iii. axiomatic design

III. Axiomatic Design

3.1 Introduction (2)

Examples of axioms in Geometry:

Points and lines are names for the elements of two (distinct) sets. Incidence is a

relationship that may (or may not) hold between a particular point and a particular line.

The followings are examples of axioms:

1) For every two points, there exists a line incident with both points.

2) For every two points, there is no more than one line incident with both points.

3) There exist at least two points incident with each line.

4) There exist at least three points. Not all points are incident with the same line.

Examples of axioms in Physics:

1) Newton’s law: F = ma

2) Thermodynamic principles

3) ……

Axiomatic design is a design methodology that was created and popularized by Professor

Suh of MIT (used be the President of KAIST).

It is a design framework that is feasible on all design disciplines.

The Independence Axiom, The Information Axiom

An axiom is a statement accepted without proof as an underlying assumption of a formal

mathematical theory. It cannot be proved. If a counter example is found for an axiom, the

axiom becomes obsolete.

Geometry, Laws in Physics, Thermodynamic principles

3.1 Introduction

Design Axioms

Axiom 1: The Independence Axiom

Maintain the independence of FRs.

Alternate Statement 1: An optimal design always maintains the independence of FRs.

Alternate Statement 2: In an acceptable design, DPs and FRs are related in such a way

that a specific DP can be adjusted to satisfy its corresponding FR without affecting

other functional requirements.

Axiom 2: The Information Axiom

Minimize the information content of the design.

Alternate Statement: The best design is a functionally uncoupled design that has

minimum information content.

3.2. Design Axioms

Usage of the axioms:

Analysis of design

Find designs that satisfy the Independence Axiom.

Determine the final design.

Is the no. of designs sufficient?

Find the best design with the Information Axiom.Multiple designs?

No

Yes

Yes

No

3.2. Design Axioms (2)

Figure 3.1. Flow chart of the application of axiomatic design

The Independence Axiom: The FR-DP relationship should be independent.

3

2

1

3

2

1

,DPDPDP

FRFRFR

DPFR

equationdesign :ADPFR

3

2

1

333231

232221

131211

3

2

1

DPDPDP

AAAAAAAAA

FRFRFR

FR: FR vector, DP: DP vector, A: design matrix

3.3 Independence Axiom

3

2

1

33

22

11

3

2

1

000000

DPDPDP

AA

A

FRFRFR 1111 DPAFR

2222 DPAFR

3333 DPAFR

3

2

1

333231

2221

11

3

2

1

000

DPDPDP

AAAAA

A

FRFRFR 1111 DPAFR

2221212 DPADPAFR

3332321313 DPADPADPAFR

3.3 Independence Axiom (2)

1) Uncoupled design: Each DP satisfies the corresponding FR independently. - diagonal matrix

2) Decoupled design: The Independence Axiom is satisfied when the design sequence is right.

- triangular matrix

(3) Coupled design: No sequences of DPs can satisfy the FRs independently. –general matrix

3

2

1

333231

232221

131211

3

2

1

DPDPDP

AAAAAAAAA

FRFRFR




Constraints (Cs): Cs can be defined regardless of the independence of the FR-DP relationship.

3.3 Independence Axiom (3)

(a) Vertically hung door (b) Horizontally hung door

Example 3.1 [Design of a Refrigerator Door]

FRs for a refrigerator door are as follows:

FR1: Provide access to the items stored in the refrigerator.

FR2: Minimize energy loss.

Which door is better between the following doors?

3.4 Application of the Independence Axiom

Figure 3.2. Refrigerator doors

2

1

2

1 0DPDP

XXX

FRFR

2

1

2

1

00

DPDP

XX

FRFR

X: nonzero value (relationship)

Example 3.1 [Design of a Refrigerator Door]

Vertically hung door in Figure 3.2(a)

DP1: Vertically hung door

DP2: Thermal insulation in the door

(2) Horizontally hung door in Figure 3.2(b)

DP1: Horizontally hung door

DP2: Thermal insulation in the door

Which one is better?

3.4 Application of the Independence Axiom (2)

Example 3.2 [Design of a Water Faucet]

Some commercial water faucets are evaluated. FRs for the faucet are defined as follows:

FR1: Control the flow of water (Q).

FR2: Control the temperature of water (T).



For the one in Figure 3.3(a)

DP1: Angle

DP2: Angle

1

2

)()(

)()(

22

11

2

1

DPDP

XXXX

TFRQFR

1

2


Figure 3.3.(a) Coupled design


For the one in Figure 3.3(b)

DP1: Angle

DP2: Angle

1

2

)()(

00

)()(

22

11

2

1

DPDP

XX

TFRQFR

Cold water Hot water

2

1


Figure 3.3.(b) Coupled design

)()(

00

)()(

2

1

2

1

DPYDP

XX

TFRQFR

Which one is the best? Is it different from what you expected?

Y


For the one in Figure 3.3(c)

DP1: Displacement Y

DP2: Angle


Figure 3.3.(c) Coupled design

3

2

1

3

2

1

00

00

DPDPDP

XXxX

X

FRFRFR

3

2

1

3

2

1

00

00

DPDPDP

XXxX

X

FRFRFR

for Figure 2.1(a)

Example 3.3 [Axiomatic Design of the Toaster in Example 2.1]

Make the design matrices for the products in Figure 2.1(a) and Figure 2.1(b)

Solution


for Figure 2.1(b)

(a) Functional domain

FR

FR1 FR2

FR11 FR12

…

… FR21 FR22 …

DP

DP1 DP2

DP11 DP12

…

… DP21 DP22 …

①

②

③

④

(b) Physical domain

For complicated systems: we need a decomposition which yields a hierarchy.

The zigzagging process


Figure 3.4. Zigzagging process between domains

FR: 서울에서부산으로가라.

지그재그과정의예 (서울에서부산가는법)

서울에서부산으로가라. 비행기

비행장으로가라. 표를사라. …비행기를타라.

서울에서부산으로가라. 기차

기차역으로가라. 표를사라. …기차를타라.

(1)

(2)

상위단계의 DP가하위단계의 FR을결정한다.

Example 3.4 [Decomposition of Example 2.2]

The top level (first level) FRs and DPs are:

FR1: Freeze food or water for long-term preservation.

FR2: Maintain food at a cold temperature for short-term preservation.

DP1: The freezer section

DP2: The chiller section



The second (first level) FRs are:

FR11: Maintain the temperature of the freezer section in the range of .

FR12: Maintain a uniform temperature in the freezer section.

FR13: Control the relative humidity to 50% in the freezer section.

FR21: Maintain the temperature of the chiller section in the range of .

FR22: Maintain a uniform temperature in the chiller section within of the preset

temperature .

C2C18

C3C2

C5.0


13

11

12

13

11

12

0000

DPDPDP

XXXX

X

FRFRFR


The second level DPs are:

DP11: Sensor/compressor system that activates the compressor when the temperature of

the freezer section is different from the preset one.

DP12: Air circulation system that blows the air into the freezer and circulates it uniformly.

DP13: Condenser that condenses the moisture in the returned air when the dew point is

exceeded.


21

22

21

22 0DPDP

XXX

FRFR


The second level DPs are:

DP21: Sensor/compressor system that activates the compressor when the temperature of

the chiller section is different from the preset one.

DP22: Air circulation system that blows the air into the chiller section and circulates it

uniformly.


21

22

13

11

12

21

22

13

11

12

00000000000000000

DPDPDPDPDP

XXX

XXXX

X

FRFRFRFRFR


The entire design equation is


Physical integration

There is a saying that a simple design is a good one.

A good design makes one DP satisfy multiple FRs?

A coupled design is better?

This is the case where multiple DPs make a physical entity. Multiple DPs satisfy FRs of the

same number.

Physical Integration: recommended


2

1

2

1

00

DPDP

XX

FRFR

Example 3.5 [Bottle-can opener]

Given FRs are

FR1: Design a device that can open bottles.

FR2: Design a device that can open cans.


DP2

DP1

12 FRs and 12 DPs: decoupled design

Example 3.6 [Beverage Can Design]

Another example for physical integration


In the designing process, the Independence Axiom should be satisfied first.

When multiple designs that satisfy the Independence Axiom are found, the Information

axiom is utilized to find the best design.

The best design is the one with minimum information.

Generally, the information is related to complexity.

How can we measure the complexity?

How can we quantify the information content?

3.5 The Information Axiom

pI /1log2

The information can be defined in various ways.

Up to now, one measure is used for the information content.

where I is the information content and p is the probability of success to satisfy an FR with a DP.

The reciprocal of p is used to make the larger probability have less information.

The logarithm function is used to enhance additivity.

The base of the logarithm is 2 to express the information content with the bit unit.

3.5 The Information Axiom (2)

3

2

1

33

22

11

3

2

1

000000

DPDPDP

AA

A

FRFRFR

Suppose p1, p2 and p3 are probabilities of satisfying FR1, FR2 and FR3 with DP1, DP2 and

DP3, respectively. The total information Itotal is

3

12

3

1total

1logi ii

i pII

For the following uncoupled design:


If p1 is the probability that DP1 satisfies FR1, then the probability that DP2 satisfies FR2 under

the satisfaction of FR1 by DP1 is a conditional probability. Suppose it is p21. Then the

probability of success p that both FR1 and FR2 are satisfied is p=p1p21.

The total information content for p is

2

1

2

1 0DPDP

XXX

FRFR

212121221122 loglog)(loglog IIpppppI

For the following decoupled design:


System range (Asr): response (output)

Design range: target, Common range: Acr

srcr / AAps

)/(log srcr2 AAI

Design range

Probability density

Probability density function

of the system

Common range

FR1 3 5 7 9

Information content can be calculated by using the probability density function in the

following figure:


Example 3.7 [An Example of Calculating Information Content]

For the bottle-can opener problem

The probability of satisfying FR1 with DP1: 0.9

The probability of satisfying FR2 with DP2: 0.85

The total information content is

bits)(3865.02345.01520.085.01log

9.01log 2221total

III

What if we do not have physical integration?

3.6 Application of the Information Axiom

city A city B

Price $45,000 - $60,000 $70,000 - $90,000

Commuting time 35-50 min 20-30 min

Example 3.9 [Calculation of the Information Content Using the Probability Density Function]

A person defines two functional requirements to buy a house as follows:

FR1: Let the price range be from $50,000 to $80,000.

FR2: Let the commuting time be within 40 minutes.

The following table shows the conditions of city A and city B.

3.6 Application of the Information Axiom (2)

For city A

59.15

15log,59.015.1log 2221

AA II

)bits(18.221 AAA III

0.112log21

BI 0.0

1010log22

BI

bits)(0.121 BBB III

Which one is better?

Design range

BiasProbability

density

FR

Target

Probability density function of the

system

Common range

Variation from the peak value

Example 3.9 [Calculation of the Information Content Using the Probability Density Function]

For city B

3.6 Application of the Information Axiom (3)

Time

Two axioms are independent of each other.

The flow in Figure 3.1 is recommended.

The numbers of FRs and DPs should be the same (ideal design).

The no. of DPs is smaller: New DPs should be added.

The no. of DPs is larger (redundant design): Some DPs are eliminated or specific DPs are

fixed.

Axiomatic design is useful in conceptual design. It can be used for creative design or

evaluating an existing design.

3.7 Discussion

Solution neutral environment

iii. axiomatic design

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