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G.N.Khalsa College
 BMM/BMS Department
  A Project Report on
  Operations Research

  {Assignment Problem}




       Group # 07
Under Prof Guide- Nimalan

                            Page | 1
“
The essence of mathematics is not to make simple things complicated, but t
                        ”

                    ~Albert Einstein




 ACKNOWLEDGEMENT

                                                       Page | 2
It was an immensely pleasurable experience in
   working on this project on the topic of,
         “Assignment Problem ”
However, our efforts alone could not have been
   sufficient for completion of this project.
 Invaluable advice & suggestion from large
 number of people have gone into this project.
The foremost among them in our guide and our
 Prof in charge BMS/BMM Nimalan,
 Guru Nanak Khalsa College (GNKC),
   whose constant corrective guidance &
 motivation enabled a focused effort towards
 completion of the project. We are also very
  grateful to our parents for being extremely
                  supportive.


        Group Members

                                                Page | 3
Page | 4
Bird’s Eye View
Sr. #                   Topic           Pg. #

1.      Assignment Problem – Theory     5-6

2.      Introduction                    7

3.      Hungarian Method                8

4.      Minimization Case study         9-10
        Example # 01
5       Example # 02                    11-13

6.      Alternate Solution Case study   14-16
        Example # 03




           ASSIGNMENT PROBLEMS

                                         Page | 5
What is an Assignment Problem?
• The assignment problem can be stated as a problem where
  different jobs are to be assigned to different machines on the
  basis of the cost of doing these jobs. The objective is to minimize
  the total cost of doing all the jobs on different machines.

• The peculiarity of the assignment problem is only one job can be
  assigned to one Machine i.e., it should be a one-to-one
  assignment

• The cost data is given as a matrix where rows correspond to jobs
  and columns to machines and there are as many rows as the
  number of columns i.e. the number of jobs and number of
  Machines should be equal

• This can be compared to demand equals supply condition in a
  balanced transportation problem. In the optimal solution there
                                                               Page | 6
should be only one assignment in each row and columns of the
   given assignment table. one can observe various situations
   where assignment problem can exist e.g., assignment of workers
   to jobs like assigning clerks to different counters in a bank or
   sales man to different areas for sales, different contracts to
   bidders.

• Assignment becomes a problem because each job requires
  different skills and the capacity or efficiency of each person with
  respect to these jobs can be different. This gives rise to cost
  differences. If each person is able to do all jobs equally efficiently
  then all costs will be the same and each job can be assigned to
  any person.

• When assignment is a problem it becomes a typical optimization
  problem it can therefore be compared to a transportation
  problem. The cost elements are given and are a square matrix
  and requirement at each destination is one and availability at
  each origin is also one.




                 INTRODUCTION
The assignment problem is one of the fundamental combinatorial
optimization problems in the branch of optimization or operations
research in mathematics. It consists of finding a maximum weight
matching in a weighted bipartite graph.
In its most general form, the problem is as follows:


                                                                 Page | 7

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Maths project

  • 1. G.N.Khalsa College BMM/BMS Department A Project Report on Operations Research {Assignment Problem} Group # 07 Under Prof Guide- Nimalan Page | 1
  • 2. “ The essence of mathematics is not to make simple things complicated, but t ” ~Albert Einstein ACKNOWLEDGEMENT Page | 2
  • 3. It was an immensely pleasurable experience in working on this project on the topic of, “Assignment Problem ” However, our efforts alone could not have been sufficient for completion of this project. Invaluable advice & suggestion from large number of people have gone into this project. The foremost among them in our guide and our Prof in charge BMS/BMM Nimalan, Guru Nanak Khalsa College (GNKC), whose constant corrective guidance & motivation enabled a focused effort towards completion of the project. We are also very grateful to our parents for being extremely supportive. Group Members Page | 3
  • 5. Bird’s Eye View Sr. # Topic Pg. # 1. Assignment Problem – Theory 5-6 2. Introduction 7 3. Hungarian Method 8 4. Minimization Case study 9-10 Example # 01 5 Example # 02 11-13 6. Alternate Solution Case study 14-16 Example # 03 ASSIGNMENT PROBLEMS Page | 5
  • 6. What is an Assignment Problem? • The assignment problem can be stated as a problem where different jobs are to be assigned to different machines on the basis of the cost of doing these jobs. The objective is to minimize the total cost of doing all the jobs on different machines. • The peculiarity of the assignment problem is only one job can be assigned to one Machine i.e., it should be a one-to-one assignment • The cost data is given as a matrix where rows correspond to jobs and columns to machines and there are as many rows as the number of columns i.e. the number of jobs and number of Machines should be equal • This can be compared to demand equals supply condition in a balanced transportation problem. In the optimal solution there Page | 6
  • 7. should be only one assignment in each row and columns of the given assignment table. one can observe various situations where assignment problem can exist e.g., assignment of workers to jobs like assigning clerks to different counters in a bank or sales man to different areas for sales, different contracts to bidders. • Assignment becomes a problem because each job requires different skills and the capacity or efficiency of each person with respect to these jobs can be different. This gives rise to cost differences. If each person is able to do all jobs equally efficiently then all costs will be the same and each job can be assigned to any person. • When assignment is a problem it becomes a typical optimization problem it can therefore be compared to a transportation problem. The cost elements are given and are a square matrix and requirement at each destination is one and availability at each origin is also one. INTRODUCTION The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. It consists of finding a maximum weight matching in a weighted bipartite graph. In its most general form, the problem is as follows: Page | 7