This document outlines the key concepts and methods for transportation and assignment models covered in Chapter 10. It introduces transportation and assignment problems, and describes specialized algorithms for finding optimal solutions, including the northwest corner rule, stepping-stone method, MODI method, and Vogel's approximation method for transportation problems. It also covers the Hungarian (matrix reduction) method for solving assignment problems optimally in a step-by-step manner. The chapter objectives are to solve both transportation and assignment problems, as well as address special cases like unbalanced supply/demand and degeneracy.

1. Quantitative Analysis
for Management
Chapter 10
Transportation and
Assignment Models
To accompany
10-1
© 2000 by Prentice

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Hungarian Method
Assignments
Person
1
Project
2
Adams
Brown
Cooper
To accompany
3
6
10
9
10-38
© 2000 by Prentice