The document discusses the Travelling Salesman Problem (TSP), which seeks the optimal route visiting a set of cities exactly once before returning to the start. It outlines the history, applications in various fields, and provides a mathematical model along with a solution using the Hungarian method. The conclusion emphasizes the TSP as a significant and challenging combinatorial optimization problem that attracts research from multiple disciplines.

1. TRAVELLING SALESMAN PROBLEM
Presented By :-- Bharat Kumar 😊, M.Sc (Hons.) Mathematics
Department of Mathematics, Panjab University, Chandigarh

2. Table Of Contents 🎯📚
1⃣ Travelling Salesman Problem Statement
2️⃣ Brief History Of Travelling Salesman Problem
3️⃣ Applications Of Travelling Salesman Problem
️4️⃣ Mathematical Model Of Travelling Salesman Problem
️5⃣ Solution Of Travelling Salesman Problem Using Hungarian Method
6⃣ Conclusion

3. The Travelling Salesman Problem asks the following question:
“Given a set of cities and the distance ( or time or cost)
between each pair of cities, what is the best possible
route that visits each city exactly once and return to the
starting city?”
Brief History of Travelling Salesmen Problem:
👉The TSP was defined in the 1800s by the Irish Mathematician W.R.
Hamilton and the British Mathematician Thomas Kirkman. It was
however first formulated as a mathematical problem in 1930 by Karl
Menger.
👉The name Travelling Salesman Problem was introduced by American
Hassler Whiteney.

4. Applications of Travelling Salesman Problem:
👉 Management:- Postal Deliveries, School Bus Routing etc.
👉Computer Wiring:- Connecting together computer components using
minimum wire length .
👉Genome Sequencing:- Arranging DNA fragments in sequence.
👉Job Sequencing:- Sequencing jobs in order to minimise total set-up time
between jobs.
👉Mission Planning:- Determining an optimal path for each army men (or
planner) to accomplish the goals of the mission in the minimum possible time.

5. Mathematical Model of Travelling Salesman Problem:
If cij is the cost (or distance or time) of travelling from city i to city j and xij=1 if the
salesman travels from city i to j and zero otherwise, then the problem is to find
X=(xij) which minimise :
Here, cii = M where M is a large positive number to ensure xii=0 (i=1,2,...,n).
👉 The total number of tours in an n city travelling salesman problem is (n-1)!

6. Solution of Travelling Salesman Problem Using Hungarian Method:
A travelling salesman, named Mr. Bharat Kumar has to visit five cities. He has to
visit each city only once and then return to the starting city. The distance between
city i and j is cij and is given in the below table. What is the shortest tour possible?
🚴♂️🤯🤯
“As every thread of gold is valuable, so is every minute of time.” --- John Mason

7. Solution: Here, the no. of rows = no. of columns = 5 ( Table 1.1)
Step 1: Find out each row minimum element and subtract it from that
row as done in ( Table 1.2 )
Step 2: Find out each column minimum element and subtract it from
that column as done in ( Table 1.3 )
( Table 1.1 ) ( Table 1.2 ) ( Table 1.3 )

8. Step3: Make assignment in the admissible cells (i.e. cells containing 0)
in the reduced table. Rowwise and Columnwise assignment is shown in
the ( Table 1.4 ).
The solution gives the sequence A->E->A and B->C->D->B which is
not a tour and hence it is not a feasible solution for TSP.
( Table 1.4)

9. Step4: The feasible solution for TSP can be obtained by bringing the minimum
non-zero element ( i.e. 1 ) into the solution. 1 occur at 3 places and we can bring
any of 1 into the solution and examine whether the changed solution is feasible
for TSP or not.
Let us make assignment in the cell (A,B)=1 and examine whether the changed
solution is feasible for TSP or not. Rowwise and columnwise assignment is
shown in ( Table 1.5 ).
(Table 1.5)

10. The solution in ( Table 1.5 ) gives the sequence A->B->C->D->E->A
which is a tour since it forms a complete cycle and hence an optimal
feasible solution for TSP.
👉 Thus, the shortest route for the travelling salesman is given in the
below table. ( Table 1.6 )
ROUTE DISTANCE
A -----> B 2
B -----> C 3
C -----> D 4
D -----> E 5
E -----> A 1
TOTAL SHORTEST DISTANCE 👉 15 🙂
( Table 1.6 )

11. 🔎 Conclusion 🎯
The Travelling Salesman Problem is one of the challenging problems in real life
and also most well studied combinatorial optimization problem. Many
researchers from different fields like Operational Research, Algorithms Design
and Artificial Intelligence etc. are attracted by it. In this presentation, this
problem has been solved by using Hungarian Method. This problem is used as
a benchmark for many optimization methods and has wide applications.