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The Application of
Simulated Annealing
for Logistics
Amber
Amber Laboratory - ADN.cn
Introduce myself
Name: Hu
Nickname: Amber
School: Fuzhou No.1 Middle School
I had taken part in Nation Olympiad in
Informatics.
Mathematic modeling
In logistics, there are many problems of
combination optimization.
Their functions are so complex that they are very
hard to solve in a expected time.
Traveling Salesman Problem (TSP)
It’s a typical and useful problem in logistics.
The simple description of TSP is that a person
wants to find the shortest plan (order) to travel
all the n cities. The person visits each city once
and only once, and returns to the starting city.
Mathematic modeling
The mathematical formulation of TSP:
Let the distance between two cities, city i
and city j be dist(i,j)
Statement: In this lecture dist(i,j) is the
Euclid distance.
Let the plan (order) to travel all n cities be
P=p1p2…pn-1pn.There doesn’t exist pi=pj.
Let p1=pn+1 for expressing simply.
Mathematic modeling
The total distance of the plan (order) P is
the function - the sum of all distances in
plan.
n
f ( P)   dist ( pi , pi 1 )
i 1
The aim of TSP is to get the minimum of
this function f(P).
Mathematic modeling
The right figure shows
how to travel all the
30 cities.
This solution is not
the best solution for
this map, namely not
the minimum of the
function.
The rule of nature
Consider the rules of nature:
Substance always closes to the lowest
energy state.
For example, consider the
configuration of extranuclear electrons.
If there are a electron
closing to a nude atom,
then that electron must be
automatically arranged to
the lowest energy orbit.
Simulated Annealing
Notice there exists the similarity between
the minimum of the function and the
lowest energy state.
Can we design an algorithm that can find the
minimum automatically as substance
closing to the lowest energy state?
Simulated Annealing
Consider annealing solids
When a physical system is at a high temperature,
the atoms in the system are in a highly
disordered state, and the total energy of the
system is also high. Lowering the temperature of
the system results in the atoms of the system
acquiring a more orderly state, thus reducing the
energy of the system. This process of cooling is
known as annealing .
Simulated Annealing
Simulated Annealing (SA) is developed to
simulate the behaviour of atoms at lowering
temperature.
It begins with an initial solution. In simulated
annealing, we generate a neighbour solution
using an interchange operation. The neighbour
is accepted if it is better. It is also accepted with
a certain probability if it is worse. The probability
of accepting an worse solution decreases as the
temperature. SA is terminated if temperature
closes to 0 or there is no better solution for
certain times.
Simulated Annealing
Simulated Annealing (SA) Algorithm:
1 Initialization: system’s prime temperature T, initial solution S0, iterating
times at each temperature stage L, Termination condition AIM
2 while (true)
2.1 for each k=1..L, do 2.1.1 to 2.1.4
2.1.1 Generate new solution SN from current solution S, get
their distinction D.
2.1.2 if (D<0 or satisfy probability exp( D/T)), then S:=SN.
2.1.3 if (current solution S<best solution SB), then SB:=S.
2.1.4 if (T closes to 0 or the iterating times that there exists no
new better solution is greater than AIM), then consider S is the best
solution and go to 3.
2.2 diminish temperature
2.3 S=SB
3 output the best solution SB
Simulated Annealing
TSPSA
This is the experiment version written by
Amber. This program uses Simulated
Annealing Algorithm to solve TSP.
The left figure of the windows shows the
current solution in Annealing every 5
iterations. We can see that the solution is
getting better and better.
Cooling Schedule
SA is controlled by Cooling Schedule. Cooling
Schedule contains the prime temperature T,
diminished rate k, iterating times at each
temperature stage L, termination condition AIM.
In order to work best, SA must be under a good
combination of Cooling Schedule values which
must be determined by large-scale experiment
Application
SA is not only used in solving TSP. It is the
general algorithm for optimization.
For example, the post offices setting
problem is that to build some post offices
in a city, minimizing the total cost of
service – the distances between each
postman and his client.
Thank you.
Simulated Annealing
For example, to grow a crystal, which is
highly ordered, the system needs to be
heated to a temperature which allows
many atomic rearrangements. Then the
system must be carefully cooled, ensuring
a thermal equilibrium is reached at each
temperature stage, until the system is
`frozen' into a good crystal.
Simulated Annealing
This process of cooling is known as
annealing. If the cooling process is
performed too quickly, extensive
irregularities can be locked into the
crystals structure, with the resulting energy
level far greater than in a perfect crystal.