The document discusses the travelling salesman problem (TSP) and assignment problems in operations research. It provides an introduction to assignment problems, describing how they involve allocating resources like people, jobs, or teachers to minimize costs. It then gives an example of an assignment problem involving assigning four men to four jobs. The document also provides an overview of the TSP, explaining that it involves finding the shortest route to visit all cities on a list once. Finally, it discusses some applications of the TSP in areas like planning, logistics, and manufacturing.

1. Operation Research(2171901)
Travelling Salesman Problem
Branch : Mechanical 7-D-1
Prepared by: Dangar Shivji S.(150123119006)
Submitted to: Prof.:-

2. Introduction
• It involves assignment of people to projects, jobs to
machines, workers to jobs and teachers to classes
etc., while minimizing the total assignment costs.
• One of the important characteristics of assignment
problem is that only one job (or worker) is assigned
to one machine (or project).
• An assignment problem is a special type of linear
programming problem where the objective is to
minimize the cost or time of completing a number of
jobs by a number of persons.

3. Examples
• Aadhunik spices company has four men available for work
on four separate jobs. Only one man can work on any one
job. The cost of assigning each man to each job is given in
the following table. The objective is to assign men to jobs
such that the total cost of assignment is minimum.

4. • Step 1
• Identify the minimum element in each row and
subtract it from every element of that row.

5. • Step 2
• Identify the minimum element in each column and
subtract it from every element of that column.

6. • Make the assignment for the reduced matrix obtain from steps 1
and 2 in the following way:

7. • Draw the minimum number of vertical and
horizontal lines necessary to cover all the zeros in
the reduced matrix obtained from last step

8. Since the number of assignments is equal to the number of rows (&
columns), this is the optimal solution.
The total cost of assignment = A1 + B4 + C2 + D3
Substitute the values from original table: 20 + 17 + 24 + 17 = 78.

9. The Travelling Salesman Problem (TSP) is an NP-hard problem
in combinatorial optimization studied in operations research and theoretical
computer science. Given a list of cities and their pairwise distances, the task is
to find a shortest possible tour that visits each city exactly once.
The problem was first formulated as a mathematical problem in 1930 and is one
of the most intensively studied problems in optimization. It is used as a
benchmark for many optimization methods. Even though the problem is
computationally difficult, a large number of heuristics and exact methods are
known, so that some instances with tens of thousands of cities can be solved.
THE TRAVELLING SALESMAN PROBLEM

10. The TSP has several applications even in its purest formulation, such
as planning, logistics, and the manufacture of microchips. Slightly
modified, it appears as a sub-problem in many areas, such as DNA
sequencing. In these applications, the conceptcity represents, for example,
customers, soldering paints, or DNA fragments, and the
concept distance represents travelling times or cost, or a similarity
measure between DNA fragments. In many applications, additional
constraints such as limited resources or time windows make the problem
considerably harder.
APPLICATIONS OF TSP

11. THANK YOU