OPERATIONAL RESEARCH

TOPIC: BUSINESS APPLICATIONS OF ASSIGNMENT MODEL AND TRAVELLING

SALESMAN MODEL

SUBMITTED TO: DR.SUBHOJIT BHATTACHARYA

SUBMITTED BY :

AKHILESH KR MISHRA

PGDM MARKETING

XAVIER INSTITUTE OF SOCIAL SERVICE[XISS];

RANCHI

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.

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.

Step 1

Identify the minimum element in each row and subtract it from every

element of that row.

Step 2

Identify the minimum element in each column and subtract it from

every element of that column.

Make the assignment for the reduced matrix obtain from steps 1 and 2 inthe following way:

Draw the minimum number of vertical and horizontal lines necessary

to cover all the zeros in the reduced matrix obtained from last step

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.

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

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

A travelling salesman of the above firm has to visit five cities. He wishes to start from a particular city, visit one city once and then return to his starting point. The travelling cost (in '000 Rupees) of each city from a particular city is given below:

Emertxe IT Firm Co Ltd

The cities are encoded as A=Vellore,B=Chennai,C=Tirupati ,D=Pondicherry ,E=Bangalore

Now the salesman thinks what should be the sequence of his visit so that the cost is minimum?

Salesman solves the given travelling salesman problem as an assignment problem by Hungarian method of assignment, an optimal solution is shown. However, this solution is not the solution to the travelling salesman problem as it gives the sequence A — E — A. This violates the condition that salesmen can visit each city only once.

The 'next best' solution to the problem which also satisfies this extra condition of

unbroken sequence to all cities, can he obtained by bringing the next (non-zero)

minimum element, i.e. 1 into the solution.

Case 1 Make the unit assignment in the cell (A, B) instead of zero

assignment in the cell (A, E) and delete row A and column so as to

eliminate the possibility of any other assignment in row A and column IN

make the assignments in the usual manner. The resulting assignments

are -:

The solution gives the sequence: A-> B,B->C,C->D, D->E, E->A.

Corresponding to this feasible solution is Rs.15,000.

Case 2: If we make the assignment in the cell (D, C) instead of (D. E), then no feasible solution is obtained in terms of zeros or which may give cost less than Rs.15,000. Hence the best solution is: A->B,B-> C,C->D,D->E,E-> A, and the total cost associated with this solution is Rs.15,000

The Rollick’s Ice Cream Company has a distribution depot in

Greater Kailash Part I for distributing ice-cream in South

Delhi. There are four vendors located in different parts of

South Delhi (call them A. B, C and D) who have to be

supplied ice-cream every day. The following matrix displays

the distances (in kilometers) between the depot and the

four vendors:

How company van follow so that the total distance travelled

is minimized?

Company solves the given travelling salesman problem as an assignment problem by using the Hungarian method of assignment

All four cases of possible solution to travelling salesman problem tried with element 0.5 as well as zero element do not provide a desired solution. Thus, we look for the 'next best' solution by bringing the next (non-zero) element 1 along, with 0.5 and zero elements into the solution. Make the assignment in cell (Vendor C, Vendor B) and delete row 4 and column 3.In the remaining matrix assignments are made using zeros and 0.5 in the cells.

The set of assignments given is a feasible solution to the travelling salesman problem. The route for the salesman is: Vendor C-> Vendor B-> Depot ->Vendor D->Vendor A ->Vendor C. The total distance (in km) to be covered in this sequence is 15 km

In this presentation, we presented the dynamic Hungarian algorithm for the

assignment problem

with changing costs and travelling salesman method for path optimization.

The goal of the algorithm is to efficiently repair an optimal

assignment when changes in the edge costs occur, as can happen in many

real-world

scenarios.

The dynamic Hungarian algorithm

is useful in any domain that requires the repeated solution of the

assignment problem

when costs may change dynamically. In future work, we will apply this

algorithm to

various transportation-related problems

CONCLUSION