412-note8c

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Computational Methods for Management and Economics

Carla Gomes

Module 8cThe Assignment Problem

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The assignment problem

• Special type of linear programming problem

where assignees are assigned to tasks

! Assignees and tas"s people# machines# plants#$ehicles# locations# etc

• Gi$en a set of tasks to be performed and a set of

assignees who are a$ailable to perform thesetas"s# the problem is to determine which assignee

should be assigned to each tas"

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The assignment problem

To fit the model for an assignment problem# thefollowing assumptions need to be satisfied%

& The number of assignees and the number of tas"s arethe same

' Each assignee is to be assigned to e(actly one tas"

) Each tas" is to be performed by e(actly one assignee

* There is a cost associated with each combination ofan assignee performing a tas"

+ The ob,ecti$e is to determine how all the assignmentsshould be made to minimi-e the total cost

.ote% These assumptions appear /uite restricti$e 0owe$er by using eg# dummy assignees

and1or tas"s we can apply it to se$eral problems

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Sellmore Company Assignment Problem

• The mar"eting manager of Sellmore Company will be holding thecompany2s annual sales conference soon

• 0e is hiring four temporary employees%

! Ann3 4an3 5oan3 Sean

• Each will handle one of the following four tas"s% ! 6ord processing of written presentations

! Computer graphics for both oral and written presentations

! Preparation of conference pac"ets# including copying andorgani-ing materials

! 0andling of ad$ance and on7site registration for the conference

Question: Which person should be assigned to which task?

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ata for the Sellmore Problem

Required Time per Task (Hours)

Temporarymployee

Word!rocessing "raphics !ackets Registrations

HourlyWage

Ann )+ *& '9 *: ;&*

4an *9 *+ )' +& &'

5oan )< += )= *) &)

Sean )' +& '+ *= &+

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The .etwor" >epresentation

A2

A1

T4A4

T3A3

T2

T1

Assignees Tasks

49 0

54 0

46 8

69 0

(Ann)

(Ian)

(Joan)

(Sean)

(Word processing

(Graphics)

(Packets)

(Registrations)

574

378560

564

384612

507 728

559

480

765

375

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5ob Shop ?Assigning Machines to

@ocations• The 5ob Shop Company has purchased three new

machines of different types

• There are fi$e a$ailable locations where the machinecould be installed

• Some of these locations are more desirable for particularmachines because of their pro(imity to wor" centers thatwill ha$e a hea$y wor" flow to these machines

Question: How should the machines be assigned tolocations?

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Materials70andling Cost ata

#ost per Hour

$ocation: % & '

*achine

& ;&) ;&= ;&' ;&* ;&+

' &+ B &) ': &=

) * 9 &: = 9

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The Assignment Problem Model

• ecision $ariables%

& if assignee i performs tas" ,

! (i, : otherwise

i#, &#' D# n

Each (i, is a binary $ariables ! it ta"es only two $alues% :

or &

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The Assignment Problem Model

jand iall for ij

xor ij

x

ni for n

j ij x

n j for n

i ij x

t s

n

i

n

j ij

xij

c Z Min

&:

##'#&&

&

##'#&&

&

& &

==

==∑

=

==∑=

∑=∑=

=

Each tas" is performed

by e(actly one assignee

Each assignee

performs

e(actly one tas"

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The Assignment Problem Model !

solution procedures

• 6e can use se$eral solution procedures%

! Standard simple( ?I

! Transportation simple( ?faster?I

! Speciali-ed algorithms that further e(ploit the

special structure of the Assignment Problem

?eg the 0ungarian Algorithm and thereforeare considerably faster than Transportation

simple( ?I ecause all the >0S are & and the matri( A only has : and & elements# the solutions

for the assignment problem are highly degenerate# which causes wasted iterations

where nothing changes e(cept for the labeling of which allocations of -ero correspondto degenerate $ariables rather than non7basic $ariables

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Algebraic Formulation@et xi, & if swimmer i swims stro"e j3 : otherwise

t i, best time of swimmer i in stro"e j

Minimi-e Time J i J j t i, xi,

sub,ect to

each stro"e swum% J i xi, & for each stro"e j

each swimmer swims &% J j xi, K & for each swimmer i

and

xi, L : for all i and j

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idding for Classes• 4n the MA program at a prestigious uni$ersity in the Pacific

.orthwest# students bid for electi$es in the second year of their program

• Each of the &: students has &:: points to bid ?total and must

ta"e two electi$es• There are four electi$es a$ailable%

! uantitati$e Methods

! Finance

! Nperations Management

! Accounting

• Each class is limited to + studentsQuestion: How should students be assigned to the classes?

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Points id for Electi$es

lecti+es

,tudentQuantitati+e

*ethods -inance.perations

*anagement /ccounting

George =: &: &: ':

Fred ': ': *: ':

Ann *+ *+ + +

Eric +: ': + '+

Susan ): ): ): &:

@i- +: +: : :

Ed 9: ': &: :

a$id '+ '+ )+ &+

Tony )+ &+ )+ &+

5ennifer =: &: &: ':

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etter Products ?Assigning Plants to Products

The etter Products Company has decided to initiate the production of four new products# usingthree plants that currently ha$e e(cess capacity

0nit #ost

!roduct: % & '

#apacity

/+ailable

!lant

& ;*& ;'9 ;'8 ;'* 9+

' *: '< B ') 9+

) )9 ): '9 '& *+

>e/uired production ': ): ): *:

Question: How to split up the production o1 the products among plants?

Management wants to consider two options% option&7 permit product splitting ! the

same product can be produced in more than one plant3 option' ! prohibit productsplitting3

tt P d t ?A i i Pl t t P d t

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etter Products ?Assigning Plants to Products

Nption &7 Transportation Formulation

Nption & ! Permit product splitting Transportation Formulation

0nit #ost

2estination(!roduct): % & ' (2) ,upply

,ource(!lant)

& *& '9 '8 '* : 9+

' *: '< M ') : 9+

) )9 ): '9 '& : *+emand ': ): ): *: 9+

Nptimal solution% (&' ):3 (&) ):3 (&+ &+3 ('+=:3()&':3()*'+ OI;)'=: ?cost a day

Plant & ! produces all of products ' and )3 Plant ' produces )9+ of product *3 Plant )%

Produces ='+ of product * and all of product &3

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etter Products ?Assigning Plants to Products Nption '

Nption ' ! Prohibit product splitting Assignment Formulation?I

0nit #ost

2estination

(!roduct): % & ' (2)

,ource(!lant)

&a 8': 8&: 8*: <=: :

&b 8': 8&: 8*: <=: :

'a 8:: 89: M <': :

'b 8:: 89: M <': :

) 9*: <:: 8&: 8*: Memand ': ): ): *: 9+

?II

?I we can also use the transportation simple( algorithm to sol$e the assignement model

?II ;*& I ': ;8':3 the other entries are obtained in a similar way3Nptimal solution%

Plant & ! produces products ' and )3 Plant ' produces product &3 Plant )%Produces product *3 OI;)'<: ?cost a day ! slightly more# but there are hidden costs of splitting

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Summary

• @inear programming model ! general framewor" that

includes a wide $ariety of specific types of problems

• Some specific types ha$e special structures which allow

for streamlined algorithms much more efficient than the

simple( method ! eg# the transportation problem and the

assignment problem

• oth the transportation problem and the assignment

problem ha$e a number of common applications it isimportant to recogni-e them when they arise and to use the

best available algorithm

412-note8c

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