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Operations Research

MBA-024

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ASSIGNMENT MODEL

UNITII

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Suppose there are 5 machines and 5 jobs to be

performed. 1 machine can do only 1 job at a time.

How the machines should be assigned to the

jobs? Given the cost of performing a job on a

machine.

An Assignment Problem (AP) is always square.

The assignment is done on a one-to-one

matching basis.

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Only one allocation is possible in a given row or

column. The AP is inherently degenerate, as the

assignment is done on a one-to-one matching

basis. (In the above 5 machine 5 job problem,

there would be 25 cells, but allocation can be

made only in 5 cells, whereas 9 cells should be

allocated.)

Total number of assignments possible = n!

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Mathematical Formulation

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Hungarian Method

MachineX Y Z

Job

A 25 31 35

B 15 20 24C 22 19 17

Machine opportunity and job opportunity cost

to be determined. The minimum element of a particular row or

column to be subtracted from all elements of

that row or colu

mn.

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Machine X Y ZJob

A 25 31 35

B 15 20 24C 22 19 17

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Machine X Y ZJob

A 25 31 35

B 15 20 24C 22 19 17

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Machine X Y Z

Job

A 25-25 31-25 35-25

B 15-15 20-15 24-15

C 22-17 19-17 17-17

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Machine X Y ZJob

A 0 6 10

B 0 5 9C 5 2 0

Job Opportunity Cost

1st Reduced Cost Matrix

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Machine X Y ZJob

A 0 6 10

B 0 5 9C 5 2 0

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Machine X Y ZJob

A 0 4 10

B 0 3 9C 5 0 0

Machine Opportunity Cost

2nd Reduced Cost (Total Opportunity Cost) Matrix

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0 4 10

0 3 95 0 0

Optimal assignment is that

assignment where totalopportunity cost is zero.

We draw horizontal and

vertical lines so as to crossall the zero elements using

the minimum number oflines.

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0 4 100 3 9

5 0 0

Optimal assignment is that

assignment where total

opportunity cost is zero. We draw horizontal and

vertical lines so as to cross

all the zero elements usingthe minimum number oflines.

If the minimum number of lines required to

do so is equal to the number of rows or

columns, then optimum allocation is possible.

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0 4 100 3 9

5 0 0

Optimal assignment is that

assignment where total

opportunity cost is zero. We draw horizontal and

vertical lines so as to cross

all the zero elements usingthe minimum number oflines.

If the minimum number of lines required to

do so is equal to the number of rows or

columns, then optimum allocation is possible.

In this case it is not possible.

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0 4 100 3 9

5 0 0

We identify the minimum

element not covered by

lines.

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0 4 100 3 9

5 0 0

We identify the minimum

element not covered by

lines. In this case it is 3.

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0 4 10

0 3 9

5 0 0 We subtract it from

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Assignment Model: Applications

Assign salespeople to sales territories.

Assign vehicles to routes.

Assign accountants to client accounts. Assign contracts to bidders through systematic

evaluation of bids from competing suppliers.

Assign naval vessels to patrol sectors. Schedule teachers to classes.

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Assignment Model: Applications

Matching men to machines according to

pieces produced per hour by each individual

on each machine.

Matching teams to projects by the expected

cost of each team to accomplish each project.