MATCHING PROBLEM

MOHAMED ASLEM JAMAL

PRASANNA BASKARAN

PRASHANTH KARTHIKEYAN

PATRIC SAMUEL PAUL

SRINATH GOWTHAM

DECISION SUPPORT FOR LOGISTICS MANAGEMENT

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

MATCHING PROBLEMGRAPHICAL MODAL ELEMENTS

THE MATCHING PROBLEMVisualization of the MP

JOO KOON

BOON LAY

KENTRIDGE

JURONG EAST

ANGMOKIOHARBORFRONT

BUKIT BATOK

MARINA BAY

BOUNA VISTA

PAYE LEBAR

THE MATCHING PROBLEMVisualization of the MP

JOO KOON

BOON LAY

KENTRIDGE

BOUNA VISTA

HARBORFRONT

MARINA BAY

ANGMOKIO

BUKIT BATOK

JURONG EAST

PAYE LEBAR

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

MATCHING PROBLEMBASIC 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 𝑐𝑖𝑗 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.

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

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

MATCHING PROBLEM

Formal Problem Model

1) 𝑚𝑖𝑛. 𝑓 𝑥 =

𝑖=1

𝑛

𝑗=1

𝑛

𝑐𝑖𝑗. 𝑥𝑖𝑗

S.T

2)

𝑗=1

𝑛

𝑥𝑖𝑗 = 1

3)

𝑖=1

𝑛

𝑥𝑖𝑗 = 1

For i=1,….,n

For j=1,….,n

4) 𝑥𝑖𝑗 Є {0,1} For all i and j

Each assignee is assigned only one task

Each task is assigned only one assignee

MATCHING PROBLEMBOTTLENECK 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.

MATCHING PROBLEMMAXIMUM 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.

MATCHING PROBLEMAGENDA

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

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

MATCHING PROBLEM

“HUNGARIAN METHOD”

• Formulate this problem as an matching problem?

• Obtain an optimal solution using the Hungarian method?

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

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

MATCHING PROBLEM“HUNGARIAN METHOD”

2. Subtract the smallest entry in each column from all the entries of its column.

COLUMNMIN

$0 $1 $0 $0

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

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

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

“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.

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.

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

“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.

Now the minimum number of lines required to cover all zeros is equal to m. Hence the solution is optimized.

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

MATCHING PROBLEM

5,“HUNGARIAN METHOD”

TAXI 1CUST 3

TAXI 2CUST4

TAXI 3CUST 1

TAXI 4CUST 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

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.

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

MATCHING PROBLEMUse 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.

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

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.

MATCHING PROBLEMANSWER

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.

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

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

MATCHING PROBLEMLITERATURE 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

THANK YOU