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CS621: Artificial Intelligence
Pushpak Bhattacharyya
CSE Dept.,
IIT Bombay
Lecture 7: Traveling Salesman Problem as
search; Simulated Annealing; Comparison
with GA
4-city TSP
dij not necessarily
Equal to dji
2
1
d12
d23
d31
d23
d14
4
3
d34
TSP: State Representation
Position
(α)
City (i)
1
1
1
2
0
3
0
4
0
`i’ varies over
cities
`α’ varies over
positions
2
0
0
1
0
3
0
1
0
0
4
0
0
0
1
Objective Functions
F1 = k1 ∑i ((∑α xiα) – 1)2 + k2 ∑β ((∑j xjβ) – 1)2
1(a)
1(b)
F2 = k3 ∑i ∑j ∑α dij (xiα xi,α+1 + xiα xi,α-1)

Minimize F = F1 + F2
2
Metropolis Algorithm
1)
2)
3)
4)
Initialize: Start with a random state matrix S.
Compute the objective function value at S. Call this
the energy of the state E(S).
The states are transformed by the application of an
operator (for TSP, inversion of adjacent cities)
Compute change the energy ΔE=Enew-Eold
if ΔE <=0, accept the new state Snew
Else, accept Snew with probability e
(‘T’ is the “temperature” and KB, the Boltzmann
constant)

5)
E ( state)
K BT
Metropolis Algorithm (contd)
6) Continue 2-5 until there is no appreciable
change in energy
7) The current state may be one of the local
minima
8) Increase the temperature and continue 2-7
until the global minimum is reached
How to probabilistically accept
a state?



Suppose the probability

e
E ( state)
K BT
=p
Generate a random number from a
uniform distribution [0,1]
Number generated is in the range [0-p]:
Accept the new state, else continue search
from the old state itself
Why?

E ( state)
K BT
The significance of p (= e
) is that if
the states are generated infinite number
of times then a proportion p of them will
be the concerned new state

(0,0)
(1,1)
Uniform distribution
Of [0,1]
(0,0)
(0,p)
(0,1)
Why? (contd)

If numbers in the range [0,1] are
generated randomly, p% of them will be
in the range [0,p]. Hence this process can
simulate the state generation process
(0,0)
(1,1)
Uniform distribution
Of [0,1]
(0,0)
(0,p)
(0,1)
Compare with Roulette Wheel Algorithm for Selection
Chromosome
1
2
3
4
5
Total
Fitness
6.82
1.11
8.48
2.57
3.08
22.0
% of total
31
5
38
12
14
100
Acknowledgement: http://www.edc.ncl.ac.uk/highlight/rhjanuary2007g02.php/
Roulette Wheel Selection
Let i = 1, where i denotes chromosome index;
Calculate P(xi) using proportional selection;
sum = P(xi);
choose r ~
U(0,1);
while sum < r do
i = i + 1; i.e. next chromosome
sum = sum + P(xi);
end
return xi as one of the selected parent;
repeat until all parents are selected
Significance of “temperature”

We have a pseudo temperature T
As T increases so does e
T is a parameter in the algorithm
When stuck in the local minima, we
increase the temperature
The probability of going to a higher
energy state increases
Shaken out of local minima

Similar to annealing of metal






E ( state)
K BT
Annealing of Metal


The metal should have a stable crystal
structure so that it is not brittle
For this it is repeatedly heated and then
cooled slowly