1.
G-L-5:
Optimisation
Advanced Modelling
Prof. Dr. Christos Spitas
Dr. Y. Song (Wolf)
Faculty of Industrial Design Engineering
Delft University of Technology
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2.
Contents
• Why optimisation?
• A case study
• The gradient descent method
• Discussions
• What did we learn today?
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3.
Optimisation@Modelling
what are you
studying?
Case brief
‘touch’ your
thoughts
choices
what
exactly...?
what do you
foresee?
Build abstract
model
System
choices
choices
what does your
model foresee?
Thought
simulation
Experience
Solve/ simulate
choices
Compare
did you both
agree?
everything you did/ saw/ felt/
remember to be true...
Finish!
Acceptable
?
Revisit
choices
do you need more
insights/ data?
Experiment
Courtesy of centech.com.pl and http://www.clipsahoy.com/webgraphics4/as5814.htm
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4.
Why Optimisation?
Product optimisation
Product Optimisation is not just making a product better in some
respect (a criterion / metric), but making it the BEST (optimal) in this
respect.
In mathematics, Optimisation refers to choosing the element from
some set of available alternatives that maximises/minimises a
specified metric.
Identify,
choose
Learn
Modelling
System
Solve
Evaluation
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5.
A case study: The coffee cups
In the new generation of Douwe Egberts® coffee
machine, the preheat coffee cup feature is introduced. In
the preheating process, water steam from the nozzle
preheats the cup to a certain temperature before the
coffee is served.
Experiments indicate that:
1. if the cup is preheated to 92°C , the best coffee can be
served;
2. 80% of water steam is condensed inside the cup, the
rest is absorbed by the air.
Besides, Douwe Egberts® also manufactures its own
coffee cups with different sizes and materials, for
example, the Hollandsche series and the Standard series.
You are asked to find the initial steam temperature and
the amount of steam that should be produced in order to
preheat either of the two types of cups as close as
possible to 92°C (minimise the deviation).
Fictional case study, for education only.
Douwe Egbert® are resisted trademarks.
Courtesy of http://www.douweegbertscoffeesystems.com/dg/OutOfHome/OurProducts/Coffee/Cafitesse/Machines/
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6.
System identification
Steam
Hollandsche
cup
Standard
cup
Fiction case study, for education only.
Douwe Egbert® are resisted trademarks.
Courtesy of http://www.douweegbertscoffeesystems.com/dg/OutOfHome/OurProducts/Coffee/Cafitesse/Machines/
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7.
Cause-effect
Superheated steam
Condensing
Condensed
Fiction case study, for education only.
Douwe Egbert® are resisted trademarks.
Courtesy of http://www.douweegbertscoffeesystems.com/dg/OutOfHome/OurProducts/Coffee/Cafitesse/Machines/
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8.
Modelling
Hollandsche cup
m
c
steam steam
(Tstea min itial 100) m steam Lsteam m steam c water (100 THfinal ) 80% mholland cholland (THfinal Tinitial )
Steam cools
down
m
c
steam steam
Latent
heat
Condensed
water cools
down
Cup is
heated
(Tstea min itial 100) m steam Lsteam m steam c water (100 TSfinal ) 80% mstandard cstandard (TSfinal Tinitial )
Standard cup
Courtesy of http://www.douweegbertscoffeesystems.com/dg/OutOfHome/OurProducts/Coffee/Cafitesse/Machines/
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9.
Modelling - Choices
Hollandsche cup
m
c
steam steam
(Tstea min itial 100) m steam Lsteam m steam cwater (100 THfinal ) 80% mholland cholland (THfinal Tinitial )
Standard cup
m
c
steam steam
(Tstea min itial 100) m steam Lsteam m steam c water (100 TSfinal ) 80% mstandard cstandard (TSfinal Tinitial )
We choose
The density of water is 1000 kg/m3;
The latent heat of water vaporization is 2,260,000 J/kg;
The specific heat of water steam is 2080 J/(kg·K);
The specific heat of water is 4181 J/(kg·K);
The initial temperatures of both cups are 20 °C;
The weight of the Hollandsche cup is 0.20 kg, the specific heat is 750 J/(kg·K);
The weight of the Standard cup is 0.13 kg, the specific heat is 1070 J/(kg·K);
The steam temperature doesn’t change before it reaches the cup;
The complete process happens in a very short time;
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10.
Solving/Simulation
Hollandsche cup final temperature
8320msteamTsteaminitial 9.8804 106 msteam 15000
THfinal (msteam , Tsteaminitial )
16724msteam 750
Design parameters ( msteam , Tsteaminitial )
8320msteamTsteaminitial 9.8804 106 msteam 13910
TSfinal (msteam , Tsteaminitial )
16724msteam 695.5
Standard cup final temperature
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11.
Our wishes
Design brief
In the new generation of Douwe Egberts® coffee
machine, the preheat coffee cup feature is introduced. In
the preheating process, water steam from the nozzle
preheats the cup to a certain temperature before the
coffee is served.
Make the deviation
THfinal (msteam , Tsteaminitial ) 92
as small as possible
Experiments indicate that:
1. if the cup is preheated to 92°C , the best coffee can be
served;
Wishes
2. 80% of water steam is condensed inside the cup, the
rest is absorbed by the air.
Besides, Douwe Egberts® also manufactures its own
coffee cups with different sizes and materials, for
example, the Hollandsche series and the Standard series.
You are asked to find the initial temperature and the
amount of steam that should be produced in order to
preheat either of the two types of cups as close as
possible to 92°C (minimize the deviation).
AND
Make the deviation
TSfinal (msteam , Tsteaminitial ) 92
as small as possible
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12.
The metrics
Example
The differences
M 1 (THfinal (msteam , Tsteaminitial ) 92) 2
M 2 (TSfinal (msteam , Tsteaminitial ) 92) 2
y ( x) x 5
2
y ( x) x 5
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13.
Formulate the objective function based
on metrics
2
f (msteam , Tsteaminitial ) Wi M i
i 1
The objective
function
Non-dimensional
We want to minimise
min f ( m steam , Tsteaminitial ) where f ( m steam , Tsteaminitial )
2
W M
i
i 1
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i
14.
Formulate the objective function based
on metrics
In our case study, we choose
Wi 0.5
min f (msteam , Tsteaminitial ) 0.5(THfinal (msteam , Tsteaminitial ) 92) 2 0.5(TSfinal (msteam , Tsteaminitial ) 92) 2
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15.
Start from a 1D problem:
Finding the minimum of a function
f ( x)
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16.
The gradient descent method
f ( x)
f ( x) Guess
Start with a point (guess)
Repeat
•Determine a decent direction
•Choose a step
•Update
Until stopping criterion is
satisfied
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17.
The gradient descent method
f ( x)
f ( x) Guess
Start with a point (guess)
1 Guess
Repeat
•Determine a decent direction
•Choose a step
•Update
Until stopping criterion is
satisfied
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18.
The gradient descent method
Step f ( x) Guess
Start with a point (guess)
f ( x)
1 Guess
Repeat
2 Next Step
•Determine a decent direction
•Choose a step
•Update
Until stopping criterion is
satisfied
xNextSep xGuess Step f ( x) Guess
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19.
The gradient descent method
Start with a point (guess)
f ( x)
f ( x) Here
1
Repeat
2 Now we
•Determine a decent direction
•Choose a step
•Update
are here
Until stopping criterion is
satisfied
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20.
The gradient descent method
Start with a point (guess)
f ( x)
1
Repeat
2
•Determine a decent direction
•Choose a step
•Update
3
4
6
5
Until stopping criterion is
satisfied
f
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21.
2D gradient descent method
x
y
f ( x, y ) xe
x2 y2
f ( x, y )
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22.
2D gradient descent method
f ( x, y )
x
f ( x, y )
y
f ( x, y )
(
f ( x, y ),
f ( x, y ))
x
y
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23.
2D gradient descent method
Start
Start with a point (guess)
Repeat
•Determine a decent direction
•Choose a step
•Update
Until stopping criterion is
satisfied
Minima
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24.
Solving our design problem:
The coffee cups
Guess
Msteam = 0.003
Tsteaminitial=110
Gradient
decent path
It is a
compromise
The minimum
value of the
objective function
is 6
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25.
Wishes vs Designs
Put real optimal parameters in side
Design brief
In the new generation of Douwe Egberts® coffee
machine, the preheat coffee cup feature is introduced.
In the preheating process, water steam from the nozzle
preheats the cup to a certain temperature before the
coffee is served.
Experiments indicate that:
1. if the cup is preheated to 92°C , the best coffee can
be served;
2. 80% of water steam is condensed inside the cup,
the rest is absorbed by the air.
Besides, Douwe Egberts® also manufactures its own
coffee cups with different sizes and materials, for
example, the Hollandsche series and the Standard
series.
You are asked to find the initial temperature and the
amount of steam that should be produced in order to
preheat either of the two types of cups as close as
possible to 92°C (minimize the deviation).
Hollandsche cup final temperature
THfinal (msteam , Tsteaminitial ) 89.5C
Output of the
optimised
design
TSfinal ( msteam , Tsteaminitial ) 94.3C
Standard cup final temperature
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26.
Generalised form
To optimise a function:
min f ( p1 , p2 ,... pn )
0
0
( p10 , p 2 ,... p n )
f
m
m
m
( p1 , p2 ,... pn )
stepm
m
m
( p1m 1 , p 2 1 ,... p n 1 )
m
m
( p1m , p 2 ,... p n ) step m ( f
f
Start with a point (guess)
m
m
m
p1 , p 2 ,... p n
m
m
m
p1 1 , p 2 1 ,... p n 1
Repeat
•Determine a decent direction
•Choose a step
•Update
)
Until stopping criterion is
satisfied
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27.
Question!
Weights in the objective function
The market share
We can emphasise
a wish by increasing
Its relative weight
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28.
Question! Weights in the objective function
Put real optimal parameters in side
min f (msteam , Tsteaminitial )
where
And compare to previous
min f (msteam , Tsteaminitial ) W1 (THfinal (msteam , Tsteaminitial ) 92) 2 W2 (TSfinal (msteam , Tsteaminitial ) 92) 2
Weight
Value = 0.25
Weight
Value = 0.75
Output of the
optimised
design
Hollandsche cup final temperature
Standard cup final temperature
THfinal (msteam , Tsteaminitial ) 88.4C
TSfinal ( msteam , Tsteaminitial ) 93.1C
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29.
Question! Square in the metrics
y ( x) x 5
M 1 (THfinal (msteam , Tsteaminitial ) 92) 2
Why square instead
of absolute value?
M 2 (TSfinal (msteam , Tsteaminitial ) 92) 2
y ( x) x 5
2
y ( x) x 5
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30.
Problem: find the maxima
With a minus
in front –f(x)
Gradient ascent
Original
function f(x)
Or
Function f(x)
Guess
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31.
Problems: Guess the starting point
The starting point
is not good
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32.
Problems: Choice of the sizes of steps
Small steps: inefficient
Large steps: potential risks
The
starting
point
The
starting
point
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33.
Advanced: Dynamic steps
f
f
m
m
m
( p1 , p 2 ,... pn )
Acquire the direction
m
m
m
( p1 , p 2 ,... pn )
Assign the initial step
If the step is too large,
reduce the step to
a certain percentage
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34.
Problems: The stopping criterion
Intuitive criterion
f
f
2
f
x
i 1
i
N
2
2
2
f
f
f
x1
x2
x N
Other criteria
Example 1
Example 2
m
m
m
m
( p1m 1 , p 2 1 ,... p n 1 ) ( p1m , p 2 ,... p n )
m
m
m
m
f ( p1m 1 , p 2 1 ,... p n 1 ) f ( p1m , p 2 ,... p n )
…
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35.
Fundamental problems: Local minima
x
f ( x, y )
y
esin( x ) cos( y ) 0.1( x 2 y 2 )
f ( x, y)
Local minima
Global minima
Top view
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36.
Fundamental problems: Local minima
Local minimum
Guess
Global minimum
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37.
The world is much larger
Quasi-Newton /
conjugate gradient methods
Box (complex) / Hook-Jeeves
Tools for
Optimisation
Genetic algorithms
(Evolution Strategy)
and many more
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38.
What did we learn today?
Formulate the objective
function (metric) of your design
Gradient descent method
is a possible way to find the minimum
of your objective function
An optimum is (almost) always
a compromise (competing metrics!)
Optimisation methods
have limitations
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39.
Success!
Prof. Dr. Christos Spitas
Dr. Y. Song (Wolf)
Faculty of Industrial Design Engineering
Delft University of Technology
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