Chapter 10
Transportation and
Assignment Models
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to
• Structure special LP problems
using the transportation and
assignment models.
• Use the N.W. corner, VAM,
MODI, and stepping-stone
method.
• Solve facility location and other
application problems with
transportation methods.
• Solve assignment problems with
the Hungarian (matrix reduction)
method
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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
10.6 Vogel’s Approximation Method
10.7 Unbalanced Transportation
Problems
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter Outline continued
10.8 Degeneracy in Transportation
Problems
10.9 More Than One Optimal Solution
10.10 Maximization Transportation
Problems
10.11 Unacceptable or Prohibited
Routes
10.12 Facility Location Analysis
10.13 Approach of the Assignment
Model
10.14 Unbalanced Assignment Models
10.15 Maximization Assignment
Problems
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Importance of Special
Purpose Algorithms
• Fewer, less complicated,
computations than with simplex
• Less computer memory required
• Produce integer solutions
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Transportation
Problem
Des Moines
(100 units)
capacity
Albuquerque
(300 units)
required
Cleveland
(200 units)
required
Evansville
(300 units)
capacity
Boston
(200 units)
required
Ft. Lauderdale
(300 units)
capacity
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Transportation Costs
To
(Destinations)
From
(Sources)
Albuquerque
Cleveland
Boston
Des Moines
$5
$4
$3
Evansville
$8
$4
$3
Fort
Lauderdale
$9
$7
$5
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Unit Shipping Cost:1Unit,
Factory to Warehouse
Albuquerque
Cleveland
Boston
Factory
(A)
(C)
(B)
Capacity
5
4
3
Des Moines
(D)
Evansville
(E)
8
4
3
Fort
Lauderdale
(F)
9
7
5
Warehouse
Req.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Total Demand and Total
Supply
Albuquerque
Boston
(A)
(B)
Cleveland
Factory
(C)
Capacity
Des Moines
(D)
100
Evansville
(E)
300
Fort
Lauderdale
(F)
300
Warehouse
Req.
300
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
200
10-10
200
700
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Transportation Table For
Executive Furniture Corp.
Albuquerque
Boston
(A)
(B)
Des Moines
(D)
5
Evansville
(E)
8
Fort
Lauderdale
(F)
9
Warehouse
Req.
Cleveland
Factory
(C)
Capacity
4
3
100
4
3
300
7
5
300
300
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
200
10-11
200
700
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-12
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Initial Solution
North West Corner Rule
Albuquerque
Boston
(A)
(B)
Des Moines
(D)
Evansville
(E)
5
4
3
100
100
8
200
Fort
Lauderdale
(F)
Warehouse
Req.
Cleveland
Factory
(C)
Capacity
4
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
300
100
9
300
3
7
5
100
200
300
200
200
700
10-13
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-14
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Stepping-Stone Method The Des Moines-toCleveland Route
Albuquerque
Boston
(A)
(B)
Des Moines
(D)
5
4
100
-
Start
3
100
+
8
Evansville
(E)
4
200
3
300
100
-
+
Fort
Lauderdale
(F)
Warehouse
Req.
Cleveland
Factory
(C)
Capacity
9
7
100
+
300
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
5
200
300
200
10-15
200
700
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Stepping-Stone Method
An Improved Solution
Albuquerque
Boston
(A)
(B)
Des Moines
(D)
Evansville
(E)
5
Cleveland
Factory
(C)
Capacity
4
3
100
100
8
100
Fort
Lauderdale
(F)
100
Warehouse
Req.
300
4
300
200
9
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
3
7
5
200
200
10-16
200
300
700
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Third and Final Solution
Albuquerque Boston
(A)
(B)
Des Moines
(D)
5
Cleveland Factory
(C) Capacity
4
3
100
100
Evansville
(E)
8
Ft Lauderdale
(F)
9
200
Warehouse
Req.
300
4
7
5
100
200
10-17
300
100
200
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
3
200
300
700
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-18
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Vogel’s Approximation
1. For each row/column of table,
find difference between two
lowest costs. (Opportunity cost)
2. Find greatest opportunity cost.
3. Assign as many units as
possible to lowest cost square in
row/column with greatest
opportunity cost.
4. Eliminate row or column which
has been completely satisfied.
4. Begin again, omitting
eliminated rows/columns.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-19
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Special Problems in
Transportation Method
• Unbalanced Problem
• Demand Less than Supply
• Demand Greater than Supply
• Degeneracy
• More Than One Optimal
Solution
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-20
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Unbalanced Problem
Demand Less than Supply
Customer
Dummy
Customer
Factory
1
2
Capacity
8
5
0
Factory 1
170
Factory 2
15
Factory 3
3
Customer
Requirements 150
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10
0
130
9
0
80
80
10-21
150
380
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Unbalanced Problem
Supply Less than Demand
Customer Customer Customer Factory
2
1
3
Capacity
Factory 1
8
Factory 2
15
Dummy
0
Customer
Requirements 150
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
5
16
170
10
7
130
0
0
80
80
10-22
150
380
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Degeneracy
Customer Customer Customer Factory
2
1
3
Capacity
Factory 1
5
4
3
100
Factory 2
100
8
4
Customer
Requirements 100
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
9
7
100
10-23
120
20
100
Factory 3
3
5
80
80
100
300
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Degeneracy - Coming Up!
Customer Customer Customer Factory
2
1
3
Capacity
Factory 1
8
5
16
70
70
15
Factory 2
50
10
9
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10
50
30
Customer
Requirements 150
130
80
3
Factory 3
7
80
10-24
50
80
280
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Stepping-Stone Method The Des Moines-toCleveland Route
Albuquerque
Boston
(A)
(B)
Des Moines
(D)
5
4 Start
3
100
100
-
+
8
Evansville
(E)
4
200
3
300
100
-
+
9
Fort
Lauderdale
(F)
Warehouse
Req.
Cleveland
Factory
(C)
Capacity
7
100
+
300
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
5
200
300
200
10-25
200
700
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Assignment Problem
Person
Adams
Brown
Cooper
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
1
$11
$8
$9
10-26
Project
2
$14
$10
$12
3
$6
$11
$7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-27
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-28
PG 10.13b
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Hungarian Method
Initial Table
Person
Project
1
2
3
Adams
11
14
6
Brown
8
10
11
Cooper
9
12
7
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-29
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Hungarian Method
Row Reduction
Person
Project
1
2
3
Adams
5
8
0
Brown
0
2
3
Cooper
2
5
0
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-30
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Hungarian Method
Column Reduction
Person
Project
1
2
3
Adams
5
6
0
Brown
0
0
3
Cooper
2
3
0
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-31
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Hungarian Method
Testing
Person
Project
1
Adams
5
Covering
Line 2
2
3
6
0
Brown
0
0
3
Cooper
2
3
0
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-32
Covering
Line 1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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 Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-33
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Hungarian Method
Testing
Person
Covering
Covering
Line 1 Project Line 3
1
2
3
3
4
0
Adams
Brown
Cooper
0
0
5
0
1
0
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
10-34
Covering
Line 2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Hungarian Method
Assignments
Person
Project
1
2
6
Adams
Brown
Cooper
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
3
10
9
10-35
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Maximization Assignment
Problem
Adams
Brown
Cooper
Davis
1
$11
$8
$9
$10
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
Project
2
3 Dummy
$14 $6
$0
$10 $11
$0
$12 $7
$0
$13 $8
$0
10-36
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Maximization Assignment
Problem
Adams
Brown
Cooper
Davis
1
$32
$6
$5
$4
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
Project
2
3 Dummy
$0
$8
$14
$4
$3
$14
$2 $77
$14
$1
$6
$14
10-37
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458