Chapter 6 Assignment Problems

Quantitative Techniques in

Management

4th Edition

N. D. Vohra

© 2010

Chapter 6

Specially Structured Linear Programmes II: Assignment Problem

Contents

1. Assignment Problem: A Variant of the Transportation Problem

2. Methods of SolutionComplete EnumerationTransportation MethodSimplex MethodHungarian Assignment Method

3. Hungarian Assignment Method (HAM)4. Special Topics

Unbalanced Assignment ProblemsConstrained Assignment ProblemsUnique vs. Multiple Optimal SolutionsMaximization Problems

5. Dual of Assignment Problem

Solution to an Assignment Problem

Can be solved by complete enumeration – ruled out since the number of alternatives is very large (equal to n!)

Can be formulated and solved as a transportation problem – not efficient because the solution would be severely degenerate

Can be formulated and solved an LPP (a 0-1 IPP) – also not an efficient way of solving the problem

Can be solved by Hungarian Assignment Method (HAM)

HAM is most efficient of all methods

Assignment Problem: A Variant of Transportation Problem

WorkerJob

SupplyA B C D

1 45 40 51 67 1

2 57 42 63 55 1

3 49 52 48 64 1

4 41 45 60 55 1

Demand 1 1 1 1 4

Assignment Problem as LPP

Min Z = 45x11 + 40x12 + 51x13 + 67x14 + 57x21 + 42x22 + 63x23 + 55x24 + 49x31 + 52x32 + 48x33 + 64x34 + 41x41 + 45x42 + 60x43 + 55x 44

Subject to x11 + x12 + x13 + x14 = 1 x21 + x22 + x23 + x24 = 1 x31 + x32 + x33 + x34 = 1 x41 + x42 + x43 + x44 = 1 x11 + x21 + x31 + x41 = 1 x12 + x22 + x32 + x42 = 1 x13 + x23 + x33 + x43 = 1 x14 + x24 + x34 + x44 = 1

xij = 1 if assignment is made xij = 0 if assignment is not made

Hungarian Assignment MethodStep 1

Balance the problem if it is unbalancedPlace an M as the cost element if some assignment is prohibitedConvert into equivalent min problem if it is a max problemFor this conversion, subtract each value of the matrix from a constant value, K ( generally the largest value in the given matrix)

Step 2Obtain zeros in every row and column by row reductions and column reductions

Hungarian Assignment Method (…continued)Step 3

Cover all zeros by least number of horizontal/ vertical linesIf number of lines = n move to step 4 else, make adjustment: select least uncovered value; subtract it from every uncovered value; and add to each one at intersection of linesAgain cover zeros with lines to find if you could go to 4

Step 4Make assignments by considering zeros

HAM: Making Assignments

Select a row/column with a single zeroMake an assignment at zero by placing a square around itCross out zero/s, if any, in the corresponding column/rowContinue in this manner until all assignments are madeIf there are multiple zeros left in all rows and columns, select a zero arbitrarily, make assignment and cancel other zeros in the row and column involvedMultiple zeros in all rows and columns remaining indicates multiple optimal solutions

HAM: An Example

PersonJob

1 2 3 4 5

A 27 18 X 20 21

B 31 24 21 12 17

C 20 17 20 X 16

D 22 28 20 16 27

To solve this problem:Balance by adding a dummy personPlace an M for each prohibited assignment

X indicates prohibited assignment

Solve the Problem: Time (in hours)

Solution to Assignment Problem

PersonJob

1 2 3 4 5

A 27 18 M 20 21

B 31 24 21 12 17

C 20 17 20 M 16

D 22 28 20 16 27

E 0 0 0 0 0

Row Reductions: RCT-1

PersonJob

1 2 3 4 5

A 9 0 M 2 3

B 19 12 9 0 5

C 4 1 4 M 0

D 6 12 4 0 11

E 0 0 0 0 0

Solution (…continued)

RCT-2

PersonJob

1 2 3 4 5

A 9 0 M 6 3

B 15 18 5 0 1

C 4 1 4 M 0

D 2 8 0 0 7

E 0 0 0 4 0

Number of lines covering zeros (=4) in RCT-1 is smaller than n (=5)So, revise the table:(the least uncovered value = 4)RCT-2 is the resulting tableIn RCT- 2, number of lines covering all zeros = 5, which is equal to n, so make assignments

Solution: Making Assignments

To make assignments in RCT-2,Make an assignment at zero by placing a square around itSelect column 1, assign E-1Cross out zeros in row ESelect column 2, assign A-2Select column 3, assign D-3Select row 2, assign B-4Cross out zero in column 4Select row 5, assign C-5

Optimal Assignment Solution:A-2; B-4; C-5; D-3 and Job 1: unassignedTotal Time = 66 hours

Solution to Example 6.14

Flight 201 202 203 204

101 24 25 6 11

102 22 23 28 9

103 16 17 22 27

104 10 11 16 21

Layover Time Matrix – Crew at Mumbai

Layover Time Matrix – Crew at New Delhi

Flight 201 202 203 204

101 20 19 14 9

102 22 21 16 11

103 28 27 22 17

104 10 9 28 23

Solution to Example 6.14 (…continued)

Flight 201 202 203 204

101 20* 19* 6 9*

102 22 21* 16* 9

103 16 17 22 17*

104 10 9* 16 21

(*) indicates that crew is based at MumbaiOther values indicate crew is based at New Delhi

The above matrix is derived by comparing various pairs of flights and taking the lower of the two layover values involved; since for a given pair of flights, crew will be based where the layover time is smaller

Layover Time Matrix – Crew at New Delhi / Mumbai

Solution to Example 6.14 (…continued)

Flight 201 202 203 204

101 14 13 0 3

102 13 12 7 0

103 0 1 6 1

104 1 0 7 12

Optimal Pairing of Flights:

Reduced Cost Table 1

Flight FlightLayover

TimeCrew-Base

101 203 6 New Delhi

102 204 9 New Delhi

103 201 16 New Delhi

104 202 9 Mumbai

Multiple Choice Questions

Assignments can be made when the minimum number of lines in a reduced cost table with which all zeros can be covered is

1. Exactly equal to n.

2. Equal to or greater than n.

3. Equal to or less than n.

4. Less than or greater than n.


Mark the correct statement:

1. Assignment problems can be solved using HAM only.

2. An assignment problem is a special type of transportation problem.

3. Unbalanced assignment problems cannot be solved.

4. An assignment problem cannot be of maximisation type.


Mark the correct statement:1. An assignment problem is said to be

unbalanced when all the workers cannot perform all the jobs involved.

2. An assignment problem may require the introduction of both: dummy jobs and dummy workers.

3. An unbalanced assignment problem is one where the number of workers is more than, or less than, the number of jobs.

4. An assignment problem with m workers and n jobs will involve a total of m n possible assignments.


The minimum number of lines covering all zeros in a Reduced Cost Table can be:

1. At most n.

2. At least n.

3. at most r + c –1 (r: rows, c: columns).

4. n – 1.


When making assignment, we begin with

1. Scanning and selecting rows with single zeros and then proceed to scan and select columns.

2. Scanning and selecting columns with single zeros and then proceed to scan and select rows.

3. Scanning and selecting rows or columns with single zeros.

4. Selecting rows and columns with single zeros in turn.


Mark the correct statement: 1. If we begin making assignments by

selecting a row/column with multiple zeros, then multiple optimal solutions would result.

2. Multiple optimal solutions may have different objective function values.

3. While making assignments, if we face a situation where all the rows/columns remaining have multiple zeros, then multiple optimal solutions are indicated.

4. Multiple optimal solutions imply that at least one worker gets more than one job assigned.


In an assignment problem involving 8 workers and 8 jobs, total number of assignments possible are:

1. 8

2. 16

3. 64

4. 8!


Mark the wrong statement: 1. An assignment problem can be

solved by writing it as a linear programming problem and using the simplex algorithm.

2. The Hungarian method of solution assumes that one worker can be assigned only one job.

3. The relative cost of assigning job j to worker i is not changed by the subtraction of a constant from either a row or a column of the cost matrix.

4. For converting a maximisation problem into a minimisation problem, a constant value, usually the largest value, is subtracted from all elements of the given matrix.


Which of the following should be preferred in making assignments?

1. A row with a single zero.

2. A column with multiple zeros.

3. A column with a single zero.

4. Either (a) or (c).

Chapter 6 Assignment Problems

Documents

Transcript of Chapter 6 Assignment Problems