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Quantitative Analysis
for Management
Chapter 10
Transportation and
Assignment Models
To accompany

10-1

© 2000 by Prentic...
Chapter Outline
10.1 Introduction
10.2 Setting Up a Transportation Problem
10.2 Developing an Initial
Solution:Northwest C...
Chapter Outline - continued
10.6 Vogel’s Approximation Method:
Another Way to Find an Initial Solution
10.7 Unbalanced Tra...
Learning Objectives
Students will be able to
♣ Structure special linear programming
problems using the transportation and
...
Learning Objectives - continued
♣ Solve the facility location problem and other
application problems with the transportati...
Specialized Problems
♦ Transportation Problem
♣ Distribution of items from several sources to
several destinations. Supply...
Transportation Problem
Cleveland
(200 units)
required

Des Moines
(100 units)
capacity
Albuquerque
(300 units)
required

E...
Transportation Costs
From
(Sources)

To
(Destinations)
Albuquerque Boston Cleveland

Des Moines

$5

$4

$3

Evansville

$...
Unit Shipping Cost:1Unit,
Factory to Warehouse
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

Des Moines
(D)

5

4

3

Evansv...
Total Demand and Total Supply
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

Factory
Capacity

Des Moines
(D)

100

Evansvill...
Transportation Table For
Executive Furniture Corp.
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

Des Moines
(D)

5

4

3

Ev...
Initial Solution Using the
Northwest Corner Rule
♦ Start in the upper left-hand cell and allocate
units to shipping routes...
Initial Solution
North West Corner Rule
Albuquerque
(A)
Des Moines
(D)
Evansville
(E)

100
200

Warehouse
Req.
To accompan...
The Stepping-Stone Method
♦ 1. Select any unused square to evaluate.
♦ 2. Begin at this square. Trace a closed path back t...
Stepping-Stone Method - The
Des Moines-to-Cleveland Route
Albuquerque
(A)
Des Moines
(D)
Evansville
(E)
Ft Lauderdale
(F)
...
Stepping-Stone Method
An Improved Solution
Albuquerque
(A)
Des Moines
(D)
Evansville
(E)

100
100

Ft Lauderdale
(F)

100
...
Third and Final Solution
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

5

4

3

Evansville
(E)

8

4

3

Ft Lauderdale
(F)

...
MODI Method: 5 Steps
1. Compute the values for each row and column: set Ri +
Kj = Cij for those squares currently used or ...
Vogel’s Approximation
1. For each row/column of table, find difference
between two lowest costs. (Opportunity cost)
1. Fin...
Vogel’s Approximation
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

Des Moines
(D)

5

4

3

Evansville
(E)

8

4

3

Ft Lau...
Vogel’s Approximation
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

Des Moines
(D)

5

4

3

Evansville
(E)

8

4

3

Ft Lau...
Vogel’s Approximation
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

5

4

3

Evansville
(E)

8

4

3

Ft Lauderdale
(F)

9

...
Vogel’s Approximation
Albuquerque
(A)

Boston
(B)

Cleveland
(C)

5

4

3

Evansville
(E)

8

4

3

Ft Lauderdale
(F)

200...
Special Problems in
Transportation Method
♦ Unbalanced Problem
♣ Demand Less than Supply
♣ Demand Greater than Supply

♦ D...
Unbalanced Problem
Demand Less than Supply
Customer 1 Customer 2

Dummy

Factory
Capacity

Factory 1

8

5

16

Factory 2
...
Unbalanced Problem
Supply Less than Demand
Customer 1 Customer 2

Customer 3

Factory 1

8

5

16

Factory 2

15

10

7

D...
Degeneracy
Customer 1 Customer 2

Customer 3

5

4

3

Factory 2

8

4

3

Factory 3

9

Factory 1

Customer
Requirements
...
Degeneracy - Coming Up!
Customer 1 Customer 2
Factory 1
Factory 2
Factory 3
Customer
Requirements
To accompany

70
50
30

...
Stepping-Stone Method - The
Des Moines-to-Cleveland Route
Albuquerque
(A)
Des Moines
(D)
Evansville
(E)
Ft Lauderdale
(F)
...
The Assignment Method
1. subtract the smallest number in each row from
every number in that row
♣ subtract the smallest nu...
The Assignment Method
continued
3. if the number of lines does not equal the number of
rows or columns
♣ subtract the smal...
Hungarian Method
Person
Adams

Initial Table
Project
1
2
11
14

3
6

Brown

8

10

11

Cooper

9

12

7

To accompany

10-...
Hungarian Method
Row Reduction
Person
Project
1
2
Adams
5
8

3
0

Brown

0

2

3

Cooper

2

5

0

To accompany

10-33

© ...
Hungarian Method
Column Reduction
Person
Project
1
2
Adams
5
6

3
0

Brown

0

0

3

Cooper

2

3

0

To accompany

10-34
...
Hungarian Method
Person
Adams

1
5

Testing
Project
2
6

Covering
Line 2

3
0

Brown

0

0

3

Cooper

2

3

Covering
Line...
Hungarian Method
Revised Opportunity Cost Table
Person
Project
1
2
3
Adams
3
4
0
Brown

0

0

5

Cooper

0

1

0

To accom...
Hungarian Method
Person
Adams

Testing
Covering
Line 1 Project
1
2
3
4

Covering
Line 3

3
0

Brown

0

0

5

Cooper

0

1...
Hungarian Method
Assignments
Person
1

Project
2

Adams
Brown
Cooper
To accompany

3
6

10
9
10-38

© 2000 by Prentice
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Transcript of "Chap 10"

  1. 1. Quantitative Analysis for Management Chapter 10 Transportation and Assignment Models To accompany 10-1 © 2000 by Prentice
  2. 2. Chapter Outline 10.1 Introduction 10.2 Setting Up a Transportation Problem 10.2 Developing an Initial Solution:Northwest Corner Rule 10.4 Stepping-Stone Method: Finding a Least-Cost Solution 10.5 MODI Method To accompany 10-2 © 2000 by Prentice
  3. 3. Chapter Outline - continued 10.6 Vogel’s Approximation Method: Another Way to Find an Initial Solution 10.7 Unbalanced Transportation Problems 10.8 Degeneracy in Transportation Problems 10.9 More Than One Optimal Solution 10.10 Facility Location Analysis To accompany 10-3 © 2000 by Prentice
  4. 4. Learning Objectives Students will be able to ♣ Structure special linear programming problems using the transportation and assignment models. ♣ Use the northwest corner method and Vogel’s approximation method to find initial solutions to transportation problems. ♣ Apply the stepping-stone and MODI methods to find optimal solutions to transportation problems. To accompany 10-4 © 2000 by Prentice
  5. 5. Learning Objectives - continued ♣ Solve the facility location problem and other application problems with the transportation model. ♣ Solve assignment problems with the Hungarian (matrix reduction) method. To accompany 10-5 © 2000 by Prentice
  6. 6. Specialized Problems ♦ Transportation Problem ♣ Distribution of items from several sources to several destinations. Supply capacities and destination requirements known. ♦ Assignment Problem ♣ One to one assignment of people to jobs, etc. Specialized algorithms save time! To accompany 10-6 © 2000 by Prentice
  7. 7. Transportation Problem Cleveland (200 units) required Des Moines (100 units) capacity Albuquerque (300 units) required Evansville (300 units) capacity Boston (200 units) required Ft. Lauderdale (300 units) capacity To accompany 10-7 © 2000 by Prentice
  8. 8. Transportation Costs From (Sources) To (Destinations) Albuquerque Boston Cleveland Des Moines $5 $4 $3 Evansville $8 $4 $3 Fort Lauderdale $9 $7 $5 To accompany 10-8 © 2000 by Prentice
  9. 9. Unit Shipping Cost:1Unit, Factory to Warehouse Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 7 Factory Capacity 5 Warehouse Req. To accompany 10-9 © 2000 by Prentice
  10. 10. Total Demand and Total Supply Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity Des Moines (D) 100 Evansville (E) 300 Ft Lauderdale (F) 300 Warehouse Req. To accompany 300 200 10-10 200 700 © 2000 by Prentice
  11. 11. Transportation Table For Executive Furniture Corp. Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 7 5 Warehouse Req. To accompany 300 200 10-11 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
  12. 12. Initial Solution Using the Northwest Corner Rule ♦ Start in the upper left-hand cell and allocate units to shipping routes as follows: ♣ Exhaust the supply (factory capacity) of each row before moving down to the next row. ♣ Exhaust the demand (warehouse) requirements of each column before moving to the next column to the right. ♣ Check that all supply and demand requirements are met. To accompany 10-12 © 2000 by Prentice
  13. 13. Initial Solution North West Corner Rule Albuquerque (A) Des Moines (D) Evansville (E) 100 200 Warehouse Req. To accompany Cleveland (C) 5 4 3 8 4 3 7 5 9 Ft Lauderdale (F) Boston (B) 100 100 300 200 10-13 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
  14. 14. The Stepping-Stone Method ♦ 1. Select any unused square to evaluate. ♦ 2. Begin at this square. Trace a closed path back to the original square via squares that are currently being used (only horizontal or vertical moves allowed). ♦ 3. Place + in unused square; alternate - and + on each corner square of the closed path. ♦ 4. Calculate improvement index: add together the unit cost figures found in each square containing a +; subtract the unit cost figure in each square containing a -. ♦ 5. Repeat steps 1 - 4 for each unused square. To accompany 10-14 © 2000 by Prentice
  15. 15. Stepping-Stone Method - The Des Moines-to-Cleveland Route Albuquerque (A) Des Moines (D) Evansville (E) Ft Lauderdale (F) + Warehouse Req. To accompany 100 200 Boston (B) 5 Cleveland (C) 4 Start 3 + 8 - 9 + 300 100 100 200 10-15 4 3 7 5 - 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
  16. 16. Stepping-Stone Method An Improved Solution Albuquerque (A) Des Moines (D) Evansville (E) 100 100 Ft Lauderdale (F) 100 Warehouse Req. Boston (B) 5 4 3 8 4 3 7 5 200 9 300 To accompany Cleveland (C) 200 10-16 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
  17. 17. Third and Final Solution Albuquerque (A) Boston (B) Cleveland (C) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 200 9 Warehouse Req. 300 Des Moines (D) To accompany 100 200 7 200 10-17 100 100 200 5 Factory Capacity 100 300 300 700 © 2000 by Prentice
  18. 18. MODI Method: 5 Steps 1. Compute the values for each row and column: set Ri + Kj = Cij for those squares currently used or occupied. 2. After writing all equations, set R1 = 0. 3. Solve the system of equations for Ri and Kj values. 4. Compute the improvement index for each unused square by the formula improvement index: Cij - Ri - Kj 5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method. To accompany 10-18 © 2000 by Prentice
  19. 19. Vogel’s Approximation 1. For each row/column of table, find difference between two lowest costs. (Opportunity cost) 1. Find greatest opportunity cost. 1. Assign as many units as possible to lowest cost square in row/column with greatest opportunity cost. 1. Eliminate row or column which has been completely satisfied. 1. Begin again, omitting eliminated rows/columns. To accompany 10-19 © 2000 by Prentice
  20. 20. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 7 Warehouse Req. 200 1 To accompany 300 3 10-20 200 5 200 Factory Capacity 100 1 300 4 300 2 700 2 © 2000 by Prentice
  21. 21. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 Warehouse Req. 100 7 300 200 1 To accompany 200 5 100 1 300 4 300 2 3 10-21 200 Factory Capacity 700 © 2000 by Prentice
  22. 22. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 Des Moines (D) Warehouse Req. 100 100 7 300 200 4 To accompany 200 5 100 1 300 300 2 3 10-22 200 Factory Capacity 700 © 2000 by Prentice
  23. 23. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 200 9 Warehouse Req. 300 Des Moines (D) To accompany 100 100 100 200 10-23 200 7 5 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
  24. 24. Special Problems in Transportation Method ♦ Unbalanced Problem ♣ Demand Less than Supply ♣ Demand Greater than Supply ♦ Degeneracy ♦ More Than One Optimal Solution To accompany 10-24 © 2000 by Prentice
  25. 25. Unbalanced Problem Demand Less than Supply Customer 1 Customer 2 Dummy Factory Capacity Factory 1 8 5 16 Factory 2 15 10 7 Factory 3 3 9 10 Customer Requirements To accompany 150 80 10-25 150 170 130 80 380 © 2000 by Prentice
  26. 26. Unbalanced Problem Supply Less than Demand Customer 1 Customer 2 Customer 3 Factory 1 8 5 16 Factory 2 15 10 7 Dummy 3 9 10 Customer Requirements To accompany 150 80 10-26 150 Factory Capacity 170 130 80 380 © 2000 by Prentice
  27. 27. Degeneracy Customer 1 Customer 2 Customer 3 5 4 3 Factory 2 8 4 3 Factory 3 9 Factory 1 Customer Requirements To accompany 100 100 100 7 100 10-27 20 80 100 5 Factory Capacity 100 120 80 300 © 2000 by Prentice
  28. 28. Degeneracy - Coming Up! Customer 1 Customer 2 Factory 1 Factory 2 Factory 3 Customer Requirements To accompany 70 50 30 Customer 3 8 5 16 15 10 7 9 10 80 3 150 80 10-28 50 50 Factory Capacity 70 130 80 280 © 2000 by Prentice
  29. 29. Stepping-Stone Method - The Des Moines-to-Cleveland Route Albuquerque (A) Des Moines (D) Evansville (E) Ft Lauderdale (F) + Warehouse Req. To accompany 100 200 Boston (B) 5 Cleveland (C) 4 Start 3 + 8 - 9 + 300 100 100 200 10-29 4 3 7 5 - 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
  30. 30. The Assignment Method 1. subtract the smallest number in each row from every number in that row ♣ subtract the smallest number in each column from every number in that column 2. draw the minimum number of vertical and horizontal straight lines necessary to cover zeros in the table ♣ if the number of lines equals the number of rows or columns, then one can make an optimal assignment (step 4) To accompany 10-30 © 2000 by Prentice
  31. 31. The Assignment Method continued 3. if the number of lines does not equal the number of rows or columns ♣ subtract the smallest number not covered by a line from every other uncovered number ♣ add the same number to any number lying at the intersection of any two lines ♣ return to step 2 4. make optimal assignments at locations of zeros within the table To accompany 10-31 PG 10.13b © 2000 by Prentice
  32. 32. Hungarian Method Person Adams Initial Table Project 1 2 11 14 3 6 Brown 8 10 11 Cooper 9 12 7 To accompany 10-32 © 2000 by Prentice
  33. 33. Hungarian Method Row Reduction Person Project 1 2 Adams 5 8 3 0 Brown 0 2 3 Cooper 2 5 0 To accompany 10-33 © 2000 by Prentice
  34. 34. Hungarian Method Column Reduction Person Project 1 2 Adams 5 6 3 0 Brown 0 0 3 Cooper 2 3 0 To accompany 10-34 © 2000 by Prentice
  35. 35. Hungarian Method Person Adams 1 5 Testing Project 2 6 Covering Line 2 3 0 Brown 0 0 3 Cooper 2 3 Covering Line 1 0 To accompany 10-35 © 2000 by Prentice
  36. 36. Hungarian Method Revised Opportunity Cost Table Person Project 1 2 3 Adams 3 4 0 Brown 0 0 5 Cooper 0 1 0 To accompany 10-36 © 2000 by Prentice
  37. 37. Hungarian Method Person Adams Testing Covering Line 1 Project 1 2 3 4 Covering Line 3 3 0 Brown 0 0 5 Cooper 0 1 0 To accompany 10-37 Covering Line 2 © 2000 by Prentice
  38. 38. Hungarian Method Assignments Person 1 Project 2 Adams Brown Cooper To accompany 3 6 10 9 10-38 © 2000 by Prentice
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