04 dr.s.p.anbuudayasankar - amirtha
TRANSCRIPT
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OPTIMISATION
Why do you need Meta Heuristics
Dr.S.P.Anbuudayasankar, M.E.., M.B.A., PhD
Associate Professor
Department of Mechanical Engineering
Amrita School of Engineering
Amrita Vishwa Vidyapeetham
Coimbatore - 641112
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Optimization vs Decision Problems
Optimization Problem
A computational problem in which the object is to find the
best of all possible solutions
(i.e. find a solution in the feasible region which
has the minimum or maximum value of the
objective function.)
Decision Problem
A problem with a “yes” or “no” answer.2
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Objective Function & Search Space
Objective Function
Max (Min) some function of decision variables
Subject to (s.t.)
equality (=) constraints
inequality ( ) constraints
Search Space
Range or values of decisions variables that will be searched
during optimization3
,,,
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A solution to an optimization problem
specifies the values of the decision variables, and therefore also the
value of the objective function
A feasible solution
satisfies all constraints
An optimal solution
is feasible and provides the best objective function value
A near-optimal solution
is feasible and provides a superior objective function value, but not
necessarily the best.
Types of Solutions
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Optimization problems can be
continuous (an infinite number of feasible solutions)
combinatorial (a finite number of feasible solutions).
Continuous vs Combinatorial
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Constraints can be
Hard (must be satisfied) or
Soft (is desirable to satisfy)
Example:
In our course schedule
a hard constraint is that no classes overlap
a soft constraint is that no class be before 10 AM
Constraints
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Objectives
Single or
Multiple
Size
number of decision variables,
range/count of possible values of decision variables
search space size
Degree of constraints
number of constraints
difficulty of satisfying constraints
proportion of feasible solutions to search space
Aspects of an Optimization Problem
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Aspects of an Optimization Problem (Contd.)
Deterministic (all variables are deterministic) or
Stochastic (the objective function and/or some decision
variables and/or some constraints are random
variables)
Decomposition decomposite a problem into series problems,
and then solve them independently
Relaxation solve problem beyond relaxation, and then can
solve it back easier
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Models
“The essence of Operations Research lies in the
construction and use of models.”
A model is a simplified representation of something real.
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Optimization need Model
Guess…..What model is this ?
min i j cijxij
s.t. j xij < si, i,
i xij > dj, j,
xij > 0, i, j.
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Transportation Model !
min i j cijxij
s.t. j xij < si, i,
i xij > dj, j,
xij > 0, i, j.
Costs PLANT Bearings
Supplier Chennai Madurai Cochin Coimbatore Available
Kumar 21 50 40 35 275,000
Ganesh 35 30 22 42 400,000
Sudhir 55 20 25 70 300,000
Gopal 43 25 37 58 500,000
Capacity 200,000 600,000 225,000 350,000
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Modeling become
P
NP
NP- hard
&
NP – complete
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What is P
P = Set of problems than can be solved in
polynomial time
What is Polynomial Time ?
Reasonable Time
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What is NP
NP = Set of problems for which a solution
can be verified in polynomial time
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What is NP Complete
Minesweeper
TSP
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What is NP-Hard
Allocation Problem
Patient 1
Patient 2
Hospital 1
Hospital 2
5
4
6
3
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? Commonly Believed
Relationship
P
NP NP-
Hard
NP Complete
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When modeling become
NP-complete
&
NP-hard
We need….
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Heuristics
and
Metaheuristics
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Heuristics
Heuristics
are rules
to search to find optimal or near-optimal solutions
Problem Specific Algorithms
Examples
FIFO
LIFO
earliest due date first
largest processing time first
shortest distance first, etc.
Vogel‟s Approximation Method to solve transportation problem22
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Heuristics
Examples of problems that are tackled with heuristic algorithms :
Cutting/Packing
Graph problems
Scheduling/Timetabling
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Packing Problem
How it works (Packing stage)
2 3 41
Gap
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How it works (Packing stage)
2 3 4
1
Gap
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How it works (Packing stage)
2 3
41 X
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How it works (Packing stage)
2 3 4
1 X
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How it works (Packing stage)
2 3 4
1 X
Gap
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How it works (Packing stage)
2
3 4
1 X
Gap
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How it works (Packing stage)
2 3
4
1 X
Gap
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How it works (Packing stage)
2 3
4
1 X
X
Gap
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How it works (Packing stage)
2 3
4
1 X
X3 X
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How it works (postprocessing)
Due to the nature of the algorithm, „towers‟ may form
„Towers‟ are boxes with large height dimensions and lower width
The tallest tower is removed from the packing, rotated
so that it is effectively „knocked down‟ and placed back into the
packing at the lowest point available
This is repeated until the solution is not improved any further
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11 4 356 7
10
129 8 3 2
1
Gap
How it works
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11 4 356 7
10
129 8 3 2
1
How it works
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11 4 356 7
10
129 8 3 2
1
How it works
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11 4 356 7
10
129 8 3 2
1
How it works
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11 4 356 7
10
129 8 3 21
old height
new heightImprovement
How it works
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Have you been paying attention?
Q) How many different orders can these boxes be placed
in?
Answer
4 x 3 x 2 x 1 = 24 different orders
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Cracking a combination lock
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How do we pack all of these?
?
(50 pieces)
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That‟s not so easy…
50 pieces means 50 x 49 x…..2 x 1 different orderings
=
304140932017133780436126081660647688443776415689605
12000000000000
That’s quite a lot really. If a computer could evaluate
1000 orderings per second it’d still take approximately:
964424568801159882154128873860501295166718720476931
506 years! (and that’s without being allowed to rotate
the pieces)
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Travelling Salesman Problem
One of the most researched Problems
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Traveling Salesman Problem (TSP)
Depot
2
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Combinatorial Explosion
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Combinatorial Explosion
Cities Routes
1 1
2 1
3 3
4 12
5 60
6 360
7 2520
8 20160
9 181440
10 1814400
11 19958400
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Combinatorial Explosion
A 10 city TSP has 181,000 possible solutions
A 20 city TSP has 10,000,000,000,000,000
possible solutions
A 50 City TSP has
100,000,000,000,000,000,000,000,000,000,00
0,000,000,000,000,000,000,000,000,000,000
possible solutions
There are 1,000,000,000,000,000,000,000
litres of water on the planet
Mchalewicz, Z, Evolutionary Algorithms for Constrained Optimization Problems, CEC 2000 (Tutorial)
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Traveling Salesman Problem
If there are n cities,
How many possible solutions exist
The number of possible routes is (n!)/2
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91.9 CPU years on a
Intel Xeon 2.8 GHz processor
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Vehicle Routing Problems (VRPs)
Proven NP – hard problem
Depot
1
9
8
4
3
2
6
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Heuristic for VRP
Depot
OVERLOAD
OK !
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Depot
OVERLOAD
Heuristic for VRP
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Depot
OK !
Heuristic for VRP
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Meta-Heuristics– what are they?
Wikipedia says
“A metaheuristic is a heuristic method for solving a
very general class of computational problems by
combining user given black-box procedures — usually
heuristics themselves — in a hopefully efficient way.”
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Meta-heuristics
Genetic algorithms
Simulated Annealing
Ant Colony Optimisation
Particle Swarm Optimisation
Tabu Search
Hill Climbing / Greedy search
… many more
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Understanding the Concept
X
f (X)
Local
Minima
Global
Minima
Local
Maxima
Global
Maxima
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Rastrigin‟s Function
Global minimum at ( 0, 0
)
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