of co-ordination

Quality control function coordinates all the activities of various departments of the

organization and customer requirements. The old concept of production cycle has

three functions: design, manufacturing and inspection. These are considered as

independent departments by old concept. But the new process model explains that

they are inter-dependent and are cyclic process.

b. Principle of prevention

The routine inspection procedure in a manufacturing unit just identifies the good and

the bad items. But the new process model approach is to eliminate the production of

defective items by taking appropriate corrective action when the process deviates

from standard process. Here SQC techniques are used to evaluate the deviations from

standards by continuous monitoring the process and taking corrective and preventive

actions.

c. Important ideas of quality control

1. Build quality into the product rather than inspecting

2. Collect information to coordinate the activities of various departments.

3. Collect data to study cause-effect relationship that can help in solving

problems.

4. Use the data and information effectively.

Areas of application

Quality control concept used for incoming material control, process control, and

production control.

1. Incoming material control.

From quality control point of view, procuring the right quality of materials and

stocking at economic levels assumes great importance. Materials purchased from

outsiders and sub contractors are incoming materials. The materials coming from

another department are also incoming materials and require quality control.

2. Process control.

A manufacturing process may have several steps. And it is necessary to assure the

quality at each stage. This helps in building the desired quality into the finished

product. So process control and techniques ensure quality economically. To achieve

process control the following steps are involved.

Planning: It is necessary to determine quality characteristics in order to practice

control, procedures should be established to determine these characteristics and

proper equipment to measure the characteristics should be available. Criteria should

be developed to judge the process control.

Analysis: The information collected should be analyzed.

Process engineering: The analysis of information reveals any poor quality problem

and a corrective action plan should be established. The process behavior pattern

obtained from information analysis should be associated with technical and operating

conditions of the process.

Process capability: It is necessary to determine the best process under given

conditions of manufacturing.

Process control involves evaluating the quality of out going product (finished product

in store and warehouse), collecting information and data from distribution (customer

reaction), improvement of product quality through constant feedback analysis.

Quality costs

The components of quality costs are preventive, appraisal and failure costs.

Preventive costs: To have quality in first time and every time it is necessary that the

process should not make defective product. The costs associated to achieve such stage

include costs such as quality control engineering, employee’s quality training and quality

maintenance of patterns and tools.

Appraisal costs: The quality appraisal methods such as inspection, tests, laboratory

acceptance examinations and audits are expenses to maintain the company quality level.

Failure costs: The products that are not conforming to company quality specifications are

rejected as defective products. These products incur losses due to scrap, spoilage, rework,

field complaints, etc.

Basic aspects of QC program

Engineering: The development of a new product is basically engineering. The

development of quality evaluation through improved inspection is also engineering.

Identifying the basic cause of defects and their rectification is through engineering.

Statistical: The behavior of a process can be studied through statistical techniques. The

idea of prevention and control and the collection of data for improvement is statistical.

Managerial: The planning to execute the statistical and engineering information is

management concept. The creation of right atmosphere for quality consciousness in the

organization depends on the policies and practices of the management. The effective

coordination of the quality control function with others is management concept.

Benefits of QC program

The quality control program helps in building information system that can help to

improve quality and reduce the cost. A better understanding of the product is possible

through the application of process behavior, cause-effect study and experimentation

involved in QC program.

Other benefits of QC program are:

1. Quality improvement.

2. Optimizing the production condition.

3. Cost reduction

4. Scrap reduction.

5. Material saving.

6. Avoid bottlenecks.

7. Scientific tolerance evaluation.

8. Reduction in inspection.

9. Maintenance of operating efficiency.

10. Quality consciousness.

11. Fewer customer complaints.

Inspection When a work is accomplished it is customary to inspect and confirm that the specified

requirements are met. In modern industry when mass production started the inspection

methods needed a complete revolution and new techniques were introduced in inspection

of finished products.

In modern production techniques, the production of a complete product is divided into its

component parts separately; each part is manufactured separately at the same site or

outside. Since parts are manufactured at different places during assembly all the parts

should fit exactly with the allowed limits.

It is not possible to manufacture a product exactly all the time due to inherent and random

sources of errors in machines and men. But the variations observed should be controlled

to certain limits and the range of allowed limits should not affect the ultimate

performance of the product. So the inspection assures that the variations in characteristics

are within allowed limits and will not hamper its ultimate fitness for use.

The aim of inspecting is to see whether the product meets the workmanship demanded by

the service specification.

Objectives of inspection

The basic objectives of inspection are given below:

1. The inspection detects and isolates faulty material before it enters the process and

thus avoids wastage.

2. The inspection brings to the attention of persons concerned the defect before the

fault becomes serious.

3. The inspection ensures that no substandard or defective product reaches the

customer.

4. The inspection promotes organizational commitment for quality and thus

complaints are minimized.

Steps of inspection work

The following steps are observed during inspection.

1. Inspection of incoming material

2. Inspection of in-process material

3. Inspection of finished product.

4. Inspection of equipment and tool.

Incoming material inspection

Incoming materials are classified as materials that directly used for making finished

product and indirect materials such as fuel, power and water.

There should be a detailed inspection procedure for inspecting the quality of incoming

materials. In addition to inspection to quality all materials should be inspected for

damage or corrosion due to transportation or storage.

The equipments used for this inspection are go-no-go gauges, plug and ring gauges,

measuring instruments, non-destructive testing equipments and test benches.

The indirect materials such as fuels and water should be checked for their quality.

The regular supply of power should be available and alternative sources should be kept at

ready as back up.

Inspection of manufacturing materials

Inspection during manufacturing consists of the following steps:

1. First piece inspection or first off inspection

2. Periodic inspection during processing

3. Inspecting every piece that leaves the department

4. Final inspection

First piece inspection

After the machine is set in its proper operating condition trial run is made to get standard

product. Once trial run started producing standard product the first product is checked for

its quality. If this one meets the standard requirement the machine is handed over to the

operator for regular production.

By this inspection wastage is minimized at the early stages to reduce the cost of

production.

The machine set up verified from the beginning itself so that defects are not produced

further.

The disadvantage for this inspection that production has to wait until first sample is

checked and approved and men and machine remain ideal during this inspection period.

Floor inspection

The inspection is conducted near the machine itself.

The production errors are detected at the early stages and waiting time is reduced.

The disadvantage for this inspection is that testing instruments have to be brought near

the machine and the inspector has to carry them with him. Dust, shop floor vibration,

noise and such other factors influence the inspection work. The shop floor people may

influence the inspection.

Centralized inspection

It is carried out in a separate room fully equipped with all measuring instruments.

Inspection work is completely separated from manufacturing so that inspection takes

place in un-biased and independent manner.

This type of inspection gives accurate results that are unbiased and independent. The

inspector is not distracted in his work and inspection is properly supervised.

The disadvantage of this inspection is that it takes time and machine errors are not

quickly revealed. The production may be delayed due to bottlenecks in the inspection

room.

Process inspection

A sample is collected at scheduled intervals and inspected for compliancy with standard.

The inspection is carried out continuously so that any defect formation can be easily

identified and rectified. The inspector keeps close watch on various process control

parameters.

Final inspection

When an assembly is completed it is inspected to verify whether all the component parts

have undergone inspection and whether the parts are free from defects. Final inspection is

carried out in every piece manufactured.

Metrology

Metrology is the science of measurement. For every kind of quantity measured there must

be a unit to measure it. For understanding by all, this unit should be a universal one.

Inspection is carried out on any component by measuring some characteristic of the

component. The results are expressed in terms of number of units and acceptability of the

product within the specified limit is ascertained.

There are different types of measuring instruments depending upon the degree of

accuracy.

Tool and equipment control

In order that the parts produced are of consistent and acceptable quality, it is necessary to

keep the production tools, measuring instruments and test equipments as accurate as

possible. Each measuring and testing equipments has its own inherent accuracy beyond

which it cannot be more accurate. But it should give its inherent accuracy consistently. So

the equipments should be maintained and controlled properly. A planned and periodic

control of the equipments should be practiced.

Calibration of test equipments

The instruments used on test equipments should be known accuracy in order that the

results obtained are meaningful. The test equipments should be calibrated at scheduled

times.

Reference standards: In every country reference standard institutions hold standards. The

sub standards are periodically taken to these institutions for calibration against the

reference standards. These standards are checked once in 20 years against natural

standards, the wavelength of Krypton.

Statistical quality control

Statistics is the science of numbers. Statistical techniques are used in industry as a tool

for achieving quality control. So statistical quality control is a branch of quality control. It

is the collection, analysis and interpretation of data to solve a particular problem.

Variations: It is very interesting to note that no two pieces of product manufactured in

identical conditions and with great care are not exactly alike. Sometime the variation may

be small. But it is true that variations do exist.

Types of variations: The variations can be within the product itself or among parts

produced during the same period of time or at different periods of time.

Reasons for variations:

1. Tool wear

2. Bearings that loosen

3. Machine vibrations

4. Faulty jigs and fixtures

5. Poor raw materials

6. Careless or untrained operators

7. Weather changes

8. Measuring errors.

Types of quality characteristics

Variable: When a measured quality characteristic is expressed in universal units then that

characteristic is a variable. Variable can have any values within limits or outside.

Attribute: When the number of items conforming or not conforming to specified

requirements is identified it is recorded by attributes. (E.g.: items can be good or bad here

the attributes are good and bad).

Sometime variables are treated as attributes. (E.g. the length of a part can be measured

and decided whether the part is good or bad.)

Terminology in frequency plot

Frequency: The number of times a particular value of a variable repeats itself in a given

set of data.

Frequency distribution: It is the number of times a given quality characteristic

measurement occurs within the sample of product being checked.

Frequency plot: Pictorial representation of variables

Uses of frequency plot

A frequency plot is a fingerprint of the process behavior that can help to find out error in

the process and take corrective action.

Terminology used in frequency distribution

Cell: The interval along the scale of measurement

Cell boundaries: Measurement values at the end of the cells

Cell interval: Difference between upper and lower boundaries of the cell

Cell midpoint: The value equidistant from the cell boundaries

Frequency: Number of observations in each of the cells

The relative frequency: Frequency of a particular cell divided by the total number of

measurements.

Construction of a frequency distribution

1. Construct a tally sheet

2. Determine the range Difference between highest and lowest values

3. Determine the cell interval: Use the formula C = R/i where C is the number of

cells; R is the range; i is the cell interval.(By assuming various odd numbers of

the interval, the number of cells and the cell interval can be obtained.)

4. Determine the cell midpoints.

5. Determine the cell boundaries

6. Cell frequency: Amount of numbers in each cell

7. Cumulative frequency: Add the frequency number in each cell.

Graphic representation of frequency distribution

Histogram: A set of rectangles with bases that represent cell midpoints and heights

that represent the frequency of a cell.

Frequency polygon: Place dot over each cell midpoint at the height indicated for each

frequency

Cumulative frequency or ogive: Frequency all values less than the upper cell

boundary.

Presenting the Data

Tables: Require much closer reading and are more difficult to interpret than diagrams.

Graphs: Have a visual appeal and are more impressive to understand.

Charts: Shows trends and comparisons.

Several frequency polygons can be drawn on the axis while each distribution should have

separate histograms.

Universe and sample in statistics

If a company manufacture one million nuts and bolts in a day then it is not easy to check

all the nuts and bolts produced in a day. In statistics a sample is collected from the

universe so that the sample represents the universe and the information collected from

this sample can be applied to the universe as a whole. So selecting a sample is to learn the

universe.

Normal distribution

The variation that takes place in any similar product or a product manufactured by the

same machine has a general tendency. If the height of Indians is considered then it is

possible to see that majority of Indian’s height cluster around the mean value while a

small percentage is extremely tall or extremely short. This is a type of normal or nature’s

distribution.

The normal distribution curve is bell shaped and symmetrical about the average value. It

is high at the middle and diminishes gradually as the distance increases away from the

average.

In actual practice no set of quality control data have obtained which conformed exactly

to the normal distribution. But if increase number of samples and made actually to

represent the universe then we can get normal distribution

Measures of central tendency

A frequency distribution or histogram is sufficient for many quality control problems.

But due to many problems in manufacturing additional information are required to solve

the problems. The statistical techniques like measure of central tendency and dispersion

give more information

A measure of central tendency of distribution is a numerical value that described the

central position of the data or how the data tend to build up in the center.

Mean: The sum of the values in a distribution is divided by number values

If χ1, χ2, χ3, χ4,………are N observed numbers then their mean is given by

Χ = χ1+χ2+χ3+……………+ χn = Σ χi

N N

Median: It is the magnitude of middle value that has half value above and half below.

Set of numbers 3, 4, 5, 6, 7, 8, 9 has median 6

Set of numbers 22, 24, 24, 24, 24, 30 has median of 24

When the number in the series is even the median is the average of the two middle

numbers.

For set of numbers 3, 4, 5, 6, 7, 8, the median is the average of 5 and 6 which is 5.5

Mode: This is the value that occurs with the greatest frequency in the set of numbers.

Set of numbers 3, 3, 4, 5, 5, 5, 7 has a mode 5

Set of numbers 22, 23, 25, 30, 32, 36 does not have a mode.

Set of numbers 105, 105, 107, 108, 109, 109, 110, 112 has two modes 105 and 109.

Comparison of mean, median & mode

Mean is the most frequently used one and is based on all observations so least affected by

sampling fluctuations. The drawbacks are that it gives importance to bigger than items

and less to smaller items. So the conclusions go wrong.

The median is not affected by extreme deviations. So when extreme values are present in

an observation median is a more satisfactory measure. But it is affected by sampling

fluctuations.

Mode is the most typical nature in the distribution. But it is not a rigidly defined

measure.

M, Md, Mn 1 2 3 1 2 3

Figure 1 Figure 2 Figure 3

In fig1 the curve is symmetrical and the mean, median and mode are represented by

central line. There is only one value.

In fig 2 the curve is positively skewed and 1 represent mean, 2 medians and 3 modes. So

the mean will have higher value than other two.

In fig 3 the curve is negatively skewed and mean will have smaller value than other two

with mode having highest value.

Measure of variability

The central tendencies cannot describe distribution and how the measurements are

arranged in relation to the center. So the variability indicators, range and standard

deviation, are used in quality control to describe and understand the frequency

distribution.

Range: It is the difference between the largest and smallest of observations.

Standard deviation: It is a measure of dispersion and is commonly used.

Standard deviation is denoted by sigma, δ, and is given by the expression:

δ = √Σ (χi – χ)2

n

The variance is the square of standard deviation.

Applications of measures of dispersions

Range is very common measure of dispersion used in quality control. It provides

knowledge of the total spread of the data. It is valuable when the amount of data is too

small or too scattered. It is not a function of central tendency. But as the number of

observations increase the accuracy of range decreases.

Standard deviation gives a more precise measure. When the data have extreme high and

low values the standard deviation is more desirable.

Relationship of mean and standard deviation

There is definite relationship between the mean, the standard deviation and the

normal curve. The normal curves with same standard deviation and different mean

are shown below. It shows that the curves just changed their location.

Mean= 2 = 0 = 3

The figure below shows curves with same mean and different standard deviations.

Larger the standard deviation the flatter the curve and smaller the standard deviation

the curve become more peaked.

Area under the normal curve

The formula for normal curve is:

f(z) = e-z2/π

√2 π

where π = 3.14159

e = 2.71828

z = χi – χ

δ

z is the standardized normal value. The normal curve in its standard form will have a

mean of zero and standard deviation of 1. The curve a asymptotic at z = -3 and z =

+3.The area under the curve is 1 or 100%.

The normal curve is symmetrical about the mean and as given above the total area is

1.00. So area of each half about the mean is 0.5. For practical purpose each half can be

taken as 3δ. So χ + 3δ contains 99.73% of the total area.

Application: The percentage of items included between any two values can be determined

using the normal distribution curve. For example suppose the mean value for the weight

for brand of cereal is 0.297 for the last one year and standard deviation is 0,024kg.

Suppose we want percentage data that are below the lower specification limit of 0,296kg.

Then calculate ‘z’ from two means and standard deviation.

(0.274-0.297)/0.024 = -0.96. From table for ‘z’ = -.0.96 the area is 0.1685 or 16.85%.

If we want percentage of the data that are above 0.347kg, then ‘z’ = (0.347-0.297)/0.024

= 2.08.For this value of ‘z’, from table the area is 0.9812. Now subtract this value from

1.000,then 1.000 – 0.9812 = 0.0188 = 1.88%. Thus 1.88% of the data are above 0.347 kg.

Conclusion

There are two types of statistics, deductive and inductive statistics. Deductive statistics

describe and analyze the subject or group. Inductive statistics determines from a limited

amount of data an important conclusion about much a larger amount of data.

X-R charts Construction

Control chart

It is a simple graphical device to know whether a process is under control at a given time.

In any manufacturing process there is some variation in characteristic from product to

product. The causes for this variation are two, chance and assignable causes. The chance

causes are inevitable in any process and assignable causes can be discovered and

corrected. For this purpose results of measurements of any characteristic are reduced to a

numerical form and a graph is plotted. These control charts are called X-R charts.

Average, X = Sum of the measurements = X1+X2+…+Xn

Number of measurements n

Range = Largest measurement – smallest measurement

R = Sum of R .

Number of samples

Preliminary considerations

Quality characteristics: It is necessary to know which of the quality characteristic should

be measured and studied during a process for constructing control chart. Inspection

records will show particular characteristics where the product failed and rejected. So

control chart may be produced based on such characteristics.

Equipment and gauge: The equipment and gauge used for measurements should give

exact and accurate results. These instruments should be checked for their accuracy

periodically and should be maintained with proper schedule. There should be proper

procedure for operating the instruments and recording data.

Selection of samples: The information provided by a control chart depends on the

selection of samples or sub groups. The choice of sub groups depends on the purpose of

the control chart. If the chart is to control the quality the selected sub group should reveal

assignable causes. So the samples should be collected in a short time from consecutive

items. If the purpose is to know the homogeneity then the final product examination can

reveal the required information. Depending upon the suspected variations the sample

groups should be classified and collected. (Because the problem may be due to machine,

operator, shift, material, environment, etc.). There is no general rule for sample size.

Generally four to five sample may be preferred. The frequency of sampling depends on

the stage of production. In initial stages of production, frequent sampling may be required

which can be reduced when a state of control is achieved. The collection of sample

should not be at a specified time because the operator will take more care during the

specified sampling time.

Control limits. The chart requires a basis for judging the results. So control limits are

used to know whether the process is under control or not. The control charts are based on

average results and not on individual results. Control limits are different from

specification limits because control limits based on average results and specification is

based on individual units.

If points lie outside the control limits the assignable cause should be looked for and

corrected.

Control limits for X chart: UCL = X + A2R; LCL = X - A2R

Control limits for R chart: UCL = R + D4R; LCL = R - D4R

The value of A and D are available from standard tables.

If any point is outside the control limit then recalculate the limits by avoiding that result.

Steps to construct the chart.

1. When the first off approval is over by inspection start collecting data.

2. Determine the frequency of taking sample.

3. Record the results on data sheet.

4. Calculate average and range for each sample taken.

5. Add all averages and ranges.

6. Find out the average of averages and the average of ranges.

7. Plot individual average and range control charts.

8. Calculate the control limits for average and range.

9. Plot UCL and LCL for average and range.

10. Then calculate R/d2 where d2 is taken from the factors table.

Control chart patterns

The pattern of plotted chart gives information on the process. A knowledge of production

process and the pattern of points on the control chart will lead to a proper interpretation

of the process. It is possible to prepare a list of causes that might give rise to a particular

pattern of points. The pattern of points on the control chart is the key to its interpretation.

Pattern: It is necessary to understand whether the pattern on the control chart is natural or

unnatural. This is the same as saying whether the process is in a state of control or not.

The ability to interpret the charts depends o n the ability to distinguish between the

natural and unnatural patterns of the processes as described on the chart.

For a natural pattern of control chart:

a) Most of the points will be near the central line;

b) A few points will be near the control limits and none of the points fall outside the

control limits.

Typical charts: Range chart

UCL

R

LCL

Average chart

UCL

X

LCL

Pattern of points

Extreme variation (Freak): freak is a point falling outside control limit. The variations

shown on a normal control chart is due to inherent variation. But when a freak is

observed on the chart then an assignable cause has to be searched.

Freaks occur due to: errors in measurement and calculation, temperature or pressure,

wrong setting, damage or mishandling of parts, or the operation is at its end or

commencement.

Shifts (Runs): If more points continuously fall on one side of the central limit then there

is an assignable cause for this. Such appearance of points continuously on one side of

control limit is known as runs.

If the run is above the central line then this indicates undesirable situation because

increase in process spread is expected. But if the run is below the central line then spread

is reduced and process could maintain closer tolerances.

The cause for this may be change in material used, change of operator, inspector,

inspection equipment, machine or setting. If the run is above the central line then a new

careless operator, loose fixing of parts, etc. may be the cause.

Trends: It is a continuous movement of points upwards or down wards. Trend results

from causes which work on process. Generally a trend is caused by tool wear or wear

of threads, deterioration of solution strength, accumulation of dirt and clogging

Cycles: Cycles are patterns that repeat themselves by showing peaks and troughs. They

are indicative of assignable causes operating on the processes due to variation that affect

them periodically. Generally this pattern is associated with seasonal effects, mechanical

reasons, shift differences etc. Some of the causes are worn out position, roller

eccentricity, fatigue, and rotation of persons. A cyclic pattern on range chart may indicate

rotation of fixtures, gauges or moulds, excessive play of tool or die or blunt tool.

Too close (Hugging): The points are very close to central line. The causes for such

Patterns are improper or irrational selection of sub groups and biases in measurement.

The causes for this effect are in the selection of samples. When several machines are

running samples are collected from each machine. Also the instrument cannot take exact

measurement or inspector does not know how to take measurements. In some cases the

recordings made without actual measurements give rise to this pattern.

Erratic fluctuation: Too much ups and downs may be due to single or group of causes.

The causes for such cases are difficult to identify. It may be due to different causes acting

at different times on the process. Frequent adjustment of machines or different types of

materials can give rise to such patterns.

Special remarks: The control chart for averages and ranges should be examined together

to judge the process for control. Satisfactory control over the process exists when both

the charts show a state of control. Where range chart shows control and average chart

does not then the process level is shifted due to external causes. This can be rectified in

many cases by adjusting or setting the machine.

Analysis of X and R chart

Process capability: A process includes machine, material, and operator and

environmental, etc. The basic characteristic of any manufacturing process is variability.

Any quality characteristics exhibit variability. The measured variations are two types,

inherent and external. Inherent variations are natural and material, machine and men

cause external variations. External variations can be traced and corrected. When external

variations are eliminated a true picture of the process capability is obtained.

Process capability may be defined as the minimum spread of a specific measurement

variation that will include 99.73% of the measurement from the given process.

Process capability studies. It is necessary to evaluate the capability of a process and under

stand its limits. In process capability study inherent variation is estimated and external

variations eliminated and thus process is brought under control.

Process capability and specification: There is always a specification for the quality

characteristic of a product. The process used to manufacture the product should produce

the items within the specified limits. The quality control is to determine the process

capability in relation to specification. The process capability is expressed as 3δ on either

side of the mean X. So this 3δ range for a given process is taken in to consideration when

setting the specification.

The specification limits should be considerably wider than the process capability. In a

normal distribution curve A the process capability is 6δ that is centered within the

specification limit 10 δ.

Upper spec limit

B

3δ A 10 δ

3δ

C

Lower spec limit

So the mean can shift within the process capability range, 3δ, still the process remain

under control and specification limits are met as in cases B and C

Condition II

In olden times the specification limit are made tight which made the process

incapable to produce the part. The case is shown below:

B Upper spec limit

A 5δ

C

Here Specification limit is 5δ. So a more accurate process with tighter process

capability is used. Usually more accurate process is more expensive. If the process is

centered exactly between the specification limits a reasonably satisfactory product

can be produced. However if the center of process is either low (curve C), or high

(curve B) a significant percentage of all pieces produced will be outside the

specification.

Process with single specification: Certain product has only one specification limit

(life of lamp has only one limit, lower limit only).

A B C

Lower spec limit

The above are ideal conditions in which the process should produce items reasonably

close together in their quality characteristics and having value in excess of the

specification. B and C show longer life.

B C

A

Lower spec limit

Actually the life of the lamps can be less than specification value as shown above.

Process capability and short runs: A short run job is one that is completed before enough

data can be collected for drawing a control chart. The best method is to determine the

machine capability.

Control charts for percent defective The X-R charts are excellent means for controlling quality. But these charts have

limitation. They cannot be used for quality characteristics like attributes. Also there are

more than 10, 000 quality characteristics and if we depend on control charts then more

than 10, 000 charts should be used to control processes. But this is impractical and in

such cases control charts for attributes can be used.

Attributes

Attribute refers to quality characteristics that classify the items as good or bad. Usually

attributes are checked where measurements are not possible such as color, scratches and

damage. There are also cases where the measurement is too much expensive.

Defect and defective

A defect is a quality characteristic that does not conform to specifications. A defective

part is one that is not usable because it has one more defects.

Types of attribute charts

A p-chart shows fraction of defective in a sample or sub-group.

100p-chart is for percent defective and np-chart is for number of defects in a sample.

Quality characteristics observed as attribute

Fraction defective, p =. Defective articles.

Total number of articles inspected

Percent defective = 100 time the fraction defective

Types of p-charts

Constant sample size: These are charts based on comparison with control limits of the

percent defective values computed from samples of constant size.

Varying sample size: These are charts made for those parts that have been 100 percent

inspected as a part of regular factory routine.

Application of p-chart

Proportion of defective articles estimated for a period of time.

Management is made aware of the quality level.

Out of control article are located and calls for action.

It can suggest where X-R chart can be used.

Shipment can be made based on acceptance criteria.

Selection of the operation

A p-chart can be used to control fraction defective of single or group quality

characteristics, parts products or number of products. It can also be used to evaluate the

performance of operator, department, work place, shift, plant or corporation.

Determination of size and frequency

As a sample size of 25 units is very popular for percent defective charts.

Larger sample size may give greater accuracy but at an expensive cost. Too small a

sample size is very dangerous as it may mislead the result. So the size should large

enough to give dependable result.

Control limits p-chart

The standard deviation is expressed as δp = √p (1-p)/n where p is the fraction defective

and n is the number in samples.

The control limits are p + δp; UCL = p+ 3 δp and LCL = p - 3δp

Significance: if points lie above UCL then the process is producing poor quality items. If

the points lie below the LCL then the process is producing better quality and the cause

should be investigated and whether it can be incorporated permanently in the process.

Plotting the control charts

1. Record all the data.

2. Calculate fraction defective and its percent.

3. Calculate average fraction defective.

4. Compute standard deviation using the above formula

5. Calculate UCL and LCL

6. Plot chart with the above data.

Plots

Percent defective:

UCL

p

LCL

Fraction defective

UCL

LCL

If any sub group is above UCL then it should be discarded and a new p should be

calculated.

p new = ∑np-npd

∑n – nd

npd = number defective in the discarded sub group

nd = number inspected in the discarded sub group

p-chart for variable sample size

Usually p-charts are developed for constant group size. Sometimes varying sample size is

practiced for inspection. If the items inspected varies only slightly then limits are

computed on average numbers. If there is vide variation the control have to be calculated

for each group size n since the control limits are a function of the group size.

Fraction defective chart with variable limits

Fraction defective chart with variable limits is very difficult to interpret. The expression

for control limits involves standard deviation that depends on number of items inspected.

In such cases Stabilized fraction defective chart is developed using standard deviation as

the ordinate.

Interpretation of p-chart

1. The process is judged for control in the same way as it is for X-R charts.

2. If a point lies above the UCL then the cause should be investigated.

3. If a point falls below the LCL it is a situation that chances for further

improvement is present in the process.

Common mistakes

Do not quickly assume that a p-chart is in-control. Break the chart in to the component

characteristics or sources and study each one separately.

If an inspector overlooks some defects then p-chart will show drop and slight changes in

the demarcation between defective and non-defective items may make surprisingly large

differences in the recorded percentages of defective.

Tracing the cause of out of control defects often leads to reducing level of other p-charts.

The principal difficulties in interpreting p-chart most often arise from including too many

characteristics in one chart.

Advantages of p-charts

1. They can be used for a wide range of problems.

2. Information are available

3. The information furnished by them can be correlated to cost.

4. They are more readily understood

5. They are easier to operate as no measurement is involved.

C-charts and Pn-charts X R charts can be used to quality characteristics that are measurable and p-chart may be

used for attributes for accepting or rejecting.

C-chart is another method of plotting attribute characteristics. It looks at the number of

defects per unit of production. But C-chart has a more restricted field of usefulness as

compared to X, R and p-charts.

Distinction between ‘A’ defect and ‘A’ defective

A defect is any characteristic that doe not conform to requirement.

A defective is a part that has more than allowable number of defects.

Examples of C-chart

1. The number of defects in rivets of an aircraft wing.

2. The number of surface imperfections observed in a galvanized sheet.

3. The number of imperfections observed in a piece of clothes.

Construction of C-chart

1. Calculate average number of defects using the formula:

C = Total number of defects in all units

Total number of units

2. The trial control limits: UCL = C + 3√C; LCL = C – 3√C

3. Plot the number of defects in the individual samples.

4. If the LCL is negative then take it as zero.

5. Draw central line at C and the control limits.

Advantages of c-chart

1. Applicable to 100% defect free inspection.

2. If a certain number of defects per unit are tolerable

3. Special studies of variations in the quality of a particular product have been

made using the C chart.

4. It has been applied to sampling acceptance procedures based on defects per

unit.

Application

C-chart can be used to plot a single quality characteristic or a group, a single machine or

a group

It has the advantage in different situations such as short studies to ascertain the variation

in quality of a characteristic or piece, periodic sampling where number of defect is

tolerable, 100% inspection where the aim is to reduce the cost of rework, for acceptance

sampling procedure based on defects per unit.

Number of defectives charts (Pn charts)

If the samples to be plotted on a chart are all of the same size it is easier to understand it

if we plot the number of defectives found in each sample instead of calculating the

percentage.

Center line of Pn chart: It is the average number of defectives per constant size sample.

This is obtained by multiplying the average percentage defective by n the constant

sample size.

Example: Suppose the lot size is 400 and number of lots are 12 and sum of no of

defectives per lot is 360, then P = 360/(12x400) = 0.075; Pn = 0.075x400 = 30

Pn = Total no of defectives per number of samples.

Control limits

UCL = Pn + 3√Pn(1-P)

LCL = Pn - 3√Pn(1-P)

Advantages of Pn chart

This chart is same as p-chart in its use and interpretation. It may be considered as a

simplified version of the p-chart. The Pn chart may not present as complete a picture as

the p-chart

Comparison of results is possible only when sample sizes are same. But average fraction

defective can be compared directly.

Sampling by attributes The word sampling is used in every day life activities like selecting some item of

purchasing. When we go to purchase rice from shop we take a handful of rice and

examine its quality and sometime s more than one handful of rice or wheat we take and

decided whether to purchase or not. Similarly in industry sampling is an inspection

method for deciding over the quality. So to accept or reject a lot the decision is taken

based on the quality of the sample taken for inspection. Some procedures and techniques

are laid down to this sampling

Sampling inspection is the evaluation of the quality of small number of units of products

drawn from the lot and decides the quality of the product.

Purpose

1. Interpret the specification

2. Measure the product

3. Modify the inspection criteria.

Advantages and disadvantages

100% inspection

1. Uneconomical

2. Not acceptable if inspection is by destructive tests

3. The inspection leads to fatigue and tiresome.

4. Sometimes fatigue ness overlook some defective material

5. Not practical when lot is large.

Sampling inspection

1. Economical.

2. There is less damage if inspection is through destructive test.

3. It causes less fatigue to the inspectors.

Disadvantages

1. There are risks of accepting bad lots and rejecting good lots

2. Extra planning and documentation are required

3. The sample gives less information about the product.

Acceptance sampling

Assumptions

The inspectors follow the prescribed sampling plan

The inspection is made without error.

The nature of the lot is uniform.

Misconception

Nothing but perfect lots can be used.

A lot can be judged by looking at fewer pieces.

The sampling inspection always uses adequate sampling.

A strong sampling plan is one that rejects a lot when a single defect is found in the

sample.

Random sampling is not required.

100% inspection is the only way to ensure good quality.

Industrial uses

Determine quality of incoming raw materials.

Decision is made on semi finished product quality for its further processing.

Determines the quality of out going product quality.

Improves and controls quality.

Lot formation

A lot is collection of items from which a sample is drawn and inspected to determine its

acceptability. Some guidelines on lot formation are given:

1. Do not mix products from different sources unless there is evidence that lot-

to-lot variation is negligible.

2. Do not accumulate products over extensive period of time.

3. Do not make use of extraneous information in lot formation.

4. Do not make lots as large as possible.

Types of inspection

Inspection by attribute: An item is classified as defective or non-defective.

Inspection by count of defects: The defects are counted.

Inspection by variables: The quality characteristics of an item are measured and

expressed in units.

Inspection by attributes

It has been found that a sampling inspection can ensure the required degree of quality,

provided the procedure is scientifically planned.

Procedure

The sampling plan designated, as ABC-Std-105 is applicable to inspection of the

following:

a) End items.

b) Components and raw materials.

c) Operations

d) Material in process

e) Supplies in storage

f) Maintenance operation

g) Data or records

h) Administrative procedures

Terminology

1. Sampling plan is a statement of sampling procedure and rules for making

decision about the lot.

2. Sampling inspection is the inspection of a portion of a lot to decide whether to

accept or reject the lot with the view that the sample analyzed represent the

lot.

3. Item is the unit of product or material over which inspection is performed.

4. Lot is a collection of items from which a sample is collected for inspection.

5. Lot size is the number of items in the lot.

6. Sample is a group of items drawn for inspection from the lot.

7. Sample size is the number of items in the sample.

8. Defect is failure to meet the requirement imposed on an item with respect to a

single characteristic.

9. Defective is an item that does not meet the specified quality requirement.

10. Acceptance number is the maximum allowable number of defectives in the

sample for acceptance of the lot.

11. Process average is the average percent defective of the product submitted by

the producer for inspection.

12. Acceptable quality level is defined as maximum percent defective that can be

considered satisfactory as a process average.

13. Lot tolerance percent defective is the percentage of defective in a lot that can

be tolerated in only specified proportion of lots (conventionally taken as

10%).

14. Average outgoing quality is the average quality of the outgoing products

including all accepted lots and all rejected lots which have been effectively

100 percent inspected and all defectives replaced by non-defectives

Sampling plans

1. Single sampling plan

2. Double sampling plan.

3. Multiple sampling plan

4. Sequential sampling plan

Single sampling:

In this sampling inspection the decision to accept or reject o lot is based on the

inspection result of a single sample. If number of defectives found in a single sample is

found equal to or less than the acceptance number then the lot is acceptable. Otherwise

rejected.

Inspect a sample of

‘n’ pieces

The number of defective found in sample

Exceeds ‘c’ Not exceeds ‘c’

Reject the lot Accept the lot

Double sampling plan:

Here C1 = acceptance number of first sample

C2 = acceptance number of first and second sample combined

Example: Let us consider a sampling plan of the following:

N = 50 n2 = 6

n1 = 3 C2 = 1

C1 = 0

Procedure:

1. Select at random from the lot (N) a sample with three pieces (n1)

2. If no defective then lot accepted

3. If there are two or more defects in the sample, reject the lot

4. If there is one defective, take a second sample of six pieces (n2)

5. If in the combined sample there is only one defective (C2) then accept the

lot

Inspect a first sample of n1

pieces

The number of defectives found in first

sample

Does not

exceed C1

Exceeds C1

Exceeds

C1 but not

C2

Inspect a second sample

of n2 pieces

Number of defectives in first

and second combined

Does not

exceed C2

Exceeds C2

Accept the lot Reject the lot

Multiple sampling plans: In this plan, take more than two samples before reaching a

decision.

Sequential sampling plans

Item sequential plan

Group by group sequential plan

Item sequential plan: In this type of sampling plan the number of items to be inspected is

not pre-determined. The items are drawn from the lot one at a time and inspected. After

inspecting each item a decision is made whether to accept the lot or reject or to continue

the inspection by taking one more item. This procedure is continued till the lot is either

accepted or rejected. The formula for the acceptance or rejection criteria is given below

for inspection by attributes.

P1 = AQL as fraction defective

P2 = LTPD as fraction defective

α= Producer’s risk

β = consumer’s risk

Let B = log (1- α)/ β

A = log (1- β)/ α

g1 = logP2/P1

g2 = log {(1-P1)/1-P2)}

m1 = B/(g1+g2)

m2 = A/(g1+g2)

S = g2/(g1+g2)

Then, (nS-m1) is the acceptance number., an. If the number of defectives found after

testing n items is less than or equal to an, the lot is accepted.

Similarly, (nS+m2) is the rejection number, rn. If the number of defectives found after

testing n items is greater than or equal to rn, the lot is rejected.

To facilitate the use of this plan the value of an and rn for different values of n can be

calculated before hand and kept ready

Group -by- group sequential plan

Sometimes it may be found more practical to inspect a sequence of items rather than a

sequence of items themselves. The items are drawn from lot in a group and inspected.

After inspection of each group the cumulative results are examined and decision is made

either to accept the lot or to reject it.

In item - by -item method points are plotted and for group – by – group the results are

cumulated by groups and plotted.

For example, if the group consists of 20 items, the results of inspecting the first 20 items,

then the first 40, the first 60, and so on are plotted.

Lots consisting of 5000 bicycle rims are submitted for inspection for thickness chromium

plating by the BNF jet test. A sequential sampling plan with AQL (P1) of 0.1, LTPD (P2)

of 0.30, producer’s risk of 0.02 and consumer’s risk of 0.04 is to be used.

B = 1.3891

A = 1.6813

G1 = 0.4771

G2 = 0.1091

M1 = 2.37

M2 = 2.868

S = 0.186

Acceptance number, an = -2.37+0.186n

Rejection number, rn = 2.868+0.186n

Then the sampling plan would be as follows:

From each lot, bicycle rims are selected at random and subjected to the BNF jet one by

one. If n is the number of rims inspected at any stage and d is the number of defective

rims observed at that stage then,

If d is less than or equal to an, the lot sampled is accepted.

If d is greater than or equal to rn, the lot sampled is rejected.

If d is between ‘an’ and ‘rn’ another bicycle rim should be selected from the lot and tested

for a possible decision.

Graphic procedure: A graph is prepared with the sample size on the x-axis and the

number of defectives on the y-axis. Two parallel lines with equations d = -2.37+ 0.186n

and 2.686+0.186n are drawn. These lines demarcate the regions acceptance, rejection and

continuance of inspection.

With the graph thus prepared, rims are selected at random and subjected to the BNF jet

test. After testing each rim the number of defective rims observed out of n rims tested till

then is plotted in the graph. A decision to accept or reject the lot will be made when a

point plotted in the graph falls in the acceptance or rejection region respectively

inspection is continued.

Comparison

1. Single sampling inspection gives only one chance of passing compared to

double and multiple sampling.

2. Number of pieces inspected per lot is high in single sampling while double

sampling is less by 50% and multiple sampling by 30% less compared to

single sampling.

3. Single sampling has the lowest cost of administration, training, etc.

Criteria for selection

1. The protection to producer and consumer.

2. Cost of inspection

3. Administrative convenience

4. Information obtained on lot quality

5. Acceptability of plan to producers

6. The difficulty in training inspectors to use multiple sampling

7. The need for quick and reliable estimates of process average

8. The availability of inspection personnel and facilities

9. An OC curve drawn for particular AQL, lot size will have the above

characteristics and hence will have the acceptability of both the producer

and the consumer

Inspection level

1&II---------- special small sample inspection

III, IV&V------inspection per lot is 1; 2 1/2; 3 1/2.

IV---------------generally adopted

V----------------acceptance of a single defective is serious matter and cost of inspection is

low

VI----------------Acceptance of defective item not serious and cost of inspection high.

If A< B< C< D are given to indicate the sample size under different inspection levels for

varying lot sizes, then the table gives information of sample size for different levels.

Lot size I II III IV V

2-8 A A A A B

9-15 A A A B C

16-25 B B B C D

26-50 B C C D E

51-100 C C C E F

101-150 C D D F G

151-300 D E E G H

301-500 D F F H J

501-1000 E F G J K

1001-3000 E G H K L

3001-10000 F G J L M

10001-35000 F H K M N

Procedure of lot acceptability-AQL plans

The acceptability of a lot of items submitted for inspection is to be determined with the

help of one of the sampling plans associated with the specified AQL. Depending on the

type of sampling plan chosen the acceptability criteria will be determined with the help of

tables already given in the information.

Usage of tables: Example: lot size is 2500 and inspection level is IV the from table the

code is K

If AQL is 2.5% then from Table two for code k and AQL = 2.5%, the acceptance number

is 7.

If there are more than seven defectives reject the lot.

Double sampling

1. Obtain code letter from the lot size and inspection levels.

2. Read sample size (n1) and cumulative sample size (n2) from table for the

code.

3. Note acceptance number for sample n1, and cumulate sample size, n2, for

given AQL and code from the table

4. Accept the lot if number of defectives in the first sample is less than or equal

to first acceptance number and rejected if it is greater than or equal to the

first rejection number.

5. If the number of defectives lies between a1 and r1 a second sample is

chosen. The number of defectives in the combined sample is compared with

second acceptance number.

Normal inspection: Inspection under a sampling plan that is in force for a particular

product and producer is called normal inspection.

Tightened inspection: Inspection can be tightened either by raising the inspection level

that is by selecting a sample size or code higher than that adopted for normal inspection

or by employing a smaller AQL.

Reduced inspection: If the quality of the submitted lots is consistently better than the

AQL chosen, reduced inspection may be resorted to.

Criteria for shifting levels

1. If two out of five consecutive lots have been rejected during normal

inspection change over to tightened inspection.

2. If during tightened inspection five consecutive lots have been accepted

changes over to normal inspection.

3. If one of the 10 consecutive lots has been rejected while on normal

inspection change over to reduced inspection.

4. If 10 consecutive lots are rejected and if at the same time the number of

defective lies between the acceptance number and rejection number on

reduced inspection change over to normal inspection

Design of sampling plans

Single sampling plan: Example: inspection level = II, AQL = 4%; lot size = 2500.

For lot size 2500 and level II the code is G

For code G and AQL 4% the sample size is 32 and acceptance number is 3

If acceptance number is 3 then rejection number is 4

If inspection is tightened then the AQL will go down that is from 4% to 2.5%

Now for code G and AQL 2.5% the sample size is again 32 and acceptance number is 2

and rejection number is 3

If inspection is reduced choose a higher AQL say 6.5%

Now for code G and AQL 6.5% the sample size is again 32 and acceptance number is 5

and rejection number is 6

Operation characteristics curves

It is necessary to compare the performance of a sampling plan over a range of quality

plans before making a final decision on sampling plan. Operation characteristic curve

provides the means to evaluate the operation of a sampling plan. Each acceptable

sampling plan has an operation characteristic curve associated with it. Mora than one

acceptance sampling plans may have the same OC curve.

An OC curve shows, for every possible fraction defective ‘p’ in a given lot submitted for

inspection, the probability ’p a’ that such a lot will be accepted by the acceptance

sampling plan that the OC curve represents. It is the graph drawn with lot fraction

defective ‘p’ on X-axis versus the probability of acceptance ‘pa’ on Y-axis.

Ideal OC curve: If it desired to accept all lots 3% or less defective and reject all lots with

a quality level greater than 3% defective., the curve is as shown:

A B

C D

1 2 3 4 5 6

X-axis represents % defectives and Y-axis represents probability of acceptance. The

ABCD represents Acceptance line with 3% defectives

Inference from the graph

1. All lots with less than 3% defective have an acceptance of 1.0 (certainty)

2. All lots greater than 3% defective have a probability of acceptance of zero.

Four parameters of OC curves

1. Producer’s risk is the probability that a good lot will be rejected by the

sampling plan. In some plans, this risk is fixed at o.o5; in others it varies

from about 0.01 to 1.0.

2. Consumer’s risk is the probability that a bad lot will be accepted by the

sampling plan.

3. Acceptable quality level is the maximum percent defective or the maximum

number of defects per hundred that for the purpose of a sampling inspection,

can be considered satisfactory as a process average. A sampling plan should

have a low producer’s risk for quality that is equal to or better than AQL.

4. Limiting quality or Lot tolerance percent defective is the level of

defectiveness that is unsatisfactory and therefore should be rejected by the

sampling plan. A consumer’s risk of 0.10 is common and LTPD has been

defined as the lot quality for which the probability of acceptance is 0.10 i.e.

10% of such lots will be accepted.

OC curve for a general plan

Example: An inspector collects 150 pieces from a large lot and to accept the lot if it has

no more than 4% defective pieces.

1.00 AQL = 1.2%

0.50

LTPD = 5.3%

0.10

1 2 3

X-axis = Percent defective, 100p

Y-axis = probability of acceptance, pa

From the curve a lot having 3% defective has 0.5 chance of being accepted.

But a lot with 3.5% defective, a bad lot, has 0.39 chance

Relationship between the parameters of OC curves

1.0

0.9 Producers risk, α

0.1

Consumer’s risk, β

2 6

From the OC curve given above the producer’s risk 10% (0.90) at an AQL of 2%.

This means that though the percent defective is agreed at 2% level in actual inspection

resulted at a producer’s risk factor of 10%, the risk that the producer takes of having a lot

rejected even though the true quality of the lot is at the AQL.

Interpretation of OC curve

Steps in the selection of sampling plan. The risk in each sampling plan can be studied

from OC curves. Compare the relative risks of two or more sampling plans for a sampling

situation. Construct sampling tables from OC curves to relate risks. Determine relative

risks in selecting incorrect sampling plan. Select sampling plan with relatively less risk

from the viewpoint of both supplier and consumer.

Rules of thumbs sampling plans

Below is given OC curves for different sampling plans

1

N = 50

n = 5

c = 0

N = 200 N = 100

n = 20 n = 10

c = 0 c = 0

N = 1000

0 n = 100; c = 0

2 4 6 8 10

The sample is 10% of the lot for each of the plans

A 4% defective is allowed for all.

If the lot is 50 then probability of acceptance is 80%

If the lot is 200 then probability of acceptance is 44%

If the lot is 1000 then probability of acceptance is 3%

Evaluation of parameters affecting acceptance-sampling plans

Sampling plans are affected by lot size, sample size and acceptance number.

The following figure shows curves for changed lot size but with constant

sample size and acceptance number

It can be noticed that lot size has little effect on the probability of acceptance.

The curves shown below explain the effect of changing sample size while all others

remain the same.

N=1000;n=20;c=0

N=200;n=20;c=0

N=100;n=20;c=0

C B A

X-axis = Percent of lots expected to be accepted

Y-axis = Quality of incoming lots in percent defective

As the sample size gets larger and larger, the operating characteristic curve becomes

steeper. The probability of acceptance becomes smaller. This simply means that the

larger the number of items in the sample, the more possibilities of finding enough

defectives to exceed the acceptance number.

The effect of change in acceptance number is shown below:

A B C

Sampling

plan

Sample

size

Accept

number

Reject

number

A 32 1 2

B 50 1 2

C 125 1 2

Sampling

plan

Sample

size

Accept

number

Reject

number

A 80 1 2

B 80 2 3

C 80 3 4

As the acceptance number increases, the operating characteristic curve gets flatter. That

means the probability of acceptance increases. As the number of allowable defectives

become larger the more the chance that an acceptance decision can be made.

The figure below shows the effect of varying both the sample size and the acceptance

number. Notice how the curve starts to approach the ideal IOC curve when the

acceptance number is changed from zero.

Sampling

plan

Sample

size

Accept

number

Reject

number

A 32 1 2

B 200 7 8

A B

This is generally the case. When the acceptance number is made larger than zero, the

sample size must be increased to achieve probability levels provide adequate protection.

The figure below shows the case where the shape of OC curve changes as the acceptance

number changes. As the acceptance number decreases the curve becomes steeper.

A B C D

X-Sampling

plan

Lot size Sample size Accept

number

A 2000 300 2

B 2000 50 0

C 2000 50 2

D 2000 50 4

axis = % defective; Y-axis = probability of acceptance

Above Figure shows change in the shape of OC curve with change in acceptance number.

As the acceptance number decreases the curve becomes steeper.

Consumer producer relationship

Producer’s risk is represented by, α, and is the probability of rejection of a good lot. This

is given as 0.05 and sometimes ranges from 0.01 to 0.1 .Associated with producer’s risk

is a numerical definition of good quality is called the Acceptable Quality level (AQL).

α = 0.05 AQL = 0.7%

β = 0.10 LTPD = 2.6%

0.7%

In the above curve N =4000; n = 300; c = 4, AQL = 0.7% for α = 0.05

The product that is 0.7% defective will have a rejection probability pf 0.05 or 5%. So by

this sampling plan 1 out of 20 lots will be rejected.

Consumer’s risk (β) is the probability of acceptance of a bad lot. Here this value is 10%.

Associated with consumer’s risk is called lot tolerance percent defective (LTPD) in the

lot that has to be tolerated by the consumer. In the above curve this value is 2.6% at β =

0.10. This means the lot with acceptance limit of 2.6% defective will have acceptance

probability of 10%. So one out of ten lots that are 2.6% defective will be accepted by the

sampling plan.

Average outgoing quality

The average outgoing quality (AOQ) is another technique for the evaluation of a

sampling plan.

AOQ = 100p.pa

The average outgoing quality is the quality that leaves the inspection operation. Any

rejected lot is rectified or sorted and returned with 100% good products. If not rectified

then AOQ is the same as the incoming quality.

Example: N = 3000; n = 89; c = 2

Process

quality,

100p’

Sample size,

n

np’ Probability of

acceptance, pa

AOQ,

100p’.pa

1.0 89 0.9 0.938 0.938

2.0 89 1.8 0.731 1.462

3.0 89 2.7 0.494 1.482

5.0 89 4.4 0.185 0.925

6.0 89 5.3 0.106 0.636

7.0 89 6.3 0.053 0.385

9.0 89 8.0 0.014 0.126

2.5 89 2.2 0.623 1.358

Curve without rectification

AOQL = 1.55% (approximate)

X-axis = Incoming process quality (percent defective)

Y-axis = AOQ (percent defective)

Analysis of the curve shows that for 2% defective the average outgoing quality is 1.46%

defective. When the incoming quality is 6% defective the AOQ is 0.64% defective. Since

rejected lots are rectified the average outgoing quality is always better than the incoming

quality. For this sampling plan as the percent defective of the incoming quality changes,

the average outgoing quality never exceeds the limit of approximately 1.55% defective.

The AOQ curve in conjunction with OC curve provides the powerful tool for describing

and analyzing acceptance sampling plan.

Quality function in industry

The quality of an article refers to those factors that determine its fitness for the intended

purpose. If the product is manufactured to meet the needs and fancies of the customer

then the product finds its way to fit into its purpose. The quality of a product extends over

its entire life span.

Fundamental factors

The quality is a company policy developed through innovations and workmanship to

create the product. Quality system is carefully planned by the organization. All steps to

manufacture the product is controlled and monitored and corrective actions taken in time.

The factors that effect quality are the following:

Markets: The customer is the deciding person a company’s product should be in the

market or deleted from the market. So the organization has to identify the customer wants

and needs.

Men: Men with specialized knowledge are required to meet organizations demands.

Money: Quality costs money. The quality costs are preventive cost, appraisal cost and

internal and external failure. Improvement also requires money.

Management responsibility. The responsibility is distributed among several groups.

Marketing does product planning function and its quality requirements. Engineering

section take the responsibility of designing. Production people develop and refine the

process to make quality product that conform specifications. Control section makes

quality measurements and assures the quality of the product.

Materials: Supply of materials required for manufacture of product with assured

specification requirements.

Machine: There is a demand for quality products with reduced cost and high productivity.

This forced manufacturers to use sophisticated machines.

Methods: Introduction of new materials and new machines will necessarily call for newer

methods of manufacture and inspection.

Miscellaneous: Increased complexity and higher performance requirements for products

have served to emphasis the importance of product reliability.

Quality function

A major function of a manufacturing company is quality or achieving fitness for use. This

function is carried out through a wide assortment of activities. From market study and

consumer’s needs an idea is developed into a product. Again the customer determines

what type of qualities is required for the particular product to function the needs of the

customer affectively. The research and development department of the company then

create a product concept that can meet these requirements. Design engineers prepare the

product and its specification. Other departments specify the processes and instruments

capable of fabricating and measuring these qualities. Purchasing section buy materials

and components possessing appropriate qualities.

Shop operators are trained to use the processes and instruments to build the specified

qualities into the product. The sales force through the distribution chain urges consumers

to buy the quality products.

Departments responsible for quality

Quality is everyone’s responsibility. The responsibility for quality is delegated to the

various departments along with the authority to make decisions about quality. The closed

loop is shown below:

Purchasing: Using quality standards established by product engineering purchasing

section procure quality materials and equipments. The incoming materials are checked

for quality.

Design: Manufacturing dept develops processes and procedures to produce quality

product. A product design review is conducted in order to anticipate the probable quality

problems. Quality problems are related to specifications and tolerances. Process selection

and development is concerned with cost, quality, time and efficiency. The sequence of

operation is developed to minimize problems. Method study is used to determine the best

way performing the operations.

Customer

Product service: ensures quality

of performance

Marketing: determines

customer quality requirements

Packaging & shipping:

Preserves& protects quality of

product

Product Engg: Translates

requirements into quality

oriented design

Inspection Y& test: Appraising

quality of conformance

Purchasing: Purchases quality

materials and equipments

Manufacturing: Transforms

design and materials into quality

products

Manufacturing Engg:

Develops quality oriented

processes and procedures

Quality

Control

Manufacturing: The quality must be built into a product. The process and procedures to

manufacture quality products should be developed.

Inspection and test: The quality of purchased and produced items should be appraised for

conformance to quality. The equipments used for measurements should be accurate and

calibrate against international standards. Controlled procedures for performing inspection

should be established.

Packaging and shipping: The quality of the product should not be affected during

distribution and shipment. So proper packing, handling, and delivering facilities should

be ensured. Proper protection should be given to the product

Product service: The responsibility of the manufacturer is not finished once product is

delivered to the customer. The customer should be satisfied that product meets all

expected requirements of the customer. Incase any after sales service is required for the

product the manufacturer should be ready for that.

Quality control: This department does not have a direct responsibility on quality. It assists

or supports the other departments as they carry out their responsibilities. Quality control

appraises the current quality, determines the quality problem areas and assists in the

correction or minimization of these.

Quality assurance system

A quality assurance system is an effective method of attaining and maintaining the

desired quality standards. It is based on the fact that quality is the responsibilities of all

functions. The functions are broken down to sub systems as given below. The quality

assurance system is an evaluation or audit of each of these sub systems to determine how

effectively the functions are being performed.

Management of quality system

Product quality and reliability development

Product and process quality planning

Supplier quality assurance

Product and process quality evaluation and control

Special quality studies

Quality information feedback

Quality measurement equipment

Quality training and manpower development

Field quality evaluation and control

Organization of quality function

Since quality is everybody’s responsibility it always turns out to be nobody’s

responsibility. The following six steps are used to organize the function:

1. Define the quality problems for the solution of which the organization is being

created.

2. Establish the objective that the organization must achieve.

3. Determine basic work elements that must be established in meeting these

objectives.

4. Combine these basic functions into job packages.

5. Consolidate the job packages into an organizational component suitable for the

requirements.

6. Locate the component in that segment of the organizations where it can do its job

and achieve its objective.

Feed back loop

The continuous feed back loop is shown:

Quality planning: It establishes the basic quality control system for the company’s

product.

Quality appraisal: It evaluates in accordance with the quality plan the conformance and

performance of the parts and products with engineering and specifications.

Quality analysis: It provides by rapid feed back of process control engineering results in

new planning. This analysis also fosters corrective action for product quality deviations.

Planning

Analyzing

Measuring