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Assignment Problem

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TABLE OF CONTENT

•Meaning•Methods• Steps•Practical Problems

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MEANING04/10/2023

Allocating various resources to various activities

On one to one basis

The time and cost are minimised

Sale and profit are maximised

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Assignment Problem

• The Hungarian Mathematician D. Konig developed simpler & more efficient method.

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Methods of Assignment Problem

Minimisation ProblemMaximisation ProblemUnbalanced ProblemProhibitive Problem

Travelling Salesman ProblemAir-Crew Problem

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Steps to follow

Solution

Re-Assignment

Making an Assignment

Column Reduction

Row Reduction

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Minimization Problem04/10/2023

Problem: Five jobs are to be assigned to 5 machines to minimize the total time required to process the jobs on machines. The time in hours for processing each job on each machine are given in the matrix below. By using assignment algorithm make the assignment for minimizing the time of processing.

Machines (time in hours)Jobs V W X Y Z

A 2 4 3 5 4B 7 4 6 8 4C 2 9 8 10 4D 8 6 12 7 4E 2 8 5 8 8

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Maximization problem• Problem: A company has five jobs V, W, X, Y and Z and

five machines A, B, C, D and E. The given matrix shows the return in Rs. of assigning a job to a machine. Assign the jobs to machines so as to maximize the total returns. Machines

Returns in Rs.

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Jobs A B C D E

V 5 11 10 12 4

W 2 4 6 3 5

X 3 12 5 14 6

Y 6 14 4 11 7

Z 7 9 8 12 5

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Unbalanced problem• Problem: A manager has 4 jobs on hand to be

assigned to 3 of his clerical staff. Clerical staff differs in efficiency. The efficiency is a measure of time taken by them to do various jobs. The manager wants to assign the duty to his staff, so that the total time taken by the staff should be minimum. The matrix given below shows the time taken by each person to do a particular job. Help the manager in assigning the jobs to the personnel.

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Jobs Men (time taken to do job in hours)X Y Z

A 10 27 16B 14 28 7C 36 21 16D 19 31 21

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Prohibitive problem• Consider a problem of assigning four clerks to four

tasks. The time (hours) required to complete the tasks is given below:

Clerk 2 cannot be assigned to task A and Clerk 3 cannot be assigned to task B. Find all the optimal assignment solution

10

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Travelling Salesman Problem• Problem: A salesman stationed at city A has to decide

his tour plan to visit cities B, C, D, E and back to city A I the order of his choice so that total distance traveled is minimum. No sub touring is permitted. He cannot travel from city A to city A itself. The distance between cities in Kilometers is given below:

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Cities A B C D E

A ∞ 16 18 13 20

B 21 ∞ 16 27 14

C 12 14 ∞ 15 21

D 11 18 19 ∞ 21

E 16 14 17 12 ∞