Maths project
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This document provides an overview of an operations research project on the assignment problem. It includes an introduction to assignment problems, describing them as problems where different jobs must be assigned to different machines in a way that minimizes total costs. It then outlines the Hungarian method for solving assignment problems and provides two case study examples applying the method to minimize costs in assignment matrices. It also describes an alternate solution case study example. The document is presented as a project report with sections on the theory of assignment problems, the Hungarian method, and three examples applying the method to different assignment matrices.
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
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- 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