Design Methods Course Summary

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Design MethodsCourse Summary

• TMR4115
• Thursday, November 27 2003
• Prof. Stein Ove Erikstad

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Agenda

• Briefly summarize what we have been through this
  fall
• Major subjects
• Syllabus, readings
• Final exam
• types of problems
• evaulation criteria

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Models and Methods in Design
Reality
4
Design Task Environment
Design Task Environment
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What characterises marine systems design?Design
Task Environment

• Complex mapping between form and function
• Multi-dimensional, partly non-monetary
  performance evaluation
• High cost of error
• Shallow knowledge structure
• Strong domain tradition
• Strict time and resource constraints on the
  design process
• Predominantly one-of-a-kind and
  engineered-to-order solutions

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Inference processes in design
Design analysis (design spiral)
Deriving design descriptions
Generating design alternatives
Acquiring design knowledge
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LP standard form
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LP formulation
max Z 2x1 3x2 x1 x2 2500 1.5x1
3x2 4500 x1 500
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• Decision variables
• x1 weight cement (t)
• x2 weight brine (t)
• Objective function coefficients
• c1 2 freight rate cement (1000 EUR)
• c2 3 freight rate brine (1000 EUR)
• Deadweight restriction
• b1 2500 max cargo deadweight (t)
• Volume restriction
• a21 1.5 specific volume cement (incl.
  tanks)(m3/t)
• a22 3 specific volume brine (incl.
  tanks)(m3/t)
• b2 3500 max available volume (m3)
• Minimum cargo requirements
• b3 500 min required cement (t)

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Graphical solutionOptimal solution
Z 2x1 3x2
x2
Feasible region
x1
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SimplexOutlining the algorithm
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SimplexCore principles

• Only Corner-Point Feasible solutions (CPF
  solutions) are considered
• Successive iterations from CPF to CPF
• Initiates at the origin
• Consider only adjacent solutions for next
  iteration
• Choose edge with the highest rate of improvement
• If all edges gives negative rate of improvement
  -gt optimality

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2. Select the leaving basic variable using the
i.e. the constraint we will hit first when
increasing the variable selected in Step 1
Z
C1
C2
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Tie-breaking Solutions
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Post-optimality analysis

• Reoptimization
• Shadow prices
• Sensitivity analysis
• Parametric programming
• (Hillier Lieberman Ch. 4.7)

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Shadow prices

• The bs in the model can be considered scarce
  resources
• but not necessarily unchangeable
• Shadow prices
• The rate at which Z changes for marginal changes
  in the bs
• Provided by simplex

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Example Our supply ship
Final table from Simplex
Z
C1
C2
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Example Our supply ship II
Z
C1
C2
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Using LINDO for Sensitivity Analysis
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You should know ...

• ... how to break ties and discover unbounded
  solutions
• ... how to handle variants of the LP standard
  form
• ... how to do simple post-solution analysis

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Selecting design alternatives
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Related to the design process
Problem statement
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Classification of selection problems
1
3
1-dimensional
1-dimensional
No uncertainty
Uncertainty
4
2
Multi- dimensional
Multi- dimensional
No uncertainty
Uncertainty
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Core concepts and notation

• Decision problem
• We must select a conceptual solution for a high
  speed passenger vessel
• Decision alternatives
• A - SES
• B - Catamaran
• C - FoilCat
• Attributes
• X1 - Max. service speed
• X2 - Motion behaviour
• X3 - Total annual cost
• Attribute values
• xA - (42, good, 8) (knots,
  , mEUR)
• xB - (34, medium, 5)
• xC - (48, excellent, 10)

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Objective and objective hierarchy

• Preferable features
• 1. Complete cover all relevant aspects
• 2. Operational measurable and relevant criteria
• 3. Decomposable splitting/grouping possible
• 4. Non-redundant avoid counting some features
  twice
• 5. As small as possible

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Objective hierarchy - example
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The form of the value function
vi(xi)
vi(xi)
xi
xi
vi(xi)
vi(xi)
xi
xi
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One-dimensional value function v i(x i)

• 1. Determine scale
• 2. Normalise
• 3. Determine form of function

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Example
1
.

Determine scale
3. Determine form of function
min
Lower limit


x

27

kn
v
max
Upper limit

x

33

kn
1.0
v
2.
Normalize value function
v
(
27
)

0
.
0
v
v
(
33
)

1
.
0
0.0
v
33
27
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The Assumption of an additiv value function
Examples of dependencies between the attributes
where
X1 max speed X2 motion behaviour X3 - total
annual cost
i.e.
we are willing to accept high annual cost for a
vessel with a high service speed IF the motion
characteristics are good. If, on the contrary,
the motion characteristics are bad, we are not
able to exploit the speed potential, and thus not
willing to pay much for a high max service speed.
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Practical approaches to selection problems

• 1) Determine an objective hierarchy with
  corresponding attributes
• 2) Determine weights for each attribute, such
  that S?i 1
• 3) Determine upper and lower limits for each
  attribute, and give these the values 1.0 and 0.0,
  respectively. Choose the form of the value
  function default is linear.
• 4) For each attribute Evaluate the different
  alternatives, use the value function to determine
  the value
• 5) Calculate for each
  alternative. Choose the alternative with the
  highest value.

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Non-Linear Programming

• TMR4115
• Friday, Sept 19 2003
• Prof. Stein Ove Erikstad

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Agenda

• Intro non-linear programming
• non-linear objectives and constraints
• concave and convex functions
• conditions for optimality
• Classes of non-linear optimization models
• One-variable unconstrained optimization
• a simple search algorithm
• Excel example
• Multi-variable unconstrained optimization
• gradient search
• example by hand

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Graphical solutionThe Linear Case
Z 2x1 3x2
x2
Feasible region
x1
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Graphical solutionNon-Linear Objective
Z non-linear
x2
Feasible region
x1
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Graphical solutionNon-Linear Objective
Z non-linear,interior optimum
Feasible region
x1
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Graphical solutionNon-linear constraints
x2
Feasible region
x1
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Classes of non-linear problems

• Unconstrained
• Linearly constrained
• Quadratic
• linearly constrained
• quadratic objective function
• Convex programming
• Separable programming
• f(x), g(x) separable
• linear approximation -gt use simplex
• Nonconvex programming
• geometric programming
• fractional programming

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Core issues

• Optimality conditions
• unconstrained problems
• concave and convex functions
• KKT-conditions for constrained optima
• Quadratic programming
• characteristics
• modelling towards a Simplex structure

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Necessary and sufficient conditions for optimality
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Heuristic Search Methods
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Optimisation in Design
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Ill-structured design problems
Perhaps something is to be learned by turning
the question around. We have generally asked how
problems can be provided with sufficient
structure that problem-solvers () can go to work
on them. We may ask instead how problem-solvers
of familiar kinds can go to work on problems that
are, in important respects, ill-structured. Simon
, 1973
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Iterative processSatisficing rather than
optimizing
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Goal-directed vsGenerative Design Methods
Understand the difference!!!
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Concept Exploration Models
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Concept Exploration Models

• Automation of the design spiral
• Simulation, parametric variation
• Different search and evaluation algorithms
• Model analysis, simplification

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Generative metoder
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CEM Basic Steps

• GENERATE one, or a set of, design description(s)
• ANALYSE the description(s) to derive the relevant
  design performance(s)
• EVALUATE the performances with respect to the
  design goals
• DECIDE whether the current best solution is
  acceptable, or whether it is necessary to
  generate additional design solutions

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RD Challenges 2Decision Support integrated in
CAD/PDM
?
?

• Bridge theory and practice

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Generative methodsShipX Concept Exploration Model
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Improving Process Efficiency

• Methods that aim to reduce the number of
  dimensions of the search space
• Methods that aim to reduce the size (extent) of
  the search space by imposing bounds and
  constraints on it.
• Methods that aim to select a subset of design
  descriptions to be interpreted
• Methods that aim to reduce the computational
  burden by reducing the complexity of the design
  interpretation

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RD Challenges 1Derive models from existing CAD
models
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Generative methodsDerive simplified, solvable
models
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Reducing search space dimensions

• Factor contribution
• Selecting design points
• Calculating performance (analysis)
• Use ANOVA to calculate factor effects
• How to
• effects of many factors simultaneously
• avoid dimensionality explotion
• Design of Experiments/ Fractional Factorial
  Designs
• Orthogonal Arrays
• Select specific points that adhere to certain
  properties
• balancing property

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Genetic Algorithms

• Search based on mechanisms from natural selection
  and genetics
• chromosomes
• genes
• alleles
• locus
• genotype
• phenotype
• generation
• population
• survival of the fittest
• The driving force in GA is the combination and
  exchange of chromosome material during breeding

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Difference from classical optimization

• Population of points
• not a single one
• Use fitness (payoff)
• not derivatives
• Probabilistic transition rules
• not deterministic

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Design Representation in Genetic Algorithms

• A design description is represented as a string
• This string may be
• a direct binary representation of a design
  variable
• a binary representation of a discrete point in an
  interval
• a combination of different types of varibles,
  e.g. numerical, boolean, lists, etc.

100100100111101010111001011000111
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Three core processes

• Reproduction
• Crossover
• Mutation

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Reproduction

• Individuals are selected for mating
• Probability of selection increases with fitness
• survival of the fittest

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Crossover

• Simple crossover by splitting strings and
  exchanging parts
• Crossover point selected randomly

1 0 1 1
0 0 0 1
1 0 1 1
1 0 1 1 0 0 1 0
0 0 1 0
0 1 1 1 0 0 0 1
0 0 1 0
0 1 1 1
0 1 1 1
0 0 0 1
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Mutation

• Arbitrary change at single point
• Typically low frequency in both natural and
  artificial systems
• But useful insurance towards premature loss of
  important notions

Mutation Flipping bit position
0 1 1 1 0 1 0 1
0 1 1 1 0 0 0 1
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A Simple GA Algorithm
We want to determine the value of the design
variable x for which the performance f(x) is
maximized

• Create initial population
• Determine range
• Create random individs
• Create associated bitstring representation
• Select for reproduction
• Determine fitness for each
• Assign propability for reproduction based on
  fitness
• Use spinning wheel to select individs for mating

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A Simple GA Algorithm 2

• Use crossover to create children
• For each pair, randomly select crossover point
• Exchange bitstrings to create two new children
• Use mutation to insert new genetic material
• For each individ, use mutation probability to
  determine whether to mutate or not
• If mutation, randomly select position to flip bit

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Some strategies for improved performance

• Normalized fitness function
• often necessary for survival of the fittest
  mechanism to work
• increases relative fitness difference
• Elitism
• preserve best individs from generation to
  generation
• avoids loosing good solutions
• may lead to premature convergence
• Parent competition
• children must also comete against parents
• quicker convergence poorer search coverage
• Tuning algorithm settings
• population size
• mutation probability

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Pro Cons

• Computationally intensive
• for complex fitness evaluation
• Gives little insight into design model behaviour
• blackbox approach
• Difficult to know about quality of solution

• Easy to adapt to existing systems
• wrap around existing analysis tools
• only link is fitness calculation (performance)
• No requirements to model behaviour
• Not trapped in local optima

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Transportation problemsDuality

• Duality
• Formulating the dual problem from the primal
• Understanding the dual What does it say?
• Complementary slackness theorem
• Transportation problem
• Formulating the problem
• Finding an intial feasible solution (NW, Vogel)
• Usind duality theory to derive an algorithm

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Network Optimization
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Identifying a network
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Vocabulary

• Nodes the platforms
• Arcs the pipelines
• Directed or undirected arc
• Directed and undirected networks
• Path - sequence of arcs connecting two nodes
• Directed and undirected path
• Cycle (directed or undirected)
• Connected/unconnected network
• Arc capacity
• Supply node, demand node, transshipment node

Directed arc
Cycle
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Problem types

• Shortest path
• Minimum spanning tree
• Maximum flow
• Minimum cost flow

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Shortest Path - Algorithm
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Minimum spanning tree

• Assumptions
• Undirected, connected network
• postive length distance between each node
• Objective
• Provide a path between each pair of nodes
• Minimize the total length
• Algorithm
• Greedy, i.e. we may make a decision without
  worrying about the effect on subsequent decisions
• 1. Select the shortest link in the network
• 2. Select the shortest link to an unconnected
  node from any currently linked node

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Maximum Flow Algorithm
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Network
Source
2
6
5
2
1
Transhipment nodes
7
4
4
3
Arc capacity
1
7
5
Sink
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• Minimum cost flow
• assumptions
• setting up/modelling the problem
• formulation as a standard LP problem (notnetwork
  LP)

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OptimizationCore subjects

• Non-linear programming
• Local and global minimum/convergence rates
• Conditions for optimality
• Convexity
• Newtons method for unconstrained minimum,
  modified/quasi Newtons Method
• Linear programming
• Multiple objective optimization
• Integer programmering
• Transport problems
• Assignment problems

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Summary
Practice

• Commercially available tools CAD, PDM
• Advanced modelling and analysis

• Advanced decision support
• optimization
• AI, KBS
• Textbook problems

Theory
80
Final exam
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Scope

• Basis
• See Syllabus
• Assignments are relevant problem types

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Typical problem outline

• Modelling
• Methods application
• Understanding
• Post-solution analysis
• What-if
• Problem variations
• Application

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Summary
Practice
The end

• Understand the industrial setting for applying
  theory
• Understand the practical challenges facing users
• Contribution from industry to academia
  Knowledge, experience, case

• Educate engineers for ndustry
• Transfer competencies to industry using graduated
  engineers

Theory