Assignment Algorithm by Hungarian Method, Slides of Optimization Techniques in Engineering

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GEM 805 1

Assignment Algorithm

(Hungarian Method)

By: J.E.T

2010-2011 ©

The Assignment Problem

Consider the problem of assigning n assignees to n tasks. Only

one task can be assigned to an assignee, and each task must be

assigned.

There is also a cost associated with assigning an assignee i to

task j , c

ij

The objective is to assign all tasks such that the total cost is

minimized.

The Assignment Problem

To fit the assignment problem definition, the following

assumptions 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^ cij associated with assignee^ i^ performing task^ j.
• (^) The objective is to determine how all n assignments should be

made to minimize the total cost.

Assignment Problem –

Flow Diagram

n

assignees

tasks

a – assignee

t – tasks a 1

a

2

a

3

t 1 t 2 t 3 t 4

c

11

c

12

n

c

nn

a

n

Assignment Problem –

Conversion to Standard Cost Matrix

1 2 3 4 1 10 9 8 7 Assignees 2 4 - 5 6 3 2 1 8 - Tasks

Consider following cost matrix, how do you convert to satisfy

the standard definition of the assignment problem?

1 2 3 4 1 10 9 8 7 Assignees 2 4 M 5 6 3 2 1 8 M 4 0 0 0 0 Tasks

Add “big M” to avoid incompatible assignments, and add a

dummy assignee (or task) to have equal assignees and tasks.

Assignment Problem –

Math Formulation

(x isbinary,foralliand j)

x 0

x 1

x 1

Z c x

ij ij n i 1 ij n j 1 ij m i 1 n j 1 ij ij

       

Minimize

s.t.

Total Cost

i

j

Does this formulation look familiar?

Is this a Linear Program?

Each supply is 1

Each demand is 1

Assignment Problem –

Hungarian Method

Consider following cost matrix.

1 2 3 4 Row Minimum

Assignees 2 2 12 6 5 2

Tasks

Assignees 2 0 10 4 3 3 4 5 0 6 4 0 2 4 8 Column Min

Reduce by Row Minimum.

Assignment Problem –

Hungarian Method

Reduced by Column Minimum:

Assignees 2 0 10 4 1

Tasks

Assignees 2 0 9 3 0

Tasks

Reduce by Minimum of uncovered cells (1):

Assignment Problem –

Hungarian Method

Summary of Hungarian Method: Step 1 – Find the minimum element in each row. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix by subtracting from each cost the minimum cost in its column. Step 2 – Draw the minimum number of lines (horizontal or vertical) that are needed to cover all the zeros in the reduced cost matrix. If n lines are required, an optimal solutions is available among the covered zeros in the matrix. If fewer than n lines are needed, proceed to step 3. Step 3 – Find the smallest nonzero element (call its value k ) in the reduced cost matrix that is uncovered by the lines drawn in Step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to step 2.

EXAMPLE CASE: SUPPLY = DEMAND

Example: Step 1(a)

Row

Column

 - Job 1 Job 2 Job 3 Job 

• Machine
• Machine
• Machine
• Machine - Job 1 Job 2 Job 3 Job
  • Machine
  • Machine
  • Machine
  • Machine - Job 1 Job 2 Job 3 Job Example: Step 1(b)
  • Machine
  • Machine
  • Machine
  • Machine
    • Job 1 Job 2 Job 3 Job
• Machine
• Machine
• Machine
• Machine
• Example: Step - Job 1 Job 2 Job 3 Job
  • Machine
  • Machine
  • Machine
  • Machine - continue to Step We need 3<4 lines, so
  • Example: Step - Job 1 Job 2 Job 3 Job
• Machine
• Machine
• Machine
• Machine - Job 1 Job 2 Job 3 Job - Machine - Machine - Machine - Machine

Example: Step 2 (again) Job 1 Job 2 Job 3 Job 4 Machine 1 10 0 3 0 Machine 2 0 9 3 0 Machine 3 5 5 0 4 Machine 4 0 1 3 5

Need 4 lines, so we have the

optimal assignment and we

stop

Zero Assignment