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Statistical Process Control Using Design of Experiments (DOE)

Student Guide

First Edition (July 1992), Second Edition (July 1995), Third Edition (December 2000) Please address any reader comments to The Quality Group, 6059 Boylston Drive, Suite 250, Atlanta, GA 30328. The Quality Group may use or distribute any of the information you supply in any way it believes appropriate without incurring any obligation. You may, of course, continue to use the information you supply. Copyright © International Business Machines Corporation 1992, 1995. All rights reserved.

Note to U.S. Government Users - Documentation related to restricted rights - Use, duplication or disclosure is subject to restrictions set forth in GSA ADP Schedule Contract with IBM Corp.

Table of Contents

Using Design of Experiments (DOE) 1

Course Group Contents Introduction to DOE

Benefits of DOE Some Important Definitions Activity 1 Types of Experiments Activity 2

Application of DOE An Approach to DOE Phases In Applying DOE Activity 1 Activity 2 The Return of DOE

Appendix A. Answers to Activities Introduction to DOE – Activity 1 Introduction to DOE -- Activity 2 Application of DOE -- Activity 1 Application of DOE -- Activity 2

Glossary

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Using Design of Experiments (DOE)

Course Group Contents

This Course Group introduces DOE--the Design of Experiments, and shows you what it is, why it is helpful, and how it can be implemented to improve the quality of products and processes. You review a few examples and a case study using DOE. This Course Group contains two courses:

Introduction to DOE

Introduction to DOE covers the what and why of DOE. In addition to a definition of Design of Experiments you review several key DOE terms and benefits.

Application of DOE

Application of DOE suggests a series of steps your team can use to apply the Design of Experiments. This is the how of DOE. The steps are divided into five phases:

• Plan

• Design

• Run

• Analysis

• Act

Whether or not you choose to use these steps exactly as shown, the who, where, and when of DOE are up to you and your team.

Page 1 Introduction to DOE

Introduction to DOE

The SPC Starter Kit and Applying SPC Course Groups introduced you to Statistical Process Control (SPC), and showed how it can be used to find and eliminate special cause variation. The Process Capability Course Group showed you how to assess the extent to which a stable process is able to meet specifications. This is the definition of process capability. But what if your process is in control and yet is not good enough? What if common cause variation needs to be reduced?

Introduction to DOE shows how DOE can benefit your processes and defines some key terms. This course contains 2 lessons:

Lesson 1: Benefits of DOE

Lesson 1 defines DOE, describes the purpose of DOE, and shows how it improves a process by reducing common cause variation. This lesson also looks at the benefits DOE brings to the production process.

Lesson 2: Some Important Definitions

Lesson 2 defines DOE terms and discusses the different types of experiments.

Introduction to DOE Page 2

Terms Defined in the Glossary

• Design of Experiments (DOE)

• Design Matrix

• Factor

• Fractional Factorial Experiment

• Full Factorial Experiment

• Informal/Ad Hoc Experiment

• Interaction Effect

• Interaction Plot

• Level

• One-factor-at-a-time Experiment

• Response Variable

• Statistically Designed Experiment

• Trial


Benefits of DOE DOE (pronounced D-O-E) stands for Design Of Experiments. You may have been involved with Experiments in the past, but those experiments may not have been statistically Designed. DOE improves processes using structured experiments. This means that as a process is run, process or tool settings are changed in an orderly and systematic way to find the best combination of settings.

For example, suppose you are baking cookies and want to improve them. The cookies are good, but they could be better. There are differences in taste and texture from one batch to another. You may try different oven temperatures, as well as different amounts of flour, sugar, and shortening in attempting to bake a better, more consistent batch. But, if you try random combinations of temperatures and ingredient amounts, your chances of hitting the best mix will be small.

DOE uses carefully planned experiments to learn as much about the process as possible with limited time and resources. These plans are usually shown in a design matrix like the following. Don't worry about how to read it for now. Just remember that the "D" in DOE comes from a careful Design.

Factors Trial

A B C D

1

2

3

4

5

6

7

8

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+

+

+

+

-

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+

Figure 1. Design Matrix


This is an example of how using DOE improves a process. Remember that the cookies you baked before using DOE were pretty good, but had too much common cause variation. The ability of DOE to help reduce common cause variation makes it an important tool for continuous improvement.

Let's look at some DOE benefits. DOE helps to:

• Reduce variation. Reducing common cause variation may require new tools, new materials, or new process settings. DOE improves and optimizes a process.

• Improve products. When there is less variation, a product is more consistent. You and your customers know what to expect.

• Reduce the number of defects and increase yield. A centered process with reduced variation has more of its output within specifications.

• Reduce costs. Rework and scrap go down as the number of defects decrease.

• Increase knowledge about the production process. Understanding process relationships helps to improve the process today and to improve the design tomorrow.

• Improve quality. An improved and better understood process produces higher quality products.


Summary: World-class companies build quality into their products and processes by using DOE during design. They know it is more efficient to prevent problems now than to fix them later. As manufacturing and design groups begin to work more closely together, the use of DOE will surely rise.

Time

DesignPhase

ProductionPhase

Also Rans

World Class

Quality Effort

Figure 2. World Class Graph

Why use DOE? What could be better for the manufacturer and the consumer than better quality and lower costs? The next lesson shows you some important definitions. Then, Application of DOE shows how using carefully planned experiments helps your team achieve some or all of the benefits previously listed.


Some Important Definitions It is much easier to communicate when everyone speaks, or at least understands, the same language. That is what Lesson 2 is all about. This lesson defines the following terms related to DOE:

• Response Variable

• Factor

• Level

• Trial

• Interaction Effect

In addition to these new terms, the following types of experiments are discussed:

• Statistically designed

• Full factorial

• Fractional factorial

• One-factor-at-a-time

Example: Suppose you are applying DOE to the process of baking cookies. Your cookie recipe looks like this:

____________________________________________________________

Ingredients:

2/3 cup shortening 2 cups all-purpose flour

3/4 cup sugar 1/2 teaspoons baking powder

1 teaspoon vanilla 1/4 teaspoon salt

1 egg

4 teaspoons milk

Directions:

Pre-heat oven to 375 degrees.

Cream shortening, sugar, and vanilla. Add egg; beat until light and fluffy. Stir in milk. Sift together flour, baking powder, and salt; blend into creamed mixture...

...Bake on greased cookie sheet for 10 minutes. Cool slightly; remove from pan.


____________________________________________________________

Definitions

Response Variable: Before starting any experiment you must have an objective and some way of measuring your success in achieving that objective. You must know where you are going and when you have arrived.

The response variable is the specific parameter used to decide if the process is improving. It determines the type of data to collect. For example, the crispness of the cookies can be one response variable. Sweetness can be another. The proportion of cookies broken could be a third.

If your main objective is to make a lightly baked but crisp cookie, then crispness would be a good response variable. In most manufacturing cases the response variable will be a well defined measurement - like thickness. In this example, though you could surely find some expensive equipment to measure hardness, you could also rate them on a scale of 1 to 10.

Factor: A factor is any variable that may influence a product or process. Selecting possible factors is a lot like picking parameters to control in an I-2 meeting. Many of the same tools can be used. For example, fishbone diagrams and brainstorming are great ways to get a list of possible factors. The following fishbone diagram shows several candidates for factors in the experiment to improve cookie crispness.

Amount

Oven

Ingredients Method Equipment

Cookie Crispness

People Environment

SugarTemperature

Time

Cookie Sheet

TrainedHumidity

Temperature

Steel

YesNo

Type SolidAir PocketBaking

DoughAluminum

Figure 3. Fishbone Diagram

If too many factors are chosen for the experiment, it will take too long to run. If too few are chosen, you may not learn enough about the process. Typically, the number of factors used in DOE varies from 3 to about 8. For this experiment a good list of factors may include amount of sugar, oven temperature, temperature of the dough before baking, baking time and the type of cookie sheet used.


Introduction to DOE

Level: A level is defined as the setting or category that a factor can take. For example, the cookie recipe calls for 3/4 cup of sugar. When designing the experiment, you could try using 1/2 cup and 1 cup of sugar. You could also try aluminum and steel cookie sheets. These are the levels of the factors.

When choosing levels for a factor be sure that they are reasonable, yet far enough apart to make a difference. A level of one teaspoon of sugar is too low to be reasonable--the cookies would never pass a taste test. Temperatures of 375°F and 380°F will not make a difference--they are too close.

Two levels for each factor are most commonly used, but you may see more. This course always uses two levels for each factor. Two levels are the easiest to work with and usually provide enough information.


Introduction to DOE -- Activity 1

Purpose: Identify possible factors and levels to be used in an experiment to improve a process.

Instructions: Use the Fishbone diagram below, complete each of the following exercises.

Amount

Oven

Ingredients Method Equipment

Cookie Crispness

People Environment

SugarTemperature

Time

Cookie Sheet

TrainedHumidity

Temperature

Steel

YesNo

Type SolidAir PocketBaking

DoughAluminum

Figure 4. Fishbone Diagram

1. Can you think of any more ideas to add to the fishbone diagram above for the cookie baking process?

2. Pick 4 factors that you think could affect the crispness of the cookies.

3. Select two levels for each of the factors.

4. If you want to try some more or if you don't know enough about baking cookies, pick another process. Pick a response variable that will tell you if your process has improved. It can be anything from gas mileage in your car to your score in a game of golf. Then, create a fishbone diagram, pick 4 factors, and select 2 levels for each factor.


Trial: One run during an experiment is called a trial. During each trial, factors are set at their chosen levels. Data (the response variable) are collected from each trial and analyzed when the experiment is finished. In the cookie baking example, one trial could be the same as one batch of cookies.

Design Matrix: A design matrix is used to keep track of the settings and levels for each trial. It is really just a map of the experiment. The design matrix is created before beginning the experiment or conducting any trials.

Below is a sample of a design matrix for baking cookies. Notice that the levels for each factor are shown as a + or a -. Since only two levels are used for each factor, it is easy to assign a + to one level and a - to the other.

Trial

Factors

A (Sugar)

B (Bake Temp)

C (Bake Time)

1 + + + 2 - + + 3 + - + 4 - - + 5 + + - 6 - + - 7 + - - 8 - - -

Figure 5. Cookie Experiment Design Matrix

Note: The trials in this design matrix are not randomized. The SPC Expert should randomize the order of the trials when designing the experiment. Then, the operator can follow the design matrix when running the experiment.


Interaction Effects: Two factors interact when the effect of one factor on the response variable depends on the level of the other factor. In other words, if two factors interact, you may not be able to decide on the best level for one factor without knowing the level of the other factor.

For example, the oven temperature and amount of sugar are two factors that may interact when baking cookies. Following is an interaction plot from our cookie baking example. This plot suggests that there is an interaction because one line is much steeper than the other. You cannot determine the proper baking temperature without knowing the amount of sugar.

High Sugar

Low Sugar

Crisp

Soft

300 degrees 400 degrees

Figure 6. Interaction plot


Types of Experiments

Statisically Designed: This is another name for DOE. Statistically designed experiments are planned in advance to include efficient combinations of factors and levels. When an experiment is statistically designed it is possible to learn much about the process without a lot of trials.

Full Factorial: Full factorial experiments test every possible combination of factors and levels. A design matrix is built to ensure that all combinations of factors are covered. The cookie experiment design matrix Figure 5 on page 11 is for a full factorial experiment. Notice that all possible combinations of + and - can be found in one of the trials.

A full factorial experiment will give you as much information as possible, but they may require too many trials. For example, if you have 6 factors you would need 64 trials.

Fractional Factorial: In a fractional factorial experiment, only selected combinations of factors and levels are tested. Although these experiments involve fewer trials and take less time than full factorial experiments, they can still give almost as much information.

In the cookie baking example it is possible to include another factor, the type of cookie sheet, without adding to the number of trials. With a careful statistical design it is possible to squeeze more information out of an experiment. The following design matrix is for a fractional factorial experiment which tests four factors in eight trials.


Trial

Factors

A (Sugar)

B (Bake Temp)

C (Bake Time)

D (Sheet)

1 + + + + 2 - + + - 3 + - + - 4 - - + + 5 + + - - 6 - + - + 7 + - - + 8 - - - -

Figure 7. Fractional Factorial Design Matrix

One-Factor-at-a-Time: In this type of experiment, the level of one factor is changed while the other factors remain the same. Then, when the best level for that factor is found, it is held constant and the next factor is adjusted. This continues until you look at all factors.

Take the cookie example. Suppose you have only two factors: amount of sugar and baking temperature. You could start by setting the oven temperature to 375° F and vary the amount of sugar.

The following chart shows this one-factor-at-a-time experiment. Suppose that 1/2 cup of sugar made the best cookies. This is Trial 3 with a result of 7. Now you would fix the amount of sugar at 1/2 cup and vary the temperature. If the best cookies now resulted from 400° F, the new recipe would read 1/2 cup sugar and bake at 400° F


Trial Number and Result (10 best)

1 cup

3/4 cup

1/2 cup

Best (10) Trial 2 (5)

Trial 1 (6)

Trial 3 (7) Trial 5 (8)Trial 4 (3)

Note: *This combination was never run in this experiment

*

350 F 375 F 400 F

Figure 94. One-Factor-at-a-Time Results

This doesn't sound like a bad way to experiment, but what if the perfect combination is 1 cup of sugar and 350° F? You never even gave that combination a chance. With DOE you could have learned the best settings without running any additional trials. The difference is in the planning and the design.

One-factor-at-a-time experiments are very common. Unfortunately they are inefficient and often lead to the wrong conclusion.

Summary: The first lesson in this unit showed you why DOE is important. This lesson showed you what DOE is and defined several key DOE terms. The next unit reviews how DOE is applied in the manufacturing environment.


Introduction to DOE -- Activity 2

Purpose: Practice reading and using a design matrix.

Instructions: Use the following design matrix to answer each of the following questions.

Trial

Factors

A (Sugar)

B (Bake Temp)

C (Bake Time)

D (Sheet)

1 + + + + 2 - + + - 3 + - + - 4 - - + + 5 + + - - 6 - + - + 7 + - - + 8 - - - -

1. Find at least two combinations of factor level (+ and -) not included in this experiment.

2. How many trials would be needed for a full factorial experiment for 4 factors, each at two levels? (One way to find out is to write them all out. There are less than 25.)

3. Which trials are run with the high level of sugar and the low level of temp?


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