Ashby Method 2.6

8/17/2019 Ashby Method 2.6

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2 - Ashby Method

2.6 - Multi-objective optimisation inselection

Outline

• Conflicting objectives

• Multi-objective optimisation

• Reaching a compromise

• Value functions and exchange constants

• Weighed-properties method

• Case studies

Resources:

• M. F. Ashby, “Materials Selection in Mechanical Design” Butterworth Heinemann, 1999

Chapter 9

• M. F. Ashby, “Multi-objective optimisation in material design and selection”

Acta Materialia, vol. 48, pp. 359-369, 2000

• M. M. Farag, “Quantitative methods of materials selection”

“Handbook of Materials Selection” (M. Kutz) Wiley & Sons, 2002, chap. 1


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Problem of conflicting objectives

• Real life often requires a compromise betweenconflicting objectives:

Price versus performance of a bike or car

• Conflict arises because the choice that optimises onemetric of performance will not in general do the same for

the others.

• Best choice is a compromise, optimising none butpushing all as close to optimum as their interdependenceallows.

Conflicting objectives in design

• Common design objectives, influencing choice of material, are:

Minimising mass (sprint bike; satellite components)

Minimising volume (mobile phone; minidisk player)

Maximising energy density (flywheels, springs)

Minimising eco-impact (packaging)

Minimising cost (everything)

• Each objective defines a performance metric. Take, as example

mass, m we wish to minimise bothcost, C (all other constraints being met)

Solutions that minimise mass seldom minimise cost,

and vice versa

Objectives


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Multi-objective optimisation: Terminology

• Non-dominated solution (B):

no one other solution is better byboth metrics

• Trade-off surface: the surface on which the non-dominated solutions lie(also called the Pareto Front)

• Three strategies for finding best compromise

• Solution: a viable choice,meeting constraints, but notnecessarily optimum by either

criterion.

• Dominated solution (A): some other solution is better by

both metrics

Cheap Metric 2: cost C Expensive L i g h t

M e t r i c 1 : m a s s m

H e a

v y

A Dominated

solution

B Non-dominatedsolution

Trade-off

surface

Finding a compromise: Strategy 1

• Make trade-off plot

• Sketch trade-off surface

• Use intuition to select asolution on the trade-off surface

Mass and cost of bicycles:

• Well defined trade-off surface

• “Solutions” on or near the surface offer thebest compromise between mass and cost

• Choose from among these; the choicedepends on how highly you value a light

bicycle -- a question of relative values


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• Sketch trade-off surface

• Reformulate one of the

objectives as constraint,

setting an upper limit for it

Cheap Metric 2: cost C Expensive L i g h t


H e a v y

Trade-off

surface

Upper limit on C

Optimum solution

minimising mMass and cost of bicycles:

• Good if you have budget limit

• Trade-off surface leads you to

the best choice within budget

• But not a true optimisation --cost has been treated as a

constraint, not an objective.


Define locally linear

Value Function V

CmV +α=

Seek material with smallest V:

• Evaluate V for eachsolution, and rank

or


plot on it contours of V

(lines of constant V have

slope -1/ α)

read off solution with lowest V

Cheap Metric 2: cost C Expensive

L i g h t


H e a v y

V1

V2V3

V4 Contours ofconstant V

Decreasingvalues of V

Optimum solution,

minimising V

α

−1

• Value lines are straight only if the scales are linear

• For logarithmic scales the value lines are curved

log (αααα m + C) ≠ log αααα m + log C


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Log scales

Cheap cost, c Expensive

Decreasing

values of V

A linear relation, on log scales,

plots as a curve V1/ αC1/ αm

CmαV

⋅+⋅−=

+=

Linear scales

L i g h t e r

m a s s , m

H e a v i e r

Decreasing

values of V

-1/αααα

Cheap cost, c Expensive

L i g h t e r

m a s s , m

H e a v i e r

Exchange Constant

The quantity αααα is called an “exchange constant” -- it measures thevalue of performance, here the value of saving 1 kg of mass.

Transport System: mass saving αααα (£ per kg)

Family car (based on fuel saving)

Truck (based on payload)Civil aircraft (based on payload)

Military aircraft (performance payload)

Space vehicle (based on payload)

0.5 to 1.5

5 to 20100 to 500

500 to 2000

1000 to 9000

Cm

V

∂

∂=αCmV +α=

Exchange constants for mass saving


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Case study: Casing for a minidisk player

• Electronic equipment -- portablecomputers, players, mobile phones-- all miniaturised; many now lessthan 12 mm overall thick

• An ABS or Polycarbonate casinghas to be > 1mm thick to be stiff

enough for protection; casing

occupies 20% of the volume

• Find best material for a stiff casing of minimum thickness and weight

minimise casing thickness

minimise casing mass

• The thinnest may not be the lightest … need to explore trade-off

Objective 1

Objective 2

Performance metrics for the casing

Function Stiff casing

t

w

L

F

Metric 1 3 / 1

3 / 13

E

1

wE4

LS

t ∝

=

Objective 2 Minimise mass m

Metric 2(from Part 2.3) 3 / 13 / 1

2

3 / 12

EEL

C

wS12m

ρ∝

ρ

=

m = massw = widthL = length

ρ = densityt = thicknessS = required stiffnessI = second moment of areaE = Youngs Modulus

Objective 1 Minimise thickness t

3L

IE48S =

Constraints

12

twI

3

=

• Adequate toughness,

Klc > 15 MPa.m1/2

• Stiffness, S

with


7/16

Relative performance metrics

• We are interested here in substitution. Suppose the casing is

currently made of a material Mo (ABS).

• The thickness of a casing made from an alternative material M,

differs (for the same stiffness) from one made of Mo by the factor

• The mass differs by the factor

• Explore the trade-off between and

3 / 1o

o E

E

t

t

=

ρ

ρ=

o

3 / 1o

3 / 1o

E.

Em

m

ot

t

om

mM0 = ABS:

• ρ0 = 1,2 Mg/m3

• E0 = 2,4 GPa

Trade-off plot

Thickness relative to ABS

0.1 1 10

M a s s r e l a t i v e t o A B S

0.1

1

10

PTFE

PC

ABS

PMMA

PP

NylonPolyester

PE

Ionomer Ni-alloys

Cu-alloys

Steels

Al-alloys

Al-SiC Composite

Ti-alloys

Mg-alloys

CFRP

GFRP

Lead

Polymer foams.

ElastomersTrade-off

surface

Thickness relative to ABS, t/to

M a s s r e l a t i v e t o A B S , m / m

o

Additionalconstraints:

• K1c > 15MPa.m1/2

Woodsuppressed


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0 .1 1 1 0


0.1

1

10

PTFE

PC

ABS

PMMA

PP

NylonPolyester

PE

Ionomer Ni-alloys

Cu-alloys

Steels

Al-alloys

Al-SiC Comp osite

Ti-alloys

Mg-alloys

CFRPGFRP

Lead

Polymer foams

.

ElastomersTrade-offsurface



o

• The four sectors of a trade-off plot for substitution

A. Better by

both metrics

C. Lighter

but thicker

D. Worse by

both metrics

B. Thinner

but heavier

Trade-off plot

• Finding a compromise: CFRP, Al and Mg alloys all offer reduction in mass and thickness

Trade-off plot


0.1 1 10


0.1

1

10

PTFE

PC

ABS

PMMA

PP

NylonPolyester

PE

Ionomer Ni-alloys

Cu-alloys

Steels

Al-alloys

Al-SiC Composite

Ti-alloys

Mg-alloys

CFRP

GFRP

Lead

Polymer foams

.

ElastomersTrade-off

surface



o

M = CFRP:

• ρ= 1,5 Mg/m3

• E = 220 GPa

• t/t0 = 0,22

• m/m0 = 0,28

M = Al alloys:

• ρ= 2,6 Mg/m3

• E = 75 GPa

• t/t0 = 0,31

• m/m0 = 0,68

• Is material cost relevant? Probably not -- the case only weighs

a few grams. Volume and weight are much more valuable.


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Case study: Air cylinders for trucks

Design goal: lighter, cheap air cylinders for trucks

Compressed air tank

Design requirements for the air cylinder

Pressure vessel

• Minimise mass

• Minimise cost

• Dimensions L, R, pressure p, given• Must not corrode in water or oil

• Working temperature -50 to +1000C

• Safety: must not fail by yielding• Adequate toughness: K1c > 15 MPa.m1/2

• Wall thickness, t;

• Choice of material

Specification

Function

Objectives

Constraints

Free

variables

R = radius

L = length

ρ = densityp = pressuret = wall thickness

L

2RPressure p

t


10/16

Performance metrics for the air cylinder

=

⋅=

θσ

−=+

−=σ

=⋅

=π

πσ

• Thin-walled pressure vessels are treated as membranes. The

approximation is reasonable when t < b/4

• The stresses in the wall do not vary significantly with radial distance, r

>⇒<

σσσσr

σσσσθθθθσσσσz

Performance metrics for the air cylinder

Metric 1

Eliminate t to give:

L

2RPressure p

t

Constraint

Objective 2

( )

+= yf2

σ

ρ

SpQ1LR2m π

mCC m=

f

y

S

σ

t

Rpσ


11/16

Finding a compromise: Value Function

Define locally linearValue Function V

CmV +α=

Seek material with smallest V:

• Evaluate V for eachsolution, and rank

or


plot on it contours of V(lines of constant V haveslope -1/ α)

read off solution with lowest V

Cheap Metric 2: cost C Expensive

L i g h t


H e a

v y

V1

V2 V3 V4 Contours of

constant V

Decreasing

values of V

Optimum solution,minimising V

α−

1

Cm

V

∂

∂=α

Exchange Constant

α = £20/kg (trucks)

Metric 2 ( Cost index)1e-5 1e-4 1e-3 0.01 0.1 1 10

M e t r i c 1 ( M a s s i n d e x )

1e-6

1e-5

1e-4

1e-3

0.01

Decreasingvalues of V

Finding a compromise: Value Function


K1c >15 MPa.m1/2

Tmax > 373 K

Tmin < 223 K

Water: good +

Organics: good +


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Relative performance metrics

• This is a problem of substitution. The tank is currently made of a plain

carbon steel.

• The mass m and cost C of a tank made from an alternative material M,differs (for the same strength) from one made of Mo by the factors

For plain carbon steel and

• Explore the trade-off between and

ρ

σ

σ

ρ=

o

o,y

yo

.m

m

ρ

σ

σ

ρ=

oo,m

o,y

y

m

o C.

C

C

C

0.03 / σρ oy,o = 0.02 / σρC oy,oom, =

omm

oCC

Trade-off plot

Price * Density / Elastic limit0.1 1 10 100

D e n s i t y / E l a s t i c l i m i t

0.1

1

10

Mild steel

High-C steel

Al-alloys

GFRP CFRP

Mg-alloys

Ti-alloys

Ni-alloys

Cu-alloys

Low alloy steel

Al-SiC Composite

Lead alloys

Zn-alloys

Cost relative to plain carbon steel, C/Co

M a s s

r e l a t i v e t o p l a i n c a r b o n s t e e l , m / m

o Trade-offsurface


K1c >15 MPa.m1/2

Tmax > 373 K

Tmin < 223 K

Water: good +

Organics: good +


13/16

Finding a compromise: the value function

• Aluminium alloy and low alloy steels offer modest reductions inmass at little or no increase in material cost (Region A - Better byboth metrics).

• The lightest solutions are GFRP, CFRP and Titanium alloys, but ata cost penalty -- is it worth it? Define a relative value function:

• The relative exchange constant, α*, is related to α by

• With mo = 10 kg, Co = £50 and α = £20/kg (trucks), α* = 4 .

(a) evaluate V* numerically and rank candidates, or

(b) plot onto relative trade-off plot (lines of slope )

ooo

*

C

C

m

m*

C

VV +α==

α=αo

o

C

m*

4

1−

+=⇒+= αα

Value function on trade-off plot

Value contour for α* = 4 (α = £20/kg)

Price * Density/ Elastic limit0.1 1 10 100


0.1

1

10

Mild steel

High-C steel

Al-alloys

CFRP

Mg-alloys

Ti-alloys

Ni-alloys

Cu-alloysZn-alloys

Lead alloys

Low alloysteel

Al-SiC CompositeGFRP

V*

Price * Density / Elastic limit0.1 1 10 100


0.1

1

10

Mild steel

High-C steel

Al-alloys

GFRP CFRP

Mg-alloys

Ti-alloys

Ni-alloys

Cu-alloysZn-alloys

Lead alloys

Lowalloysteel

Al-SiC Composite

V*

Value contour for α* = 200 (α = £1000/kg)

oo

*

C

C

m

m*V +α=

Trade-off

surfaceTrade-offsurface

• Value lines are curved because of logarithmic scales.

M a s s r e l a t i v e t o p l a i n c a r b o n s t e e l , m / m

o

M a s s r e l a

t i v e t o p l a i n c a r b o n s t e e l , m / m

o

Cost relative to plain carbon steel, C/Co Cost relative to plain carbon steel, C/Co


14/16

Multi-objective analysis: Weighted-Properties Method

• Previous selection problems involved two conflictingobjectives -- often technical performance vs.

economic performance

• Real design problems involve more than twoconflicting objectives

• Weighted-Properties Method -- Each objective isconsidered as a property to be optimised, and isassigned a certain weight depending on its importance

to the production and performance of the part in service

• A weighted-property value is obtained by multiplyingthe numerical value of the property (V) by the weightingfactor (ϕ).

• The individual weighted-property values correspondingto each material choice are then summed to give acomparative performance index for each solution (γ ).

• Solutions with the higher performance index (γ ) areconsidered more suitable for the application.

Weighted-Properties Method: Compare alternative solutions

∑=⋅ϕ=

γ where i is summed over all

the n relevant properties


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• In its simple form, the weighted-properties method hasthe drawback of having to combine unlike units, whichcould yield irrational results.

• The property with higher numerical value will have

more influence than is warranted by its weighting factor.

• This drawback is overcome by introducing scalingfactors. Each property is so scaled that its highestnumerical value does not exceed 100.


• For a given property, the scaled value (B) for a

given candidate solution is equal to:

• Comparative performance index for each solution:


∑=

⋅ϕ=

γ

100xcomparedbetosolutionsoflisttheinvalueMaximum

solutiontheforVpropertyofvalueNumerical B =

Scaled property

(property to

be maximised)

100xsolutiontheforVpropertyofvalueNumerical

comparedbetosolutionsoflisttheinvalueMinimum B =

Scaled property

(property to

be minimised)


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46,503,831,36

21,306,000,82

v [dm3] w [kg] C [ €]

BS 350

F3K20S

V1 V2 V3

γ γγ γ = ϕϕϕϕ1B1 + ϕϕϕϕ2B2 + ϕϕϕϕ3B3

B1 B2 B3

100xsolutiontheforVpropertyofvalueNumerical

comparedbetosolutionsoflisttheinvalueMinimum B =

Scaled property(property to be minimised)

(21,30/46,50) x 100(3,83/3,83) x 100(0,82/1,36) x 100

(21,30/21,30) x 100(3,83/6,00) x 100(0,82/0,82) x 100

B1 B2 B3

BS 350

F3K20S

γ γγ γ BS 350γ γγ γ F3K20S

• Performance index for each solution (γ ) can be analyzedvarying the weighting factor (ϕ) corresponding to eachscaled property (B).

• Digital Logic Method for definition of weighting factors ϕ

Weighted-Properties Method: Analysis

∑=

⋅ϕ=

γ

(Properties)

Σ ϕ = 1.0

ϕ

( 3/10 = 0.3 )

Ashby Method 2.6

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