matching problem hungarian algorithm bottleneck maximum cardinality

1. DECISION SUPPORT FOR LOGISTICS MANAGEMENT
MATCHING PROBLEM
MOHAMED ASLEM JAMAL
PRASANNA BASKARAN
PRASHANTH KARTHIKEYAN
PATRIC SAMUEL PAUL
SRINATH GOWTHAM

2. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

3. MATCHING PROBLEM
GRAPHICAL MODAL ELEMENTS

4. THE MATCHING PROBLEM
Visualization of the MP
JOO KOON
BOON LAY
KENTRIDGE
JURONG EAST
BUKIT BATOK
HARBORFRONT ANGMOKIO
MARINA BAY
BOUNA VISTA
PAYE LEBAR

5. THE MATCHING PROBLEM
Visualization of the MP
JOO KOON
BOON LAY
KENTRIDGE
BOUNA VISTA
HARBORFRONT
MARINA BAY
ANGMOKIO
BUKIT BATOK
JURONG EAST
PAYE LEBAR

6. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

7. MATCHING PROBLEM
BASIC ASSUMPTIONS
• The number of assignees and the number of tasks are the same.
• Each assignee is exactly assigned to exactly one task.
• Each task is to be performed by exactly one assignee.
• There is a cost 푐푖푗 associated with assignee i (i=1,….,n) performing task
j(j=1,….,n).
• The objective is to determine how all n assignments should be made to
minimize the total cost.

8. MATCHING PROBLEM
Definition of Variables
i:= assignee
j:= tasks to be matched to assignees
xij:= defines whether assignee i is matched to task j
xij =1 : i performs task j
xij =0 : i doesn’t perform task j
cij := defines the costs associated with assignee i for performing task j

9. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

10. MATCHING PROBLEM
Formal Problem Model
1) 푚푖푛. 푓 푥 =
푛
푖=1
푛
푗=1
푐푖푗. 푥푖푗
S.T
2)
푛
푗=1
푥푖푗 = 1
3)
푛
푖=1
푥푖푗 = 1
For i=1,….,n
For j=1,….,n
4) 푥푖푗 Є {0,1} For all i and j
Each assignee is assigned only one task
Each task is assigned only one assignee

11. MATCHING PROBLEM
BOTTLENECK PROBLEMS
• Bottleneck assignment problems occur, for instance in connection with
assigning jobs to parallel machines so as to minimize the latest completion
time.
• Let n jobs and n machines be given.
• If the machines work in parallel, we want to assign the jobs to the machines
such that the latest completion time is as early as possible.
Note :
• Bottleneck problems is not about minimising the total time taken for
completing each job. Rather it is about minimising the maximum time taken
for the jobs to be completed.
• The cost coefficient cij is the time needed for machine j to complete job i.

12. MATCHING PROBLEM
MAXIMUM CARDINALITY MATCHING
• A matching(X) within a graph G is where no node is connected to more than
one edge.
• A maximum-cardinality matching is a matching that contains the largest
possible number of matchings. There may be many maximum matching's.
• The matching number v(G) of a graph G is the size of a maximum matching.
• The following figure shows examples of maximum matchings' in the three
graphs.

13. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

14. MATCHING PROBLEM
“HUNGARIAN METHOD”
Suppose that a taxi firm has four taxis (the agents) available, and four
customers (the tasks) wishing to be picked up as soon as possible. The firm
prides itself on speedy pickups, so for each taxi the "cost" of picking up a
particular customer will depend on the time taken for the taxi to reach the
pickup point. The solution to the assignment problem will be whichever
combination of taxis and customers results in the least total cost.
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $54 $54 $51 $53
TAXI 2 $51 $57 $52 $52
TAXI 3 $50 $53 $54 $56
TAXI 4 $56 $54 $55 $53

15. MATCHING PROBLEM
“HUNGARIAN METHOD”
• Formulate this problem as an matching problem?
• Obtain an optimal solution using the Hungarian method?

16. MATCHING PROBLEM
“HUNGARIAN METHOD”
STEP 1 : Subtract the smallest entry in each row from all the entries of its row.
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $54 $54 $51 $53
TAXI 2 $51 $57 $52 $52
TAXI 3 $50 $53 $54 $56
TAXI 4 $56 $54 $55 $53
ROW MIN
$51
$51
$50
$53

17. MATCHING PROBLEM
“HUNGARIAN METHOD”
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $3 $3 $0 $2
TAXI 2 $0 $6 $1 $1
TAXI 3 $0 $3 $4 $6
TAXI 4 $3 $1 $2 $0

18. MATCHING PROBLEM
“HUNGARIAN METHOD”
2. Subtract the smallest entry in each column from all the entries of its column.
TAXI 1 $3 $3 $0 $2
TAXI 2 $0 $6 $1 $1
TAXI 3 $0 $3 $4 $6
TAXI 4 $3 $1 $2 $0
COLUMN
MIN
CUST 1 CUST 2 CUST 3 CUST 4
$0 $1 $0 $0

19. MATCHING PROBLEM
“HUNGARIAN METHOD”
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $3 $2 $0 $2
TAXI 2 $0 $5 $1 $1
TAXI 3 $0 $2 $4 $6
TAXI 4 $3 $0 $2 $0

20. MATCHING PROBLEM
“HUNGARIAN METHOD”
3, Draw the minimum number of lines (Horizontal, Vertical or both) that are needed
to cover all the zeros in the reduced cost matrix.
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $3 $2 $0 $2
TAXI 2 $0 $5 $1 $1
TAXI 3 $0 $2 $4 $6
TAXI 4 $3 $0 $2 $0

21. “HUNGARIAN METHOD”
For m X m matrix, if m lines are required to cover all zeros, then an optimal solution is available among the
covered zeros in the matrix.
CUST 1 CUST 2 CUST 3 CUST4
TAXI 1 $3 $2 $0 $2
TAXI 2 $0 $5 $1 $1
TAXI 3 $0 $2 $4 $6
TAXI 4 $3 $0 $2 $0
If the minimum number of lines required to cover all zeros is equal to m then proceed to step 5.
However, as in this case the minimum number of lines required to cover all zeros is less than m additional
step for optimization is required.

22. “HUNGARIAN METHOD”
4. Find the smallest non zero element (call its value k) in the reduced cost
matrix that is uncovered by the lines drawn in step 3. Now subtract k from
each uncovered element of the reduced cost matrix and add k to each element
that is covered by two lines.
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $4 $2 $0 $2
TAXI 2 $0 $4 $0 $0
TAXI 3 $0 $1 $3 $5
TAXI 4 $4 $0 $2 $0
Now the minimum number of lines required to cover all zeros is equal
to m. Hence the solution is optimized.

23. MATCHING PROBLEM
5,“HUNGARIAN METHOD”
TAXI 1CUST 3
TAXI 2CUST4
TAXI 3CUST 1
TAXI 4CUST 2
CUST 1 CUST 2 CUST 3 CUST 4
TAXI 1 $4 $2 $0 $2
TAXI 2 $0 $4 $0 $0
TAXI 3 $0 $1 $3 $5
TAXI 4 $4 $0 $2 $0

24. PSEUDO CODE:
STEP 1: Create a table with m rows and n columns, with assignees along the
rows and
Tasks along the columns where m=n.
IF No of rows is not equal to No of columns, add a dummy row or column such
that m=n.
STEP 2: Subtract smallest entry in each row from all entries of its row
Draw the minimum no of lines (horizontal, vertical or both) that are needed
to cover all the zeros in the table.
IF No of lines is equal to m and n, TERMINATE
ELSE
STEP 3:Subtract smallest entry in each column from all entries of its column
Draw the minimum no of lines (horizontal, vertical or both) that are needed
to cover all the zeros in the table.
IF No of lines is equal to m and n, TERMINATE
ELSE
STEP 6: Find the least value among the uncovered entries.
STEP 7: Subtract this value from all the uncovered entries and add the value
to the double covered entries (point of intersection of the lines).
STEP 8 : TERMINATE if the minimum no of lines (horizontal, vertical or both)
that are needed to cover all the zeros in the table is equal to m and n.

25. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

26. MATCHING PROBLEM
Use Cases
The matching problem is a linear programming problem where assignees are
being assigned to perform tasks. Examples are:
1,How to schedule the flights routes between two cities so that the
layover times for the crew can be minimized?
2,Regulate the arrival of trains and processing times minimize the
passengers waiting time and reduce congestion, formulate suitable
transportation policy, thereby reducing the costs and time of trans-shipment.

27. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

28. FIND THE MATCHING PROBLEM?
Exercise
1, A company has two plants producing a certain products that is to be shipped
to three distribution centers. Unit production costs are the same at the two
plants.
2, Management of returnable bottles where the filled bottles are brought to
the customers and empty bottles are return to the brewery, to be recycled
(Environmental issues).
3, Manager has four jobs in hand to be allocated to four of his clerical staff. The
staff differs in efficiency. The manager wants to allocate the duty to his staff so
that the time taken by the staff must be minimum. Help the manager in
allocating the jobs to the personnel.

29. MATCHING PROBLEM
ANSWER
3, Manager has four jobs in hand to be allocated to four of his clerical staff. The
staff differs in efficiency. The manager wants to allocate the duty to his staff so
that the time taken by the staff must be minimum. Help the manager in
allocating the jobs to the personnel.

30. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

31. MATCHING PROBLEM
AGENDA
1. Problem Illustration
2. Definitions
3. Problem Model
4. Selected Algorithms
5. Use Cases
6. Exercise
7. Solving the problem using excel
8. Literature Research

32. MATCHING PROBLEM
LITERATURE RESEARCH
• Introduction to Operations Research by Hillier and Lieberman.(Seventh Edition)
• Operation research Applications and Algorithms by Wayne L.Winston.
• A well solved class of Integer programs: Matching by Jack Edmonds and Ellis
Johnson.
• Assignment Problems - Revised Reprint, by Rainer Burkard, Mauro
Dell'Amico, Silvano Martello - Society for Industrial and Applied Mathematics,
Philadelphia, 2012

33. THANK YOU