程序代写代做代考 Excel matlab algorithm ant Welcome to Design VI

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