OPERATIONS RESEARCH ASSIGNMENT ON
Submitted to : Mrs. Anuradha
Date of Submission : 17 September, 2009
Class : M Com - 1st Semester
Submitted by : Bobby 0921606
Christ University, Bangalore
TABLE OF CONTENTS
S. NO. CONTENTS PAGE NO.
1. INTRODUCTION TO STATISTICAL PROCESS 3-6
1.1 STATISTICAL QUALITY CONTROL 3
1.2 ORIGIN AND MEANING OF STATISTICAL
PROCESS CONTROL 3–4
1.3 APPLICATION OF STATISTICAL PROCESS
2. CONTROL CHARTS 7–9
2.1 INTRODUCTION TO CONTROL CHARTS 7
2.2 3 BASIC COMPONENTS 7-8
2.3 THINGS TO LOOK FOR 8
2.4 TYPES OF ERRORS 8–9
2.5 BENEFITS 9
2.6 CAPABILITIES 9
3. BASIC VARIABLES 10 – 13
1. INTRODUCTION TO STATISTICAL PROCESS CONTROL
Quality has always been an integral part of all products including services. Consumers
decide among alternative competitive products based on the quality of the products so
quality becomes an important decision criterion. So, quality improvement has become the
key factor for the success and growth of any business organization. Investment on quality
improvement gives rich returns.
1.1 STATISTICAL QUALITY CONTROL
Statistical quality control refers to using statistical techniques for measuring and
improving the quality of processes and includes SPC in addition to other techniques, such as
sampling plans, experimental design, variation reduction, process capability analysis, and
process improvement plans. Thus SQC refers to the statistical methods used for maintenance
of quality in a continuous flow of products. One of the methods is ―Statistical Process
1.2 STATISTICAL PROCESS CONTROL
SPC is the primary analysis tool of quality improvement. It is the applied science that helps
you collect, organize and interpret the wide variety of information available to your
business. Whether you track revenues, billing errors, or the dimensions of manufactured
components, SPC can help you measure, understand and control the variables that affect
your business processes.
Statistical Process Control was pioneered by Walter A. Shewhart in the early 1920s.
W. Edwards Deming later applied SPC methods in the United States during World War II,
thereby successfully improving quality in the manufacture of munitions and other
strategically important products. Deming was also instrumental in introducing SPC methods
to Japanese industry after the war had ended.
Shewhart created the basis for the control chart and the concept of a state of statistical
control by carefully designed experiments. While Dr. Shewhart drew from pure
mathematical statistical theories, he understood that data from physical processes seldom
produces a "normal distribution curve" (a Gaussian distribution, also commonly referred to
as a "bell curve"). He discovered that observed variation in manufacturing data did not
always behave the same way as data in nature (for example, Brownian motion of particles).
Dr. Shewhart concluded that while every process displays variation, some processes display
controlled variation that is natural to the process (common causes of variation), while others
display uncontrolled variation that is not present in the process causal system at all times
(special causes of variation).
In 1989, the Software Engineering Institute introduced the notion that SPC can be usefully
applied to non-manufacturing processes, such as software engineering processes, in the
Capability Maturity Model (CMM). This notion that SPC is a useful tool when applied to
non-repetitive, knowledge-intensive processes such as engineering processes has
encountered much skepticism, and remains controversial today.
Statistical process control (SPC) involves using statistical techniques to measure and
analyze the variation in processes. Most often used for manufacturing processes, the intent
of SPC is to monitor product quality and maintain processes to fixed targets. SPC is used to
monitor the consistency of processes used to manufacture a product as designed. It aims to
get and keep processes under control. No matter how good or bad the design, SPC can
ensure that the product is being manufactured as designed and intended. Thus, SPC will not
improve a poorly designed product's reliability, but can be used to maintain the consistency
of how the product is made and, therefore, of the manufactured product itself and its as-
1.3 APPLICATION OF STATISTICAL PROCESS CONTROL
The following description relates to manufacturing rather than to the service industry,
although the principles of SPC can be successfully applied to either. SPC has also been
successfully applied to detecting changes in organizational behavior with Social Network
Change Detection introduced by McCulloh (2007). Selden describes how to use SPC in the
fields of sales, marketing, and customer service, using Deming's famous Red Bead
Experiment as an easy to follow demonstration.
In mass-manufacturing, the quality of the finished article was traditionally achieved
through post-manufacturing inspection of the product; accepting or rejecting each article (or
samples from a production lot) based on how well it met its design specifications. In
contrast, Statistical Process Control uses statistical tools to observe the performance of the
production process in order to predict significant deviations that may later result in rejected
Two kinds of variation occur in all manufacturing processes: both these types of
process variation cause subsequent variation in the final product. The first is known as
natural or common cause variation and may be variation in temperature, properties of raw
materials, strength of an electrical current etc. This variation is small, the observed values
generally being quite close to the average value. The pattern of variation will be similar to
those found in nature, and the distribution forms the bell-shaped normal distribution curve.
The second kind of variation is known as special cause variation, and happens less
frequently than the first.
For example, a breakfast cereal packaging line may be designed to fill each cereal box
with 500 grams of product, but some boxes will have slightly more than 500 grams, and
some will have slightly less, in accordance with a distribution of net weights. If the
production process, its inputs, or its environment changes (for example, the machines doing
the manufacture begin to wear) this distribution can change. For example, as its cams and
pulleys wear out, the cereal filling machine may start putting more cereal into each box than
specified. If this change is allowed to continue unchecked, more and more product will be
produced that fall outside the tolerances of the manufacturer or consumer, resulting in waste.
While in this case, the waste is in the form of "free" product for the consumer, typically
waste consists of rework or scrap.
By observing at the right time what happened in the process that led to a change, the
quality engineer or any member of the team responsible for the production line can
troubleshoot the root cause of the variation that has crept in to the process and correct the
SPC indicates when an action should be taken in a process, but it also indicates when
NO action should be taken. An example is a person who would like to maintain a constant
body weight and takes weight measurements weekly. A person who does not understand
SPC concepts might start dieting every time his or her weight increased, or eat more every
time his or her weight decreased. This type of action could be harmful and possibly generate
even more variation in body weight. SPC would account for normal weight variation and
better indicate when the person is in fact gaining or losing weight.Statistical Process Control
is easy to do. Although it involves complex mathematics, computers are ideally suited to the
task. They easily collect, organize and store information, calculate answers, and present
results in easy to understand graphs, called control charts. Computers accept information
typed in manually, read from scanners or manufacturing machines, or imported from other
computer databases. The resulting control charts can be examined in greater detail,
incorporated into reports, or sent across the Internet. A computer collecting information in
real time can even detect very subtle changes in a process, and even warn you in time to
prevent process errors before they occur.
SPC can help you understand and reduce the variation in any business process.
Greater consistency in fulfilling your customer's requirements leads to greater customer
satisfaction. Reduced variation in your internal processes leads to less time and money spent
on rework and waste. Both directly yield greater profitability and security for your business.
SPC is one of the essential tools necessary to maintain an advantage in today's competitive
The objective of process control is to quickly detect and weed out the identifiable causes of
variation in the process . This objective is achieved through the technique of Control charts.
2. CONTROL CHARTS
The Control Chart, one of the seven basic tools of quality control (along with the histogram,
Pareto chart, check sheet, cause-and-effect diagram, flowchart, and scatter diagram) was
invented by Walter A. Shewhart while working for Bell Labs in the 1920s. The control chart
is supposed to detect the presence of special causes of variation. In its basic form, the
control chart is a plot of some function of process measurements against time. The points
that are plotted on the graph are compared to a pair of control limits. A point that exceeds
the control limits signals an alarm.
An Example : Imagine a process that produces soap bars. The production manager wants to
monitor the mean weight of soap bars produced on the line. The target value of the weight
of a single soap bar is 100 gm. It is also known that an estimate of the weight standard-
deviation for a single soap bar, is 5 gm. Daily samples of 10 bars are taken, during a stable
period of the process. For each sample, the weights are recorded, and their mean/average is
Thus a control chart is a graphical representation of certain descriptive statistics for
specific quantitative measurements of the manufacturing process. The comparison detects
any unusual variation in the manufacturing process, which could indicate a problem with the
process. Several different descriptive statistics can be used in control charts and there are
several different types of control charts that can test for different causes, such as how
quickly major vs. minor shifts in process means are detected. Control charts are also used
with product measurements to analyze process capability and for continuous process
2.2 Three basic components of Control Charts
A centerline, usually the mathematical average of all the samples plotted.
Upper and lower statistical control limits that define the constraints of common cause
Performance data plotted over time.
2.3 Things to look for
The point of making control charts is to look at variation, seeking special causes and
tracking common causes. Special causes can be spotted using several tests:
1 data point falling outside the control limits
6 or more points in a row steadily increasing or decreasing
8 or more points in a row on one side of the centerline
14 or more points alternating up and down
The simplest interpretation of the control chart is to use only the first test listed. The others
may indeed be useful (and there are more not listed here), but be mindful that, as you apply
more tests, your chances of making Type I errors, i.e. getting false positives, go up
2.4 Types of errors
Control limits on a control chart are commonly drawn at 3s from the center line because 3-
sigma limits are a good balance point between two types of errors:
Type I or alpha errors occur when a point falls outside the control limits even though
no special cause is operating. The result is a witch-hunt for special causes and
adjustment of things here and there. The tampering usually distorts a stable process as
well as wasting time and energy.
Type II or beta errors occur when you miss a special cause because the chart isn't
sensitive enough to detect it. In this case, you will go along unaware that the problem
exists and thus unable to root it out.
All process control is vulnerable to these two types of errors. The reason that 3-sigma
control limits balance the risk of error is that, for normally distributed data, data points will
fall inside 3-sigma limits 99.7% of the time when a process is in control. This makes the
witch hunts infrequent but still makes it likely that unusual causes of variation will be
If your process is in control, is that good enough? No. You have to start by removing
special causes, so that you have a stable process to work with. But then comes the real fun,
and often the most substantial benefits: it is time to improve the process, so that even
common cause variation is reduced.
Provides surveillance and feedback for keeping processes in control
Signals when a problem with the process has occurred
Detects assignable causes of variation
Accomplishes process characterization
Reduces need for inspection
Monitors process quality
Provides mechanism to make process changes and track effects of those changes
Once a process is stable (assignable causes of variation have been eliminated),
provides process capability analysis with comparison to the product tolerance
All forms of SPC control charts
o Variable and attribute charts
o Average (X— ), Range (R), standard deviation (s), Shewhart, CuSum,
combined Shewhart-CuSum, exponentially weighted moving average (EWMA)
Selection of measures for SPC
Process and machine capability analysis (Cp and Cpk)
Quality problem solving
3. BASIC VARIABLE CHARTS – X CHART AND R CHART
The two control charts most often associated with Statistical Process Control, (SPC),
are the X-Bar and Range or R charts. X-Bar is just a fancy word for mean, also known as
average. Range charts are just that, the range that the sample lay between. You should
always set up the Range or R-Bar control charts first.
X-bar & Range Charts are a set of control charts for variables data (data that is both
quantitative and continuous in measurement, such as a measured dimension or time). The X-
bar chart monitors the process location over time, based on the average of a series of
observations, called a subgroup. The Range chart monitors the variation between
observations in the subgroup over time. The standard chart for variables data, X-bar and R
charts help determine if a process is stable and predictable
X charts and R charts are used if the data is a variable like weight, length ( data which
have real numbers in measurement). Data which has qualitative characteristics are called
attributes like pass or fail, leaks,small, medium , large etc for which P charts are used.
An X-bar represents the average/ mean of the sample and R (range) chart represents the
range that the sample lies in between . They are control charts used with processes that
have a subgroup size of two or more.
The charts are compared with control limits. Control limits, are typically calculated
based on the number of original samples, or first samples, taken on any given process . The
samples collected should be a sufficient representation of the population of items
The samples are taken by the means of stratified random sampling plan, by dividing
the process into subgroups and a representative sample is drawn from the subgroup and
control limites are established by using a formula.
Once you have established your control limits, periodically you take more samples,
usually 2 to 5 more sample pieces, and calculate the mean and range and plot them on the
The X-bar chart shows how the mean or average changes over time and the R chart shows
how the range of the subgroups changes over time.
When to Use an X-bar / R Chart
X-bar / Range charts are used when you can rationally collect measurements in groups
(subgroups) of between two and ten observations. Each subgroup represents a "snapshot" of
the process at a given point in time. The charts' x-axes are time based, so that the charts
show a history of the process. For this reason, you must have data that is time-ordered; that
is, entered in the sequence from which it was generated. If this is not the case, then trends or
shifts in the process may not be detected, but instead attributed to random (common cause)
For subgroup sizes greater than ten, use X-bar / Sigma charts, since the range statistic is a
poor estimator of process sigma for large subgroups. In fact, the subgroup sigma is
ALWAYS a better estimate of subgroup variation than subgroup range. The popularity of
the Range chart is only due to its ease of calculation, dating to its use before the advent of
computers. For subgroup sizes equal to one, an Individual-X / Moving Range chart can be
used, as well as EWMA or Cu Sum charts.
X-bar Charts are efficient at detecting relatively large shifts in the process average, typically
shifts of +-1.5 sigma or larger. The larger the subgroup, the more sensitive the chart will be
to shifts, providing a Rational Subgroup can be formed. For more sensitivity to smaller
process shifts, use an EWMA or Cu Sum chart.
Interpreting an X-bar / R Chart
Always look at the Range chart first. The control limits on the X-bar chart are derived
from the average range, so if the Range chart is out of control, then the control limits on the
X-bar chart are meaningless.
After reviewing the Range chart, interpret the points on the X-bar chart relative to the
control limits and Run Tests. Never consider the points on the X-bar chart relative to
specifications, since the observations from the process vary much more than the subgroup
Interpreting the Range Chart
On the Range chart, look for out of control points. If there are any, then the special
causes must be eliminated. Brainstorm and conduct Designed Experiments to find those
process elements that contribute to sporadic changes in variation. To use the data you have,
turn Auto Drop ON, which will remove the statistical bias of the out of control points by
dropping them from the calculations of the average Range, Range control limits, average X-
bar and X-bar control limits.
Also on the range chart, there should be more than five distinct values plotted, and no
one value should appear more than 25% of the time. If there are values repeated too often,
then you have inadequate resolution of your measurements, which will adversely affect your
control limit calculations. In this case, you'll have to look at how you measure the variable,
and try to measure it more precisely.
Once you've removed the effect of the out of control points from the Range chart, look at the
Interpreting the X-bar Chart
After reviewing the Range chart, look for out of control points on the X-bar Chart. If
there are any, then the special causes must be eliminated. Brainstorm and conduct Designed
Experiments to find those process elements that contribute to sporadic changes in process
location. To use the data you have, turn Auto Drop ON, which will remove the statistical
bias of the out of control points by dropping them from the calculations of the average X-bar
and X-bar control limits.
Look for obviously non-random behavior. Turn on the Run Tests, which apply
statistical tests for trends to the plotted points.
If the process shows control relative to the statistical limits and Run Tests for a
sufficient period of time (long enough to see all potential special causes), then we can
analyze its capability relative to requirements. Capability is only meaningful when the
process is stable, since we cannot predict the outcome of an unstable process.