SEVEN TOOLS

Sabtu, 2008 November 29
Assembly line balancing and group working
Assembly line balancing and group working: a heuristic procedure for workers' groups operating on the same product and workstation


In this paper, we examine an assembly line balancing problem that differs from the conventional one in the sense that there are multi-manned workstations, where workers' groups simultaneously perform different assembly works on the same product and workstation. This situation requires that the product is of sufficient size, as for example in the automotive industry, so that
the workers do not block each other during the assembly work. The proposed approach here results in shorter physical line length and production space utilization improvement, because the same number of workers can be allocated to fewer workstations. Moreover, the total effectiveness of the assembly line, in terms of idle time and production output rate, remains the same. A heuristic assembly line balancing procedure is thus developed and illustrated. Finally, experimental results of a real-life automobile assembly plant case and well-known problems from the literature indicate the effectiveness and applicability of the proposed approach in practice.



[1] Ghosh S, Gagnon RJ. A Comprehensive literature review and analysis of the design, balancing and scheduling of assembly systems. International Journal of Production Research 1989;27:637-70.

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[2] Johnson RV. Balancing assembly lines for teams and workgroups. International Journal of Production Research 1991;29:1205-14.

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[3] Bukchin J, Masin M. Multi-objective design of team oriented assembly systems. European Journal of Operations Research 2004;156:326-52.

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[4] Becker B, Scholl A. A survey on problems and methods in generalized assembly line balancing, European Journal of Operational Research, doi: 10.1016/j.ejor.2004.07.023, in press, [corrected proof, available online 11 September 2004].

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[5] Engstrom T. Intra-group work patterns in final assembly of motor vehicles. International Journal of Operations & Production Management 1994;14:101-13.

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[6] Psoinos D, Georgiadis P, Koskosidis D, Asteriadis T, Dimitriadis S. Organising the production line of military jeep-type vehicles (Mercedes 240 GD) of Hellenic Vehicles Industry S.A. (EL.B.O.). Research report, Industrial Management Division, Aristotle University of Thessaloniki, Greece, 1987 [in Greek].

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[7] Engstrom T, Jonsson D, Johansson B. Alternatives to line assembly: some Swedish examples. International Journal of Industrial Ergonomics 1996;17:235-45.

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I Baybars, A survey of exact algorithms for the simple assembly line balancing problem, Management Science, v.32 n.8, p.900-932, Aug.1986

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[9] Scholl A, Becker B. State-of-the-art exact and heuristic solution procedures for simple assembly line balancing. European Journal of Operational Research, doi: 10.1016/j.ejor.2004.07.022, in press [corrected proof, available online 30 December 2004].

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F B Talbot , J H Paterson , W V Gehrlein, A comparative evaluation of heuristic line balancing techniques, Management Science, v.32 n.4, p.430-454, April 1986

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[11] Scholl A. Balancing and sequencing of assembly lines. Physica Verlag: Heidelberg; 1999.

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[12] Baxey GM. Assembly line balancing with multiple stations. Management Science 1974;20:1010-21.

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[13] Akagi F, Osaki H, Kikuchi S. A method for assembly line balancing with more than one worker in each station. International Journal of Production Research 1983;21:755-70.

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[14] Pinto PA, Dannenbring DG, Khumawala BM. Branch and bound and heuristic procedures for assembly line balancing with paralleling of stations. International Journal of Production Research 1981;19:565-76.

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[15] Bartholdi JJ. Balancing two-sided assembly lines: a case study. International Journal of Production Research 1993;31: 2447-61.

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[16] Hoffmann TR. Assembly line balancing with a precedence matrix. Management Science 1963;9:551-62.

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[17] Dimitriadis S, Georgiadis P. Assembly line balancing with smoothed workstation assignments. Yugoslav Journal of Operations Research 1995;5:259-70.

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[18] Scholl A, Klein R. Balancing assembly lines effectively--a computational comparison. European Journal of Operational Research 1999;114:50-8.

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[19] Hu TC. Parallel sequencing and assembly line problems. Operations Research 1961;9:841-8.

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[20] Cheng TCE, Sin CCS. A state of the art review of parallel machine scheduling research. European Journal of Operational Research 1990;47:271-92.

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[21] Baker KR. Introduction to sequencing and scheduling. New York: Wiley; 1974.

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[22] Talbot FB, Patterson JH. An integer programming algorithm with network cuts solving the assembly line balancing problem. Management Science 1984;30:85-9.

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[23] Fleszar K, Hindi KS. An enumerative heuristic and reduction methods for the assembly line balancing problem. European Journal of Operational Research 2001;145:606-20.
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Rabu, 2008 November 19
The meaning of Balance
Main Entry: 1bal·ance
Pronunciation:
Function: noun
Etymology: Middle English, from Anglo-French, from Vulgar Latin *bilancia, from Late Latin bilanc-, bilanx having two scalepans, from Latin bi- + lanc-, lanx plate
Date: 13th century
1: an instrument for weighing: as a: a beam that is supported freely in the center and has two pans of equal weight suspended from its ends b: a device that uses the elasticity of a spiral spring for measuring weight or force
2: a means of judging or deciding
3: a counterbalancing weight, force, or influence
4: an oscillating wheel operating with a hairspring to regulate the movement of a timepiece
5 a: stability produced by even distribution of weight on each side of the vertical axis b: equipoise between contrasting, opposing, or interacting elements c: equality between the totals of the two sides of an account
6 a: an aesthetically pleasing integration of elements b: the juxtaposition in writing of syntactically parallel constructions containing similar or contrasting ideas
7 a: physical equilibrium b: the ability to retain one's balance
8 a: weight or force of one side in excess of another b: something left over : remainder c: an amount in excess especially on the credit side of an account
9: mental and emotional steadiness
— bal·anced \-lən(t)st\ adjective
— in the balance or in balance : with the fate or outcome about to be determined
— on balance : with all things considered



Main Entry: 2balance
Function: verb
Inflected Form(s): bal·anced; bal·anc·ing
Date: 1588
transitive verb1 a (1): to compute the difference between the debits and credits of (an account) (2): to pay the amount due on : settle b (1): to arrange so that one set of elements exactly equals another (2): to complete (a chemical equation) so that the same number of atoms and electric charges of each kind appears on each side
2 a: counterbalance , offset b: to equal or equalize in weight, number, or proportion
3: to weigh in or as if in a balance
4 a: to bring to a state or position of equipoise b: to poise in or as if in balance c: to bring into harmony or proportion
intransitive verb
1: to become balanced or established in balance
2: to be an equal counterpoise
3: waver 1

Main Entry: balance of power
Date: 1701
: an equilibrium of power sufficient to discourage or prevent one nation or party from imposing its will on or interfering with the interests of another


Main Entry: nitrogen balance
Function: noun Date: 1944
: the difference between nitrogen intake and nitrogen loss in the body or the soil


Main Entry: platform balance
Function: noun
Date: 1811
: a balance having a platform on which objects are weighed —called also platform scale
Main Entry: balance of payments
Date: 1844
: a summary of the international transactions of a country or region over a period of time including commodity and service transactions, capital transactions, and gold movements RUN CHARTS/TIME PLOT/TREND CHART

TUTORIAL OUTLINE


To access the information that you are interested in, select the topic from the Tutorial outline. At the end of each section is hypertext to enable a user to jump to any section of the tutorial, or if they wish, they can continue on to the next section by scrolling down.
OVERVIEW
INSTRUCTIONS
RUN CHART EXAMPLE
SOFTWARE AVAILABLE
SOURCES AND RELATED TOPICS

OVERVIEW
PURPOSE
In-depth view into Run Charts--a quality improvement technique; how Run charts are used to monitor processes; how using Run charts can lead to improved process quality
USAGE
Run charts are used to analyze processes according to time or order. Run charts are useful in discovering patterns that occur over time.
KEY TERMS
Trends:
Trends are patterns or shifts according to time. An upward trend, for instance, would contain a section of data points that increased as time passed.
Population:
A population is the entire data set of the process. If a process produces one thousand parts a day, the population would be the one thousand items.
Sample:
A sample is a subgroup or small portion of the population that is examined when the entire population can not be evaluated. For instance, if the process does produce one thousand items a day, the sample size could be perhaps three hundred.
HISTORY
Run charts originated from control charts, which were initially designed by Walter Shewhart. Walter Shewhart was a statistician at Bell Telephone Laboratories in New York. Shewhart developed a system for bringing processes into statistical control by developing ideas which would allow for a system to be controlled using control charts. Run charts evolved from the development of these control charts, but run charts focus more on time patterns while a control chart focuses more on acceptable limits of the process. Shewhart's discoveries are the basis of what as known as SQC or Statistical Quality Control.
| INSTRUCTIONS | RUN CHART EXAMPLE | SOFTWARE | SOURCES & RELATED TOPICS |

INSTRUCTIONS FOR CREATING A CHART


Step 1 : Gathering Data
To begin any run chart, some type of process or operation must be available to take measurements for analysis. Measurements must be taken over a period of time. The data must be collected in a chronological or sequential form. You may start at any point and end at any point. For best results, at least 25 or more samples must be taken in order to get an accurate run chart.

Step 2 : Organizing Data
Once the data has been placed in chronological or sequential form, it must be divided into two sets of values x and y. The values for x represent time and the values for y represent the measurements taken from the manufacturing process or operation.

Step 3 : Charting Data
Plot the y values versus the x values by hand or by computer, using an appropriate scale that will make the points on the graph visible. Next, draw vertical lines for the x values to separate time intervals such as weeks. Draw horizontal lines to show where trends in the process or operation occur or will occur.

Step 4 : Interpreting Data
After drawing the horizontal and vertical lines to segment data, interpret the data and draw any conclusions that will be beneficial to the process or operation. Some possible outcomes are:
Trends in the chart
Cyclical patterns in the data
Observations from each time interval are consistent
| OVERVIEW | RUN CHART EXAMPLE | SOFTWARE | SOURCES & RELATED TOPICS |

RUN CHART EXAMPLE

Problem Scenario
You have just moved into a new area that you are not familiar with. Your desire is to arrive at work on time, but you have noticed over your first couple of weeks on the job that it doesn't take the same amount of time each day of the week. You decide to monitor the amount of time it takes to get to work over the next four weeks and construct a run chart.

Step 1: Gathering Data
Collect measurements each day over the next four weeks.Organize and record the data in chronological or sequential form.
M T W TH F

WEEK 1 33 28 26.5 28 26
WEEK 2 35 30.5 28 26 25.5
WEEK 3 34.5 29 28 26 25
WEEK 4 34 29.5 27 27 25.5

Step 2: Organizing Data
Determine what the values for the x (time, day of week) and day (data, minutes to work) axis will be.

Step 3: Charting Data
Plot the y values versus the x values by hand or by computer using the appropriate scale. Draw horizontal or vertical lines on the graph where trends or inconsistencies occur.

Step 4: Interpreting Data
Interpret results and draw any conclusions that are important. An overall decreasing trend occurs each week with Mondays taking the most amount of time and Fridays generally taking the least amount of time. Therefore you accordingly allow yourself more time on Mondays to arrive to work on time.

FLOW CHARTS
OVERVIEW Brief description and understanding of flow charts.
HISTORY A glimpse into the past of flow charting.
INSTRUCTIONS How to gather data and develop a process flow.
INTERPRETATION How to read a flow chart.
EXAMPLE Simple example of flow chart and necessary symbols.
SUGGESTED SOFTWARE
RELATED TOPICS & SOURCES

OVERVIEW
Quality Improvement Tool: Flow charts used specifically for a process.
A flow chart is defined as a pictorial representation describing a process being studied or even used to plan stages of a project. Flow charts tend to provide people with a common language or reference point when dealing with a project or process.
Four particular types of flow charts have proven useful when dealing with a process analysis: top-down flow chart, detailed flow chart, work flow diagrams, and a deployment chart. Each of the different types of flow charts tend to provide a different aspect to a process or a task. Flow charts provide an excellent form of documentation for a process, and quite often are useful when examining how various steps in a process work together.
When dealing with a process flow chart, two separate stages of the process should be considered: the finished product and the making of the product. In order to analyze the finished product or how to operate the process, flow charts tend to use simple and easily recognizable symbols. The basic flow chart symbols below are used when analyzing how to operate a process.

In order to analyze the second condition for a flow process chart, one should use the ANSI standard symbols. The ANSI standard symbols used most often include the following:
Drive Nail, Cement, Type Letter.
Move Material by truck, conveyor, or hand.
Raw Material in bins, finished product on pallets, or filed documents.
Wait for elevator, papers waiting, material waiting
Read gages, read papers for information, or check quality of goods.
Any combination of two or more of these symbols show an understanding for a joint process.
| HISTORY | INSTRUCTIONS | INTERPRETATION | EXAMPLE | SOFTWARE | RELATED TOPICS |

HISTORY AND BACKGROUND
As a whole, flow charting has been around for a very long time. In fact, flow charts have been used for so long that no one individual is specified as the "father of the flow chart". The reason for this is obvious. A flow chart can be customized to fit any need or purpose. For this reason, flow charts can be recognized as a very unique quality improvement method.
| OVERVIEW | INSTRUCTIONS | INTERPRETATION | EXAMPLE | SOFTWARE | RELATED TOPICS |

INSTRUCTIONS
Step-by-Step process of how to develop a flow chart.
Gather information of how the process flows: use a)conservation, b)experience, or c)product development codes.
Trial process flow.
Allow other more familiar personnel to check for accuracy.
Make changes if necessary.
Compare final actual flow with best possible flow.

Note: Process should follow the flow of Step1, Step 2, ... , Step N.
Step N= End of Process
CONSTRUCTION/INTERPRETATION tip for a flow chart.
Define the boundaries of the process clearly.
Use the simplest symbols possible.
Make sure every feedback loop has an escape.
There is usually only one output arrow out of a process box. Otherwise, it may require a decision diamond.
| OVERVIEW | HISTORY | INTERPRETATION | EXAMPLE | SOFTWARE | RELATED TOPICS |

INTERPRETATION
Analyze flow chart of actual process.
Analyze flow chart of best process.
Compare both charts, looking for areas where they are different. Most of the time, the stages where differences occur is considered to be the problem area or process.
Take appropriate in-house steps to correct the differences between the two seperate flows.
| OVERVIEW | HISTORY | INSTRUCTIONS | EXAMPLE | SOFTWARE | RELATED TOPICS |

EXAMPLE
Process Flow Chart- Finding the best way home
This is a simple case of processes and decisions in finding the best route home at the end of the working day.


Process Flow Chart- How a process works
(Assembling a ballpoint pen)


Histograms

Overview, Purpose &, Key Terms
History & Background
Creating a Histogram
Interpretation
Example
Software
Related Topics

Overview


Overview: This histogram tutorial will provide information on how to construct and interpret histograms for use in quality process control (Q.C.). The main areas that will be covered in this tutorial are the following:
Tutorial Instructions
Histogram Background
Creating a Histogram (interactively by example)
Interpreting Histograms
Recommended Additional Q.C. Topics and Software
Purpose: The purpose of this tutorial is to let you become familiar with graphical histograms which are used widely in quality control (Q.C.). Histograms are effective Q.C. tools which are used in the analysis of data. They are used as a check on specific process parameters to determine where the greatest amount of variation occurs in the process, or to determine if process specifications are exceeded. This statistical method does not prove that a process is in a state of control. Nonetheless, histograms alone have been used to solve many problems in quality control.
Key Terms:
Histogram -
a vertical bar chart of a frequency distribution of data
Q.C. Methodology -
a statistical tool used in the analysis and determination of possible solutions to quality control problems in industry
Frequency Distribution -
a variation in a numeric sample of data
| HISTORY | CONSTRUCTION | INTERPRETATIONS | EXAMPLE | SOFTWARE | RELATED TOPICS |

History & Background

The histogram evolved to meet the need for evaluating data that occurs at a certain frequency. This is possible because the histogram allows for a concise portrayal of information in a bar graph format.
The histogram is a powerful engineering tool when routinely and intelligently used. The histogram clearly portrays information on location, spread, and shape that enables the user to perceive subtleties regarding the functioning of the physical process that is generating the data. It can also help suggest both the nature of, and possible improvements for, the physical mechanisms at work in the process.
| OVERVIEW | CONSTRUCTION | INTERPRETATIONS | EXAMPLE | SOFTWARE | RELATED TOPICS |

Creating a Histogram


1.Determine the range of the data by subtracting the smallest observed measurement from the largest and designate it as R.
2. Example:
3. Largest observed measurement = 1.1185 inches
4. Smallest observed measurement = 1.1030 inches
5.
6. R = 1.1185 inches - 1.1030 inches =.0155 inch
7.Record the measurement unit (MU) used. This is usually controlled by the measuring instrument least count.
8. Example: MU = .0001 inch
9.Determine the number of classes and the class width. The number of classes, k, should be no lower than six and no higher than fifteen for practical purposes. Trial and error may be done to achieve the best distribution for analysis.
10. Example: k=8
11.Determine the class width (H) by dividing the range, R, by the preferred number of classes, k.
12. Example: R/k = .0155/8 = .0019375 inch
13.The class width selected should be an odd-numbered multiple of the
14.measurement unit, MU. This value should be close to the H value:
15. MU = .0001 inch
16. Class width = .0019 inch or .0021 inch
17.Establish the class midpoints and class limits. The first class midpoint should be located near the largest observed measurement. If possible, it should also be a convenient increment. Always make the class widths equal in size, and express the class limits in terms which are one-half unit beyond the accuracy of the original measurement unit. This avoids plotting an observed measurement on a class limit.
18. Example: First class midpoint = 1.1185 inches, and the
19.class width is .0019 inch. Therefore, limits would be
20. 1.1185 + or - .0019/2.
21.Determine the axes for the graph. The frequency scale on the vertical axis should slightly exceed the largest class frequency, and the measurement scale along the horizontal axis should be at regular intervals which are independent of the class width. (See example below steps.)
22.Draw the graph. Mark off the classes, and draw rectangles with heights corresponding to the measurement frequencies in that class.
23.Title the histogram. Give an overall title and identify each axis.
Now you have a histogram!!

| OVERVIEW | HISTORY | INTERPRETATIONS | EXAMPLE | SOFTWARE | RELATED TOPICS |

Interpretations

When combined with the concept of the normal curve and the knowledge of a particular process, the histogram becomes an effective, practical working tool in the early stages of data analysis. A histogram may be interpreted by asking three questions:
1.Is the process performing within specification limits?
2.Does the process seem to exhibit wide variation?
3.If action needs to be taken on the process, what action is appropriate?
The answer to these three questions lies in analyzing three characteristics of the histogram.
1.How well is the histogram centered? The centering of the data provides information on the process aim about some mean or nominal value.
2.How wide is the histogram? Looking at histogram width defines the variability of the process about the aim.
3.What is the shape of the histogram? Remember that the data is expected to form a normal or bell-shaped curve. Any significant change or anomaly usually indicates that there is something going on in the process which is causing the quality problem.
Examples of Typical Distributions
NORMAL

Depicted by a bell-shaped curve
most frequent measurement appears as center of distribution
less frequent measurements taper gradually at both ends of distribution
Indicates that a process is running normally (only common causes are present).
BI-MODAL

Distribution appears to have two peaks
May indicate that data from more than process are mixed together
materials may come from two separate vendors
samples may have come from two separate machines.
CLIFF-LIKE

Appears to end sharply or abruptly at one end
Indicates possible sorting or inspection of non-conforming parts.
SAW-TOOTHED

Also commonly referred to as a comb distribution, appears as an alternating jagged pattern
Often indicates a measuring problem
improper gage readings
gage not sensitive enough for readings.
SKEWED

Appears as an uneven curve; values seem to taper to one side.
It is worth mentioning again that this or any other phase of histogram analysis must be married to knowledge of the process being studied to have any real value. Knowledge of the data analysis itself does not provide sufficient insight into the quality problem.
OTHER CONSIDERATIONS
Number of samples.
For the histogram to be representative of the true process behavior, as a general rule, at least fifty (50) samples should be measured.
Limitations of technique.
Histograms are limited in their use due to the random order in which samples are taken and lack of information about the state of control of the process. Because samples are gathered without regard to order, the time-dependent or time-related trends in the process are not captured. So, what may appear to be the central tendency of the data may be deceiving. With respect to process statistical control, the histogram gives no indication whether the process was operating at its best when the data was collected. This lack of information on process control may lead to incorrect conclusions being drawn and, hence, inappropriate decisions being made. Still, with these considerations in mind, the histogram's simplicity of construction and ease of use make it an invaluable tool in the elementary stages of data analysis.

| OVERVIEW | HISTORY | CONSTRUCTION | EXAMPLE | SOFTWARE | RELATED TOPICS |

Example


The following example shows data collected from an experiment measuring pellet penetration depth from a pellet gun in inches and the corresponding histogram:
Penetration depth (inches)
__________________________

2
3
3
3
3
4
4
4
5
5
6
6

Some important things to remember when constructing a histogram:
Use intervals of equal length.
Show the entire vertical axes beginning with zero.
Do not break either axis.
Keep a uniform scale across the axis.
Center the histogram bars at the midpoint of the intervals (in this case, the penetration depth intervals).
Scatter Diagrams
(A Brief Tutorial)

How to Use Tutorial
This tutorial is designed to allow the user to develop and interpret scatter diagrams. Other additional information is presented within the History and Key Terms sections of this tutorial so the user will have a better understanding of scatter diagrams.
The user can venture through the tutorial by clicking on the desired topic in one of the menus, or by using the scroll on the right side of the screen to move through the page.
Several examples are also furnished in this tutorial to enable the user to develop a more clear understanding of the information being presented. When the scatter diagram has been plotted from the data, the user can view several different graphs within the Interpretations sections of the tutorial, read the interpretation of the diagrams pattern, and be able to draw conclusions about the plotted diagram by comparing it to one of the five possible graph patterns.

Overview
Key Terms
History
Construction of Scatter Diagrams
Interpretations
Examples
Related Topics


Overview
Scatter diagrams are used to study possible relationships between two variables. Although these diagrams cannot prove that one variable causes the other, they do indicate the existance of a relationship, as well as the strength of that relationship.
A scatter diagram is composed of a horizontal axis containing the measured values of one variable and a vertical axis representing the measurements of the other variable.
The purpose of the scatter diagram is to display what happens to one variables when another variable is changed. The diagram is used to test a theory that the two variables are related. The type of relationship that exits is indicated by the slope of the diagram.

| KEY TERMS | HISTORY | CONSTRUCTION | INTERPRETATIONS | EXAMPLES | RELATED TOPICS |

Key Terms
 Variable - a quality characteristic that can be measured and expressed as a number on some continuous scale of measurement.

 Relationship - Relationships between variables exist when one variable depends on the other and changing one variable will effect the other.

 Data Sheet - contains the measurements that were collected for plotting the diagram.

 Correlation - an analysis method used to decide whether there is a statistically significant relationship between two variables.

 Regression - an analysis method used to identify the exact nature of the relationship between two variables.

| OVERVIEW | HISTORY | CONSTRUCTION | INTERPRETATIONS | EXAMPLES | RELATED TOPICS |

History
Commonly, while a cause-effect diagram has been used to describe the relationship between two variables, the histogram was used to visualize the structure of the data. However, a means of observing the kinds of relationships between variables was needed. Using the theory of linear regression which originated from studies performed by Sir Francis Galton (1822-1911), the scatter diagram was developed so that intuitive and qualitative conclusions could be drawn about the paired data, or variables. The concept of correlation was employed to decide whether a significant relationship existed between the paired data. Furthermore, regression analysis was used to identify the exact nature of the relationship.
The Guide to Quality Control and The Statistical Quality Control Handbook, written by a Japanese quality consultant named Kaoru Ishikawa are useful in providing an understanidng on how to use and interpret a scatter diagram. Ishikawa believed that there was no end to qualithy improvement and in 1985 suggested that seven base tools be used for collection and analysis of qualtiy data. Among the tools was the scatter diagram.

| OVERVIEW | KEY TERMS | CONSTRUCTION | INTERPRETATIONS | EXAMPLES | RELATED TOPICS |

Construction of Scatter Diagrams
 Collect and construct a data sheet of 50 to 100 paired samples of data, that you suspect to be related. Construct your data sheet as follows:
Car Age(In Years) Price(In Dollars)
1 2 4000
2 4 2500
3 1 5000
4 5 1250
: : :
: : :
: : :
: : :
100 7 1000
 Draw the axes of the diagram. The first variable (the independent variable) is usually located on the horizontal axis and its values should increase as you move to the right. The vertical axis usually contains the second variable (the dependent variable) and its values should increase as you move up the axis.

 Plot the data on the diagram. The resulting scatter diagram may look as follows:

 Interpret the diagram. See interpretation section of tutorial.
| OVERVIEW | KEY TERMS | HISTORY | INTERPRETATIONS | EXAMPLES | RELATED TOPICS |

Interpretations
The scatter diagram is a useful tool for identifying a potential relationship between two variables. The shape of the scatter diagram presents valuable information about the graph. It shows the type of relationship which may be occurring between the two variables. There are several different patterns (meanings) that scatter diagrams can have. The following describe five of the most common scenerios :
1.The first pattern is positive correlation, that is, as the amount of variable x increases, the variable y also increases. It is tempting to think this is a cause/effect relationship. This is an incorrect thinking pattern, because correlation does not necessarily mean causality. This simple relationship could be caused by something totally different. For instance, the two variables could be related to a third, such as curing time or stamping temperature. Theoretically, if x is controlled, we have a chance of controlling y.


2.Secondly, we have possible positive correlation, that is, if x increases, y will increase somewhat, but y seems to be caused by something other than x. Designed experiments must be utilized to verify causality.



3.We also have the no correlation category. The diagram is so random that there is no apparent correlation between the two variables.



4.There is also possible negative correlation, that is, an increase in x will cause a tendency for a decrease in y, but y seems to have causes other than x.



5.Finally, we have the negative correlation category. An increase in x will cause a decrease in y. Therefore, if y is controlled, we have a good chance of controlling x.



Key Observations
*A strong relationship between the two variables is observed when most of the points fall along an imaginary straight line with either a positive or negative slope.
*No relationship between the two variables is observed when the points are randomly scattered about the graph.
| OVERVIEW | KEY TERMS | HISTORY | CONSTRUCTION | EXAMPLES | RELATED TOPICS |

Example 1
Situation: The new commissioner of the American Basketball League wants to construct a scatter diagram to find out if there is any relationship between a players weight and her height. How should she go about making her scatter diagram?
1.Collect the data (Remember to use 50-100 paired samples).



2.Draw and label your x and y axes.



3.Plot the data on the diagram.



4.Interpret your chart.
According to this scatter diagram the new commisioner was right. There does seem to be a positive correlation between a player's weight and her height. In other words, the taller a player is the more she tends to weight.

Quality Control Charts

General Purpose
General Approach
Establishing Control Limits
Common Types of Charts
Short Run Control Charts
Short Run Charts for Variables
Short Run Charts for Attributes
Unequal Sample Sizes
Control Charts for Variables vs. Charts for Attributes
Control Charts for Individual Observations
Out-of-Control Process: Runs Tests
Operating Characteristic (OC) Curves
Process Capability Indices
Other Specialized Control Charts

General Purpose
In all production processes, we need to monitor the extent to which our products meet specifications. In the most general terms, there are two "enemies" of product quality: (1) deviations from target specifications, and (2) excessive variability around target specifications. During the earlier stages of developing the production process, designed experiments are often used to optimize these two quality characteristics (see Experimental Design); the methods provided in Quality Control are on-line or in-process quality control procedures to monitor an on-going production process. For detailed descriptions of these charts and extensive annotated examples, see Buffa (1972), Duncan (1974) Grant and Leavenworth (1980), Juran (1962), Juran and Gryna (1970), Montgomery (1985, 1991), Shirland (1993), or Vaughn (1974). Two recent excellent introductory texts with a "how-to" approach are Hart & Hart (1989) and Pyzdek (1989); two recent German language texts on this subject are Rinne and Mittag (1995) and Mittag (1993).


General Approach
The general approach to on-line quality control is straightforward: We simply extract samples of a certain size from the ongoing production process. We then produce line charts of the variability in those samples, and consider their closeness to target specifications. If a trend emerges in those lines, or if samples fall outside pre-specified limits, then we declare the process to be out of control and take action to find the cause of the problem. These types of charts are sometimes also referred to as Shewhart control charts (named after W. A. Shewhart who is generally credited as being the first to introduce these methods; see Shewhart, 1931).
Interpreting the chart. The most standard display actually contains two charts (and two histograms); one is called an X-bar chart, the other is called an R chart.

In both line charts, the horizontal axis represents the different samples; the vertical axis for the X-bar chart represents the means for the characteristic of interest; the vertical axis for the R chart represents the ranges. For example, suppose we wanted to control the diameter of piston rings that we are producing. The center line in the X-bar chart would represent the desired standard size (e.g., diameter in millimeters) of the rings, while the center line in the R chart would represent the acceptable (within-specification) range of the rings within samples; thus, this latter chart is a chart of the variability of the process (the larger the variability, the larger the range). In addition to the center line, a typical chart includes two additional horizontal lines to represent the upper and lower control limits (UCL, LCL, respectively); we will return to those lines shortly. Typically, the individual points in the chart, representing the samples, are connected by a line. If this line moves outside the upper or lower control limits or exhibits systematic patterns across consecutive samples (see Runs Tests), then a quality problem may potentially exist.


Establishing Control Limits
Even though one could arbitrarily determine when to declare a process out of control (that is, outside the UCL-LCL range), it is common practice to apply statistical principles to do so. Elementary Concepts discusses the concept of the sampling distribution, and the characteristics of the normal distribution. The method for constructing the upper and lower control limits is a straightforward application of the principles described there.
Example. Suppose we want to control the mean of a variable, such as the size of piston rings. Under the assumption that the mean (and variance) of the process does not change, the successive sample means will be distributed normally around the actual mean. Moreover, without going into details regarding the derivation of this formula, we also know (because of the central limit theorem, and thus approximate normal distribution of the means; see, for example, Hoyer and Ellis, 1996) that the distribution of sample means will have a standard deviation of Sigma (the standard deviation of individual data points or measurements) over the square root of n (the sample size). It follows that approximately 95% of the sample means will fall within the limits ± 1.96 * Sigma/Square Root(n) (refer to Elementary Concepts for a discussion of the characteristics of the normal distribution and the central limit theorem). In practice, it is common to replace the 1.96 with 3 (so that the interval will include approximately 99% of the sample means), and to define the upper and lower control limits as plus and minus 3 sigma limits, respectively.
General case. The general principle for establishing control limits just described applies to all control charts. After deciding on the characteristic we want to control, for example, the standard deviation, we estimate the expected variability of the respective characteristic in samples of the size we are about to take. Those estimates are then used to establish the control limits on the chart.


Common Types of Charts
The types of charts are often classified according to the type of quality characteristic that they are supposed to monitor: there are quality control charts for variables and control charts for attributes. Specifically, the following charts are commonly constructed for controlling variables:
X-bar chart. In this chart the sample means are plotted in order to control the mean value of a variable (e.g., size of piston rings, strength of materials, etc.).
R chart. In this chart, the sample ranges are plotted in order to control the variability of a variable.
S chart. In this chart, the sample standard deviations are plotted in order to control the variability of a variable.
S**2 chart. In this chart, the sample variances are plotted in order to control the variability of a variable.
For controlling quality characteristics that represent attributes of the product, the following charts are commonly constructed:
C chart. In this chart (see example below), we plot the number of defectives (per batch, per day, per machine, per 100 feet of pipe, etc.). This chart assumes that defects of the quality attribute are rare, and the control limits in this chart are computed based on the Poisson distribution (distribution of rare events).

U chart. In this chart we plot the rate of defectives, that is, the number of defectives divided by the number of units inspected (the n; e.g., feet of pipe, number of batches). Unlike the C chart, this chart does not require a constant number of units, and it can be used, for example, when the batches (samples) are of different sizes.
Np chart. In this chart, we plot the number of defectives (per batch, per day, per machine) as in the C chart. However, the control limits in this chart are not based on the distribution of rare events, but rather on the binomial distribution. Therefore, this chart should be used if the occurrence of defectives is not rare (e.g., they occur in more than 5% of the units inspected). For example, we may use this chart to control the number of units produced with minor flaws.
P chart. In this chart, we plot the percent of defectives (per batch, per day, per machine, etc.) as in the U chart. However, the control limits in this chart are not based on the distribution of rare events but rather on the binomial distribution (of proportions). Therefore, this chart is most applicable to situations where the occurrence of defectives is not rare (e.g., we expect the percent of defectives to be more than 5% of the total number of units produced).
All of these charts can be adapted for short production runs (short run charts), and for multiple process streams.


Short Run Charts
The short run control chart, or control chart for short production runs, plots observations of variables or attributes for multiple parts on the same chart. Short run control charts were developed to address the requirement that several dozen measurements of a process must be collected before control limits are calculated. Meeting this requirement is often difficult for operations that produce a limited number of a particular part during a production run.
For example, a paper mill may produce only three or four (huge) rolls of a particular kind of paper (i.e., part) and then shift production to another kind of paper. But if variables, such as paper thickness, or attributes, such as blemishes, are monitored for several dozen rolls of paper of, say, a dozen different kinds, control limits for thickness and blemishes could be calculated for the transformed (within the short production run) variable values of interest. Specifically, these transformations will rescale the variable values of interest such that they are of compatible magnitudes across the different short production runs (or parts). The control limits computed for those transformed values could then be applied in monitoring thickness, and blemishes, regardless of the types of paper (parts) being produced. Statistical process control procedures could be used to determine if the production process is in control, to monitor continuing production, and to establish procedures for continuous quality improvement.
For additional discussions of short run charts refer to Bothe (1988), Johnson (1987), or Montgomery (1991).
Short Run Charts for Variables
Nominal chart, target chart. There are several different types of short run charts. The most basic are the nominal short run chart, and the target short run chart. In these charts, the measurements for each part are transformed by subtracting a part-specific constant. These constants can either be the nominal values for the respective parts (nominal short run chart), or they can be target values computed from the (historical) means for each part (Target X-bar and R chart). For example, the diameters of piston bores for different engine blocks produced in a factory can only be meaningfully compared (for determining the consistency of bore sizes) if the mean differences between bore diameters for different sized engines are first removed. The nominal or target short run chart makes such comparisons possible. Note that for the nominal or target chart it is assumed that the variability across parts is identical, so that control limits based on a common estimate of the process sigma are applicable.
Standardized short run chart. If the variability of the process for different parts cannot be assumed to be identical, then a further transformation is necessary before the sample means for different parts can be plotted in the same chart. Specifically, in the standardized short run chart the plot points are further transformed by dividing the deviations of sample means from part means (or nominal or target values for parts) by part-specific constants that are proportional to the variability for the respective parts. For example, for the short run X-bar and R chart, the plot points (that are shown in the X-bar chart) are computed by first subtracting from each sample mean a part specific constant (e.g., the respective part mean, or nominal value for the respective part), and then dividing the difference by another constant, for example, by the average range for the respective chart. These transformations will result in comparable scales for the sample means for different parts.
Short Run Charts for Attributes
For attribute control charts (C, U, Np, or P charts), the estimate of the variability of the process (proportion, rate, etc.) is a function of the process average (average proportion, rate, etc.; for example, the standard deviation of a proportion p is equal to the square root of p*(1- p)/n). Hence, only standardized short run charts are available for attributes. For example, in the short run P chart, the plot points are computed by first subtracting from the respective sample p values the average part p's, and then dividing by the standard deviation of the average p's.


Unequal Sample Sizes
When the samples plotted in the control chart are not of equal size, then the control limits around the center line (target specification) cannot be represented by a straight line. For example, to return to the formula Sigma/Square Root(n) presented earlier for computing control limits for the X-bar chart, it is obvious that unequal n's will lead to different control limits for different sample sizes. There are three ways of dealing with this situation.
Average sample size. If one wants to maintain the straight-line control limits (e.g., to make the chart easier to read and easier to use in presentations), then one can compute the average n per sample across all samples, and establish the control limits based on the average sample size. This procedure is not "exact," however, as long as the sample sizes are reasonably similar to each other, this procedure is quite adequate.
Variable control limits. Alternatively, one may compute different control limits for each sample, based on the respective sample sizes. This procedure will lead to variable control limits, and result in step-chart like control lines in the plot. This procedure ensures that the correct control limits are computed for each sample. However, one loses the simplicity of straight-line control limits.
Stabilized (normalized) chart. The best of two worlds (straight line control limits that are accurate) can be accomplished by standardizing the quantity to be controlled (mean, proportion, etc.) according to units of sigma. The control limits can then be expressed in straight lines, while the location of the sample points in the plot depend not only on the characteristic to be controlled, but also on the respective sample n's. The disadvantage of this procedure is that the values on the vertical (Y) axis in the control chart are in terms of sigma rather than the original units of measurement, and therefore, those numbers cannot be taken at face value (e.g., a sample with a value of 3 is 3 times sigma away from specifications; in order to express the value of this sample in terms of the original units of measurement, we need to perform some computations to convert this number back).


Control Charts for Variables vs. Charts for Attributes
Sometimes, the quality control engineer has a choice between variable control charts and attribute control charts.
Advantages of attribute control charts. Attribute control charts have the advantage of allowing for quick summaries of various aspects of the quality of a product, that is, the engineer may simply classify products as acceptable or unacceptable, based on various quality criteria. Thus, attribute charts sometimes bypass the need for expensive, precise devices and time-consuming measurement procedures. Also, this type of chart tends to be more easily understood by managers unfamiliar with quality control procedures; therefore, it may provide more persuasive (to management) evidence of quality problems.
Advantages of variable control charts. Variable control charts are more sensitive than attribute control charts (see Montgomery, 1985, p. 203). Therefore, variable control charts may alert us to quality problems before any actual "unacceptables" (as detected by the attribute chart) will occur. Montgomery (1985) calls the variable control charts leading indicators of trouble that will sound an alarm before the number of rejects (scrap) increases in the production process.
Control Chart for Individual Observations
Variable control charts can by constructed for individual observations taken from the production line, rather than samples of observations. This is sometimes necessary when testing samples of multiple observations would be too expensive, inconvenient, or impossible. For example, the number of customer complaints or product returns may only be available on a monthly basis; yet, one would like to chart those numbers to detect quality problems. Another common application of these charts occurs in cases when automated testing devices inspect every single unit that is produced. In that case, one is often primarily interested in detecting small shifts in the product quality (for example, gradual deterioration of quality due to machine wear). The CUSUM, MA, and EWMA charts of cumulative sums and weighted averages discussed below may be most applicable in those situations.


Out-Of-Control Process: Runs Tests
As mentioned earlier in the introduction, when a sample point (e.g., mean in an X-bar chart) falls outside the control lines, one has reason to believe that the process may no longer be in control. In addition, one should look for systematic patterns of points (e.g., means) across samples, because such patterns may indicate that the process average has shifted. These tests are also sometimes referred to as AT&T runs rules (see AT&T, 1959) or tests for special causes (e.g., see Nelson, 1984, 1985; Grant and Leavenworth, 1980; Shirland, 1993). The term special or assignable causes as opposed to chance or common causes was used by Shewhart to distinguish between a process that is in control, with variation due to random (chance) causes only, from a process that is out of control, with variation that is due to some non-chance or special (assignable) factors (cf. Montgomery, 1991, p. 102).
As the sigma control limits discussed earlier, the runs rules are based on "statistical" reasoning. For example, the probability of any sample mean in an X-bar control chart falling above the center line is equal to 0.5, provided (1) that the process is in control (i.e., that the center line value is equal to the population mean), (2) that consecutive sample means are independent (i.e., not auto-correlated), and (3) that the distribution of means follows the normal distribution. Simply stated, under those conditions there is a 50-50 chance that a mean will fall above or below the center line. Thus, the probability that two consecutive means will fall above the center line is equal to 0.5 times 0.5 = 0.25.
Accordingly, the probability that 9 consecutive samples (or a run of 9 samples) will fall on the same side of the center line is equal to 0.5**9 = .00195. Note that this is approximately the probability with which a sample mean can be expected to fall outside the 3- times sigma limits (given the normal distribution, and a process in control). Therefore, one could look for 9 consecutive sample means on the same side of the center line as another indication of an out-of-control condition. Refer to Duncan (1974) for details concerning the "statistical" interpretation of the other (more complex) tests.
Zone A, B, C. Customarily, to define the runs tests, the area above and below the chart center line is divided into three "zones."

By default, Zone A is defined as the area between 2 and 3 times sigma above and below the center line; Zone B is defined as the area between 1 and 2 times sigma, and Zone C is defined as the area between the center line and 1 times sigma.
9 points in Zone C or beyond (on one side of central line). If this test is positive (i.e., if this pattern is detected), then the process average has probably changed. Note that it is assumed that the distribution of the respective quality characteristic in the plot is symmetrical around the mean. This is, for example, not the case for R charts, S charts, or most attribute charts. However, this is still a useful test to alert the quality control engineer to potential shifts in the process. For example, successive samples with less-than-average variability may be worth investigating, since they may provide hints on how to decrease the variation in the process.
6 points in a row steadily increasing or decreasing. This test signals a drift in the process average. Often, such drift can be the result of tool wear, deteriorating maintenance, improvement in skill, etc. (Nelson, 1985).
14 points in a row alternating up and down. If this test is positive, it indicates that two systematically alternating causes are producing different results. For example, one may be using two alternating suppliers, or monitor the quality for two different (alternating) shifts.
2 out of 3 points in a row in Zone A or beyond. This test provides an "early warning" of a process shift. Note that the probability of a false-positive (test is positive but process is in control) for this test in X-bar charts is approximately 2%.
4 out of 5 points in a row in Zone B or beyond. Like the previous test, this test may be considered to be an "early warning indicator" of a potential process shift. The false- positive error rate for this test is also about 2%.
15 points in a row in Zone C (above and below the center line). This test indicates a smaller variability than is expected (based on the current control limits).
8 points in a row in Zone B, A, or beyond, on either side of the center line (without points in Zone C). This test indicates that different samples are affected by different factors, resulting in a bimodal distribution of means. This may happen, for example, if different samples in an X-bar chart where produced by one of two different machines, where one produces above average parts, and the other below average parts.


Operating Characteristic (OC) Curves
A common supplementary plot to standard quality control charts is the so-called operating characteristic or OC curve (see example below). One question that comes to mind when using standard variable or attribute charts is how sensitive is the current quality control procedure? Put in more specific terms, how likely is it that you will not find a sample (e.g., mean in an X-bar chart) outside the control limits (i.e., accept the production process as "in control"), when, in fact, it has shifted by a certain amount? This probability is usually referred to as the (beta) error probability, that is, the probability of erroneously accepting a process (mean, mean proportion, mean rate defectives, etc.) as being "in control." Note that operating characteristic curves pertain to the false-acceptance probability using the sample-outside-of- control-limits criterion only, and not the runs tests described earlier.

Operating characteristic curves are extremely useful for exploring the power of our quality control procedure. The actual decision concerning sample sizes should depend not only on the cost of implementing the plan (e.g., cost per item sampled), but also on the costs resulting from not detecting quality problems. The OC curve allows the engineer to estimate the probabilities of not detecting shifts of certain sizes in the production quality.
Process Capability Indices
For variable control charts, it is often desired to include so-called process capability indices in the summary graph. In short, process capability indices express (as a ratio) the proportion of parts or items produced by the current process that fall within user-specified limits (e.g., engineering tolerances).
For example, the so-called Cp index is computed as:
Cp = (USL-LSL)/(6*sigma)
where sigma is the estimated process standard deviation, and USL and LSL are the upper and lower specification (engineering) limits, respectively. If the distribution of the respective quality characteristic or variable (e.g., size of piston rings) is normal, and the process is perfectly centered (i.e., the mean is equal to the design center), then this index can be interpreted as the proportion of the range of the standard normal curve (the process width) that falls within the engineering specification limits. If the process is not centered, an adjusted index Cpk is used instead. For a "capable" process, the Cp index should be greater than 1, that is, the specification limits would be larger than 6 times the sigma limits, so that over 99% of all items or parts produced could be expected to fall inside the acceptable engineering specifications. For a detailed discussion of this and other indices, refer to Process Analysis.


Other Specialized Control Charts
The types of control charts mentioned so far are the "workhorses" of quality control, and they are probably the most widely used methods. However, with the advent of inexpensive desktop computing, procedures requiring more computational effort have become increasingly popular.
X-bar Charts For Non-Normal Data. The control limits for standard X-bar charts are constructed based on the assumption that the sample means are approximately normally distributed. Thus, the underlying individual observations do not have to be normally distributed, since, as the sample size increases, the distribution of the means will become approximately normal (i.e., see discussion of the central limit theorem in the Elementary Concepts; however, note that for R, S¸ and S**2 charts, it is assumed that the individual observations are normally distributed). Shewhart (1931) in his original work experimented with various non-normal distributions for individual observations, and evaluated the resulting distributions of means for samples of size four. He concluded that, indeed, the standard normal distribution-based control limits for the means are appropriate, as long as the underlying distribution of observations are approximately normal. (See also Hoyer and Ellis, 1996, for an introduction and discussion of the distributional assumptions for quality control charting.)
However, as Ryan (1989) points out, when the distribution of observations is highly skewed and the sample sizes are small, then the resulting standard control limits may produce a large number of false alarms (increased alpha error rate), as well as a larger number of false negative ("process-is-in-control") readings (increased beta-error rate). You can compute control limits (as well as process capability indices) for X-bar charts based on so-called Johnson curves(Johnson, 1949), which allow to approximate the skewness and kurtosis for a large range of non-normal distributions (see also Fitting Distributions by Moments, in Process Analysis). These non- normal X-bar charts are useful when the distribution of means across the samples is clearly skewed, or otherwise non-normal.
Hotelling T**2 Chart. When there are multiple related quality characteristics (recorded in several variables), we can produce a simultaneous plot (see example below) for all means based on Hotelling multivariate T**2 statistic (first proposed by Hotelling, 1947).

Cumulative Sum (CUSUM) Chart. The CUSUM chart was first introduced by Page (1954); the mathematical principles involved in its construction are discussed in Ewan (1963), Johnson (1961), and Johnson and Leone (1962).

If one plots the cumulative sum of deviations of successive sample means from a target specification, even minor, permanent shifts in the process mean will eventually lead to a sizable cumulative sum of deviations. Thus, this chart is particularly well-suited for detecting such small permanent shifts that may go undetected when using the X-bar chart. For example, if, due to machine wear, a process slowly "slides" out of control to produce results above target specifications, this plot would show a steadily increasing (or decreasing) cumulative sum of deviations from specification.
To establish control limits in such plots, Barnhard (1959) proposed the so-called V- mask, which is plotted after the last sample (on the right). The V-mask can be thought of as the upper and lower control limits for the cumulative sums. However, rather than being parallel to the center line; these lines converge at a particular angle to the right, producing the appearance of a V rotated on its side. If the line representing the cumulative sum crosses either one of the two lines, the process is out of control.
Moving Average (MA) Chart. To return to the piston ring example, suppose we are mostly interested in detecting small trends across successive sample means. For example, we may be particularly concerned about machine wear, leading to a slow but constant deterioration of quality (i.e., deviation from specification). The CUSUM chart described above is one way to monitor such trends, and to detect small permanent shifts in the process average. Another way is to use some weighting scheme that summarizes the means of several successive samples; moving such a weighted mean across the samples will produce a moving average chart (as shown in the following graph).

Exponentially-weighted Moving Average (EWMA) Chart. The idea of moving averages of successive (adjacent) samples can be generalized. In principle, in order to detect a trend we need to weight successive samples to form a moving average; however, instead of a simple arithmetic moving average, we could compute a geometric moving average (this chart (see graph below) is also called Geometric Moving Average chart, see Montgomery, 1985, 1991).

Specifically, we could compute each data point for the plot as:
zt = *x-bart + (1-)*zt-1
In this formula, each point zt is computed as (lambda) times the respective mean x-bart, plus one minus times the previous (computed) point in the plot. The parameter (lambda) here should assume values greater than 0 and less than 1. Without going into detail (see Montgomery, 1985, p. 239), this method of averaging specifies that the weight of historically "old" sample means decreases geometrically as one continues to draw samples. The interpretation of this chart is much like that of the moving average chart, and it allows us to detect small shifts in the means, and, therefore, in the quality of the production process.
Regression Control Charts. Sometimes we want to monitor the relationship between two aspects of our production process. For example, a post office may want to monitor the number of worker-hours that are spent to process a certain amount of mail. These two variables should roughly be linearly correlated with each other, and the relationship can probably be described in terms of the well-known Pearson product-moment correlation coefficient r. This statistic is also described in Basic Statistics. The regression control chart contains a regression line that summarizes the linear relationship between the two variables of interest. The individual data points are also shown in the same graph. Around the regression line we establish a confidence interval within which we would expect a certain proportion (e.g., 95%) of samples to fall. Outliers in this plot may indicate samples where, for some reason, the common relationship between the two variables of interest does not hold.

Applications. There are many useful applications for the regression control chart. For example, professional auditors may use this chart to identify retail outlets with a greater than expected number of cash transactions given the overall volume of sales, or grocery stores with a greater than expected number of coupons redeemed, given the total sales. In both instances, outliers in the regression control charts (e.g., too many cash transactions; too many coupons redeemed) may deserve closer scrutiny.
Pareto Chart Analysis. Quality problems are rarely spread evenly across the different aspects of the production process or different plants. Rather, a few "bad apples" often account for the majority of problems. This principle has come to be known as the Pareto principle, which basically states that quality losses are mal-distributed in such a way that a small percentage of possible causes are responsible for the majority of the quality problems. For example, a relatively small number of "dirty" cars are probably responsible for the majority of air pollution; the majority of losses in most companies result from the failure of only one or two products. To illustrate the "bad apples", one plots the Pareto chart,

which simply amounts to a histogram showing the distribution of the quality loss (e.g., dollar loss) across some meaningful categories; usually, the categories are sorted into descending order of importance (frequency, dollar amounts, etc.). Very often, this chart provides useful guidance as to where to direct quality