2.13 Quality Control
We are all living in a competition age. In order to combat competition, every manufacturer would like to
maintain the quality of his products so that it may be liked by the users in preference to products of his
competitors. Reputation for uniformity and dependability of the quality of a product is one of the most
important assets which a manufacturing concern should try to acquire or not to lose if already acquired.
There is an old saying heard very often in business circles to the effect that one is reminded of quality long
after price is forgotten. Yet the management often fails to appreciate fully the truth of this statement. At the
time when production schedules are pressing, the tendency quite often is to sacrifice quality in favour of
quantity. The tendency is reflected in customer ill-will and reduced sales volume. Poor quality or reduced
quality costs the concern much indirectly.
So, in order to maintain the quality of the product, quality control is applied at the work place. Quality of
raw materials and the parts and the machines and equipment should be inspected quite strictly because
quality of product depends much upon the quality of raw materials. During the process of production,
several checks are done through inspection of the semi-finished product and finished product and it is
confirmed whether the quality of the product conforms to the standards and specifications already set. For
this purpose, several devices or techniques are applied. If the quality of the product, it is found during
quality control process, is found not conforming to the specifications, the steps should be taken by the
production department to improve the quality of the defected articles by reworking on those articles and if
it, anyhow, is not possible, other measures to sell those sub-standard items should be taken such as selling
such items at reduced rates as ‘B’ quality items.
This topic deals with the various devices or techniques of quality control (including statistical quality
control and inspection).
Quality Control – Meaning, Objects and Importance
Meaning of quality control : The term ‘quality control’ consists of two words ‘quality’ and ‘control’. Quality
is that characteristic or a combination of characteristics that distinguishes one article from the other or
Week
1st 2nd 3rd 4th
Actual output achieved 383 442 350 318
Achievement percentage
(Actual output/ Standard output) × 95.75% 110.5% 87.5% 79.5%
100 (383/400) × 100
Wages at piece rate at Rs. 5/- Rs. 1,915 Rs. 2,210 Rs. 1,750 Rs. 1,590
Dearness allowance 120 120 120 120
2,035 2,330 1,870 1,710
Incentive % 121/2
% 17 1/2
% 10% Nil
Incentive earned by group Rs. 254.38 Rs. 470.75 Rs. 187 Nil
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Production Planning and Productivity Management
goods of one manufacturer from that of competitors or one grade of product from another when both are
the outcome of the same factory. The main characteristics that determine the quality of an article may
include such elements as design, size, materials, chemical composition, mechanical functioning, electrical
properties, workmanship, finish and appearance. The quality of a product may be defined as the sum of a
number of related characteristics such as shape, dimension, composition, strength, workmanship,
adjustment, finish and colour.
‘Control’ may be referred to as the comparison of the actual results (finished product) with the predetermined
standards and specifications. It locates the deviations and tries to remove them. Control is the correction in
the quality of the produce when deviations in the quality are more than expected in the process. Control
consists in verifying whether everything occurs in conformity with the plan adopted, the instructions
issued and principles established. It has for object to point out weaknesses and errors in order to rectify
them and prevent recurrence. It operates on everything – things, people and action.
Thus, by the term quality control, we mean the process of control where the management tries to conform
the quality of the product in accordance with the pre-determined standards and specifications. It is a
systematic control of those variables that affect the excellence of the ultimate product.
Quality control may be defined as that industrial management technique or group of techniques by means
of which products of uniform acceptable quality are manufactured.
Quality control refers to the systematic control of those variables encountered in a manufacturing process
which affect the excellence of the end product. Such variables result from the application of materials,
men, machines and manufacturing conditions.
Thus quality control is a technique of scientific management which has the object of improving industrial
efficiency by concentrating on better standards of quality and on controls to ensure that these standards
are always maintained. In this way, for quality control purposes, first standards and specifications are
established and then to see whether the product conforms to those standards.
Objectives of quality control: The following are the main objectives of quality control programme:
1. To assess the quality of the raw materials, semi-finished goods and finished products at various stages
of production process.
2. To see whether the product conforms to the predetermined standards and specifications and whether
it satisfies the needs of the customers.
3. If the quality of the products deviates from the specifications, able to locate the reason for deviations
and to take necessary remedial steps so that the deviation should not be recurred.
4. To suggest suitable improvements in the quality or standard of goods produced without much increase
or no increase in the cost of production. New techniques in machines and methods may be applied for
this purpose.
5. To develop quality consciousness in the various sections of the manufacturing unit.
6. To assess the various techniques of quality control, methods and processes of production and suggest
improvement in them to be more effective.
7. To reduce the wastage of raw materials, men and machine during the process of production.
Importance or advantages of quality control system. The programme of quality control is advantageous to
producers and consumers both. A quality product will satisfy the customer’s needs on the one hand and
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consequently the demand of the product will increase resulting in large-scale production. On the other
hand, the goodwill of the firm increases as the producer of quality goods. It helps the producer in increasing
the market for the goods. The importance of quality control lies in the following facts:
• Reduction in costs: An efficient quality control system reduces the cost of production of the product
due to (i) reduction in wastage of raw materials, semi-finished and finished goods; (ii) large-scale
production of standard quality product; (iii) rework cost of the substandard goods is the minimum.
• Improvement in the morale of employees: By quality control programme, the employees become
quality-conscious. They understand the standards of the product well and try to improve the standards
and produce the quality goods to the best of their efforts. Thus it improves the morale of the employees.
• Maximum utilisation of resources: By establishing the quality control system, the necessary control
over the machines, equipment, men and materials and all other resources of the company is exercised.
The system will also control the misuse of facilities, wastages of all types and low-standard production.
Thus, the resources of the company are put to maximum use.
• Increase in sales: Increase in sales of the product is the main objective of the quality control system.
By introducing quality control programme in manufacturing process, a quality product is made
available to the consumers and that is too at lower rates because of lower cost of production. It in
turn increases the demand of company’s product.
• Consumer’s satisfaction: Consumers always get the quality products of standard specifications which
they find to their utmost satisfaction.
• Study of variations: It is a well-known fact that some variations are bound to exist in the nature of
production inspite of careful planning. The magnitude of variations depends upon the production
process namely machines, materials, operations etc. The techniques of quality control helps in the
study of these variations in quality of the product, and serves as a useful tool for the solution of many
manufacturing problems which cannot be solved so well by any other method.
Thus quality control is an important technique in the hands of management to maintain the quality of the
product.
Meaning of inspection: Inspection is an important and essential tool of quality control that ascertains and
controls the quality of a product. The main purpose of quality inspection is to safeguard quality by comparing
materials, workmanship and pro ducts with the set standards. Inspection is a method by which the inspector
may decide how much of the total work done conforms with the pre-determined standards or how much
is below-standards so that a decision may be taken for its approval or rejection. If it conforms to specifications,
it will be accepted otherwise the whole lot or a part of it which does not conform to standards will be
rejected. If defects in the items are beyond acceptable limits, a corrective action can also be suggested so
that the future production should be strictly according to the specification because, the fundamental purpose
of inspection is to have an effective check on the production of defective items to ensure the quality of the
product and to lower down the cost of production. Thus inspection can also be termed as a sorting process
on the basis of which products can be classified into acceptable or unacceptable.
Inspection is the art of comparing materials, products or performances with established standards. There
can be no intelligent inspection without definite standards. In any such items that are to be inspected,
some will fall outside a liberal allowance of variation from the standards, some will be well within the
limits of error, and others will be very close to the limits. Inspection is the art of selecting these three classes
of product which will be satisfactory for the work.
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Production Planning and Productivity Management
Objectives of inspection: The main objectives or functions of inspection are as follows:
• Maintenance of quality: The fundamental purpose of inspection is to maintain the quality of the
product. This function is performed by comparing materials, semi-finished or finished products, men
and machines and tools with the established standards. Items which conform to the specifications or
are within the acceptable limits are accepted and other items are rejected.
• Improving the product quality: By comparing the quality of the products against the set standards,
the defective items are located and probable reasons for the defects are established. Necessary
adjustments are done for future by removing the reasons for defects and thus the quality of the product
is improved steadily and regularly. It helps in safeguarding the prestige and confidence of the
organisation in the eyes of the consumers.
• Reduction in costs: As raw materials are inspected to see whether they are as per standards or not the
defective raw materials are thus not allowed to be used in production. Thus it saves the organisation
from loss if any and reduces the costs of production.
Statistical Quality Control:
Inspection is an integral part of the scheme of production control. By inspection, quality of the product is
conformed with the standards and specifications during or at the end of the production process. It is
imperative to have certain disturbances which cause the product to deviate slightly from the desired
standards. This type of variability is inherent in the process and is known as variability due to chance
causes. Causes of deviations like conditions of production process, nature of raw materials, the behaviour
of operations etc. Such causes are assignable causes. Due to these causes, defective items are produced
which lower the quality of the product.
Inspection has rather a limited use. It does not provide the extent to which certain products requiring a
high degree of precision conform to the strict standards set in this regard. Similarly, it does not provide all
the information on the basis of which sub-standard finished products can be scrapped in the case of mass
production industries, producing standardised products. These imperfections of inspection in quality control
make the use of statistical quality control indispensable in these cases.
Statistical quality control is the application of statistical techniques to -determine how far the product
conforms to the standards of quality and precision and to what extent its quality deviates; from the standard
quality. The purpose of statistical quality control is to discover and correct only those forces which are
responsible for variations outside the stable pattern. The standard quality is pre-determined through careful
research and investigation.
It is quite impracticable to adhere strictly to the standards of precision, especially in cases where human
factor dominates over the machine factor. Some deviation is therefore, allowed or tolerated. They are
referred to as tolerances. Within the limits set by these tolerances, the product is considered to be of standard
quality. SQC brings to light the deviations outside these limits.
Techniques of statistical quality control. The techniques of statistical quality control can be divided into
two major parts:
1. Control charts, and 2. Acceptance sampling:
1. Control chart: Control chart is the most important quality control technique. It is a chart and depicts
three lines on the chart. One line is the central line showing the average size. The other two lines, one
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below the central line and the other above the central line, indicate the limits of tolerances, within
which deviations from standards are permissible. The actual measurement of the whole lot or a sample
is plotted on the chart. Those measurement values which fall outside the tolerance limits are considered
to be out-of-control points and assignable cause may be said to exist. This will enable die manufacturers
to know the causes of variation or causes of trouble which he can amend.
2. Acceptance sampling: It can be described as the post-mortem of the quality of the product that
hasalready been produced. Under this technique, a sample is selected at random to examine whether
it conforms to the standards laid down. It can be assumed that a certain percentage of goods will not
conform to the standards, so a certain percentage of defective products in a lot may be specified. This
technique has all the limitations of sampling technique. There are two limiting levels of quality in an
acceptance sampling plan:
(i) the acceptable quality level (AQL) that represents the lowest percentage of defectives which a
buyer is expected to accept and seller is expected to supply, and
(ii) the lot tolerance percentage defective (LPTD) that represents a limit at which the buyer wants to
be quite certain that the lot will not be passed.
The two limiting levels decide the risks to be borne. The AQL involves the producer’s risk or the risk that
a lot with an acceptable quality will be rejected on the basis of sample inspection. The greater the risk of the
producer, the higher will be the charges for the product. On the other hand, risks attached to LPTD are
called the consumer’s risk. Since only a sample is inspected under the plan, it is quite likely that an item
which does not conform to standards may be accepted on the presumption that the sample is typical of the
whole lot. The acceptance sampling inspection may be of any of the following types:
(i) Inspection by attributes,
(ii) Inspection by variable, and
(iii) Inspection by number of defects per unit.
Importance or benefits of statistical quality control: The technique of statistical quality control has become
very popular since the days of World War II. In modem industry, it becomes a necessity since it offers the
following benefits:
• It saves on rejection: In the absence of statistical quality control technique, many products may be
found defective and worthless at the end of manufacturing process and may be thrown away as
scrap. SQC avoids such a situation and saves the cost of labour and material involved in the production
of defective items. It measures the extent of defect and defective products may be improved by reworking
to the level of acceptable standards. It also saves the loss which will arise out of re-working
on the items rejected outrightly or which cannot be brought to the acceptable standard.
• It maintains, high standard of quality: The statistical quality control ensures the maintenance of
high standard of quality because lower standard of quality products are not put to the market. They
are improved, if possible or rejected outrightly but in no case, sub-standard item is sold in the market.
Thus, the concern gains in goodwill.
• Reduced expenses of inspection: It reduced the expenses of inspection to a great extent and enables
the product to be manufactured at lower cost.
• Ensures standard price: If certain products are not up to the desired standard of quality and cannot
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Production Planning and Productivity Management
be improved without much expense, they can be downgraded and sold cheaper. SQC maintains the
standard price for all standard products. Thus, it increases the profitability of the concern.
• Feelings of responsibility among workers: Among workers, a feeling of responsibility develops
because they begin to understand that their work is being inspected very minutely hence they work
carefully. It helps increasing their moral.
Benefits of SQC:
(i) It saves on rejection
(ii) It maintains high standard of quality
(iii) It reduces expenses of inspection
(iv) It ensures standard price for all products
(v) Feelings of responsibility among workers development
Control Charts:
The concept of control charts and their application in production process was evolved for the first time in
1924. This concept is based on the division of observations in rational sub-groups. The sub-groups are
formed in such a way that the variation within each sub-group is attributable due to chance causes only
and variation between the sub-groups are attributed due to assignable causes. The most obvious basis for
the selection of sub-group is the order of production. The problem of process control can be solved by
applying a method which helps in finding out the quality characteristics of various sub-groups. If the
quality characteristics of various sub-groups are identical, the process is said to be in control otherwise out
of control.
Statistical basis for control charts: Suppose Q is the quality characteristic of a product. This characteristic
should be measured for a number of sub-groups. If Q follows some standard distribution (Normal, Poisson
or Binomial) with mean MQ and standard deviation óQ. If the sample size is sufficiently large then every
distribution will be normal and due to the assumption of normality the interval MQ± 3 σ Q will cover
approximately 99.73% values of Q. If all the observed values of Q lie within the limits MQ+3 σ Q, this may
be taken as the indication of absence of assignable cause. If any value of Q lies outside the limits, the
presence of assignable cause can be presumed and the process should be corrected before it is allowed to
continue.
The control chart is a horizontal chart where time variable is taken on X-axis. A control line (CL) is drawn
along the X-axis at MQ point. Time for two other limits, lower control limits LCL (MQ-3 σ Q) and upper
control limit UCL (MQ+3 σ Q) are drawn taking the CL in the centre. The values of quality characteristics
Q are also plotted on group for different observations from sample to sample or according to time variables.
The following observations can be had from the control chart.
(i) If all the plotted points for Q lie within the control limits, the process is said to be in control otherwise
it can be said to be out of control with respect to the quality characteristic Q.
(ii) If the points on a control chart lie close to one of the control lines or the points show some special
trend, then it is very difficult to pay anything about the process control. A typical control chart can be
drawn in the following figure—
X
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Thus control chart is a graph showing the range of expected variability. The following points should be
considered while drawing a control chart.
(i) If two comparable control charts are to be drawn on the same paper, it is better to take the same scale
for both the charts.
(ii) The charts should be narrow in comparison to their length.
(iii) So many charts should be avoided on the same sheet.
Types of Control Charts: Control charts are of two types: (i) control chart for variables, and (ii) control
chart for attributes.
If the quality characteristic Q is capable of quantitative measurement, the control chart is known as control
chart for variables and if the quality characteristic Q cannot be measured quantitatively the control chart is
known as control chart for attributes. The items can only be classified as defective or non-defective etc.
1. Control Chart for Variables. In this case, quality characteristic Q is capable of direct quantitative
measurement and is a continuous variable. In such cases the average and the standard deviation of
the quality characteristic can be calculated from a number of samples each having fixed number of
components, taken at random over a period of time from any process. The following situations can be
encountered in practice:
(i) The process may be in control.
(ii) The mean of the characteristic is out of control but standard deviation is not. For this purpose, control
chart for means (X-chart) are prepared.
(iii) The standard deviation is out of control and not the mean. This is studied by control chart for range
(R-chart). Here range is the measure of variability because it can be easily determined.
(iv) Both mean and standard deviation are out of control. This is studied by X and R charts simultaneously.
(A) Control chart for mean ( X -chart):
By this chart, it is found whether the mean of the characteristic can be determined by the following formula:
Central line =
=
=
ΣX
X
K
UCL (MQ + 3 σ Q)
CL (MQ)
LCL (MQ – 3 σ Q)
0 1 2 3 4 5 6 7 8 9 10 11 X
TIME VARIABLE OR SAMPLE NUMBER
FIG. CONTROL LIMITS
QUALITY CHARATERISTIC (Q)
Y
X
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Production Planning and Productivity Management
Where =
X = Mean of the sample means
Σ X = Sum of the sample means
K = No. of samples.
By this method lower control limit (LCL) and upper control limit (UCL) are determined as follows:
(i) Where standard deviation is known –
UCL =
= + 3σP
X
N
+ and LCL =
= − 3σP
X
N
where UCL = upper control limit
LCL = lower control limit
=
X = mean of the sample means
σP = standard deviation of population
N = total number of items in a sample.
(ii) Where standard deviation is not given –
The standard deviation can be determined by the following formula:
2
σP = R
d
Where R = mean of sample range
d2 = quality control factor.
(iii) Where d2 is not given, then
UCL = =
X + A2 R
LCL = =
X - A2 R
where A2 = quality control factor.
(B) Control chart for range or range chart (R-chart). This chart is used to see whether the standard deviation
of the characteristic Q is in control or not. Here can be two situations, viz, (i) where σ P is known. (ii)
where σ P is unknown.
(i) Where σ P is known. In this case central line (CL) and upper and lower limits can be determined
as follows:
Range (R) = Highest value – lowest value
Mean of R or R =
ΣR
K
Central line = R = d2 σP where K = total no. of samples.
UCL= d2 σP + 3 σP .d3
LCL= d2 σP – 3 σP d3
Where d2 and d3 can be determined from statistical tables.
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(ii) Where σ P is unknown
CL = R =
R Sumof samplerange
K No.ofSamples
=
Σ
UCL = D4 R
LCL = D3 R
Where D3 and D4 can be read from the statistical table for control charts. As a rule if LCL is negative, it is
taken to be zero.
Practical limitations of X and R-charts: X and R charts are very important tools for the diagnosis of
quality problems in a manufacturing process but they have the following practical limitations:
1. These charts can be used where quality characteristic can be measured quantitatively.
2. Quantitative measurement of quality characteristic is a costly proposition, hence its use is uneconomical
Interpretation of X and R-charts. X and R-chart can be used simultaneously to judge the quality of the
process as follows:
(i) Where R - chart shows all the points within control limits:
(a) If in X - chart, the points lie beyond one of the control limits, it shows that process level has
shifted.
(b) If in X - chart, the points lie beyond both of the control limits, it shows that the process level is
changing at random and needs frequent adjustment.
(ii) Where R-chart shows variability out of control:
(a) If X - chart shows points beyond one of the control limits, it signifies that both process level and
variability have changed.
(b) If X -chart shows points beyond both of the control limits, then this implies that variability has
increased.
(iii) If points in X and R-charts are too close to the central line, then it shows that there exist systematic
difference within sub-groups.
2. Control chart for attributes: Where the nature of product is such that the quality characteristic cannot
be measured quantitatively, the items are classified only defectives and non-defectives at the time of
final inspection.
There can be a number of factors responsible for defining any item to be defective and the separate record
for each cause maybe out of question. The defects can be measured in airy of the following two ways:
(i) Number of defective items are taken in different samples. Here P or n P charts is used.
(ii) Number of defects in one item. In this case C chart is used.
Control chart for fraction defectives ( P or n P charts )
These charts are constructed by recording at least 20 successive inspections. The percentage of defective
items is then calculated. The limits for P-chart are given by:
CL or =
×
ΣP
P
n k
UCL or upper control limit =
P(1 P)
3
n
+ − P
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Production Planning and Productivity Management
LCL or Lower control limit =
P(1 P)
3
n
− − P
Here P = central line
P = No. of defective items in a sample
K = Total no. of samples
n = Sample size (no. of items in a sample),
nP chart = In situations where the number of items inspected in each sample is the same, nP charts are
prepared. The various control limits in this case are
CL = n
ΣP
P =
K
UCL = nP + 3 P(1−P )
LCL = nP - 3 nP (1− P )
P and nP charts can be used in the case of variables also.
C-charts. C = charts are prepared where defective items are taken out by the number of defects in one
item. Items which are according to specifications are termed as standard items. Items which have one
or more defects, it means they do not fulfils one or more of the given specifications. All defects are not
of the same value. So, we may like to control defects per unit. The quality characteristic in such cases
is the number of defects per unit. C-chart is just an improvement over P-chart. C-chart follows Poisson
distribution and various limits are calculated as follows:
Central line =
Totaldefects
TotalNo.of itemsinspected
C =
Upper control limit (UCL) = C + 3C
Lower control limit (LCL) = C – 3C
Problems and Solutions
Problem 1 :
Draw the control charts for X (mean) and R (Range) from the following data relating to 20 samples, each
of size 5. Only the control line and the upper and lower control limits may be drawn in each chart.
Sample No. X R Sample No. X R
1 38.2 15 11 32.6 31
2 33.8 1 12 22.8 12
3 24.4 22 13 21.6 29
4 36.6 24 14 28.8 22
5 27.4 18 15 28.8 16
6 30.6 33 16 24.4 19
7 31.2 21 17 30.4 20
8 27.0 29 18 25.4 34
9 24.0 29 19 37.8 19
10 29.4 18 20 31.4 17
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(For sample of size 5-d2 = 2.326, d3 = 0.864)
Solution: (1)
X -chart (mean chart)
Here ΣX = 587 ΣR = 420 K = 20 N = 5
CL or X :
K or No.of samples
= ΣX
UCL = X
= + 2
3
N
R
d = 29.35 + 3 ×
2
2.326
5
= 29.35 +
27.09
2.24
= 29.35 +12.09 = 41.44
LCL = X
= – 2
3
N
R
d = 29.35 –
27.09
2.24
= 29.35 – 12.09 = 17.26
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Production Planning and Productivity Management
(2) R-Chart
210 205 210 212 211 209 219 204 212 209
212 215 208 214 210 204 211 211 203 211
CL or R =
ΣR
K
=
420
20
= 21
UCL = R + 3 σ R* = 21+ (3×7.802) = 21 + 23.41 = 44.41
LCL = R - 3 σ R = 21 – (3 × 7.802) = 21 – (3 ×7.802) = 21 – 23.41 = -2.41 or zero
* σ R = σ P × d3, or
2
R
d × d3=
21
2.326
× .864 = 9.03 × 0.864 = 7.802
Problem: 2
The following table gives the average daily production figure for 20 months each of25 working days. Given
that the population standard deviation of daily production is 35 units, draw a control chart for the mean.
Sample Range 1 cm = 5
O
189
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Solution:
X -chart
Conclusion:
All the points lie within the control limits. The process is well within control.
CL or =
X =
4200
210
20
= =
ΣX
K
UCL = =
X +
3σP
N
= 210 +
3 35
25
×
= 210 +
105
5
= 231
LCL = =
X –
3σP
N
= 210 –
3 35
25
×
= 210 –
105
5
or 210 – 21 = 189
Problem 3:
15 Samples of size 4 each were taken and the observed values are given below:
Samples Observed values
1 32 20 33 6
2 42 36 52 50
3 25 15 52 63
4 22 33 34 23
5 29 30 27 31
6 30 34 26 16
7 34 31 28 34
8 11 21 20 16
9 11 22 28 31
10 36 30 35 26
11 34 16 37 26
12 27 36 51 53
13 26 35 32 37
14 25 36 37 24
15 10 28 14 13
Calculate UCL and LCL for X Chart and R chart. Also prepare the chart on graph paper. For a sample size
4 the control factors are —
A2 = 0.729, d2 = 2.059, d3 = 0.880, d4 = 2.282.
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Production Planning and Productivity Management
Solution:
Samples Observed Values X or Mean R (Higest = Lowest)
1 32 20 33 06 91/4 = 22.75 33 -6 = 27
2 42 36 52 50 180/4 = 45.00 52 -36 = 16
3 25 15 52 63 155/4 = 38.75 63 - 15 = 48
4 22 33 34 23 112/4 = 28.00 34 - 22 = 12
5 29 30 27 31 126/4 = 31.50 39 - 27 = 12
6 30 34 26 16 106/4 = 26.50 34 - 16 = 18
7 34 31 28 34 127/4 = 31.75 34 - 28 =06
8 11 21 20 16 68/4 =17.00 21 - 11 = 10
9 11 22 28 31 92/4 = 23.00 31 - 11 = 20
10 36 30 35 26 127/4 = 31.75 36 - 26 = 10
11 34 16 37 26 113/4 = 28.25 37 - 16 = 21
12 27 36 51 53 167/4 = 41.75 53 - 27 = 26
13 26 35 32 37 130/4 = 32.50 37 - 26 = 11
14 25 36 37 24 122/4 = 30.50 37 - 24 = 13
15 10 28 14 13 65/4 = 16.25 28 - 10 = 18
X - Chart-
CL =
445.25
29.68
KorNo.of samples 15
= =
ΣX
UCL = X + A2 R = 29.68 + (0.729×17.867) = 29.68+13.02 = 42.70
LCL = 16.66
FALSE BASE LINE
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Sample number No. of defectives Sample number No. of defectives
1 6 11 10
2 2 12 4
3 4 13 6
4 1 14 11
5 20 15 22
6 6 16 8
7 10 17 0
8 19 18 3
9 4 19 23
10 21 20 10
Conclusion: Sample no. 2 is outside UCL and sample no. 15 is outside LCL.
LCL = =
X – A2R =29.68 – (0.729 x 17.867) = 16.66
R-chart –
CL or
R 268
17.867
KorNo.of samples 15
= =
Σ
R =
UCL = d4×R = 2.282 × 17.867 = 40.77
LCL = d3 x R = 0 × 17.867 = 0
Conclusion : Sample 3 is outside control limits.
Problem: 4
The following table gives the result of inspection of 20 samples of 100 items each taken in 20 working days.
Draw a P-chart. What conclusion do you draw from the chart about the process?
UCL = 40.77
CL = 17.86
LCL = 0
50
45
40
35
30
25
20
15
10
5
0
SAMPLE RANGE 1 CM = 5
2 4 6 8 10 12 14 16 18 20
SAMPLE NO. 1CM = 2 SAMPLES
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Production Planning and Productivity Management
Solution:
Total No. of items inspected = No. of samples × units inspected in each sample
= 20 × 100 =2000 units
Average fraction defectives =
Totalno.of defectives 200
0.10
Totalno.of items inspected 2000
= =
P or CL =0.10
UCL =
(1 0.10(1 0.10)
0.10 3 0.10 3 0.0009
100
⎛ − ⎞ ⎛ − ⎞ ⎜ ⎟ = + ⎜ ⎟ = +
⎝ ⎠ ⎝ ⎠
P P)
P +
n
= 0.10 + 3 × 0.03 = 0.10 + 0.09 = 0.19
LCL =
(1 0.10(1 0.10)
0.10 3 0.10 3 0.0009
100
⎛ − ⎞ ⎛ − ⎞ ⎜ ⎟ = − ⎜ ⎟ = +
⎝ ⎠ ⎝ ⎠
P P)
P –
n
= 0.10 – 3 × 0.03 = 0.10 – 0.09 = 0.01
Conclusion: Four points, i.e., sample numbers 5, 10, 15 and 19 lie outside the control limits.
Problem: 5
The following table gives the result of inspection of 20 samples of 100 items each taken on 20 working
days. Draw a P-chart. What conclusion would you draw from the chart?
UCL = 19
CL = 10
LCL = 01
24
22
20
18
16
14
12
10
08
06
04
02
0
2 4 6 8 10 12 14 16 18 20
FRACTION DEFECTIVE 1 CM = 02
SAMPLE NO. 1 CM = 2 SAMPLES
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Conclusion: Only sample number 15 is beyond control limits.
Sample No. 1 2 3 4 5 6 7 8 9 10
No. of defectives 9 17 8 7 12 5 11 16 14 15
Sample No. 11 12 13 14 15 16 17 18 19 20
No. of defectives 10 6 7 18 16 10 5 14 7 13
Solution:
Total no. of items inspected = No. of samples × units inspected in each sample
= 20 × 100 = 2000 units.
Average fraction detectives or
Total no.of defectives 80
= 0.04
Totalno. items inspected 2000
P = =
Hence P or CL= 0.04
UCL =
(1 ) .04(1 .04)
+ 3 0.04 3
100
− = + − P P
P
n
= 0.04 + 3 0.000384 = 0.04 + 3×0.0196 =0.04+0.0588 =.0988
LCL =
(1 ) .04(1 04)
– 3 0.04 3
100
− = − − P P .
P
n
= 0.04 – 3 0.000384 = 0.04 – 3 × .0196 = 0.04 – 0.0588 =–0.0188=0
Problem: 6
The following table gives the result of inspection of 20 samples of 100 items each taken on working days.
Draw a P-chart. What conclusion would you draw from the chart?
UCL = 09.88
CL = 04
LCL = 0
SAMPLE NO. 1 CM = 2 SAMPLES
2 4 6 8 10 12 14 16 18 20
20
18
16
14
12
10
08
06
04
02
0
FRACTION DEFECTIVE 1 CM = 02
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Production Planning and Productivity Management
Conclusion : No. point is beyond limit
UCL = 20.39
CL = 11
LCL = 0.1613
2 4 6 8 10 12 14 16 18 20
20
18
16
14
12
10
08
06
04
02
0
X
Y
FRACTION DEFECTIVE 1 CM = 02
SAMPLE NO. 1 CM = 2
Sample No. 1 2 3 4 5 6 7 8 9 10
No. of Defectives 0 2 4 6 6 4 0 2 4 8
Sample No. 11 12 13 14 15 16 17 18 19 20
No. of Defectives 8 0 4 6 14 0 2 2 6 2
Solution:
Total number of items inspected = No. of samples × units in each sample
= 20 ×100 = 2000 units
Average traction defectives or
Totalno.of defectives 220
0.11
Totalno.items inspected 2000
P = = =
(i) CL = 0.11
(ii) UCL =
(1 ) 11(1 .11) .0979
3 0.11 3 0.11 3
100 100
− − ⎛ ⎞ + = + = + ⎜ ⎟
⎝ ⎠
P P .
P
n
= 0.11 + 3 × 0.03129 = 0.11 + 0.09387 = 0.20387 = 0.2039
(iii) LCL =
(1 )
3
− − P P
P
n
- 3 = 0.11 - 3 × 0.03129 = 0.11-0.09387 = 0.01613
Problem 7. 18 carpets had defects in their finish as follows. Supposing the defects follow the Poisson Law.
draw a control chart for the number of defects.
No. of Defects 0 1 2 3 4 5 6
No. of carpets 0 1 2 4 3 5 3
having specified No. of Defects
Solution.
C =
Totalno.of defectives
Totalno.of carpetsexamin ed
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=
(0 0) (1 1) (2 2) (3 4) (4 3) (5 5) (6 3)
(0 1 2 4 3 5 3)
× + × + × + × + × + × + ×
+ + + + + +
=
0 1 4 12 12 25 18 72
or 4
18 18
+ + + + + + = ... CL = C = 4
UCL = C + 3 ( C )= 4 + 3 4 or 4 + 3 × 2 = 4 + 6 = 10
LCL = C – 3 ( C )= 4 – 3 ( 4 ) or 4 – 3 × 2 = -2 = 0
Part Nominal Dimension
(cm)
Total Tolerance
(cm)
A 0.600 0.002
B 0.750 0.004
C 1.250 0.006
D 0.700 0.002
Total 3.300cm 0.014cm
2 4 6 8 10 12 14 16 18 20
UCL = 10
CL = 4
LCL = 0
SAMPLE NO.
20
18
16
14
12
10
08
06
04
02
0
SCALE = 1 CM 2 DEFECTS
DEFECTS PER UNIT
Problem: 8
In the following example —
E
A B C D
F
0.600 + 0.750 + 0.002 1.250 + 0.003 0.700 + 0.001
Parts A, B, C and D must fit within the space EF. What should be the dimension and tolerance of EF?
Solution:
The typical engineering approach is as follows:
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Production Planning and Productivity Management
The tolerance would be half the total tolerance of 0.014 or 0.007 cm., and the dimension specification
would be 3.3000 ± 0.007.
One statistical approach uses the formula
T t = 2 2 2
T1 + T2 + ......... +Tn
in which T is the individual tolerance and Tt is the total tolerance. Using this concept, the tolerance for
the example would be:
Tt = (0.002)2 + (0.004)2 + (0.005)2 + (0.002)2 = 0.00775or 0.008
The dimension specification would then be expressed as 3.300 ± 0.004 cm. with a total tolerance of
0.008 cm.
It would be noted that the statistical tolerance of 0.008 is approximately 55% of the conventional
tolerance of 0.014 cm. Now if the EF dimensions were as much as 0.014 cm or even 0.010 cm. one or
several of the parts tolerance can be widened and still be within good practice.
Problem: 9
Incoming steel to be used in processing is tested to see that it is of the right chemical composition before it
is machined. Dimensions of the machined parts are inspected, prior to the heat treating operation. An
automatic heat treatment furnace is set at a temperature which hardens the parts. The temperature is set so
that the average force required to break the part is 32000 unit wt. The inherent variability of the heat
treating process produces a standard deviation of the breaking force at 3000 unit wt. Establish the control
limits for x so that α = 0.10 when sample of size n = 4 are taken.
Solution:
3000
4
σ = σ =
n
= 1500 Unit-wt.
Z(1–
2
α
) = 1.645
UCL x = μ + Z σ x = 32000 + 1.645(1500) = 34467.5 unit-wt.
UCL x = μ – Z σ x = 32000 - 1.645(1500) = 29532.5 unit-wt.
Problem: 10
The radiology department of a large hospital has an average retake rate of 8.8%; i.e. 8.80% of its x-rays
must be repeated because the picture is not sufficiently clear. Errors can occur because of incorrect patient
measurement, improper calibration or setting of the machine, poor film quality, incorrect film processing
or other reasons. During the past month, 9000 x-rays were taken, and 11.2% had to be repeated. Does the
process appear to be within its 3σ limits or does it appear that there may be some assignable cause for
variation?
Solution:
p = 0.088
xα
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xα
4 4 5 3
6 2 2 4
5 3 4 2
3 2 4 5
5 7 5 3
Control limits for p when a sample of 9000 is taken are:
p ± 3 3
(1 0.088(1 0.088)
0.088 0.09070and0.0790
9000 9000
− = ± − = p)
The value of 11.2% defective is very much outside the 3σ limits for the process.
Problem: 11
Twenty samples were taken from a cable-weaving machine while it is being operated under closely
controlled conditions. The number of defects per 100 meters for the samples is recorded in the chart below.
Determine the control chart limits for the machine.
Solution:
z =
78
3.9
20
= =
Σc
n
Control limits for the number of defects per 100 meters are: c ± 3 c
UCL = 3.9 + 3 3.9 = 9.8
LCL = 3.9 – 3 3.9 = negative or, 0, whichever is greater.
Therefore LCL = 0
Problem: 12
An attribute control chart exists for part 3146B shows the average fraction defective 0.125, upper control
limit 0.200 and lower control limit 0.050, based on two months of daily data. Recently, 12 units were
sampled each day for six days with units defective 2, 1, 2, 0, 3 & 3.
(a) Construct a control chart for management, carefully labelling the chart, and interpret it for
management.
(b) What is the significance of the fraction defective for day 4 being below the lower control limit?
Solution:
The solution for part (a) is shown in the following figure. Fraction defective for the recent six days are:
2/12 = 0.166 1/12 = 0.083 2/12 = 0.166
0/12 = 0 3/12 = 0.250 3/12 = 0.250
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