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17

QUALITY CONTROL

17.1 INTRODUCIONThe basic purpose of manufacturing is to produce engineering materials and

parts to specified shape, size and finish. The specifications for the shapes,sizes and finishes are furnished to the shop by part drawings or manufactur-

ing drawings. These specifications are often called quality characteristics.The measured quality of manufactured product is always subjected to a

certain amount of variation as a result of chance. Some stable sys-tem of chance causes is inherent in any particular scheme of pro-

duction and inspection. The reasons for variation outside this stable

system should be discovered and corrected to avoid wastage and,finally to improve quality.

17.2 QUALITY CLASSIFICATIONThe workquality as used in manufacturing, implies the best for the money

invested and does not necessarily mean the best. Quality is a relative termand is generally explained with reference to the end use of a product. A

productis said to be ofgood quality if it works well in a particular situationfor which it is meant while in other situations it may not work well and it is

said to of bad quality. A particular quality level of any product manufac-tured in production shop may occur by intent or by chance. The quality may

be classified in several ways. They are :Quality of design : Quality of design is determined before the prod-

uct is produced. This is the level of quality which the designer or engineerintended the product to have. Generally, quality of design is expressed in

terms of the tolerable variations that may be allowed for the end product.For example, suppose two items are to be produced with the same nominal

dimension. The specifications of two items are:

Item 1. Nominal size = 50 mm, tolerance = 0.5 mmItem 2. Nominal size = 50 mm, tolerance = 0.2 mm

Thus item 2 is said to have a higher quality of design than item 1.


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58 PRODUCTION AND OPERATIONS MANAGEMENT

Quality of conformance : Quality of conformance means producing a prod-uct to meet the specifications. When a product conforms, the product is con-

sidered to be a quality product. For example, two machines are to produce

an item with nominal size 50 mm and tolerance 0.1 mm. Machine 1pro-duces a range of 0.05 mm and Machine 2 produces a range of 0.02mm, then it is said that machine 2 has a higher quality of conformance. The

components of control costs and reject costs are shown hereunder:

Table 17.1 Costs of quality

Cost category Components

Control cost

Prevention cost

Appraisal costs

1. Quality planning cost: Preparing plans, manuals etc.

2. New product review: Review or prepare quality specifi-cations for new products.

3. Quality data: Collection, analysis, and reporting of data.

4. Incoming material inspection.

5. Process inspection.

6. Final goods inspection.

7. Quality laboratories: Cost of operating laboratories to in-spect material, work-in-process finished goods.

Rejection cost

Internal failure

External failure

1. Scrap: Wastage of productive hours and materials.

2. Rework.

3. Downtime.

4. Warranty: Cost of refunds, replacing or repairing theproduct.

5. Returned goods.

Quality of performance : Quality of performance is the ability of the productto perform its technical function in most economical manner. It has three

components : availability, reliability, and maintainability. Each of thesecomponents has a time dimension and must reflect the fitness of the product

during its life. Availability of a product is measured as the ratio of total up-time of a product and the combined time of total uptime and total down-

time.Reliability refers to the length of time that the product will perform itstechnical function satisfactorily under a given set of conditions. The relia-

bility of a product is often expressed in terms of mean time between failure


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QUALITY MANAGEMENT 59

(MTBF) which is the average time that the product functions from one fail-ure to the next. If the MTBF is longer, the product is said to be more reli-

able.Maintainability refers to the correction needed to a product to bring itto service once it has failed. Maintainability is measured by mean time to

repair(MTTR). If a product is immediately brought for correction and therepairpersons work on it, then the MTTR equals to downtime of the project.

17.3 INSPECTION AND QUALITY CONTROL

Inspection is very closely associated with quality control. So they should be

explained side by side.

Inspection: It is often said that no two things can ever be exactly alike. This

also holds true with manufactured parts. Even though certain variations are

accepted, parts are liable to rejection if the deviations go beyond the speci-fied quality.

Some procedure must, therefore, be set up to detect errors so that themanufacture of faulty parts does not go uncorrected.

The philosophy of inspection is only preventive and not remedial.In short, inspection is the method of measuring and/or checking the quality

of a product in terms of specified standard.There are three basic areas of inspection : (1) receiving inspection,

(2) in- process inspection and (3) final inspection.In the receiving inspection, inspections are performed on all incom-

ing materials and purchased parts. In the in-process inspection the productsare inspected as they are in process. In thefinal inspection, all finished prod-

ucts are finally inspected prior to sending them to the customer.

Quality control : The word quality as used in manufacturing implies the

best for the money invested and does not necessarily mean the best. Qual-ity is a relative term and generally explained with reference to the end use

of a product. A component is said to be of good quality if it works well in aparticular situation for which it is meant while in other situation it may not

work well and it is said to be of bad quality. The word control implies reg-ulation, and regulation implies observations and manipulation. It suggests

when to inspect, how often to inspect and how much to inspect.

The basic philosophy of quality control is both preventive and remedi-al. It is through quality control that some measures are taken to see that de-fective items are not produced at all. When they do occur, corrective action

must be taken to prevent further recurrence.


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Inspection is considered to be a tool of quality control. It checks theproducts while quality control attempts to bring the variable factors under

control.

17.4 STATISTICAL QUALITY CONTROL

Any attempt to inspect quality in the finished product by usual inspectiondevices is time consuming and costly. Besides, in a continuous production

process this 100 per cent inspection may not be found practicable.Certain statistical techniques have been devised to evaluate machines,

materials and processes by observing capabilities and trends in variations sothat continual analysis predictions may be made to control the desired quali-

ty level. These statistical techniques are called statistical quality control

methods. These methods or tools are :

1. The Shewhart Control Charts for measurable quality characteristics.

These are described as charts for variables, or as charts for X and R

(average and range) and charts for X and (average and standard de-viation).

2. The Shewhart Control Chart for fraction defective. This is described as

thep chart.3. The Shewhart Control Chart for number of defects per unit. This is de-

scribed as c chart.4. Sampling plans dealing with the quality protection.

17.5 BASIC TERMINOLOGYTo understand control charts it is necessary to understand some fundamen-

tals of statistics associated with them. These are briefly explained below :

1. Variables and attributes : All manufactured products must meet certain

requirements, either express or implied. All these requirements may be stat-ed either as variables or attributes.

When a record is made on an actual measured quality characteristic

such as dimension, hardness, temperature, etc., expressed in their units, thequality is said to be expressed by variables. Most specifications of variables

give both upper and lower limits for the measured value. Inspection using

variables is mostly done on the shop floor and is important in quality con-trol. Variables are dealt within the control chart for averages of measure-

ment, i.e., for X and R, and X and .Many requirements are necessarily stated in terms ofattributes rather

than variables. This applies to things that may be judged only by visual ex-amination. For example, the surface finish of a piece of furniture either


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presents a satisfactory appearance or it does not. In general, the thing exam-ined either conforms or does not conform to the specifications and implies

good or bad, acceptable or unacceptable.

2. Variability : Variation seems inevitable in nature. Manufacturing pro-cesses are no exception to this. Variability thus appears to be an inherent

characteristic of all processes. The amount of this basis variability dependson various characteristics of production process such as machines, the mate-

rials, the operators. The purpose of statistical analysis is to examine thisvariability and to detect when and to what extent external factors enter into

the process to alter the variability pattern.There are two types of variation:

Variations due to chance factors.

Variations due to assignable causes.

The chance causes of variations occur in a random way. For exam-ple, a little play between a nut and screw may lead to back-lash in a ma-

chine causing the cutting tool to operate differently and producing therebydifferent measurements of a specified size. The non-homogeneity in a mate-

rial may result different surface quality or finish in machining.. So it is saidthat chance variation is inherent with the system and very difficult to con-

trol even under best conditions of productions. In the quality control, chancevariations are seldom considered since variations in measurements are usu-

ally small compared to actual variations in quality characteristics. It hasbeen established that if the variations are only due to chance factors, the ob-

servations will follow normal curve described in the next article.Various assignable causes leading to variations in dimensions may

due to poor quality raw material, machine condition, changing working con-dition, mistake on the part of a worker, difference in the skill of operators

etc. Variations due to assignable causes possess greater magnitude as com-pared to those due to chance causes, of course, they can be readily con-

trolled by eliminating the causes from the system. The control chart tells

when and in some instances suggests where, to look.

Patterns of variations : In a manufacturing process no two parts can be pro-duced with identical measurements and there will be variations in the mea-

sured sizes of parts. It follows that it is necessary to have a simple methodof describing patterns of variation by using frequency distribution.

4. Normal curve : Frequency curves of may different shapes may be found.The most useful of these curve is the normal curve shown in Fig. 17.1. This


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curve is popularly known as probability curve or normal distribution curve.This symmetrical bell-shaped type of distribution is typical of

many engineering processes, the majority of the component measurements

being grouped close to the mean, and fewer measurements being eithersmaller or larger than the mean.

The horizontal scale in Fig. 17.1, represents values of standard devi-

ation above and below themean X . If the plot of mea-

surements taken approaches the

normal distribution curve, 99.73per cent of all measured values

will fall within X 3. Only0.27 per cent are expected to

fall beyond those limits. Simi-larly, 95.46 per cent of the val-

ues fall within the limits X

2, and 68.26 per cent withinX 1. Other values of areafor various values of standard

deviations may be found in text books on statistics.For example, in the manufacturing specification of a sample if the

standard deviation is 0.002, the mean X is 0.625 and tolerances is 0.006 representing 3 then from the normal distribution curve, only 0.27per cent will be expected to come under rejection as 99.73 per cent will be

expected to fall within this area. If however, a tolerance of 0.002 or 1 isgiven then the expected rejection rate will be 31.74 per cent as 68.26 per

cent of all manufactured parts will be expected to fall within this area.

Therefore, to keep a tolerance of 0.002 either the process is to be im-proved or a more accurate method of manufacture is to be used.

It may be noted that most distributions found in industrial inspection

activities are not normal, but many approach normality giving approximate-

ly same results.

17.6 CONTROL CHARTSProcess control charts are commonly used in quality control to maintain a

continuous evaluation of the manufacturing process. A control chart is sim-ply a frequency distribution of the observed values plotted as points in order

of occurrence so that each value has its own identity relative to the time ofobservation. Points on the control charts may or may not be connected. The

Figure 17.1 A typical normal

distribution curve


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chart is provided with limit lines, called control limits, having, in general,one upper control limitand one lower control limit.

A process is said to be in control if the observed values are influ-enced only by chance causes fall within the limits, and out of controlwhen

assignable causes seem to be operating in the system and the observed valuefall outside the limits. It should be remembered that the control limits of the

process control charts do not represent the performance limit or limits of themanufacturing process nor do they represent the specification limits of the

manufacturing drawing. The performance limits of the process are the limit-ing dimensions within which practically all parts fall. If the distribution is

normal or near normal, there are 3 limits of the total distribution. In fact,3 limits form the basis of quality control. However, points which fall out-side of the control limits do not necessarily represent rejected material butonly signal that some corrective action is required to prevent manufacturing

faulty parts.

17.7 CONTROL CHARTS FOR VARIABLES ( X andR Chart)

The chart for average values, the X chart and the chart for ranges, the R

chart, are used for a manufactured part which the inspector checks by mea-surement and not by gauging. Fig. 17.2 shows an example of these two

charts.

The chart for average values, the X chart and the chart for ranges,

theR chart, are used for a manufactured part, which the inspector checks, bymeasurement and not by gauging. Fig. 17.2 shows an example of these twocharts.

XChart : Samples of consecutive parts are taken from the machine

at frequent time intervals and recorded on the chart in their sequence of

manufacture. A sample consists of subgroups, the size of which should be

carefully measured. Generally, the subgroup size is five to ten. It is alwaysbetter to take frequent small subgroups than infrequent large subgroups.

The parts are measured and the average of these measurements is plotted on

the X chart. Average values are plotted instead of individual readings be-

cause sample averages tend to approach the normal distribution curve more

closely than do individual vales. The central line, X shown on the X chart

is average of the averages or grand average of the subgroup (5 to 10) aver-age. This is expressed as

XX

K=


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Where X is the average value of each subgroup and K is the number of

subgroups. The control limits are set at three standard deviations (3) of thesample averages from the mean X , and are called upper control limit. (UCL

x = X +3 x ) and the lower control limit(LCL x = X3 x ), where

xX X

K=

( )2

To shorten the calculation of control limits, the formulas for 3sigma

control limits may be replaced as :

Figure 17.2

Typical X andR control charts


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UCLx X A R

LCLx X A R

= +

=

2

2

where A2 is a constant factor the values of which are found in Table 17.2

and R is the average range.

The average value X being the arithmetic mean of the values in

each group, the control chart detects any change of the arithmetic mean or

central tendency, which may be caused by tool, wear, changed machine ad-

justments, temperature increase, new work materials, or similar causes.Rchart : The chart for ranges, the R chart is obtained from the same

sample groups that were used in determining the values of X . The central

line or the mean, of the R chart represents the average of the subgroupranges. Control limits on range charts can be calculated as :

UCL D R

LCL D R

R

R

=

=

4

3

whereD3 and D4 are constant factors the values of which are found in Table

17.2 and R represents the average of the ranges of each subgroup.

This control chart will help to detect any changes in the processwhich causes the spread of the curve of normal distribution to increase.

Any change of variability shown on the R chart is difficult to ac-count for, but may be due to wear in machine bearings or slides, etc.

In both X and R control charts, as already stated, all values will fall

within the control limits, if the variations are only due to chance causes. If

either the X or R values fall outside of the control limits there are assigna-

ble causes and some corrective action must be taken.

Observation : The X chart in Fig. 17.2 indicates that only at a cer-

tain inspection period a point went beyond the upper control limit otherwise

the process average remained in a state of statistical control. Investigationinto the process showed that the out of control point was due to the cutting

action of a worn cutting tool. Replacement of the tool, however, caused sub-sequent points to fall within the control limits.

Analysis of R chart (Fig 17.2) shows that the entire process remainedin a state of statistical control with all points falling inside the control limits.

Sudden drifting of two consecutive points well above and below the averagerange might have been caused by a number of chance causes.


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It may, therefore, be concluded from the observations that theprocess is under control at a satisfactory level and the product meets specifi-

cation.Objectives of the charts : In general, where control charts for vari-

ables, either X or R, are undertaken, some or all of the following purposes

are present :

1. To secure information to be used in establishing or changing specifica-

tions or in determining whether a given process can meet specification.

2. To secure information to be used in establishing or changing produc-

tion procedures by either elimination of assignable causes of variationor fundamental changes in the procedure.

3. To secure information to be used in establishing or changing inspec-tion procedures or acceptance procedures or both.

4. To provide a basis for decisions during the process as to when to huntfor causes of variation and take necessary corrective action.

5. To provide a basis for decisions on acceptance or rejection of manu-factured or purchased product.

Problem 17.1. Subgroups of 5 items each are taken from a manufacturing process at

regular intervals. A certain quality is measured, and X and R values are computed for

each subgroup. After 10 subgroups, X =76 and R =26. Compute the control-chart

limits.

X = =76

107 6.

andR = =

26

102 6.

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

7 6 0 58 2 6 9 11

7 6 058 2 6 6 09

211 2 6 548

0 2 6 0

. ( . . ) .

. ( . . ) .

. . .

.

17.8 CONTROL CHARTS FOR ATTRIBUTES (p charts)

X and R charts have many advantages but they are charts forvariables, i.e.,for quality characteristics that can be measured and expressed in numbers.

Many quality characteristics can be measured only as attributes, i.e., byclassifying each item inspected into one of two classes: either good or bad,

acceptable or unacceptable, conforming or nonconforming to the specifica-tion.


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TABLE 17.2 FACTORS USED IN X AND R CHARTS

Sample size(Number of sam-

ples in a sample)

n

Limit average

A2

Range lowerlimit

D3

Rangeupper

limit

D42

3

4

5

6

7

8

9

1012

1.88

1.02

0.75

0.58

0.48

0.42

0.37

0.34

0.310.27

0.00

0.00

0.00

0.00

0.00

0.08

0.14

0.18

0.220.28

3.27

2.57

2.28

2.11

2.00

1.92

1.86

1.82

1.781.72

The control charts for fraction defective p is used to quality charac-teristics that can be observed only as attributesfor example, dimensions

checked by go and non-go gauges even though they might have been mea-sured as variables. As long as the result of inspection is a classification of

an article as accepted or rejected, a single pchart may be applied to one

quality characteristic or a dozen or a hundred. As a result, it may also be ap-plied as a tool for saving the cost of computing and charting quality criteria.

The p-chart has somewhat the same objectives as the X and R chart.

It discloses the presence of assignable causes of variation, even though it is

much inferior to those charts as an instrument for actual diagnosis of causesof variation. It is used effectively in the improvement of quality by immedi-ate correction in the process before large quantities of scrap are produced.

In effect, the chart gives advance warning of the commencement of a trendtoward the production of an increasing number of defective articles. In addi-


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tion to this, the p-control chart provides management with a useful recordofquality history about changes that may occur in the quality level.

Fraction defective, p, may be defined as the ratio of the number ofdefective articles found in any inspection or series of inspections to the

number of articles actually inspected. Fraction defective is nearly alwaysexpressed as a decimal fraction. This may be expressed as:

number of defectives in subgroupp =

number inspected in subgroup (n)

_ total number of defectives during periodand p =

total number inspected during period (n)

Wherever practicable, it is desirable to have data for at least 25 sub-

groups before computing p and establishing control limits.

Per cent defective is 100p i.e. 100 times the fraction defective. As for

example, if p=0.0092, the per cent defective is 100p=0.92 per cent. For ac-tual calculation of control limits, it is necessary to use the fraction defective.

For charting, and for general presentation, the fraction defective is generallyconverted to per cent defective.

Just as in the case of the control charts for X and R, the control chart

forp has a central line indicating average value p , and upper and lower

control limits, which are equally distant from the central line. A typicalp-

chart is shown in Fig. 17.3.


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The formula for 3-sigma control limits are :

UCL p

p

p p

n

LCL p

pp p

n

p p

p p

= +

= +

=

=

3

3

1

3

31

( )

( )

The value of 3 1p p( ) can be computed once to apply to all cal-

culations of control limits. The value ofn is computed for each day and

divided into 3 1p p( ) to get the value of 3p for the day.

Observation : The process shows a drift towards running out of controlwhich if continued would require action, thus saving an accumulation of

scrap.Purpose of the p chart: As applied to 100 per cent inspection a control

chart for fraction defective may have any or all of the following purposes :

Figure 17.3

Typicalp control chart


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1. To discover the average proportion of defective articles or parts sub-mitted for inspection over a period of time.

2. To bring to the attention of management may changes in the averagequality level.

3. To discover those out-of-control high spots that call for action to iden-tify and correct causes of bad quality.

4. To discover those out-of-control low spots that indicate either relaxedinspection standard or erratic causes of quality improvement.

5. To suggest places for the use of X and R chart to diagnose quality

problems.

6. To afford a basis for judgment whether successive lots may be consid-ered as representative of a process.

Chart for p and chart for np : Whenever subgroup size is variable, the con-

trol chart must show the fraction defective or proportion defective ratherthan the actual number of defectives. However, if subgroup size is constant,

the chart for actual number of defectives, called np orpn chart, may beused.

The construction ofnp chart is identical to p. After finding out thevalue ofp this is multiplied with n to get np which indicates the central

line. The upper and lower control limits are :

UCL np

LCL np

np np

np np

= +=

3

3

where, 3np = 3 1np p( )

17.9 CONTROL CHART FOR DEFECTS (cCHART)

Thep ornp chart applies to the number of defectives in subgroups of con-stant size, while the c chart applies to the number ofdefects in subgroups of

constant size. A defective is an article whereas articles lacking conformity

to specification is a defect. Every defective may contain one or more de-fects, for example a cast part may have blow holes and surfaces cracks at

the same time.

In most cases, however, each subgroup for the c chart consists of asingle article; the variables c consists of the number of defects observed in

one article. But it is necessary that the subgroup for the c chart be a singlearticle.


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The c chart is preferred for large and complex parts. Such parts be-ing few and limited, the use ofc chart, as compared top charts, is restricted

to limited use.The c chart is plotted in the same manner as p chart except that the

control limits are based on Poisson distribution which describes more ap-propriately the distribution of defects.

The value of c is first computed. The 3-sigma limits are :

UCL c c

LCL c c

c

c

= +

=

3

3

For example, if there are 200 defects in 25 machines, the average c is

200/25=8.0. Control limits computed from the average are as follows:

UCL c c

LCL c c

c

c

= + = + =

= = =

3 8 3 8 165

3 8 3 8 0

.

Whenever calculations give a negative value of the lower controllimit of a c chart, that limit is recorded as zero.

17.10 APPLICATION OF CONTROL CHARTS

The following examples are given to show the applications of control chartsin controlling the quality characteristics of components :

1. Final assemblies (Attribute charts).

2. Manufactured components, such as shafts, spindles, balls, pin holes,slots, etc., (Variable charts).

3. Bullets and shells (Attribute charts).

4. Welded and soldered joints (Attribute charts).5. Cast and forged parts ; long lengths of cloth, rubber, etc., defects in

parts made of glass ; large and complex products like bomber engines,

turbines, I.C. engines ; etc., (c charts).

6. Punch press works, forming, spot welding, etc., (Attribute charts).7. Studying tool wear (Variables charts).8. Incoming materials (Attributes or variable charts).


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Problem 17.2. Ten sample of parts were taken from a production line for 100 per centinspection, each sample containing 300 parts. The total number of defec-tives was 350. Compute upper and lower control limits.

total number of defectives

p = total number of pieces inspected

p = 350/(10 300) = 0.1167

n = number of pieces inspected every day

p p

n

( ) . .1 01167 08833

300

=

= 0.01852

and 3p p

n

( )1= 3 0.01852 = 0.05556

Thus UCLp = p + 3

p p

n

( )1= 0.1167 + 0.0556

= 0.1723

LCLp = p 3p p

n

( )1= 0.1167 0.0556

= 0.0611

Problem 17.3. Ten castings were inspected in order to locate defects in them. After in-spection total 37 defects were found. Compute c-control limits.

c = 37/10 = 3.7

__

Therefore, UCLc = c + 3c = 3.7 + 3 3.7 = 9.472__

LCLc = c 3c = 3.7 3 3.7


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= 2.072 = 0 (Zero, as negativevalue is not feasible)

17.11 ACCEPTANCE SAMPLING

Control over quality can be established in two ways :1. By controlling the actual processes that make the part, and

2. By controlling the level of outgoing quality through proper inspec-tion procedure.

While the first one is the direct application of statistical control chart,

thesecondone applies statistical procedure.Acceptance sampling, determines through the use of samples whether

to accept or reject a manufactured lot of final products or components.

In the simplest case of acceptance sampling, a random sample of size n

is drawn from the total lotN. The items are inspected and on the basis of therejects of inspection, a decision is taken either to accept or reject the lot. If

the decision is taken to reject the lot , it may then either be subjected to 100percent inspection or be returned to the original supplier.

Acceptance sampling plan is normally designed to discriminate effec-tively between good and bad lots. This statistical procedure should accept

lots of poor quality with low probability. Thus acceptance sampling encour-ages the vendors to improve quality at the first instance and vendors are

likely to take the attitude make it right the first time. This psychological as-pect of acceptance sampling is thus of major importance.

In general, acceptance sampling is applicable when :

1. The inspection process is costly and the possible loss arising out ofpassing defective items is not high.

2. The inspection process results destruction of the product.3. 100 percent inspection does not yield significantly better result than a

planned sampling procedure.The acceptance sampling can further be divided in two types : (i) ac-

ceptance sampling by attributes, and (ii) acceptance sampling by variables.

17.12 ACCEPTANCE SAMPLING BY ATTRIBUTES

Acceptance sampling by attributes is based on the classification of parts ei-ther good or bad, acceptable or unacceptable. For inspection of componentsthis procedure is often accomplished by the use of snap gauges. Snap

gauges classify the parts either bad or good without differentiating how bador how good a component may be.


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17.13 OPERATING CHARACTERISTIC CURVE

The operating charac-

teristic curve or OCcurve describes in

graphical form the probabilities associat-

ed with accepting lotsof varying quality for a

particular sampling plan. An operating

characteristic curvecan be constructed by

plotting individual per-

cent defective values

against their corre-sponding probabilities

of acceptance. The Yaxis of the OC curve

represents the probability of accepting a lot, while the Xaxis shows the per-centage of defective units in the lot. For example, suppose an operating

characteristic curve is to be constructed for n=100 and c=3, for lots. As-sume that lot percent defective is 2% then the probability of acceptance will

be equal to the sum of probability of getting zero, one, two, and three defec-tives. Binomial distribution probability of acceptance for this case can be

expressed as:

Pa =0

c

n= P(n/N, p) where Pa = probability of acceptance, n = number of

items inspected, N = sample size

Pa =P(0/100, 0.02) +P(1/100,0.02) +P(3/100,0.02)

or,Pa= 0.135 + 0.271 + 0.271 + 0.18 = 0.857

A table of probability of acceptance for various quality level for asampling plan ofn = 100 and c = 3 is shown below. The values of such ta-

ble are derived as approximate from the Poissons cumulative table. For ex-ample, forp = 0.02, np or u is 0.02 100 or 2 and probability of acceptance

is 0.857 forc = 3 from the Poissons distribution table.

Table 17.3 Probability of acceptance for n = 100 and c = 3

Fig.17.4 Operating characteristic

curve for n = 100, c = 3


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np 0.35 0.80 2.0 2.8 3.2 3.6 4.4 5.4 6.6 7.0

p in%

0.35 0.80 2.0 2.8 3.2 3.6 4.4 5.4 6.6 7.0

Pa 1.00 0.99 0.86 0.69 0.60 0.51 0.36 0.21 0.11 0.08

In Table 17.3, or np is the actual percent defective in lot,

p is calculated by di-

viding np values by n.Pa value is available for

Poisson cumulativedistribution table.

Fig.17.4 shows thegraph corresponding to

Table 17.3.

17.14 COMPARI-

SON OF OC

CURVES

Different sampling

plans will have different OC curves. Fig.17.5 shows the OC curves for sam-ple sizes 100 and 300 with the acceptance number 1 and 3. Here the accep-

tance numbers are re-

maining proportion tothe sample size. From

Fig.17.5, it is seen thatthe OC curve becomes

steeper as the sample

size increases. If acomparison is madefor the discriminating

power of the two OCcurves, it can be seen

that both of them

Figure 17.5 Change in OC curve with

change in sample size (n/c remains con-

stant)

Figure17.6 Change in OC curve with a

change in acceptance number


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would accept lots of about 0.8 percent defective about 84 percent of thetime. It can also be seen that if the actual quality falls to 2 percent defective,

the plan with n = 100 will accept about 40 percent of the times and the planwith n = 300 will accept about 15 percent of the time.

In other words, if the acceptance number is kept proportional to sam-ple size, the plans with large sample sizes will definitely be more effective.

Fig.17.6 shows various OC curves for a sample size n = 100 with differentacceptance number (c

= 0, 3, 5). It is seenthat with increase ofc

values, the OC curvesmove toward right

side in Fig. 17.6.

However OC curves

become steeper forlower acceptance

number indicating thatthe plan is tighter.

With same percentagedefective probability

of acceptance is lowerwith lower c values

and as such lessernumber of defectives

will pass through the inspection point with lowerc values if sample size iskept constant Fig.17.7 shows that with any particular acceptance number (c

= 5) discriminating power of OC curve increases with sample size.

Figure 17.7 Discriminating power of OC

curve with variation of sample size


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An ideal sampling plan will beone that discriminates perfectly be-

tween good and bad lots. This planwill have its OC curve as a vertical

line indicating that all good qualitylots are accepted with probability 1

and all bad quality lots are acceptedwith zero probability. Fig.17.8 shows

such ideal sampling plan. Unfortu-nately, this ideal plan is a hundred

percent inspection plan and is verycostly. Thus for selecting a plan, it is

obvious that a balancing should be made between the cost of inspection and

other costs due to passing of defective items.

17.15 PRODUCERS AND CONSUMERS RISKS

There are two points on the OC

curve that are of special impor-tance. These two points corre-

spond to two acceptable errorlevels : (i) rejecting a lot which is

considered to be of good qualityand (ii) accepting a lot which is

considered to be of bad quality.These two errors are termed ac-

ceptable quality level (AQL) andlot tolerance percent defective

(LTPD).

Acceptable quality level : The

term acceptable quality level(AQL) refers to the level of qual-

ity of a product that is judged bythe consumer to be good in terms

of the percentage of defectiveitems. As the quality level is

good, it is expected that these lots will be accepted all the time. However

the decision based on sampling may be erroneous and as such it is desirablethat probability of acceptance of these lots can be kept high. Fig.17.9 showsAQL value to 1 percent and its corresponding probability of acceptance is

95%.

Figure 17.8 An ideal sampling

plan

Figure 17.9 Producers and con-

sumers risk


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Producers risk : Producers risk () refers to the probability that lots of theacceptable quality level will not be accepted. A producer will always likethat the risk involved for rejecting a good lot is reduced. Normally for AQL

level it is specified as and can be equated to 1Pa. In Fig.17.9 producerrisk is shown as 5%.

Lot tolerance percent defective (LTPD): LTPD is the dividing line made

by consumer to differentiate between good and bad lots. The consumer con-

siders lots of this level as poor and likes to keep low probability of accep-

tance. In Fig.17.9 LTPD is shown as 5%.

Consumers risk : Consumers risk () is the probability that the lots of thequality level LTPD will be accepted. Usually consumers risk () is taken as

10%.

17.16 DOUBLE AND SEQUENTIAL SAMPLING PLAN

Figure 17.10 Structure of a double sampling plan


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The sampling can discussed earlier considers sampling only once. The deci-sion to reject or to accept the lot is derived by seeing the outcome of the sam-

pling. A single sampling plan requires a larger sample size and as such thecost of inspection is comparatively high. For this reason double and sequential

sampling plans are used to make sampling plans more cost effective.

Double sampling plan : Double sampling plan follows a scheme, structure

of which is shown in Fig.17.10. In this scheme, initially a sample on n1items is taken and the sample is inspected. The number of defects found in

the sample is compared to two acceptance compared to two acceptance

numberc1 and c2. Here c1


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17.17 ACCEPTANCE SAMPLING BY VARIABLES

In acceptance sampling be variables, a record is made on the actual mea-

surement of the parameters instead of simply classifying items as bad orgood. In acceptance sampling by variables, after measuring each of the

items, the sample mean and variance (or range) are computed. The lot is ei-ther rejected or accepted after analysing these statistical measurements.

It is interesting to note that most of the times the acceptance sam-pling is done by attribute and not by variables. The reason of not selecting

acceptance sampling by variables may be one or more than one of the rea-sons, listed hereunder.

1. Snap gauges can compare a parameter and as such actual measure-ment is not needed.

2. A proper measuring tool is not available.

3. The cost of actual measurement is high.

However, acceptance sampling by variable have certain potential ad-vantages. The first is that smaller sample can adequately achieve a given

level of quality protection. Secondly, the information on the degree of varia-tion can be utilised for cause analysis.

17.17 ACCEPTANCE SAMPLING BY VARIABLES

In acceptance sampling be variables, a record is made on the actual mea-

surement of the parameters instead of simply classifying items as bad or

good. In acceptance sampling by variables, after measuring each of the

items, the sample mean and variance (or range) are computed. The lot is ei-ther rejected or accepted after analysing these statistical measurements.

It is interesting to note that most of the times the acceptance sam-pling is done by attribute and not by variables. The reason of not selecting

acceptance sampling by variables may be one or more than one of the rea-sons, listed hereunder.

Snap gauges can compare a parameter and as such actual measurementis not needed.

A proper measuring tool is not available.The cost of actual measurement is high.

However, acceptance sampling by variable have certain potential ad-vantages. The first is that smaller sample can adequately achieve a given

level of quality protection. Secondly, the information on the degree of varia-

tion can be utilised for cause analysis.

REVIEW QUESTIONS

1. Why is inspection of manufactured part necessary, and what is the primary


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2. responsibility of the inspection department ?3. Explain the difference between (1) inspection, (2) quality control and (3) statistical

quality control.

4. Describe briefly (1) variables, (2) attributes in statistical concept.

5. State why normal distribution curve is usually taken into consideration for statisti-cal quality control.

6. State why are average values ( X ) are plotted instead of individual values (X) on a

control chart.

7. In process control for variable why it is necessary to use both a chart for averagesand ranges. State what these charts signify ?

State whyp and c control chart are used. What they signify ? State the advantages of a pchart over a X chart.

What do you mean by acceptance sampling? State their usefulness over 100 per cent in-

spection.Subgroups of 5 items each are taken from a manufacturing process at regular intervals.

A certain quality characteristic is measured, and X andR values are

computed for each subgroup. After 25 subgroups, X =357.50 and R=8.80.Compute the control chart limits. All points on both charts fall within these limits.

If the specification limits are 14.40 0.40, what conclusions can be draw about theability of the existing process to produce items within these specifications? Sug-gest possible ways in which the situation could be improved.

The resistance in ohms of a certain electrical device is specified as 200 15. A controlchart is run on the manufacturing process with samples of 4 taken from the produc-

tion line every hour for 20 hr. X =4,140. R=288. What should be the 3- con-trol limits on the X andR charts?

8. Ap chart is to be used to analyze the September record for 100 per cent inspectionof certain radio transmitting tubes. The total number inspected during the monthwas 2,196, and the total number of defectives was 158

Date Number inspected Number of defectives

Sept. 14 54 8

15 162 2416 213 3

01quality_controlf_

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