Assignment Problems (1)

8/3/2019 Assignment Problems (1)

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Assignment problem

• It a special case of the transportationproblem in which the objective is to assigna number of origins to the equal number of

destinations at a minimum cost ormaximum profit. The assignment is to bemade of one-to-one basis.



• assignment problem which occurs wheneach supply is 1 and each demand is 1. Inthis case, the integrality implies that every

supplier will be assigned one destinationand every destination will have onesupplier. The costs give the charge for

assigning a supplier and destination toeach other.



• Consider m laborers to whom n tasks are assigned. No laborer caneither sit idle or do more than one task. Every pair of person andassigned work has a rating. This rating may be cost, satisfaction,penalty involved or time taken to finish the job. There will be N2 such combinations of persons and jobs assigned. Thus, theoptimization problem is to find such man- job combinations that

optimize the sum of ratings among all. Thus m=n.• The formulation of this problem as a special case of transportation

problem can be represented by treating laborers as sources and thetasks as destinations . The supply available at each source is 1 andthe demand required at each destination is 1.The cost of assigning(transporting) laborer i to task j is cij .

• It is necessary to first balance this problem by adding a dummylaborer or task depending on whether m<n or m>n, respectively. Thecost coefficient cij for this dummy will be zero.



• The problem is to find Xij , where

Xij = 1,if the ith labour is assigned to jthtask

= 0 ,if the ith labour is not assignedto jth task



• MinimizeΣ Xijcij

Subject to

ΣXij = 1 i=1,2,……………,n

ΣXij = 1 j=1,2,…………….,n

Xij =0 or 1



Assignment Problems• Hungarian method

Step 1 In a given matrix subtract the smallest element in

each row from every element of that row and do thesame in the column.Step2 In the reduced matrix obtain from step 1,subtract the smallest element in each column from

every element of that column



• Step 3 Make the assignment for the reduced

matrix obtained from step 1 and step 2

(all the zeros in rows/columns are either marked(□) or (x) and there is exactly one assignment ineach row and each column. In such a caseoptimum assignment policy for the given

problem is obtained.If there is row or column with out an assignment

go to the next step.



• Step 4 Draw the minimum number of vertical andhorizontal lines necessary to cover all the zeros in thereduced matrix obtained from step 3 by adopting thefollowing procedure.

• (i) mark(√) all rows that do not have assignments

• (ii) Mark (√) all columns (not already marked) which havezeros in the marked rows

• (iii) Mark (√) all rows (not already marked) that haveassignments in marked columns

• (iv) Draw straight lines through all unmarked rows andmarked columns



• Step 5 If the number of lines drawn are equal to

the number of rows or columns, then it is an optimumsolution ,otherwise go to step 6

• Step 6 Select the smallest element among all theuncovered elements. Subtract this smallest element from

all the uncovered elements an add it to the elementwhich lies at the intersection of two line. Thus we obtainanother reduced matrix for fresh assignments.

• Step 7 go the step 3 and repeat the procedure until thenumber of assignment become equal to the number ofrows or columns. In such a case, we shall observe thatrow/column has an assignment. Thus, the currentsolution is an optimum solution.



Example

• Consider the problem of assigning five machines tofive locations at which these machines are to beinstalled. Material handling costs in Rs per hr is given

Machines/locati

ons

1 2 3 4 5

M1 8 4 2 6 1

M2 0 9 5 5 4

M3 3 8 9 2 6

M4 4 3 1 0 3

M5 9 5 8 9 5



• Assign the machines to locations such thatthe total material handling cost isminimum.

• Also find the total minimum materialhandling cost.



EXAMPLE

• A manufacturer of dies for the automotive component industryutilises a job shop structure. During a planning horizon, themanufacturer needs to first five jobs to the five machines that are inone section of the factory. The table below has details on theprocessing time of the jobs in each of these machines if assigned.

Use the assignment method to solve the loading problem.

Machine 1 Machine 2 Machine 3 Machine 4 Machine 5

Job 1 15 17 16 22 18

Job 2 16 19 20 19 16

Job 3 19 20 17 19 20

Job 4 17 23 22 18 14

Job 5 20 19 24 16 17



Example

• A department has four subordinates andfour tasks are to be performed. Thesubordinates differ in efficiency and the

tasks differ in their intrinsic difficulties.The estimate of time ( in man-hours) eachman would take to perform each task is

given by:



Subordinate/task

A B C D

1 18 26 17 11

2 13 28 14 26

3 38 19 18 15

4 19 26 24 10



• How should the tasks be allotted to menso as to optimize the total man-hours?



EXAMPLE

The following information is available regarding four different jobs to be

performed and about the clerks capable of performing the jobs:

Clerk II cannot be assigned to job A and clerk III cannot be assigned to job B. Find the optimum assignment schedule. Also find the total timetaken to perform the jobs. Also find whether the given problem hasmore than one optimal assignment schedule.

Clerks Jobs (Time taken in hours)

A B C D

I 4 7 5 6

II - 8 7 4

III 3 - 5 3

IV 6 6 4 2



Unbalanced assignment problem

• There are four machines W, X,Y and Z. Three jobs are to beassigned to the 3 machines out of total 4 machines. The cost ofassignment (in Rs.) is given below. Find out the optimal assignment.

W X Y Z

A 18 24 28 32

B 8 13 17 18

C 10 15 19 22



Unbalanced assignment problem

Job/

machineA B C D E F

1 13 13 16 23 19 9

2 11 19 26 16 17 18

3 12 11 4 9 6 10

4 7 15 9 14 14 13

5 9 13 12 8 14 11



• Here the cost of processing any jobs onany machine is given. The assignment of jobs to machines must be on a one-to-one

basis. Assign the jobs to machines sothat the total cost is minimum. Find theminimum total cost.





• A company has 5 jobs to be done. The

following matrix shows the return in Rs. Ofassigning ith machine to the jth machinesso as to maximize the total expected profit:



9 3 4 2 10

12 10 8 11 9

11 2 9 0 8

8 0 10 3 7

7 5 6 2 9

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Assignment Problems (1)