Welcome to Design VI
Introduction to
design optimization
1
Goal of Week 1: To become familiar with the concept of mathematical
optimization, see some applications, & begin forming teams and topics
ME 564/SYS 564
Wed Aug 29, 2018
Steven Hoffenson
Optimization is trendy
Source: Google ngrams
Usage of the word
“optimization” in books, as
a percentage of all words
2
What is design optimization?
Improving a design
Increasing
efficiencyFinding a solution that
satisfies everyone
Finding the best
possible solution
Strictly speaking, design optimization is about
mathematically finding the best possible design
solution for given models and an objective
3
Every real-life problem
is an optimization problem
4
Design an optimal automobile for you
• Speed
• Efficiency
• Safety
• Capacity (people)
• Capacity (cargo)
• Sales
5
How to optimize
1. Formulate the problem
a) Define system boundaries
b) Develop analytical models
c) Explore/reduce the problem space
d) Formalize optimization problem
2. Solve the problem
a) Choose the right approach/algorithm
b) Solve (by hand, code, or software)
c) Interpret the results
d) Iterate if needed
6
𝐱𝑘+1 = 𝐱𝑘 − 𝐇(𝐱𝑘)
−1𝛁𝑓 𝐱0
(Weeks 1-2, 4, 9-12)
(Weeks 3, 5-8, 12)
1. Formulate the problem
a) Define system boundaries
What are we including? What are we assuming fixed? What are our
objectives, constraints, variables, and parameters?
b) Develop analytical models
Are they theoretical (equation-based) or empirical (data-based)? Do
they take a long time to evaluate? Can we use surrogate models?
c) Explore/reduce the problem space
Is there a feasible solution? Can we reduce the number of variables or
constraints? Is the space convex? Are there local optima?
d) Formalize optimization problem
Write it out mathematically. Then, ask and adjust based on: Is it multi-
disciplinary? Is it multi-objective? Is there uncertainty?
7
a) Define system boundaries
8
Compressor Turbine Power to compressor
Combustor
Air
Heat/fuel
Gas
Power
Air
Where you draw your box defines the problem space
and ultimately the design solution
Objectives, constraints, variables, parameters
9
Managers might
say…
Designers
might say…
What it means Car examples
Key performance
indicators
(KPIs/KPPs)
Objectives
What we want to
maximize/minimize
Seek best possible cost
or performance (e.g.,
speed, efficiency)
Requirements
Hard
constraints
Must-haves, with
specific thresholds
Must pass FMVSS
government crash test
Desirements,
Targets
Soft
constraints
Wants, with specific
thresholds
At least 36 miles per
gallon (35 wouldn’t
invalidate the project)
Decisions Variables
Things we can change
and want the
optimizer to change
Sizes, material choices,
layout, capacity
Environment Parameters
Quantities that we
can’t or won’t change
Material properties,
e.g., strength of steel
Example: Stigler diet
• Objective: Minimize cost
• Constraints: Meet 9 nutrients’ RDAs
• Variables: Amounts of each food
• Parameters: 77 foods included;
nutrient content and cost of each
food; moderately active 154-lb man
• Models: Linear equations of
nutrients and costs per unit of food
10
Stigler, G. J. (1945). The cost of subsistence. Journal of farm economics, 27(2), 303-314.
What is the lowest possible cost of a diet for a moderately-active,
154-pound male, that meets the National Research Council’s
1943 Recommended Dietary Allowances (RDA) of 9 nutrients?
Example: Crash safety
11
• Objective: Minimize injury probability
• Constraints: Pass FMVSS tests
• Variables: Thicknesses of structural elements;
stiffness of seat belt; airbag inflation rate
• Parameters: Vehicle shape; material
properties; size of mid-size male dummy;
definition of “serious injury” on Abbreviated
Injury Scale (AIS); crash test specs
• Models: Physics-based simulations
What is the lowest probability of serious injury that we can
achieve through structural and restraint system design for a
mid-sized male crash test dummy in a 35-mph crash with a
rigid barrier?
Hoffenson, S., Reed, M. P., Kaewbaidhoon, Y., & Papalambros, P. Y. (2013). On the impact of the regulatory frontal crash test
speed on optimal vehicle design and road traffic injuries. International Journal of Vehicle Design, 63(1), 39-60.
Example: Topology of a chair
12
• Objective: Maximize stiffness
• Constraints: Mass
• Variables: Material in each coordinate position (yes/no)
• Parameters: Material properties; loading direction of weight
• Model: Finite element simulation
How can we design the shape of a fixed-mass chair to hold the
maximum weight possible?
http://mocosubmit.com/generico-chair/
Exercise
In groups, come up with an objective, constraints, variables,
and parameters for designing a battery pack for an electric car
13
Quantity What it means
Objectives
What we want to
maximize/minimize
Hard constraints
Must-haves, with specific
thresholds
Soft constraints
Wants, with specific
thresholds
Variables Things we can change
Parameters
Quantities that we can’t or
won’t change
Quantity What it means Battery examples
Objectives
What we want to
maximize/minimize
Maximize capacity in kWh
Hard constraints
Must-haves, with specific
thresholds
Must meet safety standards
Soft constraints
Wants, with specific
thresholds
Weigh no more than 200 lb;
Capacity of at least 30 kWh; Volume
no more than 15 ft3; Cost no more
than $3,000
Variables Things we can change Dimensions, material choice, layout
Parameters
Quantities that we can’t or
won’t change
Material properties, e.g., density of
a particular lithium-ion battery;
thresholds of soft constraints
Note: This is what you need to do for the “optimization in the real world” assignment!
1. Formulate the problem
a) Define system boundaries
What are we including? What are we assuming fixed? What are our
objectives, constraints, variables, and parameters?
b) Develop analytical models
Are they theoretical (equation-based) or empirical (data-based)? Do
they take a long time to evaluate? Can we use surrogate models?
c) Explore/reduce the problem space
Is there a feasible solution? Can we reduce the number of variables or
constraints? Is the space convex? Are there local optima?
d) Formalize optimization problem
Write it out mathematically. Then, ask and adjust based on: Is it multi-
disciplinary? Is it multi-objective? Is there uncertainty?
14
b) Develop analytical models
• How do we represent our system mathematically?
• Inputs: Variables & parameters
• Outputs: Objectives & constraints
• Three ways to do this
1. Chemical/physical/mathematical equations
2. Simulation models (FEA, CFD, etc.)
3. Experimental data
• Design of experiments
• Metamodeling
15
Design of Experiments (DOE)
When we have experimental or computationally
expensive simulation data, we need to sample the
space efficiently
16
Week 4
Metamodeling
Fit an analytical model to data
17
Week 4
1. Formulate the problem
a) Define system boundaries
What are we including? What are we assuming fixed? What are our
objectives, constraints, variables, and parameters?
b) Develop analytical models
Are they theoretical (equation-based) or empirical (data-based)? Do
they take a long time to evaluate? Can we use surrogate models?
c) Explore/reduce the problem space
Is there a feasible solution? Can we reduce the number of variables or
constraints? Is the space convex? Are there local optima?
d) Formalize optimization problem
Write it out mathematically. Then, ask and adjust based on: Is it multi-
disciplinary? Is it multi-objective? Is there uncertainty?
18
c) Explore/reduce the problem space
Once we’ve framed the problem and defined the models,
we can ask:
• Does an optimal solution exist?
• Is the problem well-bounded?
• Are the constraints active?
• Are the functions monotonic?
• Are the functions differentiable?
• Are the functions convex?
• Can the formulation be simplified?
Answering these questions can help detect formulation
errors, save time, and potentially find the solution!
19
Week 2
1. Formulate the problem
a) Define system boundaries
What are we including? What are we assuming fixed? What are our
objectives, constraints, variables, and parameters?
b) Develop analytical models
Are they theoretical (equation-based) or empirical (data-based)? Do
they take a long time to evaluate? Can we use surrogate models?
c) Explore/reduce the problem space
Is there a feasible solution? Can we reduce the number of variables or
constraints? Is the space convex? Are there local optima?
d) Formalize optimization problem
Write it out mathematically. Then, ask and adjust based on: Is it multi-
disciplinary? Is it multi-objective? Is there uncertainty?
20
d) Formulate optimization problem
21
Variables
Objective
function
Parameters
Constraints
“negative null” form
2. Solve the problem
a) Choose the right approach/algorithm
What are the different types of algorithms (pattern
search, gradient-based, population-based)? How do they
work? When do we apply each?
b) Solve (by hand, code, or software)
Apply the chosen algorithm to the formulated problem
c) Interpret the results
Do the outputs make sense? How do we choose among
multi-objective results?
d) Iterate if needed
Use findings to update the formulation or algorithm
22
Why do we need algorithms?
• We don’t always know the shape of a function
• Too many dimensions to visualize
• Not enough data points
• Most algorithms take us from a starting point or
points, and then move in directions of improvement
23
𝑓(𝑥)
f(
x)
x
𝑓(𝑥)
Two-variable contour map
24
V
a
ri
a
b
le
1
Variable 2
Gradient-based algorithms
Use derivatives to find the optimal solution
25
zero
slope
increasing
slope
𝑥
𝑓
Optimality conditions (min)
First-order:
𝜕𝑓
𝜕𝑥
𝑥∗ = 0
Second-order:
𝜕2𝑓
𝜕𝑥2
𝑥∗ > 0
𝑥∗
Extension to multi-variable problems:
First-order: 𝛁𝑓 𝑥∗ = 𝟎
Second-order: 𝐇 𝑥∗ is positive
definite
Weeks 5-7
Population-based
e.g., genetic/evolutionary
algorithms, particle swarm,
ant colony
Gradient-free algorithms
Pattern search
e.g., Hooke-Jeeves
direct search, DIRECT,
Nelder-Meade
26
Week 3 Week 8
Business-oriented optimization
What is the most common objective in design?
27
max
𝛼,𝑃
π = 𝑄 𝛼, 𝑃 𝑃 − 𝐶 𝛼
profit sales
quantity
cost
product attributes
price
We need 2 models:
1. cost, C
2. demand, Q
Multi-objective optimization
28
f(
x)
xxopt
fmin
g
(x
)
xxopt
gmin
min
𝑥
𝑓 𝑥 , 𝑔(𝑥)
g
(x
)
f(x)
Pareto frontier
This is useful for trade-off analysis
Week 9
System design
Decomposition-based Strategy
(Partitioning and Coordination)
All In One
(AIO)
Body ElectronicsPowertrain Chassis
Transmission Engine Driveline
Valvetrain Cylinder Block
VS.
If the system-level problem is difficult to solve all at once, you
may need to decompose the problem into subsystems
Week 10
Handling uncertainty
Ensure the solution isn’t too close to a constraint
30
Week 11
Software support
MATLABExcel Specialty optimization packages
Commercial CAD software integration
Week 12
Important takeaways
• What is design optimization?
• What are the major steps of formulating
and solving an optimization problem?
• How do we set system boundaries?
• What are objectives and constraints?
• What are variables and parameters?
• How do we write a formal optimization
problem?
32
Questions
33