Transcript ENGN 8101 Modelling and Optimisation

ENGN 8101
Modelling and Optimisation
Professor Qinghua Qin
Email :[email protected]
Tel: 6125 8274
Office: Room R228 Ian Ross Building
MODELLING OF ENGINEERING SYSTEMS
Engineering?
“Profession in which knowledge of maths & natural
science gained by study and practice is used with
judgment to develop ways to utilise economically,
material and forces of nature for the benefit of
mankind”
Definition by : ABET
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Design?
“Process of converting customer requirements into
detailed plans (drawings and specifications) from
which the product, process or system can be put
together”
Product/process/system must meet all customer
requirements and be best with respect to chosen
goodness measure(s).
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DESIGN OPTIMIZATION
Selecting the “best” design within the available means
What is our criterion for “best”design?
What are the available means?
Objective function
Constraints (design requirements)
How do we describe different designs?
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Design Variables
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Optimization?
Minimise
f(x);
f  l; x  n Objectives
Subject to :
h(x) = 0; h  m
Equality
Constraints
g(x)  0; g  p
Inequality
constraints
xL  x  x U
Side constraints
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Generic Approach
• Choose an initial design
– Identify design variables, X
– Assign values to X
• Assess system for acceptance
• Modify X to improve design
• Iterate till design is acceptable
Subjective?
Heuristic?
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Objective?
Optimum design?
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Issues?
• Problem statement. Identify
– Design variables
– Constraints
– Objectives
• Choose validated analysis for function evaluation
• Optimization
– Solution procedure: Golden Section Method/ Conjugate
Gradient Method…
– Sensitivity
– Surrogate building
• Software integration
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System engineering in Practice
Design of Boeing 777
• Program launched in
Oct 29,1990
• First flight on June
1994
• 777 has 132,500
engineered, unique
parts and a total of
three million+ total
parts
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System engineering in practice
Design of the Boeing 777
• Program launched in
October 29, 1990
• First flight on June
1994
• 777 has 132,500
engineered, unique
parts and a total of
three million + total
parts
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Text Books
• “Optimization concepts and applications in
engineering”- A.D. Belegundu & T.R.
Chandrupatla - Required
• “Engineering Methods for Robust Product
Design” – W.Y. Fowlkes & C. M. Creveling
– Strongly Recommeded
• Available at ANU Bookshop
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Course Schdule
• Lecturers :
Monday 1-3
Tuesday 1-3
• Lab
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Assessment
• Problem sets(20%)
– modelling and optimization problems
• EXTEND DES + design assignment –
20%
• Experimental design exercise – 20%
• Final Exam(40%)
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Expectations
• Every student to master all fundamental
concepts
• You must spend time with the material
– It will be worth it in the future!
• I promise to do my best to provide a
robust learning environment
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Academic Honesty
• Personal and corporate integrity is an
essential
element
of
any
quality
organization. Accordingly, I expect every
student to avoid even the appearance of
cheating, and to claim credit only for his or
her own work.
• I promise the same level of personal
integrity that I expect. Cheating of any kind
simply will not be tolerated!
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MODELLING OF ENGINEERING SYSTEMS
“Real phenomena that are subject to uncertainty can be modelled using the
language of probability. This model can be parameterized using real data with
predictive behaviour subsequently generated”
Barry Nelson 1995
i.e. whose state varies with time
i.e. dynamic phenomena
e.g. an engineering system
What is a system?
- not for now!
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“Real phenomena that are subject to uncertainty can be modelled using the language of
probability. This model can be parameterized using real data with predictive behaviour
subsequently generated”
i.e. convert system into smaller
entities whose state can be modelled
using a probability distribution
e.g. what is the probability that after
an hour, a machine is still working?
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“Real phenomena that are subject to uncertainty can be modelled using the language of
probability. This model can be parameterized using real data with predictive behaviour
subsequently generated”
Take actual readings of variables and
plug them into the probabilistic model
to generate key performance
characteristics
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“Real phenomena that are subject to uncertainty can be modelled using the language of
probability. This model can be parameterized using real data with predictive behaviour
subsequently generated”
come up with statistically robust
estimates of future system behaviour
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Furthermore:
“Investigate how parametric design affects key parts of the system”
i.e. obtain combinations of parameters
that give maximum desirability in a key
performance measure
i.e.
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OPTIMIZATION
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Good engineering practice……
Take a system and use modelling and
optimization techniques to present a
theoretically robust design
NEVER design using trial and error!
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Brief example:
Probability that a light bulb breaks
States = ‘working’ or ‘not working’ over a time ‘x’
When x=0 + δx – the light bulb has just been switched on
Can be modelled as an
exponential distribution
Often used in failure
analysis
Light bulb most likely to fail when switched on!
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Purpose of modelling –
“to deduce statements about the performance of a real
or conceptual engineering system”
i.e.
Dynamic systems that are subject to uncertainty
Models = 100% predictable
Systems = never!
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Key to successful modelling of a system….
Scale down into manageable entities then
model behaviour through
probabilistic/statistical techniques
i.e. Discrete Event Modelling
An example of numerical modelling
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EXTEND – typical DE modelling software
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Also – simple
analytical modelling
i.e. analytical solutions to linear, differential and partial
differential equations
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Numerical Modeling
• Finite Element
Models, Matlab,
EXTEND
Engineering/Physical principles
Discrete model
Post-process the results
Solution of linear/nonlinear equations
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Also -
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Structural/Numerical Modelling
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Some Modelling Issues…
• Most analytical and numerical models tend to be discipline
specific (exceptions include multi-body physics models)
• Some of these modelling techniques need expert training
• There are several practical applications where
analytical/numerical modelling techniques are not effective
(cost, time, expert training or limitations in coping with the
complexity of the problem)
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MAJOR FOCUS OF ENGN8101
How to model and thus optimize an
engineering system to improve it
‘Improve’ could mean ‘enhance the quality of the output’
In terms of quality –
Possible to model a system without formally optimizing it
Also – may have to optimize a system without a model
Data generation not always possible
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Special note on Taguchi Methods
•These are methods that use quality as a performance measure
•They are not limited to any specific discipline area
•Often used when analytical and numerical modelling are
ineffective
•Often used in synergy with other modelling strategies
•Widely used in industrial settings
•More like a philosophy than a technique
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The course in a nutshell….
• Introduction
• Discrete Event Modelling
• Review of the nature of statistics
• Basic probability
• Statistical models in simulation
• Modelling of queueing processes
• Quality Engineering
• Introduction to quality engineering
• Design of experiments (orthogonals, factorial, interactions)
• Parametric design (S/N ratios, 2-step optimization)
• Importance of loss functions
• Classical Optimization
• Philosophical issues
• Mathematical foundations
• Unconstrained optimization
• Constrained optimization
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A first modelling exercise…..
• What is “best”?
• Which option “best”?
• 100% of the time? – neither will be!
Dynamic system with variables (customer type etc.)
One option best x% of the time
One option best y% of the time
If x»y – find the first option and live with the uncertainty
Uncertainty = randomness = stochastic
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Essential first step……
collect data!
What is this data giving us?
CASE STUDY 1
The general office of a large company has as one of its responsibilities, the
photocopying area. Currently, they have one photocopying machine and one
operator. Employees needing some copying work wait in a single line until called by
the operator. Some jobs involve mere copying, whereas others are more complicated,
requiring collation, stapling etc.
Employees are now complaining that they are waiting too long, so the office is
considering expanding its photocopying service. There are two options. The first is
to purchase another copier and a second operator, and the second option is to have a
second copier for self-service jobs only (i.e. no operator). Which option is best?
To assist with the task, the following data were collected on one morning period:
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Time
9:00
9:12
9:14
9:17
9:19
9:21
9:22
9:38
9:39
9:41
9:43
9:45
9:52
9:57
9:58
10:00
10:01
10:08
10:11
10:13
10:14
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Customer arrival/completion data
What happened Assistance? Time
What happened
Open
10:16 C12 in
C1 in
10:19 C13 in
C2 in
10:24 C10 out
C3 in
10:26 C11 out
C1 out
yes
10:28 C14 in
C2 out
10:36 C12 out
C3 out
10:36 C15 in
C4 in
10:38 C13 out
C5 in
10:39 C14 out
C4 out
10:42 C15 out
C6 in
10:45 C16 in
C7 in
10:47 C16 out
C8 in
10:48 C17 in
C5 out
yes
10:50 C17 out
C9 in
10:53 C18 in
C6 out
10:58 C19 in
C7 out
11:00 C20 in
C8 out
11:01 C18 out
C9 out
yes
11:07 C19 out
C10 in
11:09 C20 out
C11 in
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Assistance?
yes
yes
yes
yes
yes
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Or in a more meaningful structure……
Customer (i)
i.e. a
queueing
scenario
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Service time
(si)
Waiting time
(wi)
Interarrival
time gap (gi)
1*
7
0
12
2
2
5
2
3
1
4
3
4
3
0
21
5*
16
2
1
6
3
14
4
7
1
15
2
8
7
9
7
9*
3
10
6
10*
11
0
15
11
2
10
1
12
10
10
2
13
2
17
3
14
1
10
9
15*
3
3
8
16
2
0
9
17
2
0
3
18*
8
0
5
19*
6
3
5
20*
2
7
2
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a very
common
system model
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Data presentation methods…..Histogram
-Graphically visualize the data spread and hint at any pattern
- Divide data set into classes (ranges of values)
- sample size = n – no. of classes = √n
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
50.04
49.96
50.01
49.95
50.00
50.02
50.01
50.02
50.06
49.96
50.01
50.04
49.97
50.00
49.97
49.98
50.03
49.98
50.07
49.99
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Observations X (5 per sample)
50.03
50.02
50.00
49.99
50.03
50.01
50.01
50.01
50.00
49.97
50.02
50.10
50.01
50.00
50.00
50.05
49.97
50.02
49.99
49.96
49.99
50.00
50.04
50.02
49.93
49.99
49.99
49.93
50.08
49.92
49.96
49.98
50.00
49.94
50.00
50.03
49.90
49.98
50.01
50.01
49.95
49.97
49.98
50.03
50.08
50.00
49.97
49.96
50.04
50.03
50.01
49.98
49.99
50.05
50.00
50.02
49.99
50.06
49.95
49.99
49.94
49.98
49.92
50.02
50.09
50.09
50.00
50.00
49.95
50.03
50.02
49.92
49.95
49.94
49.96
49.97
50.01
50.00
49.93
50.02
i.e. 100 observations of the
inside diameter of a metal
sleeve
20 samples of 5 specimens
To create a histogram…
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Cumulative
Classes
Midpoint
49.89 ≤ X < 49.91
49.91 ≤ X < 49.93
49.93 ≤ X < 49.95
49.95 ≤ X < 49.97
49.97 ≤ X < 49.99
49.99 ≤ X < 50.01
50.01 ≤ X < 50.03
50.03 ≤ X < 50.05
50.05 ≤ X < 50.07
50.07 ≤ X < 50.09
50.09 ≤ X < 50.11
49.90
49.92
49.94
49.96
49.98
50.00
50.02
50.04
50.06
50.08
50.10
Tally
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||||
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Relative
Cumulative
Relative
Frequency
1
3
6
11
14
23
21
11
4
3
3
Frequency
0.01
0.03
0.06
0.11
0.14
0.23
0.21
0.11
0.04
0.03
0.03
Frequency
Frequency
0.01
0.04
0.1
0.21
0.35
0.58
0.79
0.9
0.94
0.97
1.00
100
1.00
1
4
10
21
35
58
79
90
94
97
100
25
Frequency
20
15
10
5
0
49.90 49.92 49.94 49.96 49.98 50.00 50.02 50.04 50.06 50.08 50.10
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%
Service time
(minute)
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Conclusions…
Look at the average service times:
 problem?
full service = 7
self service = 3
all service
= 4.6
 make second copier a self-service?
- not yet!
-Service time = wrong variable to base decisions on
- most likely – 2nd copier will have no effect on service times
Need to differentiate between
variables under the customer’s control
variables under the company’s control
&
Uncontrollable v. controllable factors!
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More appropriate to use a controllable factor
e.g.
delay
(time waiting in queue)
CASE STUDY 1
data – lead to sample paths
2 parts:
a) Customer characteristics (arrival times, service times)
b) Company characteristics (one-at-a-time, FCFS)
a) = system inputs
b) = system logic
Concentrate on logic – not inputs
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Sample path method – a graphical system model
SAMPLE PATH –
record of the time-dependent behaviour
of a system
SAMPLE PATH DECOMPOSITION –
SIMULATION –
generates new sample paths without building
a new system
SAMPLE PATH ANALYSIS –
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represents a sample path
as inputs and logic
extracts system performance
measures from sample paths
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2-copier system alternatives:
Full-service + self-service
2 x Full-service
Self-service system: - define the system logic…
2 queues – 1 for full-service, 1 for self-service
no swapping of queues
Customers always join appropriate queue
Service times – same as before (uncontrollable (noise) factor
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Performance measure – waiting time
i.e.
3 system events:
• customer arrival
• customer finish (full-service)
• customer finish (self-service)
Now – let’s run a simulation
based on the previous data…..
Delay
CASE STUDY 1
CURRENT TIME: 0
Full service
Self-Service
NEXT SYSTEM EVENT
TIME
customer arrival
0 + 12 = 12
full-service finish
self-service finish
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First four events:
system starts
customer 2 arrives
CURRENT TIME: 0
Full service
customer 1 arrives
customer 2 finishes
etc…
CURRENT TIME: 12
Self-Service
Full service
Self-Service
1
NEXT SYSTEM EVENT
TIME
NEXT SYSTEM EVENT
TIME
customer arrival
0 + 12 = 12
customer arrival
12 + 2 = 14
full-service finish
full-service finish
12 + 7 = 19
self-service finish
self-service finish
CURRENT TIME: 14
CURRENT TIME: 16
Full service
Self-Service
Full service
1
2
1
Self-Service
NEXT SYSTEM EVENT
TIME
NEXT SYSTEM EVENT
TIME
customer arrival
14 + 3 = 17
customer arrival
17
full-service finish
19
full-service finish
19
self-service finish
14 + 2 = 16
self-service finish
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Dear students……. answers
Continue this analysis on the sheets on the table
– then answer the following:
The first non-zero delay is at t = ?
It occurs for customer number ?
It is a delay of ? minutes
At t = ? two events occur simultaneously
The simulation ends at t = ?
45
7
1
76
129
Which customers experience delays?
How big are these delays?
7&13
1&7
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Dear students…….answers!
Continue this analysis on the sheets on the table
– then answer the following:
The first non-zero delay is at t = 45
It occurs for customer number 7
It is a delay of 1 minutes
At t = 76 two events occur simultaneously
The simulation ends at t = 129
Which customers experience delays? 7 & 13
How big are these delays 1 & 7
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How does this compare to the alternative?
i.e. 2 copiers, both full-service
System logic:
1 queue – service delivered as FCFS
3 system events:
• customer arrival
• customer finish (left-hand machine)
• customer finish (right-hand machine)
CURRENT TIME: 0
i.e.
Full service
NEXT SYSTEM EVENT
Full-Service
TIME
customer arrival
full-service finish (left)
full-service finish (right)
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RESULTS
(do it for yourself…?)
Simulation end time
Delayed customers
Delays
Total delay
1
129
7,13
1,7
8
2
124
7,13,20
1,5,1
7
Not much in it – but second model appears quicker
Unexpected??
Need to consider the “goodness” of the data set (the sample )
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DATA QUALITY
Must ensure the data is reliable
i.e. statistically realistic and representative
Sample size?
Sample time? etc..
The statistics of sampling – sampling theory
Also –
relationship between sample and population
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KEY ELEMENTS
MODELS
PERFORMANCE CRITERIA
OPTIMIZATION
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PREDICTIVE MODELS
• Predictive models provide an engineer with a quantitative
understanding of a given problem
• They provide a means to trial multiple designs without full
implementation (e.g. simulation)
• An engineer should know that a design will work before it is
built
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PERFORMANCE OPTIMIZATION
• Determining good performance criteria is a key aspect of
engineering a good system
• Quantitative performance criteria provide a means to
measure performance of a system against the design goals
• Once performance criteria can be specified then the system
design can be modified to optimize these criteria
• An engineer should design systems that optimize sensible
and practical performance criteria
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