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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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Finding a compromise: Strategy 2
• Make trade-off plot
• 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
M e t r i c 1 : m a s s m
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.
Finding a compromise: Strategy 3
Define locally linear
Value Function V
CmV +α=
Seek material with smallest V:
• Evaluate V for eachsolution, and rank
or
• Make trade-off plot
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
M e t r i c 1 : m a s s m
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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Finding a compromise: Strategy 3
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
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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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Thickness relative to ABS
0 .1 1 1 0
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 Comp osite
Ti-alloys
Mg-alloys
CFRPGFRP
Lead
Polymer foams
.
ElastomersTrade-offsurface
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
• 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
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
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
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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
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Finding a compromise: Value Function
Define locally linearValue Function V
CmV +α=
Seek material with smallest V:
• Evaluate V for eachsolution, and rank
or
• Make trade-off plot
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
M e t r i c 1 : m a s s m
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
Additionalconstraints:
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
Additionalconstraints:
K1c >15 MPa.m1/2
Tmax > 373 K
Tmin < 223 K
Water: good +
Organics: good +
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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
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
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
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-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
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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.
Weighted-Properties Method: Compare alternative solutions
• For a given property, the scaled value (B) for a
given candidate solution is equal to:
• Comparative performance index for each solution:
Weighted-Properties Method: Compare alternative solutions
∑=
⋅ϕ=
γ
100xcomparedbetosolutionsoflisttheinvalueMaximum
solutiontheforVpropertyofvalueNumerical B =
Scaled property
(property to
be maximised)
100xsolutiontheforVpropertyofvalueNumerical
comparedbetosolutionsoflisttheinvalueMinimum B =
Scaled property
(property to
be minimised)
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Weighted-Properties Method: Compare alternative solutions
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 )