Chapter 10 Introduction to Axiomatic Design

Chapter 10

Introduction to

Axiomatic Design

Suh, N. P. Axiomatic Design: Advances and Applications. New York:

This presentation draws extensively on materials from [Suh 2001]:

Oxford University Press, 2001. ISBN: 0195134664.

Example: Electrical Connector

Figure by MIT OCW.

Male connector

Female connector

Plastic

overmolding

Plastic

overmolding

Compliant pin

(for permanent connection)

Multiple layers will be stacked together

to obtain an entire connector.

Axiomatic Design Framework

The Concept of Domains

Four Domains of the Design World.

The {x} are characteristic vectors of each domain.

Figure by MIT OCW. After Figure 1.2 in [Suh 2001].

Customer domain Functional domain Physical domain Process domain

Mapping Mapping Mapping

{CAs} {FRs} {DP} {PVs}

Characteristics of the four domains of the design

world

Domains Character

Vectors

Customer Domain

{CAs}

Functional Domain

{FRs}

Physical Domain {DPs} Process Domain {PVs}

Manufacturing Attributes which

consumers desire

Functional

requirements

specified for the

product

Physical variables

which can satisfy the

functional

requirements

Process variables that

can control design

parameters (DP

)

Materials Desired performance Required Properties Micro-structure Processes

Software Attributes desired in

the software

Output Spec of

Program codes

Input Variables or

Algorithms Modules

Program codes

Sub-routines machine

codes compilers

modules

Organization Customer satisfaction Functions of the

organization

Programs or Offices

or Activities

People and other

resources that can

support the programs

Systems Attribute desired of

the overall system

Functional

requirements of the

system

Machines or

components,

sub-components

Resources (human,

financial, materials,

etc.)

Business ROI Business goals Business structure Human and financial

resource

Table by MIT OCW. After Table 1.1 in [Suh 2001].

Definitions

�Axiom:

Self-evident truth or fundamental truth for

which there is no counter examples or

exceptions. It cannot be derived from other

laws of nature or principles.

Corollary:

Inference derived from axioms or propositions

that follow from axioms or other propositions

that have been proven

Functional Requirement:

Functional requirements (FRs) are a minimum set of

independent requirements that completely

characterizes the functional needs of the product (or

software, organizations, systems, etc.) in the

functional domain. By definition, each FR is

independent of every other FR at the time the FRs are

established.

Constraint:

Constraints (Cs) are bounds on acceptable solutions.

There are two kinds of constraints: input constraints

and system constraints. Input constraints are

imposed as part of the design specifications. System

constraints are constraints imposed by the system in

which the design solution must function.

Definitions - cont’d

Design parameters (DPs) are the key physical

(or other equivalent terms in the case of

software design, etc.) variables in the physical

domain that characterize the design that

satisfies the specified FRs.

Process variable:

Process variables (PVs) are the key variables

(or other equivalent term in the case of

software design, etc.) in the process domain

that characterizes the process that can

generate the specified DPs.

Definitions - cont’d

Design parameter:

The Design Axioms

Maintain the independence of the

functional

requirements (FRs).

Axiom 2: The Information Axiom

Minimize the information content of the

design.

Axiom 1: The Independence Axiom

Example: Beverage Can Design

Consider an aluminum beverage

can that contains carbonated

drinks.

How many functional

requirements must the can

satisfy?

See Example 1.3 in [Suh 2001].

How many physical parts does it

have?

What are the design parameters

(DPs)? How many DPs are there?

Design Matrix

The relationship between {FRs} and {DPs} can be

written as

{FRs}=[A] {DPs}

form as

{dFRs}=[A] {dDPs}

[A] is defined as the Design Matrix given by

elements :

∂FRi/∂DPi

When the above equation is written in a differential

Aij =

Example

For a matrix A:

⎡

A11 A12 A13

⎤

[]=

⎢

A21 A22 A23

⎥

⎢

⎣

A31 A32 A33

⎦

⎥

Equation (1.1) may be written as

FR1 = A11 DP1 + A12 DP2 + A13 DP3

FR2 = A21 DP1 + A22 DP2 + A23 DP3 (1.3)

FR3 = A31 DP1 + A32 DP2 + A33 DP3

Uncoupled, Decoupled, and Coupled Design

Uncoupled Design

(1.4)

⎡

A11 0 0

⎤

[]=

⎢

0 A22 0

⎥

⎢

⎣

0 0 A33

⎦

⎥

Decoupled Design

⎡

11 0 0

⎤

A[]=

⎢

A21 A22 0

⎥

(1.5)

⎢

⎣

A31 A32 A33

⎦

⎥

Coupled Design

All other design matrices

Design of Processes

{DPs}=[B] {PVs}

[B] is the design matrix that defines the

characteristics of the process design and

is similar in form to [A].

Axiomatic Design Theory

Functional Requirement (FR) –‘What’ we want to achieve

A minimum set of requirements a system must satisfy

Design Parameter (DP) –‘How’ FRs will be achieved

Key physical variables that characterize design solution

Functional

Domain

{FR}

Physical

Domain

{DP}

Mapping

FR1

FR11 FR12

FR111 FR112 FR121 FR122

FR1111 FR1112 FR1211 FR1212

DP1

DP11 DP12

DP111 DP112 DP121 DP122

DP1111 DP1112 DP1211 DP1212

Decomposition – ‘Zigzagging’

Process of developing detailed

requirements and concepts by moving

between functional and physical

domain

Hierarchical FR-DP structure

Independence Axiom

Maintain the independence of FRs

Information Axiom

Minimize the information content

Design Axioms

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

⎭

⎬

⎫

⎩

⎨

⎧

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

⎭

⎬

⎫

⎩

⎨

⎧

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

⎭

⎬

⎫

⎩

⎨

⎧

Uncoupled Decoupled Coupled

p.d.f.

f(FR)

System Range,

p.d.f. f(FR)

Design

Range

|sr|

Common

Range, A

|dr|

Information content for functional requirement i = - log

Independence Axiom: Maintain the independence of FRs

Information Axiom

: Minimize the information content

FR must be satisfied within the design

range.

Design

Prob.

Density

range

To satisfy the FR, we have to map FRs in the

physical domain and identify DPs.

Design

Prob.

Density

range

System

range

Design Range, System Range, and Common

Range

Probab.

Density

( )

Design Rang e

Syst em

Rang e

Area o f

Common

Rang e Ac

Bia s

Target

Varia tion

from t he

pea k valu e

What happens when there are many FRs?

each level of the system hierarchy.

The relationship between the FRs determines how

desired certainty and thus complexity.

Most engineered systems must satisfy many FRs at

difficult it will be to satisfy the FRs within the

If FRs are not independent from each other,

the following situation may exist.

FR1

Pro b. De n s i ty

De si g n

Ra n g e

Syste m

Ra n g e

Pro b. De n s ity

De si g n

Ra n g e

Syste m

Ra n g e

FR2

Coupling decreases the design range

and thus robustness!!

Uncoupled Decoupled

⎤

⎧

DP1

⎫

⎧

⎫

⎡

11 0 0

⎤

⎧

⎫

⎧

FR1

⎫

⎡

A11 0

⎪

⎥

⎪

⎪ ⎪

⎢ ⎥

⎪

⎨

⎬

⎢

22 0

⎥

⎨

⎬

⎨

FR2

⎬

⎢

A21 A22 0

⎥

⎨

DP2

⎬

⎪

⎩

⎪

⎢

⎣

0 0

⎦

⎥

⎩

⎭

⎩

FR3

⎭

⎢

⎣

A31 A32 A33

⎥

⎪

⎭

⎪

⎪ ⎪

⎦

⎩

DP3

⎭

∆

∆DP1 =

A11

∆FR2

∆DP2 =

A22

∆FR3

∆DP3 =

2321313

1212

DPADPAFR

DPAFR

∆⋅−∆⋅−∆

=∆

∆⋅−∆

=∆

∆

=∆

What is wrong with conventional connectors?

It violates the Independence Axiom, which

states that

“Maintain the independence of Functional

Requirements (FRs)”.

It is a coupled design.

What is the solution?

Tribotek connector: A woven connector

Tribotek Electrical Connectors

(Courtesy of Tribotek, Inc. Used with permission.)

Performance of “Woven” Power Connectors

Power density => 200% of conventional

connectors

Insertion force => less than 5% of

conventional connectors

Electric contact resistance = 5 m ohms

Manufacturing cost

Capital Investment

TMA Projection System

Photos removed for copyright reasons.

What are the FRs of a face seal that must

isolate the lubricated section from the

abrasives of the external environment?

There are many FRs.

They must be defined in a solution neutral

environment.

Is this knob a good design or a poor

design?

Injection

molded knob

Shaft

with flat

milled

surface

Section A-A

Injecti on

molded k nob

Shaft

with flat

milled

surface

Sect ion A-A

Which is a better design?

(a)

(b)

Me tal

Sh af t

In ec tio n

mold ed

nylo n K n ob

M ill ed Flat

En d o f th e

sh aft

Slot

M ill ed Flat

En d o f th e

sh aft

Se cti on view A A

History

Goal

To establish the science base

for areas such as design and

manufacturing

How do you establish science

base in design?

Axiomatic approach

Algorithmic approach

References

N. P. Suh, Axiomatic Design: Advances

and Applications. New York: Oxford

University Press, 2001

N. P. Suh, The Principles of Design. New

York: Oxford University Press, 1990

Axiomatic Design

Axiomatic Design applies to all

designs:

•Hardware

•Software

•Materials

•Manufacturing

•Organizations

Axiomatic Design

Axiomatic Design helps the

design decision making process.

•Correct decisions

•Shorten lead time

•Improves the quality of products

•Deal with complex systems

•Simplify service and maintenance

•Enhances creativity

Axiomatic Design

•Axioms

•Corollaries

•Theorems

•Applications --

hardware, software,

manufacturing,

materials, etc.

•System design

•Complexity

Introduction

Stack of modules

Robot

Station

Unloading

station

Track

Xerography machine design– See Example 9.2 in [Suh 2001].

System integration

Stac k o f modul es

Stack of modules

Robo t

Station

Trac k

S t a c k o f mo du l e s

Mac hi ne A Mac hi ne B

A cluster of two machines that are physically coupled

to manufacture a part.

Introduction (cont’d)

Example1 Xerography-based Printing Machine

Light

Original

image

Image is

created here

Paper

Feed Roll

Roll

Selenium

coated Al.

cylinder

Paper

Wiper

Toner

Toner is coated

container

on surfaces of

Selenium with

electric charges

Schematic drawing of the xerography based printing machine.

Who are the Designers?

How do we design? What is design?

Is the mayor of Boston a designer?

Design Process

1. Know their

2. Define the problem they must solve to satisfy the

needs.

3. , which

analysis

5. Check the resulting design solution to see if it meets

the original customer needs.

"customers' needs".

Conceptualize the solution through synthesis

involves the task of satisfying several different

functional requirements using a set of inputs such as

product design parameters within given constraints.

4. Perform to optimize the proposed solution.

Definition of Design

Design is an interplay between

what we want to achieve and

how we want to achieve them.

Definition of Design

to to

"What

we want

achieve"

"How

we want

achieve

them"

Example: Refrigerator Door Design

Figure ex.1.1.a Vertically hung refrigerator door.

Ultimate Goal of Axiomatic Design

The ultimate goal of Axiomatic Design is to

establish a science base for design and to

improve design activities by providing the

designer with a theoretical foundation based

on logical and rational thought processes and

tools.

Creativity and Axiomatic Design

Axiomatic design enhances creativity

by eliminating bad ideas early and

designers .

thus, helping to channel the effort of

Axioms are truths that cannot be derived but for which

there are no counter-examples or exceptions.

Many fields of science and technology owe their

advances to the development and existence of axioms.

(1) Euclid's geometry

(2) The first and second laws of thermodynamics

are axioms

(3) Newtonian mechanics

Historical Perspective on Axiomatic

Design

Constraints

What are constraints?

Constraints provide the bounds on the

acceptable design solutions and differ from

the FRs in that they do not have to be

independent.

There are two kinds of constraints:

input constraints

system constraints.

Example: Shaping of Hydraulic Tubes

To design a machine and a process that can

achieve the task, the functional requirements

can be formally stated as:

FR1= bend a titanium tube to prescribed

curvatures

FR2= maintain the circular cross-section of

the bent tube

Tube Bending Machine Design (cont’s)

Given that we have two FRs,

how many DPs do we need?

Example: Shaping of Hydraulic Tubes

Fixed set of

counter-rotating

grooved rollers

Pivot

axis

Flexible set of

counter-rotating

grooved rollers

for bending

Tube between

the two rollers

See Example 1.6 in [Suh 2001].

Example: Shaping of Hydraulic Tubes

DP1= Differential rotation of the bending rollers to bend the tube

DP2= The profile of the grooves on the periphery of the bending

rollers

axis

Fixed set of

counter-rotating

grooved rollers

Pivot

Flexible set of

counter-rotating

grooved rollers

for bending

Tube between

the two rollers

Tube bending apparatus

Example: Van Seat Assembly

(Adopted from Oh, 1997)

See Example 2.6 in [Suh 2001].

Schematic drawing of a van seat that can be removed

and installed easily using a pin/latch mechanism

Example: Van Seat Assembly

Solution

The FR of the seat engagement linkage is that the

distance between the front leg and the rear latch

when the seat engages the pins must be equal to

the distance between the pins, which is 340 mm.

The linkages [see Figures E2.6.b and c in Suh

2001] determine the FR = F. The following table

shows the nominal lengths of the linkages.

Example: Van Seat Assembly

Traditional SPC Approach to Reliability and Quality

The traditional way of solving this kind of problem has been to do

the following:

(a) Analyze the linkage to determine the sensitivity of the

error.

Table a Length of linkages and sensitivity analysis

Links Nominal Length (mm) Sensitivity (mm/mm)

L12 370.00 3.29

L14 41.43 3.74

L23 134.00 6.32

L24 334.86 1.48

L27 35.75 6.55

L37 162.00 5.94

L45 51.55 11.72

L46 33.50 10.17

L56 83.00 12.06

L67 334.70 3.71

Decomposition, Zigzagging and Hierarchy

Zigzagging to decompose in the functional and the physical domains and create

the FR- and DP hierarchies

Figure by MIT OCW. After Figure 1.3 in [Suh 2001].

FR1

FR11 FR12

FR121 FR122 FR123

FR1231

Functional Domain Physical Domain

FR1232

FR2

DP1

DP11 DP12

DP121 DP122 DP123

DP1231 DP1232

DP2

Equivalent Design:

of the highest-level FRs but have different

hierarchical architecture, the designs are

defined to be equivalent designs.

Identical Design:

of FRs and have the identical design

architecture, the designs are defined to be

identical designs.

Identical Design and Equivalent Design

When two different designs satisfy the same set

FR1 = Freeze food for long-term preservation

FR2 = Maintain food at cold temperature for

short-term preservation

compartments is designed. Two DPs for this

refrigerator may be stated as:

DP1 = The freezer section

DP2 = The chiller (i.e., refrigerator) section.

Example: Refrigerator Design

To satisfy these two FRs, a refrigerator with two

FR1 = Freeze food for long-term preservation

FR2 = Maintain food at cold temperature for short-term

preservation

DP1 = The freezer section

DP2 = The chiller (i.e., refrigerator) section.

Example: Refrigerator Design

FR 1

FR 2

⎧

⎨

⎩

⎫

⎬

⎭

0 X

⎡

⎣

⎢

⎤

⎦

⎥

DP 1

DP 2

⎧

⎨

⎩

⎫

⎬

⎭

Example: Refrigerator Design

Having chosen the DP1, we can now decompose

FR1 as:

FR11 = Control temperature of the freezer

FR12 = Maintain the uniform temperature

throughout the freezer section at the

preset temperature

FR13 = Control humidity of the freezer section to

relative humidity of 50%

section in the range of -18 C +/- 2 C

Example: Refrigerator Design

FR11 =

DP11 = Sensor/compressor system that turn on and

off the compressor when the air temperature is

higher and lower than the set temperature in

the

DP12 = Air circulation system that blows air into the

freezer section and circulate it uniformly

throughout the freezer section at all times

returned air when its dew point is exceeded

Control temperature of the freezer section in the range of -18 C +/- 2 C

FR12 = Maintain the uniform temperature throughout the freezer section at the preset temperature

FR13 = Control humidity of the freezer section to relative humidity of 50%

freezer section, respectively.

DP13 = Condenser that condenses the moisture in the

Example: Refrigerator Design

Similarly, based on the choice of DP2 made, FR2

may be decomposed as:

FR21 = Control the temperature of the

chilled section in the range of 2 to 3 C

Maintain a uniform temperature

C of a preset temperature

FR22 =

throughout the chilled section within 1

Example: Refrigerator Design

and off the compressor when the air

temperature is higher and lower than the

set temperature in the chiller section,

respectively.

DP12 = Air circulation system that blows air into

the freezer section and circulate it

at all times

FR21 = Control the temperature of the chilled section in the range of 2 to 3 C

FR22 = Maintain a uniform temperature throughout the chilled section within 1 C of

a preset temperature

DP11 = Sensor/compressor system that turn on

uniformly throughout the freezer section

Example: Refrigerator Design

Figures removed for copyright reasons.

See Example 1.7 in [Suh 2001].

Example: Refrigerator Design

The design equation may be written as:

⎧

FR12

⎫

⎡

⎤

⎧

DP12

⎫

⎪ ⎪ ⎪

⎪

⎨

FR11

⎬

⎢

XXO

⎥

⎨

DP11

⎬

⎪ ⎪

⎥

⎪

⎩

FR13

⎭

⎣

⎢

XOX

⎦

⎩

DP13

⎭

Equation (a) indicates that the design is a decoupled design.

DP22 DP21

FR22 X 0

FR21 X X

Full DM of Uncoupled Refrigerator Design

DP1

DP12 DP11 DP13 DP22 DP21

FR21 X 0 0 0 0

FR1 FR11 X X 0 0 0

X 0 X 0 0

FR2 FR22 0 0 0 X 0

0 0 0 X X

DP2

____________________________________________________

FR13

_____________________________________________________

FR21

_____________________________________________

Full DM of Uncoupled Refrigerator Design

DP1

DP12 DP11 DP13 DP22 DP21

FR12 X 0 0 0 0

FR1 FR11 X X 0 0 0

FR13 X 0 X 0 0

FR2 FR22 X 0 0 0 0

0 0 0 X 0/X

DP2

____________________________________________________

_____________________________________________________

FR21

_____________________________________________

Analysis

When do we perform analysis

during the design process?

Requirements for Concurrent Engineering

[A ] [B] [C] = [A] {B]

1. Bot h diago nal [\ ] [\ ] [\]

2. D iag x Full [\ ] [X] [X]

3. D iag x tria ng. [\ ] [LT] [LT]

4. Tria. x Triang [LT] [LT] [LT]

5. Tria. x Triang [LT] [U T] [X]

6. Fu ll x Full [X] [X] [X]

Table 1.3 The characteristic of concurrent engineering matrix [C].

Ideal Design, Redundant Design, and Coupled

Design -

A Matter of Relative Numbers of DPs and FRs

Case 1:

Design

number of functional requirements, we always have a

coupled design. This is stated as Theorem 1.

Design

Axiom.

Depending on the relative numbers of DPs and FRs the design

can be classified as coupled, redundant and ideal designs.

Number of DPs < Number of FRs: Coupled

When the number of des gn parameters is less than the

Case 2: Number of DPs > Number of FRs: Redundant

When there are more design parameters than the functional

requirements, the design is called a redundant design. A

redundant design may or may not violate the Independence

The Second Axiom: The Information

Axiom

Axiom 2: The Information Axiom

Minimize the information content.

Information content I is defined in terms of the probability of

satisfying a given FR.

I = log

=−log

In the general case of n FRs for an uncoupled design, I may be

expressed as

∑

log

=−

∑

I = log

= 1 = 1 i i

Design Range, System Range, and Common Range

Probab.

Den sity

Design Rang

Syst em

Rang e

Ar ea o f

Com m on

Rang e ( Ac)

Bi a s

Tar get

Varia tion

from t he

pe a k valu e

Design Range, System Range, and Common Range in a plot of the

probability density function (pdf) of a functional requirement. The deviation

from the mean is equal to the square root of the variance. The design

range is assumed to have a uniform probability distribution in determining

the common range.

Measure of Information Content in Real Systems

The probability of success can be computed by specifying the Design

Range (dr) for the FR and by determining the System Range (sr) that the

proposed design can provide to satisfy the FR.

I = log

(1.9)

where A

denotes the area under the System Range and A

is the area of

the Common Range.

Furthermore, since A

= 1.0 in most cases (since the total area of the

probability distribution function is equal to the total probability, which is

one) and there are n FRs to satisfy, the information content may be

expressed as

I =

∑

log

(1.10)

i=1

FR1 =

to 30 minutes.

colleges.

good over 340 days a year.

less than $650,000.

Example: Buying a House

Commuting time for Prof. Wade must be in the range of 15

FR2 = The quality of the high school must be good, i.e., more than

65 % of the high school graduates must go to reputable

FR3 = The quality of air must be good, i.e., the air quality must be

FR4 = The price of the house must be reasonable, i.e., a four bed

room house with 3,000 square feet of heated space must be

Example: Buying a House

They looked around towns A, B, C and collected the following

data:

Tow n FR1=Comm. FR2=Qualit y FR3=Qualit y FR4=Price [$]

time [min] of school [%] of air [days]

A 20 to40 50 to70 300 to 320 450k to 550k

B 20 to 30 50 to 75 340 to 350 $450k to 650k

C 25 to45 50 to80 350 and up $600k to 800k

Which is the town that meets the requirements of the Wade

family the best? You may assume uniform probability

distributions for all FRs.

Example: Buying a House

Solution

10 20 30 40

Prob.

Dist.

Design Range

Commo

Range

System

Range

FR1 = Commuting Time (min).

Probability distribution of commuting time

Example: Buying a House

Prob.

Dist.

Design Range

System Range

Common Range

20 40 60 80

Quality of

(%)

Probability distribution of the quality of schools

Example: Buying a House

The information content of Town A is infinite since it cannot satisfy

FR3, i.e., the design range and the system range do not overlap at

all. The information contents of Towns B and C are computed

using Eq. (1.8) as follows:

Town I

[bits]

A 1.0 2.0 Infi nite 0 Infini te

B 0 1.32 0 0 1.32

C 2.0 1.0 0 2.0 5.0

1.8 Common Mistakes Made by Designers

i. Due to Insufficient Number of DPs

(Theorem 1)

ii. Not Recognizing a Decoupled Design

iii. Having more DPs than the number of FRs

iv. minimizing the

bias and reduction of variance

v. --

Importance of Establishing and Concentrating on FR.

Coupling

Not creating a robust design -- not

information content through elimination of

Concentrating on Symptoms rather than Cause