# 12 multiple constraints

```MME445: Lecture 12
Multiple Constraints and
Conflicting Objectives
Part 1: Selection with multiple constraints
Learning Objectives
Knowledge &amp;
Understanding
Knowledge on analytical and graphical selection method in design
with multiple constraints
Skills &amp; Abilities
Ability to select systematically when there are multiple constraints
Values &amp;
Attitudes
Understand the fact that “the most restrictive constraint wins”
in selection methods with multiple constraints
Resources
• M F Ashby, Materials Selection in Mechanical Design, 4th Ed., Ch. 07
1
Outline of today’s lecture
 Introduction and synopsis
 Selection with multiple constraints
 Summary and conclusions
Introduction &amp; Synopsis
 Most decisions in life involve trade-offs.
 Sometimes the trade-off is to cope with conflicting constraints:
I must pay this bill but I must also pay that one
— you pay the one that is most pressing
 At other times the trade-off is to balance divergent objectives:
I want to be rich but I also want to be happy
— and resolving this is more difficult since you must balance the two,
and wealth is rarely measured in the same units as happiness.
2
 Solution of the problem having one design objective with many
non-conflicting constraints is straightforward:
 Apply the constraints in sequence, rejecting at each step the materials
that fail to meet them.
 The survivors are viable candidates.
 Rank them by their ability to meet the single objective
and then explore documentation for the top-ranked candidates.
 The selection of materials and processes often must satisfy several
conflicting constraints.
 In the design of an aircraft wing spar, weight must be minimized,
with constraints on stiffness, fatigue strength, toughness and geometry.
 In the design of a disposable hot-drink cup, cost is what matters; it must be
minimized subject to constraints on stiffness, strength, and thermal conductivity.
 A second class of problem involves more than one objective,
and here the conflict is more severe.
 Nature being what it is, the choice of materials that best meets
one objective will not usually be that which best meets others.
 The designer charged with selecting a material for a wing spar that must
be both light and cheap faces an obvious difficulty: The lightest materials
are not always the least expensive, and vice versa.
 To make any progress, the designer needs a way of trading weight
against cost — a problem we have not encountered until now.
 We shall deal these type of selection problems in the next class.
3
Selection with Multiple Constraints
 Nearly all material selection problems are over-constrained,
 “Should not deflect more than something, must not fail by yielding, by fatigue,
by fast-fracture …”
 There are more constraints than free variables !!
 Recapitulating, we identify the constraints and the objective
imposed by the design requirements, and apply the following steps.
 Screen, using each constraint in turn.
 Rank, using the performance metric describing the objective
(often mass, volume, or cost) or simply by the value of the material index
that appears in the equation for the metric.
 Seek documentation for the top-ranked candidates and use this
to make the final choice.
Design with Multiple Constraints
and Multiple Objectives
Function
One Objective:
Multiple Objectives:
the performance metric
several performance metrics
One
Constraint
Rank by
performance
metric
Many
Constraints
Rank by
most restrictive
performance metric
One
Constraint
Many
Constraints
Penalty
function
method
Combination
of methods
4
Simplest case:
Design with one objective, meeting a single constraint
Tie rod
Function
Minimise mass
Carry force F
without yielding;
Length given
One Objective:
Multiple Objectives:
one performance metric
several performance metrics
One
Constraint
Rank by
performance
metric
Many
Constraints
Rank by
most restrictive
performance metric
One
Constraint
Many
Constraints
Penalty
function
method
Combination
of methods
Or several non-conflicting constraints, such as
melting point, corrosion resistance, etc.
Selection with multiple non-conflicting constraints (a) and a single objective (b).
Screen using the constraints and rank using the objective.
5
 The previous example is a special case of a single objective
that can be limited by more than one non-conflicting constraints.
 But what if the requirements for a tie-rod of minimum mass might
specify both stiffness and strength, leading to two independent
equations for the mass?
One notch up in complexity:
Single objective / Conflicting Constraints
Tie rod
Function
Minimise mass
No yield &amp;
Given deflection
No corrosion
Tmax &gt; 100 &deg;C
One Objective:
Multiple Objectives:
one performance metric
several performance metrics
One
Constraint
Rank by
performance
metric
Many
Constraints
Rank by
most restrictive
performance metric
One
Constraint
Many
Constraints
Penalty
function
method
Combination
of methods
The most restrictive constraint determines the performance metric (mass)
6
material for a
stiff, light tie rod
E
r
sy
r
material for a
strong, light tie rod
A= area, L*= length, ρ = density, S*= stiffness, E = Young’s modulus, F*f = collapse load, σy= yield strength or elastic limit
Conflicting Constraints: Strong /Stiff Light Tie Rod
Evaluate competing constraints and performance metrics
Requires stiffer
material
Max. deflection
Must not yield
Stiffness
constraint
Strength
constraint
 s

 1%; 0.1%
L E
sy = yield strength
σ  σy
m1 
= deflection
E = elastic modulus
F

r
L2  
E
 r 
m2  FL 
s y 
Competing
performance metrics
Rank by the more restrictive of the two, meaning…?
7
 If stiffness is the dominant constraint, the mass of the rod is m1;
if it is strength, the mass is m2.
 If the tie is to meet the requirements on both, its mass has to be
the greater of m1 and m2. Writing
~ = max (m1, m2)
m
(7.1)
~
we search for the material that offers the smallest value of m.
 This is an example of a “min–max” problem, not uncommon in the
world of optimization.
 We seek the smallest value (min) of a metric that is the larger (max)
of two or more alternatives.
 Can be solved either analytically or graphically
The analytical method
 Analytical solution in 3 steps:
Rank by the more restrictive of the constraints
1. evaluate both m1 and m2 for each member of the population,
2. assign the larger of the two to each member, and then
3. rank the members by the assigned value, seeking a minimum.
8
1. Calculate m1 and m2 for given L and F
m1 
 r 
m2  FL 
s y 
r
L2  
 E
F
 = 0.5%
L=1m
F = 10 kN
r, Mg/m3
E, GPa
sy, MPa
m1 (E), g
Epoxy-Aramid Fibre
1.38
80
1400
34.5
9.9
Glass-Epoxy UD Composite
1.95
45
1100
86.7
17.7
Low alloy steel
7.90
217
2445
72.8
32.3
Epoxy-Carbon Fibre (SMC)
1.70
150
345
22.7
49.3
Al 6061-T6
2.73
74
320
73.8
85.3
Titanium
4.45
117
1138
76.1
39.1
m2 (TS), g
~
2. Find the largest of every pair of m’s (which is m)
3. Find the smallest of the larger ones
The most restrictive constraint requires a larger mass
and thus becomes the controlling or active constraint.
 When there are 3,000 rather than 3 materials to choose from,
simple computer codes can be used to sort and rank them.
 But this numerical approach
1. yields a solution that is not general (as it depends on F and L)
2. lacks visual immediacy and the stimulus for creative thinking
that a more graphical method allows.
9
The graphical method
 Graphical solution using indices and bubble charts:
• More general/powerful.
• Allows for a visual while physically based selection.
• Involves all available materials.
• Incorporates geometrical constraints through coupling factors.
• For a population of materials, m1
is plotted against m2.
• Each bubble represents a
material.
• All the variables in both
equations for m1 and m2 are
specified except the material
so the only difference between one
bubble and another is the material
FIGURE 7.4
The graphical approach to min–max problems. (a) Coupled
selection using performance metrics (here, mass m).
10
 To minimize mass, the best
choices lie somewhere near
the bottom left.
 But where, exactly?
 If stiffness is paramount and
strength is unimportant the
choice must surely differ from
that if the opposite were true.
FIGURE 7.4
The graphical approach to min–max problems.
(a) Coupled selection using performance metrics (here, mass m).
 The line m1 = m2 separates
the chart into two regions.
• In one, m1 &gt; m2
and constraint 1 (stiffness) is dominant.
• In the other, m2 &gt; m1
and constraint 2 (strength) dominates
 In region 1 our objective is to
minimize m1, since it is the larger
of the two; in region 2 the opposite
is true.
• This defines a box-shaped selection
envelope with its corner on the
m1 = m2 line.
 The nearer the box is pulled to
~
the bottom left, the smaller is m.
• The best choice is the last material
left in the box.
FIGURE 7.4
The graphical approach to min–max problems.
(a) Coupled selection using performance metrics (here, mass m).
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- We need a new chart.
Index M2
• What will happen if these values
change?
Larger
• Earlier graph (Fig. 7.4(a)) was
plotted with m1 and m2 as axes,
using specific single values of
L*,S*; and Ff*
Smaller
 Now, the data is re-plotted
against materials indices
M1 = E/r and M2 = sy/r.
 Each bubble still represents
a material, but now its
position depends only on
material properties, not on
the values of L*,S*; and Ff*
Smaller
Index M1
Larger
FIGURE 7.4
(b) A more general approach: coupled selection using material indices M
and a coupling constant Cc.
m1 = L*2 S*
r
M1 =
E
r
m2 = L* F *f s
y
M2 =
r
E
r
sy
For the minimisation
problem, the RHS
factor is not altered !!
m1 = m2
L*2 S*
r
E
r
= L* F *f s
y
L*2 S* M1 = L* F *f M2
M2 =
L* S*
F *f
Coupling constant, CC
M1
log M2 = log M1 + log
L* S*
F *f
Coupling line
12
Larger
Coupling lines
• This describes a line of
slope 1, in a position that
depends on the value of
L*S*/Ff*.
Smaller
Index M2
M2 = (CC) M1
Smaller
Larger
Index M1
A box, with its
corner on the
coupling line, is
pulled down toward
the bottom left.
Large CC
Constraint 2 dominant
Coupling lines
M2 = CC M1
Index M2
 Now, the selection
strategy remains the
same:
Larger
FIGURE 7.4
(b) A more general approach: coupled selection using material indices M
and a coupling constant Cc.
Small CC
 Changing either one of
these, or the geometry
of the component (here
described by L*) moves
the coupling line and
changes the selections.
Smaller
 But the chart is now more
general, covering all
values of L*, S* and Ff*.
Decreasing
~
m
Smaller
Constraint 1 dominant
Index M1
Larger
FIGURE 7.4
(b) A more general approach: coupled selection using material indices M
and a coupling constant Cc.
13
Summary &amp; Conclusions
 Real designs are over-constrained and many have multiple
objectives
 Method of maximum restrictiveness copes with conflicting
multiple constraints
 Analytical method useful but depends on the particular conditions
set and lacks the visual power of the graphical method
 Graphical method produces a more general solution
Next Class
MME445: Lecture 24
Multiple Constraints and Conflicting Objectives
Case studies with multiple constraints
14
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