matching v 1

7/27/2019 Matching v 1

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Assignment Problem:Hungarian Algorithm and Linear Programming

collected from the Internet and edited by

Longin Jan Latecki


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Introduction to Assignment Problem

Let Cbe an nxn matrix representing the costs of each ofn workers to perform any ofnjobs. The assignment problem is to assign jobs to workers so as to minimize the total

cost. Since each worker can perform only one job and each job can be assigned to only

one worker the assignments constitute an independent setof the matrix C.

An arbitrary assignment is shown above in which workera is assigned job q, workerbis assigned jobs and so on. The total cost of this assignment is 23.

Can you find a lower cost assignment?

Can you find the minimal cost assignment?

Remember that each assignment must be unique in its row and column.


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Assignment Problem

An assignment problem seeks to minimize the total cost

assignment of m workers to m jobs, given that the costof worker i performing jobj is cij.

It assumes all workers are assigned and each job isperformed.

An assignment problem is a special case of atransportation problem in which all supplies and alldemands are equal to 1; hence assignment problemsmay be solved as linear programs.

The network representation of an assignment problemwith three workers and three jobs is shown on the nextslide.


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Assignment Problem

Network Representation

2

3

1

2

3

1c11

c12

c13

c21c

22

c23

c31c

32

c33

WORKERS JOBS


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Assignment Problem

Linear Programming Formulation

Min SScijxiji j

s.t. Sxij = 1 for each resource (row) ij

Sxij = 1 for each job (column)ji

xij = 0 or 1 for all i andj.

Note: A modification to the right-hand side of the firstconstraint set can be made if a worker is permitted to workmore than 1 job.


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Linear Programming (LP) problems can be solved on

the computer using Matlab, and many others.

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

Efficient solutions methods exist to solve AP.

AP can be formulated as an LP and solved by generalpurpose LP codes.

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

which take advantage of the problem networkstructure.


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A brute-force algorithm for solving the assignment problem involves generating all

independent sets of the matrix C, computing the total costs of each assignment and a

search of all assignment to find a minimal-sum independent set. The complexity of this

method is driven by the number of independent assignments possible in an nxn matrix.There are n choices for the first assignment, n-1 choices for the second assignment and

so on, giving n! possible assignment sets. Therefore, this approach has, at least, an

exponential runtime complexity.

As each assignment is chosen that row and column are eliminated from

consideration. The question is raised as to whether there is a better algorithm. In fact

there exists a polynomial runtime complexity algorithm.


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From en.wikipedia.org/wiki/Hungarian_algorithm

The Hungarian method is a combinatorial optimizationalgorithm which solves the assignment problem in polynomial

time. It was developed and published by Harold Kuhn in 1955,who gave the name "Hungarian method" because thealgorithm was largely based on the earlier works of twoHungarian mathematicians: Dnes Knigand Jen Egervry.

James Munkres reviewed the algorithm in 1957 and observedthat it is (strongly) polynomial. Since then the algorithm hasbeen known also as Kuhn-Munkres or Munkres assignmentalgorithm. The time complexity of the original algorithm wasO(n4), however later it was noticed that it can be modified to

achieve an O(n3) running time.In 2006, it was discovered that Carl Gustav Jacobi had solvedthe assignment problem in the 19th century, and publishedposthumously in 1890 in Latin.


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Hungarian Method

The Hungarian method solves minimization

assignment problems with m workers and m jobs.

Special considerations can include:

number of workers does not equal the number ofjobs add dummy workers/jobs with 0 assignment

costs as needed worker i cannot do jobj assign cij = +M

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

maximum profit for that job before beginning theHungarian method


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Hungarian Method

Step 1: For each row, subtract the minimum number in

that row from all numbers in that row. Step 2: For each column, subtract the minimum number

in that column from all numbers in that column.

Step 3: Draw the minimum number of lines to cover all

zeroes. If this number = m, STOP an assignment canbe made.

Step 4: Determine the minimum uncovered number(call it d).

Subtract d from uncovered numbers.

Add d to numbers covered by two lines.

Numbers covered by one line remain the same.

Then, GO TO STEP 3.


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Hungarian Method

Finding the Minimum Number of Lines and

Determining the Optimal Solution Step 1: Find a row or column with only one unlined

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

Step 2: If the circle is in a row with one zero, draw aline through its column. If the circle is in a columnwith one zero, draw a line through its row. Oneapproach, when all rows and columns have two ormore zeroes, is to draw a line through one with the

most zeroes, breaking ties arbitrarily. Step 3: Repeat step 2 until all circles are lined. If this

minimum number of lines equals m, the circlesprovide the optimal assignment.


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Example 1: AP

A contractor pays his subcontractors a fixed fee plus mileage forwork performed. On a given day the contractor is faced with three

electrical jobs associated with various projects. Given below arethe distances between the subcontractors and the projects.

Project

A B C

Westside 50 36 16

Subcontractors Federated 28 30 18Goliath 35 32 20

Universal 25 25 14

How should the contractors be assigned to minimize total costs?

Note: There are four subcontractors and three projects. We create adummy project Dum, which will be assigned to one subcontractor(i.e. that subcontractor will remain idle)


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Example 1: AP

Network Representation (note the dummy project)

50

36

160

2830

18

0

3532

20

0

25 25

14

0

West.

Dum.

C

B

A

Univ.

Gol.

Fed.


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Example 1: AP

Initial Tableau Setup

Since the Hungarian algorithm requires that therebe the same number of rows as columns, add a Dummycolumn so that the first tableau is (the smallest elementsin each row are marked red):

A B C Dummy

Westside 50 36 16 0

Federated 28 30 18 0

Goliath 35 32 20 0

Universal 25 25 14 0


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Example 1: AP

Step 1: Subtract minimum number in each row from all

numbers in that row. Since each row has a zero, wesimply generate the original matrix (the smallestelements in each column are marked red). This yields:

A B C DummyWestside 50 36 16 0


Goliath 35 32 20 0



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Example 1: AP

Step 2: Subtract the minimum number in each column

from all numbers in the column. For A it is 25, for B itis 25, for C it is 14, for Dummy it is 0. This yields:

A B C Dummy

Westside 25 11 2 0

Federated 3 5 4 0

Goliath 10 7 6 0

Universal 0 0 0 0


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Example 1: AP

Step 3: Draw the minimum number of lines to cover all zeroes

(called minimum cover). Although one can "eyeball" thisminimum, use the following algorithm. If a "remaining" row hasonly one zero, draw a line through the column. If a remainingcolumn 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), thecurrent solution is not optimal.

A B C Dummy

Westside 25 11 2 0

Federated 3 5 4 0

Goliath 10 7 6 0

Universal 0 0 0 0

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


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Example 1: AP

Step 5: Subtract 2 from uncovered numbers; add 2 to all

numbers at line intersections; leave all other numbersintact. This gives:

A B C Dummy

Westside 23 9 0 0

Federated 1 3 2 0

Goliath 8 5 4 0

Universal 0 0 0 2


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Example 1: AP

Step 3: Draw the minimum number of lines to cover all

zeroes. Since 3 (# of lines) < 4 (# of rows), the currentsolution is not optimal.

A B C Dummy

Westside 23 9 0 0

Federated 1 3 2 0

Goliath 8 5 4 0

Universal 0 0 0 2

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


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Example 1: AP

Step 5: Subtract 1 from uncovered numbers. Add 1 to

numbers at intersections. Leave other numbers intact.This gives:

A B C Dummy

Westside 23 9 0 1

Federated 0 2 1 0

Goliath 7 4 3 0

Universal 0 0 0 3


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Example 1: AP

Find the minimum cover:

A B C Dummy

Westside 23 9 0 1

Federated 0 2 1 0Goliath 7 4 3 0

Universal 0 0 0 3

Step 4: The minimum number of lines to cover all 0's isfour. Thus, the current solution is optimal (minimumcost) assignment.


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Example 1: AP

The optimal assignment occurs at locations of zeros

such that there is exactly one zero in each row and eachcolumn:

A B C Dummy

Westside 23 9 0 1

Federated 0 2 1 0

Goliath 7 4 3 0

Universal 0 0 0 3


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Example 1: AP

The optimal assignment is (go back to the original table

for the distances):

Subcontractor Project Distance

Westside C 16

Federated A 28

Universal B 25

Goliath (unassigned)

Total Distance = 69 miles


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Example 1: AP via LP

In our example the LP formulation is:

Min z = 50x11 + 36x12 + 16x13 + 0x14 + 28x21 + 30x22 + 18x23 + 0x24 +

35x31 + 32x32 + 20x33 + 0x34 + + 25x41 + 25x42 + 14x43 + 0x44s.t.

x11 + x12 + x13 + x14 = 1 (row 1)

x21 + x22 + x23 + x24 = 1 (row 2)

x31 + x32 + x33 + x34 = 1 (row 3)x41 + x42 + x43 + x44 = 1 (row 4)

x11 + x21 + x31 + x41 = 1 (column 1)

x12 + x22 + x32 + x42 = 1 (column 2)

x13 + x23 + x33 + x43 = 1 (column 3)

x14 + x24 + x34 + x44 = 1 (column 4)

xij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3, 4 (nonnegativity)


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The solver formulation is:

Distances From

Contractors to Projects at:

Contractor A B C Dummy

West Side 50 36 16 0


Goliath 35 32 20 0


Assignment of Contractors

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 0Capacity 1 1 1 1

Total Distance: 0

The Assignment Problem


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The solver solution is:

Distances From

Contractors to Projects at:

Contractor A B C Dummy

West Side 50 36 16 0


Goliath 35 32 20 0


Assignment of Contractors

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

The Assignment Problem

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