Unit 6 (2017)
Apunte InglésUniversidad  Universidad de Barcelona (UB) 
Grado  Administración y Dirección de Empresas  3º curso 
Asignatura  Dirección de operaciones 
Año del apunte  2017 
Páginas  11 
Fecha de subida  04/07/2017 
Descargas  3 
Subido por  jramos 
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Topic 6: Quality control
Quality control
Evaluates the results of the process by comparing to the ideal results. If there is any
difference between them, then the objective is to minimise it. Apart from separating correct
products (the ones that comply specifications) from defective products that must be redone,
includes the prevention concept (actions to guarantee the expected results).
All the efforts dedicated to obtain products or services that comply design specifications at
minimum cost.
Seven Ishikawa’s basic tools


Pareto chart. Based on the idea that, in general, most defects in an article can be
attributed to a reduced number of causes (Pareto Law 2080). It classifies few vital
causes from the rest of trivial causes. Pareto diagrams identify the causes of a quality
problem rapidly and easily. Example: A company produces an article which presents
several manufacturing defects. The objective is to remove them. Management wishes
to know which are the causes of most defective items.
If the two main causes of the problem are removed (grated surface and arm rupture),
84% of defective articles are avoided.
Causeandeffect diagrams. They are also known as Ishikawa diagram or fishbone, is
used to classify and clear the causes that originate an effect. It is necessary to identify
and face the causes (and NOT the effects) to solve a problem. The basic structure of
these diagrams is a central arrow and the studied effect is placed on the right.
Consequently, firstly the quality problem must be defined and the effect that
measures it. Then the causes are classified. The causes are placed tidily in the main
branches:
Inside these main branches, causes are placed in little branches. A brainstorming
session can be performed previously to identify the causes.

Final comment: the identified and classified causes are potential causes. This diagram
is the starting point to verify and confirm the real causes.
Check sheets. Check sheets are printed sheets that allow data collection in a simple
and precise way so that collection tasks are easier for the operators. The fundamental
objectives are: To ease data collection and organize data for further analysis. There
are different type of templates according to its application.
Check sheets for defective articles
They’re used to detect the type of defects and their frequency percentages in
defective products in order to reduce them.
Check sheets for defects location
Sketches of the manufactured piece where defects are located. They allow to detect if
defects are always placed in the same place.
Check sheets to control the distribution of the production process.
They’re used to collect data of continuous variables such as weight, diameter, volume,…
Then, it is possible to draw a histogram to study the distribution of the production process,
and calculate the average and dispersion.

Histograms. A histogram is a graphical representation of the distribution of data. Data
is organised to study frequency of occurrence. Example: Consider 100 measurements
of the diameter of a cylindrical piece. The number of measurements n should be (at
least) between 50 and 100 to study a certain characteristic. Firstly, data is divided in 10
groups of 10 measurements. For each set of data, we determine the maximum and
minimum values.
Secondly, the amplitude of all data is determined:
Maximum value – Minimum value = 7,58 – 7,19 = 0,39
This value is divided by k = 10 to obtain the number of classes (number of groups or
bars) of the graphic:
𝑀𝑎𝑥 − 𝑀𝑖𝑛 0,39
ℎ=
=
= 0,039 → 0,04 → 0,05
𝑘
10
The interval h is the unit to adjust the horizontal axis (bar width). In this case, we
consider 0,05. The number of classes k depends on data collection:
The value that limits the first bar is fixed taking into account the extreme of the
amplitude + half of the accuracy of collected data. In this case, the minimum value is
7,19 accuracy of real data is 0,01. Consequently, the limits of first bar is : 7,19 0,01/2 = 7,185
Rest of limits of bars will be: 7,185  7,235; 7,235 – 7,285; 7,285 – 7,335; ...
Then, the frequencies table must be calculated accounting data that belongs to each
interval:
These data allows to draw the histogram. In Cartesian axis, horizontal axis represents a
quality characteristic and the vertical axis represents the frequency (number of data
inside one bar).
Each bar limit is a class. Bar width is a class interval. The central value is the average
value. Consider that the diameter tolerances are between 7,15 cm and 7,55 cm. In this
case the process is offcentre. A certain number of pieces are produced outside
specifications limits.
These two cases show the same frequency distribution BUT case 1 is centered inside
the tolerance limits. Nevertheless, case 2 is an offcentre distribution and shows that
some articles are produced outside specifications (grey bars).
Case 3 is a bellshaped distribution that represents a variability due to random causes.
Ideal distribution. Case 4 shows a rightskewed histogram. The right tail is longer and
mass of distribution is concentrated on the left. This indicates that data does NOT
follow normal law.

Case 5 is a bimodal histogram because it presents two peaks. In some cases indicates
that data can be divided in two subsets of data that differ from each other in some
way. Case 6 is distribution that shows a small peak on the right. This indicates defects
or errors because these data does not follow the general behaviour. Probably an
assignable cause can be determined.
Functions of the histograms:
1) Verify if production is inside specifications.
2) Determine the behaviour of the distribution of data by observing the histogram
shape.
3) Analyse if stratification is necessary due to interference of different factors that
can affect variability. In this case, data are separated is subsets to differentiate
causes of dispersion and to identify the origin of the problem easily.
Scatter diagram. These diagrams are useful to analyse whether a quality characteristic
and a factor are related. Also, they’re called correlation diagrams.
Steps to make a scatter plot:
1. Identify the factors that seem to be correlated
2. Take 50 pairs of data approximately
3. Draw Cartesian axis to place the pairs of data
4. The quality characteristic is located in Yaxis
Variable X is a possible cause and Y is the quality characteristic that seems correlated.
Both variables present certain positive correlation. In other words, the quality
characteristic is related to the cause as suspected.
It is possible to calculate the correlation coefficient quantitively:
𝑟=
𝑆𝑥𝑦
√𝑆𝑥𝑦 , 𝑆𝑦𝑦
𝑛
: 𝑆𝑥𝑥
𝑛
= ∑(𝑋𝑖 − 𝑋̅)2 = ∑ 𝑋𝑖2 −
𝑖=1
𝑖=1
𝑛
𝑛
𝑆𝑦𝑦 = ∑(𝑌𝑖 − 𝑌̅) = ∑ 𝑌𝑖2 −
2
𝑖=1
𝑛
𝑆𝑥𝑦
𝑖=1
𝑛
= ∑(𝑋𝑖 − 𝑋̅) ∗ (𝑌𝑖 − 𝑌̅) = ∑ 𝑋𝑖 𝑌𝑖 −
𝑖=1
𝑖=1
(∑𝑛𝑖=1 𝑋𝑖 )2
𝑛
(∑𝑛𝑖=1 𝑌𝑖 )2
𝑛
(∑𝑛𝑖=1 𝑋𝑖 ) ∗ (∑𝑛𝑖=1 𝑌𝑖 )
𝑛
In the example:
SXX = 71,56; SYY = 122,53; SXY = 74,46; r = 0,7952 (positive correlation)
The correlation coefficient takes values between 1 and 1. If the resulting value is:

A. close to 1, this indicates there is a strong positive correlation.
B. close to 1, then the correlation is negative
C. close to 0, then the correlation is weak.
Stratification. It’s a method to identify the origins of variability of collected data. For
example, when an article is manufactured by different machines, by different
operators or using different materials, then it’s advisable to classify the data by
machinery, operators or materials. This way, it’s possible to identify the origin of the
problem. Maybe this could not be detected if all data is mixed.
Stratification is one of the seven basic Ishikawa tools. Normally complements the rest
of methodologies.
Statistical Process Control (SPC).
Is the application of statistical techniques to measure and analyse the variations of a
production process. Causes of these variations can be: random and assignable.
a. Random causes: They cannot be controlled and they appear at random. They
affect all production processes and always they can be considered.
b. Assignable causes: They can be studied and are the ones that contribute most to
the variability of the process. Usually, they’re due to tiredness of workers,
different grade of experience/training, different behaviour of materials,… so that
it’s NOT possible to obtain identical products. The same happens in service
industries: a cook CANNOT obtain two identical dishes or a lecturer CANNOT
repeat two identical classes,…
Process capacity
Process capacity is defined as the variability amplitude of a process when this is under control;
in other words, when variability is not due to assignable causes. When a product is designed,
the nominal value and a tolerance margin is defined. The tolerances interval defines the limits
inside which the product is considered correct. This interval of tolerances is limited by an
upper tolerance limit (LTS) and lower tolerance limit (LTI). For example:
Control graphics
The objective of a control graphic is to differentiate variations due to random causes or due to
assignable causes. The vertical axis represents the range of the studied quality characteristic.
Also, the lower tolerance limit, the upper tolerance limit and the average are placed on the
graphic. The horizontal axis represents time. This way, the evolution of the quality
characteristic can be observed vs. time. Also, it can be compared with respect to established
control limits.
Types:

Control graphic by variables. This means that measurements belong to a continuous
quality characteristic such as weigth, length, speed, density, volume,… In this case, the
control graphics for the average and range will be plotted. To start with, at least 100150 elements must be measured (and grouped in samples of 45 units) to guarantee
that the sample is significant.
25 samples of 5 measurements.
Examples:
∑𝑛
𝑖=1 𝑋𝑖
o
Sample average:
o
Sample range: 𝑅 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒. Example: 1513=2
o
∑
∑
𝑋
𝑅
Global average for X and R: 𝑋̅ = 𝑘=1 𝑖 𝑅̅ = 𝑘=1 𝑖 . Example:
𝑘
𝑘
14 + 14 + 13,2 + ⋯ + 14,8 + 13 + 15,2
𝑋̅ =
= 14,56
25
2+ 4+ 4+ ⋯+5+ 5+ 4
𝑅̅ =
= 3,72
25
𝑛
. Example: 14
𝑘
̅
𝑘
Graphic X: shows variation of the average of the process. Graphic R: shows variations of the
process dispersion. Consequently, control graphics are very interesting because they show
variations of the average and dispersion at the same time and indicate process anomalies.
In this example, both average and range graphics show that the process is under control
because:
1. All the points are inside the control limits.
2. Points do not follow any particular tendency.

Control graphic by attributes. An attribute is a quality characteristic that cannot be
measured. Control graphics by attributes classify the product as Acceptable or
Defective. For example, if a piece presents spots, we wish to know if the product
shows this characteristic to determine if it is acceptable or defective. Since a defective
products can have several defects, these graphics monitor the number of detected
defects or the number of defects. According to this criteria and the sample, there are
several types of graphics.
Acceptance sampling
The acceptance sampling determines the percentage of products that verify specifications. It is
used to inspect elements that the company purchase to suppliers (raw material,
components,…) or also pieces that have been processed in one step of the process and are
evaluated before going into the next step. This technique involves taking random lots of
products, measuring a certain characteristic and then comparing it with a established
standard. This is much cheaper than 100% inspection. The quality of the sample is used to
judge the quality of the complete production lot. Sampling can be defined by variables or by
attributes.
The acceptance sampling is performed through a sampling plan. A simple sampling plan is
defined by the sample size, n, and by the acceptance number, c, ( maximum number of defects
that can be found in the sample before rejecting the lot),...

If the inspected sample has a number of defects < c , the lot is accepted.
If the number of detected defects is higher, then the lot is rejected or then it’s
performed a 100% inspection.
Example: Consider that we wish to accept all production lots whose number of defective
products < 2,5% and reject the rest. Imagine a lot of 1.000 pieces that has a 4% of defective
products. The inspection has taken a sample of 20 pieces and none is defective Lot is
accepted. In this case, the sampling plan gives a wrong result. The 20 pieces sample could have
showed at random, one, two, three,…defective pieces. This fact is fundamental in sampling
plans. It’s possible to reject good production lots or accept defective lots.
Simple sampling plan by attributes. Characteristic curve
Each sampling plan has associated a characteristic curve that describes the ability of the plan
to distinguish between correct lots and defective lots. Indicates the probability that the plan
accepts lots of different qualities.
AOQ, Average Output Quality: AOQ = w . Pa (w)
AQL, Acceptable Quality Level:

Maximum number of defective articles inside a lot so that it is considered acceptable.
Lowest level of quality that the company accepts.
Lots with this level of quality will be accepted.
LTPD, Lot Tolerance Percent Defective:

Level of quality of a defective lot.
Lots with this quality level will be rejected.
From the producer perspective, a good sampling plan is one that has a low probability to
reject correct lots.
Producer risk (): Probability that the lot is rejected even though the number of defective
products is lower than AQL. From customers or consumers perspective, the sampling plan,
should have a low probability to accept defective lots:
Consumer risk (): Probability that the lot is accepted although the number of defective
products ≥ LTPD
Sampling plan. Considerations
Normally, the sampling plans are designed considering a producer risk of 5% ( = 0,05) and a
consumer risk of 10% ( = 0,10).
The selection of specific values AQL, LTPD, and is an economic decision based on costs,
company policies or contract requirements.
...