Linear Programming - Assignment Problem (1)

5/28/2018 Linear Programming - Assignment Problem (1)

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Linear ProgrammingLinear Programmingas a ann ng ooas a ann ng oo

Slide 1 of 27Slide 1 of 27Dr. Thomas Thompson IIDr. Thomas Thompson II


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Introduction to Assignment ProblemIntroduction to Assignment Problem

LetLetCCbebe anannxnnxnmatrixmatrix representingrepresenting thethe costscosts ofof eacheach ofofnnworkersworkers toto performperform anyany ofofnn

obsobs.. TheThe assi nmentassi nment roblemroblem isis toto assi nassi n obsobs toto workersworkers soso asas toto minimizeminimize thethe totaltotal

costcost.. SinceSince eacheach workerworker cancan performperform onlyonly oneone jobjob andand eacheach jobjob cancan bebe assignedassigned toto onlyonly

oneone workerworker thethe assignmentsassignments constituteconstitute ananindependentindependent setsetofof thethe matrixmatrixCC..

AnAn arbitraryarbitrary assignmentassignment isis shownshown aboveabove inin whichwhich workerworkeraaisis assignedassigned jobjob qq,, workerworkerbb

isis assignedassigned jobjobssandand soso onon.. TheThe totaltotal costcost ofof thisthis assignmentassignment isis2323..CanCan youyou findfind aa lowerlower costcost assignment?assignment?

CanCan youyou findfind thethe minimalminimal costcost assignment?assignment?

Slide 2 of 27Slide 2 of 27

RememberRemember thatthat eacheach assignmentassignment mustmust bebe uniqueunique inin itsits rowrow andand columncolumn..


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Assignment ProblemAssignment Problem

AnAn assignment problemassignment problem seeks to minimize the total costseeks to minimize the total costassignment ofassignment of mm workers toworkers to mmjobs, given that the costjobs, given that the cost

o wor ero wor er per orm ng oper orm ng o ss ccijij

..

It assumes all workers are assigned and each job isIt assumes all workers are assigned and each job is

An assignment problem is a special case of aAn assignment problem is a special case of a

transportation problem in which all supplies and alltransportation problem in which all supplies and alleman s are equa to 1; ence assignment pro emseman s are equa to 1; ence assignment pro ems

may be solved as linear programs.may be solved as linear programs.

with three workers and three jobs is shown on the nextwith three workers and three jobs is shown on the nextslide.slide.



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Network RepresentationNetwork Representation

1111

cc1111

cc1212

cc

22 22

cc2121

cc2222

cc2323

cc cc3232

33 33

cc3333


WORKERSWORKERS JOBSJOBS


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Linear Programming FormulationLinear Programming Formulation

MinMin ccii xxiii ji j

s.t.s.t. xxijij = 1 for each resource (row)= 1 for each resource (row) ii

xxijij = 1 for each job (column)= 1 for each job (column)jj

iixxijij = or or a= or or a anan ..

Note:Note: A modification to the rightA modification to the right--hand side of the firsthand side of the first

constraint set can be made if a worker is permitted to workconstraint set can be made if a worker is permitted to workmore than 1 job.more than 1 job.



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Linear Programming (LP) problems can be solved onLinear Programming (LP) problems can be solved onthe computer using Matlab, and many others.the computer using Matlab, and many others.

There are special classes of LP problems such as theThere are special classes of LP problems such as theassignment problem (AP).assignment problem (AP).

. .

AP can be formulated as an LP and solved by generalAP can be formulated as an LP and solved by general

purpose LP codes.purpose LP codes.

However, there are many computer packages, whichHowever, there are many computer packages, whichcontain separate computer codes for these modelscontain separate computer codes for these models

structure.structure.



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From en.wikipedia.org/wiki/From en.wikipedia.org/wiki/Hungarian_algorithmHungarian_algorithm

The Hungarian method is a combinatorial optimizationThe Hungarian method is a combinatorial optimizationa gorit m w ic so ves t e assignment pro em in po ynomiaa gorit m w ic so ves t e assignment pro em in po ynomiatime. It was developed and published by Harold Kuhn in 1955,time. It was developed and published by Harold Kuhn in 1955,who ave the name "Hun arian method" because thewho ave the name "Hun arian method" because the

algorithm was largely based on the earlier works of twoalgorithm was largely based on the earlier works of twoHungarian mathematicians: DnesHungarian mathematicians: Dnes KnigKnig and Jenand JenEgervryEgervry..

amesames un resun res rev ewe e a gor m n an o serverev ewe e a gor m n an o servethat it is (strongly) polynomial. Since then the algorithm hasthat it is (strongly) polynomial. Since then the algorithm has

been known also as Kuhnbeen known also as Kuhn--MunkresMunkres oror MunkresMunkres assignmentassignmentalgorithm. The time complexity of the original algorithm wasalgorithm. The time complexity of the original algorithm wasO(nO(n44)), however later it was noticed that it can be modified to, however later it was noticed that it can be modified toachieve anachieve an O nO n33 runnin time.runnin time.

In 2006, it was discovered that Carl Gustav Jacobi had solvedIn 2006, it was discovered that Carl Gustav Jacobi had solvedthe assignment problem in the 19th century, and publishedthe assignment problem in the 19th century, and published


post umous y in 1890 in Latin.post umous y in 1890 in Latin.


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Hungarian MethodHungarian Method

TheThe Hungarian methodHungarian method solves minimizationsolves minimizationassignment problems withassignment problems with mm workers andworkers and mmjobs.jobs.

Special considerations can include:Special considerations can include: number of workers does not equal the number ofnumber of workers does not equal the number of

costs as neededcosts as needed

workerworker ii cannot do jobcannot do jobjj assignassign ccii = += +MM

maximization objectivemaximization objective create an opportunity losscreate an opportunity lossmatrix subtracting all profits for each job from thematrix subtracting all profits for each job from the

Hungarian methodHungarian method



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Step 1:Step 1: For each row, subtract the minimum number inFor each row, subtract the minimum number inthat row from all numbers in that row.that row from all numbers in that row.

ep :ep : or eac co umn, su rac e m n mum num eror eac co umn, su rac e m n mum num er

in that column from all numbers in that column.in that column from all numbers in that column. Step 3:Step 3: Draw the minimum number of lines to cover allDraw the minimum number of lines to cover all

zeroes. If this number =zeroes. If this number = mm, STOP, STOP an assignment canan assignment canbe made.be made.

Ste 4:Ste 4: Determine the minimum uncovered numberDetermine the minimum uncovered number(call it(call it dd).).

SubtractSubtract dd from uncovered numbers.from uncovered numbers.

o num ers covere y wo nes.o num ers covere y wo nes. Numbers covered by one line remain the same.Numbers covered by one line remain the same.

Then GO TO STEP 3.Then GO TO STEP 3.



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Finding the Minimum Number of Lines andFinding the Minimum Number of Lines andDetermining the Optimal SolutionDetermining the Optimal Solution

ep :ep : n a row or co umn w on y one un nen a row or co umn w on y one un ne

zero and circle it. (If all rows/columns have two orzero and circle it. (If all rows/columns have two ormore unlined zeroes choose an arbitrary zero.)more unlined zeroes choose an arbitrary zero.)

Step 2:Step 2: If the circle is in a row with one zero, draw aIf the circle is in a row with one zero, draw aline through its column. If the circle is in a columnline through its column. If the circle is in a columnwith one zero, draw a line through its row. Onewith one zero, draw a line through its row. Oneapproach, when all rows and columns have two orapproach, when all rows and columns have two ormore zeroes, is to draw a line through one with themore zeroes, is to draw a line through one with themost zeroes, breaking ties arbitrarily.most zeroes, breaking ties arbitrarily.

Step 3:Step 3: Repeat step 2 until all circles are lined. If thisRepeat step 2 until all circles are lined. If thisminimum number of lines equalsminimum number of lines equals mm, the circles, the circles

rovide the o timal assi nment.rovide the o timal assi nment.



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Example 1: APExample 1: AP

con rac or pays s su con rac ors a xe ee p us m eage orcon rac or pays s su con rac ors a xe ee p us m eage orwork performed. On a given day the contractor is faced with threework performed. On a given day the contractor is faced with threeelectrical jobs associated with various projects. Given below areelectrical jobs associated with various projects. Given below arethe distances between the subcontractors and the ro ects.the distances between the subcontractors and the ro ects.

ProjectProjectAA BB CC

estsideestside 50 36 1650 36 16

SubcontractorsSubcontractors FederatedFederated 28 30 1828 30 18

GoliathGoliath 35 32 2035 32 20

How should the contractors be assigned to minimize total costs?How should the contractors be assigned to minimize total costs?

..dummy project Dum, which will be assigned to one subcontractordummy project Dum, which will be assigned to one subcontractor(i.e. that subcontractor will remain idle)(i.e. that subcontractor will remain idle)



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Network Representation (note the dummy project)Network Representation (note the dummy project)

3636

161600

West.West.

28283030

1818BBFed.Fed.

3535 3232

2020

00

2525 2525

..


1414

00

Dum.Dum.Univ.Univ.


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Initial Tableau SetupInitial Tableau Setup

Since the Hungarian algorithm requires that thereSince the Hungarian algorithm requires that therebe the same number of rows as columns, add a Dummybe the same number of rows as columns, add a Dummycolumn so that the first tableau is (the smallest elementscolumn so that the first tableau is (the smallest elements

AA BB CC DummyDummyWestsideWestside 5050 3636 1616 00

FederatedFederated 2828 3030 1818 00

GoliathGoliath 3535 3232 2020 00UniversalUniversal 2525 2525 1414 00



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Step 1:Step 1: Subtract minimum number in each row from allSubtract minimum number in each row from allnumbers in that row. Since each row has a zero, wenumbers in that row. Since each row has a zero, wes mp y generate t e or g na matr x t e sma ests mp y generate t e or g na matr x t e sma est

elements in each column are marked red). This yields:elements in each column are marked red). This yields:

AA BB CC DummyDummy

Westside 50 36 16 0Westside 50 36 16 0Federated 28 30 18 0Federated 28 30 18 0

Goliath 35 32 20 0Goliath 35 32 20 0

UniversalUniversal 2525 2525 1414 00



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Step 2:Step 2: Subtract the minimum number in each columnSubtract the minimum number in each columnfrom all numbers in the column. For A it is 25, for B itfrom all numbers in the column. For A it is 25, for B its , or t s , or ummy t s . s y e s:s , or t s , or ummy t s . s y e s:

Westside 25 11 2 0Westside 25 11 2 0

Federated 3 5 4 0Federated 3 5 4 0

Goliath 10 7 6Goliath 10 7 6 00

Universal 0 0 0 0Universal 0 0 0 0



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Step 3:Step 3: Draw the minimum number of lines to cover all zeroesDraw the minimum number of lines to cover all zeroes(called minimum cover). Although one can "eyeball" this(called minimum cover). Although one can "eyeball" thisminimum, use the following algorithm. If a "remaining" row hasminimum, use the following algorithm. If a "remaining" row hasonly one zero, draw a line through the column. If a remainingonly one zero, draw a line through the column. If a remaining

column has only one zero in it, draw a line through the row. Sincecolumn has only one zero in it, draw a line through the row. Sincethe number of lines that cover all zeros is 2 < 4 (# of rows), thethe number of lines that cover all zeros is 2 < 4 (# of rows), the

..

AA BB CC DummyDummy


Goliath 10 7 6 0Goliath 10 7 6 0

Step 4:Step 4: The minimum uncovered number is 2 (circled).The minimum uncovered number is 2 (circled).



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Step 5:Step 5: Subtract 2 from uncovered numbers; add 2 to allSubtract 2 from uncovered numbers; add 2 to allnumbers at line intersections; leave all other numbersnumbers at line intersections; leave all other numbersntact. s g ves:ntact. s g ves:

Westside 23 9 0 0Westside 23 9 0 0



Universal 0 0 0Universal 0 0 0 22



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Step 3:Step 3: Draw the minimum number of lines to cover allDraw the minimum number of lines to cover allzeroes. Since 3 (# of lines) < 4 (# of rows), the currentzeroes. Since 3 (# of lines) < 4 (# of rows), the currentso ut on s not opt ma .so ut on s not opt ma .

AA BB CC DummyDummy



Universal 0 0 0 2Universal 0 0 0 2

Step 4:Step 4: The minimum uncovered number is 1 (circled).The minimum uncovered number is 1 (circled).



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Step 5:Step 5: Subtract 1 from uncovered numbers. Add 1 toSubtract 1 from uncovered numbers. Add 1 tonumbers at intersections. Leave other numbers intact.numbers at intersections. Leave other numbers intact.

s g ves:s g ves:

Westside 23 9 0Westside 23 9 0 11



Universal 0 0 0Universal 0 0 0 33



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Find the minimum cover:Find the minimum cover:

AA BB CC DummyDummyWestside 23 9 0Westside 23 9 0 11



n versan versa

''four. Thus, the current solution is optimal (minimumfour. Thus, the current solution is optimal (minimumcost) assignment.cost) assignment.



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The optimal assignment occurs at locations of zerosThe optimal assignment occurs at locations of zerossuch that there is exactly one zero in each row and eachsuch that there is exactly one zero in each row and eachco umn:co umn:

Westside 23 9Westside 23 9 00 11

FederatedFederated 00 2 1 02 1 0

Goliath 7 4 3Goliath 7 4 3 00

Universal 0Universal 0 00 00 33



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The optimal assignment is (go back to the original tableThe optimal assignment is (go back to the original tablefor the distances):for the distances):

SubcontractorSubcontractor ProjectProject DistanceDistance

es s ees s e

Federated A 28Federated A 28

Goliath (unassigned)Goliath (unassigned)

Total Distance = 69 milesTotal Distance = 69 miles



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Example 1: AP via LPExample 1: AP via LP

n our examp e e ormu a on s:n our examp e e ormu a on s:Min z = 50xMin z = 50x1111 + 36x+ 36x1212 + 16x+ 16x1313 + 0x+ 0x1414 + 28x+ 28x2121 + 30x+ 30x2222 + 18x+ 18x2323 + 0x+ 0x2424 ++

35x35x3131 + 32x+ 32x3232 + 20x+ 20x3333 + 0x+ 0x3434 + + 25x+ + 25x4141 + 25x+ 25x4242 + 14x+ 14x4343 + 0x+ 0x4444s.t.s.t.

xx1111 + x+ x1212 + x+ x1313 + x+ x1414 = 1= 1 (row 1)(row 1)

xx + x+ x + x+ x + x+ x = 1= 1 (row 2)(row 2)xx3131 + x+ x3232 + x+ x3333 + x+ x3434 = 1= 1 (row 3)(row 3)

xx4141

+ x+ x4242

+ x+ x4343

+ x+ x4444

= 1= 1 (row 4)(row 4)

1111 2121 3131 4141xx1212 + x+ x2222 + x+ x3232 + x+ x4242 = 1= 1 (column 2)(column 2)

xx1313 + x+ x2323 + x+ x3333 + x+ x4343 = 1= 1 (column 3)(column 3)

xx1414 + x+ x2424 + x+ x3434 + x+ x4444 = 1= 1 (column 4)(column 4)xxijij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3, 4 (nonnegativity)>= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3, 4 (nonnegativity)



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The solver formulation is:The solver formulation is:

Distances FromContractors to Projects at:

Contractor A B C Dummy

West Side 50 36 16 0

e ss gnmen ro em

Federated 28 30 18 0

Gol iath 35 32 20 0

Universal 25 25 14 0

Assignm ent o f Contracto rs

to Projects at:

Contractor A B C Dummy Assigned Available

West Side 0 0 0 0 0 1

Federated 0 0 0 0 0 1

Goliath 0 0 0 0 0 1

Universal 0 0 0 0 0 1

Assigned 0 0 0 0

Capacity 1 1 1 1

Total Distance: 0



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The solver solution is:The solver solution is:

Distances FromContractors to Projects at:

Contractor A B C Dummy

West Side 50 36 16 0

e ss gnmen ro em

Federated 28 30 18 0

Goliath 35 32 20 0

Universal 25 25 14 0

Assignm ent o f Contracto rs

to Projects at:

Contractor A B C Dummy Assigned Available

West Side 0 0 1 0 1 1

Federated 1 0 0 0 1 1

Goliath 0 0 0 1 1 1

Universal 0 1 0 0 1 1

Assigned 1 1 1 1

Capacity 1 1 1 1

Total Distance: 69


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Linear Programming - Assignment Problem (1)

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