Chapter 08S – The Transportation Model
From:
To:
A
B
C
Supply
10
14
10
1
80
130
210
12
17
20
2
140
140
11
11
12
3
60
90
150
18
8
13
Toledo
160
160
Demand
220
220
220
660 \ 660
(g) Test for degeneracy in the table above.
The number of occupied cells must equal R + C – 1.
The number of occupied cells = 6.
(h) Obtain an index number of each row and column. Do this using only occupied cells. Index
for Row 1 = 0. For other rows and columns, the following holds true:
Row Index + Column Index = Cell Cost
From:
To:
A
(Index = 10)
B
(Index = 9)
C
(Index = 10)
Supply
10
14
10
1 (Index = 0)
80
130
210
12
17
20
2 (Index = 2)
140
140
11
11
12
3 (Index = 2)
60
90
150
18
8
13
Toledo
(Index = –1)
160
160
Demand
220
220
220
660 \ 660
Chapter 08S – The Transportation Model
8S–32
(1) Row 1 Index = 0.
(2) Shift to Cell 1-A:
Row 1 Index + Column A Index = 10
0 + Column A Index = 10
Column A Index = 10 – 0 = 10.
Use the Column A Index to determine the Row 2 Index:
Row 1 Index + Column C Index = 10
0 + Column C Index = 10
Column C Index = 10 – 0 = 10.
(3) There are no other occupied cells in Row 1, so shift to Row 3, Cell 3-C:
Row 3 Index + Column C Index = 12
Column B Index = 11 – 2 = 9.
(4) There are no other occupied cells in Row 3, so shift to Row Toledo, Cell Toledo-B:
Row Toledo Index + Column B Index = 8
Row Toledo Index + 9 = 8
Row Toledo Index = 8 – 9 = –1.
(i) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-B
14 – (0 + 9) = 5
2-B
17 – (2 + 9) = 6
2-C
20 – (2 + 10) = 8
3-A
11 – (2 + 10) = –1
Toledo-A
18 – (-1 + 10) = 9
Toledo-C
13 – (-1 + 10) = 4
One cell has a negative evaluation: Cell 3-A (-1). Shift as many units as possible to Cell 3-
A. The stepping stone path for Cell 3-A is shown below.
To:
10
14
10
12
17
20
2
140
140
11
11
12
3
60
10
150
8
13
160
160
Cell 3-A Stepping Stone Path
From:
To:
A
B
C
Supply
(-)
10
14
(+)
10
1
80
130
210
12
17
20
2
140
140
(+)
11
11
(-)
12
3
60
90
150
18
8
13
Toledo
160
160
Demand
220
220
220
660 \ 660
Chapter 08S – The Transportation Model
8S–34
(k) Obtain an index number of each row and column. Do this using only occupied cells. Index
for Row 1 = 0. For other rows and columns, the following holds true:
Row Index + Column Index = Cell Cost
From:
To:
A
(Index = 9)
B
(Index = 9)
C
(Index = 10)
Supply
10
14
10
1 (Index = 0)
210
210
12
17
20
2 (Index = 3)
140
140
11
11
12
3 (Index = 2)
80
60
10
150
18
8
13
Toledo
(Index = –1)
160
160
Demand
220
220
220
660 \ 660
(1) Row 1 Index = 0.
(2) Shift to Cell 1-C:
Row 1 Index + Column C Index = 10
0 + Column C Index = 10
Column C Index = 10 – 0 = 10.
(3) There are no other occupied cells in Row 1, so shift to Row 3, Cell 3-C:
2 + Column A Index = 11
Column A Index = 11 – 2 = 9.
(4) There are no other occupied cells in Row 3, so shift from Cell 3-A to Row 2, Cell 2-
A:
Row 2 Index + Column A Index = 12
Row 2 Index + 9 = 12
Chapter 08S – The Transportation Model
(l) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-A
10 – (0 + 9) = 1
1-B
14 – (0 + 9) = 5
2-B
17 – (3 + 9) = 5
2-C
20 – (3 + 10) = 7
Toledo-A
18 – (-1 + 9) = 10
Toledo-C
13 – (-1 + 10) = 4
Because no cell evaluations are negative, we have found the minimum cost solution for
Toledo (repeated below).
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
140
140
11
11
12
3
80
60
10
150
18
8
13
Toledo
160
160
Demand
220
220
220
660 \ 660
Chapter 08S – The Transportation Model
Cincinnati Option
From:
To:
A
B
C
Supply
10
14
10
1
210
12
17
20
2
140
11
11
12
3
150
7
17
13
Cinci
160
Demand
220
220
220
660 \ 660
Step 1: Initial Solution with Intuitive Lowest-Cost Approach:
(a) Check to see if supply and demand are equal. They are equal—no dummy is necessary.
To:
10
14
10
1
210
12
17
20
2
140
11
11
12
3
150
7
17
13
Chapter 08S – The Transportation Model
From:
To:
A
B
C
Supply
10
14
10
1
60
150
210 150
12
17
20
2
140
11
11
12
3
150
150
7
17
13
Cinci
160
160
Demand
220 60
220 70
220 70
660 \ 660
(f) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 2-B has the next lowest cost ($17). Assign as many units as
Supply
10
14
10
12
17
20
2
70
11
11
12
3
150
150
7
17
13
Demand
Chapter 08S – The Transportation Model
From:
To:
A
B
C
Supply
10
14
10
1
60
150
210
12
17
20
2
70
70
140
11
11
12
3
150
150
7
17
13
Cinci
160
160
Demand
220
220
220
660 \ 660
Step 2: Evaluate empty cells with the MODI method:
(a) Test for degeneracy in the table above.
The number of occupied cells must equal R + C – 1.
The number of occupied cells = 6.
R + C – 1 = 4 + 3 – 1 = 6.
10
14
10
60
150
210
12
17
20
11
11
12
7
17
13
8S–40
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
(1) Row 1 Index = 0.
(2) Shift to Cell 1-A:
Row 1 Index + Column A Index = 10
0 + Column A Index = 10
Column A Index = 10 – 0 = 10.
Shift to Cell 1-C:
Row 1 Index + Column C Index = 10
0 + Column C Index = 10
Column C Index = 10 – 0 = 10.
Row 2 Index + Column B Index = 17
10 + Column B Index = 17
Column B Index = 17 – 10 = 7.
(4) There are no other occupied cells in Row 2, so shift to Row 3, Cell 3-C:
Row 3 Index + Column B Index = 11
Row Cinci Index + 10 = 7
Row Cinci Index = 7 – 10 = –3.
(c) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-B
14 – (0 + 7) = 7
2-A
12 – (10 + 10) = –8
3-A
11 – (4 + 10) = –3
3-C
12 – (4 + 10) = –2
Cinci-B
17 – (-3 + 7) = 13
Cinci-C
13 – (-3 + 10) = 6
Three cells have negative evaluations: Cell 2-A (-8), Cell 3-A (-3), & Cell 3-C (-2). Cell
2-A has the largest negative value, so we shift as many units as possible to Cell 2-A. The
stepping stone path for Cell 2-A is shown below.