OPTIMISING THE ANSWER BY USING TSP

1. By,
SHAILESH JADHAV
RAHUL CHAUGULE
v VIRAJ PATIL

2. HISTORY
 In the 1800s, Travelling salesman problems were looked by Sir William
Rowan Hamilton and Thomas Kirkman.
 Hassler Whitney at Princeton University introduced the name
travelling salesman problem
 In the 1950s and 1960s, the problem became increasingly popular in
scientific circles in Europe and the USA.
 Richard M. Karp showed in 1972 that the Hamiltonian cycle problem
was NP-complete.

3. INTRODUCTION
 The travelling salesman problem consists of a salesman and a set of cities.
The salesman has to visit each one of the cities starting from a certain one
(e.g. the hometown) and returning to the same city. The challenge of the
problem is that the travelling salesman wants to minimize the total length of
the trip.
 Travelling salesman problem is one of the most extensively studied
optimization problem that is used to find the shortest possible route.
 TSP has many applications including following:
1. The delivery of meals to office persons.
2. Manufacture of microchips.
3. The routing of courier trucks.
4. The routing of any salesman.

4. SOLVING METHODS
 Try every possibility: (n-1)!Possibilities- Takes longer time.
 Optimising Methods: obtain guaranteed optimal solution.
 Heuristic Method: obtain ‘good’ solution ‘quickly’ by intuitive methods
.No guarantee of optimality.

5. Objective function
The mathematical formulation of the problem can be as in Eq.
Where,
d(i,j)- Distance travelled from city ‘i’ to city ‘j’.
x(i.j) –cost of travel from city ‘i’ to city ’j’.
min
x
x(i, j)d(i, j)
j 1
n
i 1
n

s.t.
x(i, j) 1, i  1,2,...,n
j 1
n

x(i, j) 1, j  1,2,...,n
i 1
n

x(i, j)  S  1, S  {1,2,...,n}
i , jS
n

x(i, j) {0,1}

6. TYPES OF PROBLEM
1.SYMMETRIC TSP
2.ASYMMETRIC TSP
1 2 3 4
1 -- 30 44 56
2 20 -- 38 68
3 25 60 -- 25
4 75 35 45 --

7. CONCLUSION
 Travelling salesman problem is one of the most extensively studied
optimization problem that is used to find the shortest possible route.
 By knowing or solving the Travelling salesman problem we get the
optimal travelling distance/path cost.
 We should know which appropriate method to be used while solving
TSP.