Assignment Model Mba 2012

8/3/2019 Assignment Model Mba 2012

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The Assignment Problem


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MODULE IIMODULE II

ySensitivity analysis ;allocation problems

Assignment anddistribution problems


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The Assignment ProblemyIn many business situations, management needsto assign - personnel to jobs, - jobs to machines, -machines to job locations, or - salespersons toterritories.

yConsider the situation of assigning n jobs to nmachines.

yWhen a job i (= 1, 2,...., n) is assigned to machine j

(= 1, 2, ....., n) the cost incurred is Cij.yThe objective is to assign the jobs to machines so

that the total cost is minimised.


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yThis situation is a special case of theTransportation Model and it is known as

the assignment problem.yHere, jobs represent sources and

machines represent destinations.

yThe supply available at each source is 1unit And demand at each destination is 1unit.


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"One machine can do the work offifty ordinary men. No machine can

do the work of one extraordinaryman."

-- Elbert Hubbard


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The assignment model can be expressed

mathematically as follows:

Xij= 0, if the job j is not assigned to machine i

1, if the job j is assigned to machine i


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yAssignment problem can be either formulatedas a linear programming model, or it can beformulated as a transportation model.

yHowever, An algorithm known as HungarianMethodhas proven to be a quick and efficient

way to solve such problems.


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Hungarian Method Example

The following table gives the costs incurred in assigning

jobs to certain machines in a factory. Determine the

assignment that minimizes the cost.


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Step 1: Select the smallest value in each row.

Subtract this value from each value in that row

Step 2: Select the smallest value in each column.

Subtract this value from each value in that

column.


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Step 3 : (i) Examine the rows successively starting from thefirst, until a row with exactly one zero element is found. Puta box around the zero as an assigned element and cross outall other zeroes in that column. Proceed in this manner

until all the rows have been examined.I

f there are morethan one zero in any row, then do not consider that row butproceed to the next.

(ii) Examine the columns successively starting from the

first, until a column with exactly one zero element is found.Put a box around the zero as an assigned element and crossout all other zeroes in that row. Proceed in this manneruntil all the columns have been examined. If there are morethan one zero in any column, then do not consider that rowbut proceed to the next.


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Step 4 : If each row and column contain exactly oneassigned zero then the assignment is optimal

.

y Therefore, we assign job 1 to machine 1; job 2 tomachine 3, and job 3 to machine 2.

y Total cost is 5+12+13 = 30.

y It is not always possible to obtain a feasibleassignment as in here.


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PROBLEM 1y Assign Jobs 1, 2, 3, 4 to inspectors A, B, C, D so that the cost is

minimised using the following data.

JOBS

INSPECTORS

1 2 3 4

5 7 9 8

2 3 4 5

7 8 9 1

5 3 2 4


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PROBLEM 2

y A department has 5 employees and 5 jobs to be performed. The time( in hours) that each employee takes is given below. Determine the

job allocation that will minimise the man hours.

E

M

P

L

O

Y

E

E

S

Jobs

1 2 3 4 5

A 8 4 2 6 1

B 0 9 5 5 4

C 3 8 9 2 6

D 4 3 1 0 3

E 9 5 8 9 5


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HOMEWORK 1y Find the optimal assignment schedule given the following data.

JOBS

EMPLOYEES

1 2 3 4

10 5 13 15

3 9 8 3

10 7 3 2

5 11 9 7


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Hungarian Method Example 2


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y A feasible assignment is not possible at this moment.

y In such a case, The procedure is to draw a minimum

number oflines through some of the rows andcolumns, Such that all zero values are crossed out.


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Hungarian Method Example 2y (i) Put a tick mark to those rows where no

assignments have been made.

y (ii) Put a tick mark to those columns which havezeros in the marked rows.

y (iii) Put a tick mark to those rows (not already

marked) which have assignments (boxed zeroes)in marked columns.

y (iv) Repeat (ii) and (iii) until no more rows andcolumns can be checked.

y Draw lines through unmarked rows and markedcolumns. If the number of these lines is equal tothe order of the matrix then the solution isoptimal. Else go to the next step.


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The next step is to select the smallest uncrossed outelement. This element is subtracted from every uncrossed out

elementand added to every element at the intersectionof two lines.


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y We can now easily assign to the zero values.Solution is to assign (1 to 1), (2 to 3), (3 to 2) and(4 to 4).

y If drawing lines do not provide an easysolution, then we should perform the task ofdrawing lines one more time.

y

Actually, we should continue drawing linesuntil a feasible assignment is possible.


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Hungarian Algorithm

y

Step 1: Check whether the number of rows = thenumber of columns in the cost matrix. If so theassignment problem is said to be balanced and we canproceed to Step 2. If it is not balanced and the number

of rows is less than the number of columns adddummy rows with zero cost to balance the problem. Ifit is not balanced and the number of columns is lessthan the number of rows add dummy columns with

zero cost to balance the problem.y Step 2: Select the smallest value in each row. Subtract

this value from each value in that row.

y Step 3: Select the smallest value in each column.

Subtract this value from each value in that column.


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Step 4 : (i) Examine the rows successively starting from thefirst, until a row with exactly one zero element is found. Puta box around the zero as an assigned element and cross outall other zeroes in that column. Proceed in this manner

until all the rows have been examined.I

f there are morethan one zero in any row, then do not consider that row butproceed to the next.

(ii) Examine the columns successively starting from the

first, until a column with exactly one zero element is found.Put a box around the zero as an assigned element and crossout all other zeroes in that row. Proceed in this manneruntil all the columns have been examined. If there are morethan one zero in any column, then do not consider that

column but proceed to the next.


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Hungarian Algorithm

y

Step 5 : If each row and column contain exactly one assignedzero then the assignment is optimal. If either a row or acolumn does not contain exactly one assigned zero then theassignment is not optimal, go to Step 6.

y Step 6: Cover all the zeros by drawing minimal number oflines as follows :

y (i) Put a tick mark to those rows where no assignments havebeen made.

y (ii) Put a tick mark to those columns which have zeros in the

marked rows.y (iii) Put a tick mark to those rows (not already marked) which

have assignments (boxed zeroes) in marked columns.

y (iv) Repeat (ii) and (iii) until no more rows and columns can

be checked.


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Hungarian Algorithmy Step 7 : Draw lines through unmarked rows and

marked columns. If the number of these lines is equalto the order of the matrix then the solution isoptimal. Else go to the next step.

y Step 8: Determine the smallest cost element notcovered by the straight lines. Subtract this elementfrom all the uncrossed elements, add it to theelements lying in the intersection of the straight linesand do not change the remaining elements on thestraight lines.

y Go to step 4 and repeat the procedure until anoptimal solution is reached.


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PROBLEM 1

y A department has 5 employees and 5 jobs to be performed. The time( in hours) that each employee takes is given below. Determine the

job allocation that will minimise the man hours.

E

M

P

L

O

Y

E

E

S

Jobs

1 2 3 4 5

A 10 5 13 15 16

B 3 9 18 13 6

C 10 7 2 2 2

D 7 11 9 7 12

E 7 9 10 4 12


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PROBLEM 2y The processing time ( in hours) for jobs when allocated

for different machines are indicated. Determine the joballocation that will minimise the total processing time.

M

A

C

H

I

N

E

S

JOBS

1 2 3 4 5

A 9 22 58 11 19

B 43 78 72 50 63

C 41 28 91 37 45

D 74 42 27 49 39

E 36 11 57 22 25


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TheTraveling Salesman Problemy In the traveling salesman problem, there are m

locations (or nodes)y

And unit costs (Cij) are associated withtraveling between locations i and j.y The goal is to find the cycle that minimizes the

total (traveling) distance required to visit alllocations (nodes) without visiting any

location twice.y The Traveling salesman begins its journey from

his/her home city And visits other cities (in noparticular order) before returning home.

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Assignment Model Mba 2012