Chapter 08S – The Transportation Model
8S–61
Cell 1-FI Stepping Stone Path
From:
To:
A
B
FI
Supply
(-)
15
9
(+)
7
1
160
500
660
(+)
10
7
(-)
6
2
240
100
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200
The quantities in the cells that have – signs are potential candidates for shifting units. Cell
1-A has 160 units and Cell 2-FI has 100 units. Therefore, 100 units can be shifted. The
result is shown below.
From:
To:
A
B
FI
Supply
15
9
7
1
60
500
100
660
10
7
6
2
340
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200
(g) Test for degeneracy in the table above.
(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
Chapter 08S – The Transportation Model
From:
To:
A
(Index = 15)
B
(Index = 9)
FI
(Index = 7)
Supply
15
9
7
1 (Index = 0)
60
500
100
660
10
7
6
2 (Index = – 5)
340
340
14
18
5
3 (Index = –2)
200
200
Demand
400
500
300
1200 \ 1200
(1) Row 1 Index = 0.
(2) Shift to Cell 1-A:
Row 1 Index + Column A Index = 15
0 + Column A Index = 15
Column A Index = 15 – 0 = 15.
Shift to Cell 1-B:
Row 1 Index + Column B Index = 9
0 + Column B Index = 9
Column B Index = 9 – 0 = 9.
Row 3 Index + Column FI Index = 5
Row 3 Index + 7 = 5
Row 3 Index = 5 – 7 = -2.
(4) There are no other occupied cells in Row 3, so shift from Cell 1-A to Row 2, Cell 2-
A:
Row 2 Index + Column A Index = 10
8S–63
FI Solution
From:
To:
A
B
FI
Supply
15
9
7
1
60
500
100
660
10
7
6
2
340
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200
Total cost = (60 x 15) + (500 x 9) + (100 x 7) + (340 x 10) + (200 x 5) = $10,500.
Chapter 08S – The Transportation Model
8S–64
LH Option
From:
To:
A
B
LH
Supply
15
9
5
1
660
10
7
5
2
340
14
18
6
3
200
Demand
400
500
300
1200 \ 1200
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.
(b) Find the cell in the table above that has the lowest unit transportation cost. Cell 1-LH and
From:
To:
A
B
LH
Supply
15
9
5
1
300
660 360
10
7
5
2
340
14
18
6
3
200
Demand
400
500
300
1200 \ 1200
8S–65
From:
To:
A
B
LH
Supply
15
9
5
1
300
660 360
10
7
5
2
340
340
14
18
6
3
200
Demand
400
500 160
300
1200 \ 1200
possible to Cell 1-B: minimum of 360 & 160 = 160. This exhausts the Column B total, so
cross out 160, and cross out the cell costs for Column B. Revise the Row 1 total to 200.
The result is shown below.
From:
To:
A
B
LH
Supply
15
9
5
1
160
300
660 360
200
10
7
5
2
340
340
14
18
6
3
200
Demand
400
500 160
300
1200 \ 1200
(e) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 3-A has the next lowest cost ($14). Assign as many units as
The result is shown below.
Chapter 08S – The Transportation Model
From:
To:
A
B
LH
Supply
15
9
5
1
160
300
660 360
200
10
7
5
2
340
340
14
18
6
3
200
200
Demand
400 200
500 160
300
1200 \ 1200
(f) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 1-A has the next lowest cost ($15). Assign as many units as
To:
15
9
5
10
7
5
14
18
6
3
200
200
Demand
400
500
300
1200 \ 1200
8S–67
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
(b) 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 = 15)
B
(Index = 9)
LH
(Index = 5)
Supply
15
9
5
1 (Index = 0)
200
160
300
660
10
7
5
2 (Index = –2)
340
340
14
18
6
3 (Index = –1)
200
200
Demand
400
500
300
1200 \ 1200
(1) Row 1 Index = 0.
(2) Shift to Cell 1-A:
Row 1 Index + Column A Index = 15
0 + Column A Index = 15
Column A Index = 15 – 0 = 15.
Row 1 Index + Column B Index = 9
0 + Column B Index = 9
Column B Index = 9 – 0 = 9.
(3) There are no other occupied cells in Row 1, so shift to Row 2, Cell 2-B:
Row 2 Index + Column B Index = 7
Row 3 Index = 14 – 15 = -1.
Chapter 08S – The Transportation Model
(c) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
2-A
10 – (-2 + 15) = –3
2-LH
5 – (-2 + 5) = 2
3-B
18 – (-1 + 9) = 10
3-LH
6 – (-1 + 5) = 2
One cell has a negative evaluation: Cell 2-A (-3). Shift as many units as possible to Cell 2-
A. The stepping stone path for Cell 2-A is shown below.
Cell 2-A Stepping Stone Path
From:
To:
A
B
LH
Supply
(-)
15
9
5
1
200
160
300
660
(+)
10
(-)
7
5
2
340
340
14
18
6
3
200
200
Demand
400
500
300
1200 \ 1200
The quantities in the cells that have – signs are potential candidates for shifting units. Cell
Supply
15
9
5
10
7
5
2
200
140
340
14
18
6
Chapter 08S – The Transportation Model
(d) Test for degeneracy in the table above.
The number of occupied cells must equal R + C – 1.
The number of occupied cells = 5.
R + C – 1 = 3 + 3 – 1 = 5.
Chapter 08S – The Transportation Model
(f) Evaluate the empty cells using the following formula: