The Assignment Problem (2)

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The Assignment ProblemThe Assignment Problem The model essentially tries to address the issue of getting the best

from every allocation.

The use of this method allows a manger to address the following

questions:

• How to assign jobs to workers according to their experience and to ensurethe best efficiency of output

• How to deal with situations wherein some allocations are not possible due to

technical reasons or in case of absence of some workers.• hich field staff is to be allotted to which territory in order to maximise the

returns.• How to get the best schedule for inbound and outbound transport

combination where the least crew layover time is an important decisioncriteria.



The Assignment ProblemThe Assignment Problem

To fit the assignment problem definition, the followingassumptions must be satisfied:

• The number of assignees and the number of tasks are the same(denoted by n).• Each assignee is to be assigned to exactly one task.• Each task is to be assigned to exactly one assignee.

• There is a cost c ij associated with assignee i performing task j.• The ob ecti!e is to determine how all n assignments should bemade to minimi"e the total cost.



Assignment Problem – Flow DiagramAssignment Problem – Flow Diagram

#

$

%

#

$

%

n

assignees tasks

a & assigneet & tasks a#

a%

a$

t#

t%

t$

t'

c##

c#%

n cnn

an



Assignment Problem – Cost MatrixAssignment Problem – Cost Matrix

! " # n! c !! c!" # c !n

$ssignees " c"! c"" # c "n

# #

n cn! cn" # c nn

Tasks

et the following represent the standard assignment problemcost matrix, c:



%onsider the matrix shown which shows the cost ofprocessing each of the jobs $& '& and % on the machines(& )& and * respectively. The cost are in rupees and theobjective here is to make allocations in such a mannerthat the total cost of the allocation is optimal& that is& theminimum cost option.

Job MachineXX Y Y ZZ

AA 20 0 !

"" 2# $ %

CC &0 ' 2



Assignment Problem –()ngarian Metho*Assignment Problem –()ngarian Metho*

Summary of Hungarian Method:

Step 1 – Ensure that the given problem is a minimisation problem andthe matrix is a square matrix.

Step – !ind the minimum element in each ro". #onstruct a ne"

matrix by subtracting from each cost the minimum cost in its ro" $ro"operation%. $&f any ro" has 'ero as an element( no need to perform ro"operation%.Step ) * !or this ne" matrix( find the minimum cost in each column$column operation%. #onstruct a ne" matrix by subtracting from eachcost the minimum cost in its column. %. $&f any column has 'ero as anelement( no need to perform column operation%

Step +* Examine the ro"s successively until a ro" "ith exactly oneunmar,ed 'ero is found. Ma,e an assignment to this 'ero by putting asquare around it( and cross all other 'eros appearing in the

corresponding column. -roceed in the same "ay for all the ro"s.



• Step * Examine the columns successively until a column "ith exactlyone unmar,ed 'ero is found. Ma,e an assignment to this 'ero by puttinga square around it( and cross all other 'eros appearing in thecorresponding ro". -roceed in the same "ay for all the columns.

• Step /* 0epeat step + and until all the 'eros in ro"s columns are eithersquared or crossed and there is exactly one assignment in each columnand each ro". &n such a case optimal assignment policy is obtained. &fthere is a ro" or column "ithout assignment( i.e. the total number ofsquared 'eros is less than the order of the matrix( then go to step 2.

• Step 2 – 3ra" the minimum number of lines $hori'ontal or vertical% thatare needed to cover all the 'eros in the reduced cost matrix. &f n lines arerequired( an optimal solutions is available among the covered 'eros inthe matrix. &f fe"er than n lines are needed( proceed to step 2.

• Step 4 – !ind the smallest non'ero element $call its value k % in thereduced cost matrix that is uncovered by the lines dra"n in Step 2. 5o"subtract k from each uncovered element of the reduced cost matrix andadd k to each element that is covered by t"o lines. 0eturn to step +.



$ computer centre has got four expert programmers. The centre needsfour application programmes to be developed. The head of the

computer centre& after studying carefully the programmes to bedeveloped& estimates the computer time +in minutes, required by therespective experts to develop the application programmes as follows:

-rogrammes

-rogrammers

6 7 # 3

1 1 8 188 48 98

48 98 118 28

) 118 1+8 1 8 188

+ 98 98 48 98



company is producing a single product and is selling it through fi!eagencies situated in different cities. ll of a sudden, there is a demand forthe product in another fi!e cities not ha!ing any agency of the company.The company is faced with the problem of deciding on how to assign theexisting agencies to despatch the product to needy cities in such a way thatthe tra!elling distance is minimised. The distances (in kms) between thesurplus and deficit cities are gi!en in the following distance*matrix:

3eficit #ities

Surplus#ities

& && &&& &

6 1/8 1)8 12 198 88

7 1) 1 8 1)8 1/8 12

# 1+8 118 1 128 14

3 8 8 48 48 118

E ) 28 48 18



• $n airline operating seven days a week has the schedule shown inthe table below for its flights between -ew elhi and /umbai. Thecrew should have a stopover of minimum 0 hours between flights.

1btain the pairing of flights that minimi2es the layover time of thecrew away from the home. 3or any given pairing& the crew will bebased in the city that results in the smallest layover.

+ew Delhi,M)mbai M)mbai,+ew Delhi

3light eparture $rrival 3light eparture $rrival

!4! 0 $/ 5 $/ "4! 5 $/ 6 $/

!4" 5 $/ 6 $/ "4" 7 $/ !4 $/

!48 ! 9/ 8 9/ "48 ! 9/ 8 9/

!4 5 9/ 6 9/ "4 ; 9/ 7 9/



Maximisation in an 6ssignment -roblemMaximisation in an 6ssignment -roblem• +i!e different machines can do any of the fi!e re uired obs with different

profits resulting from each assignment as shown in the following table. findout maximum profit possible through optimal assignment.

Machine

;ob

6 7 # 3 E1 )8 )2 +8 4 +8

+8 + 2 1 )/

) +8 ) )) )8 )

+ )4 +8 )/ )/

9 / +1 )+ )9



<nbalanced 6ssignment -roblem<nbalanced 6ssignment -roblem• Solve the follo"ing unbalanced assignment problem of minimi'ing total time

for doing all the =obs.

;ob

>perations

1 ) +

1 / ) /4 2 2

) 2 4 / 9 4

+ / ) +

9 ) 4 9 2/ + 2 + / 4



0estrictions on 6ssignments0estrictions on 6ssignments• -n the modification of a plant layout of a factory four new machines #, %, $ and ' are to

be installed in a machine shop. There are fi!e !acant places ,/,0,1 and E a!ailable. /ecause

of limited space, machine % cannot be placed at 0 and cannot be placed at . The cost oflocating of machine i to place in rupees is shown below. +ind optimal assignment schedule:

A " C D -

M! ' $ 0

M2 2 ' , 0 '

M& , % !M% % . 2 ! .

The Assignment Problem (2)

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