Chapter 08S – The Transportation Model
8S–121
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
18
16
22
27
0
Baltimore
2
50 48
Demand
41
34 2
35 28
60 4
16
186 \ 186
cross out 28 and cross out the cell costs for Column C. Revise the Row Baltimore total to
20. The result is shown below.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
18
16
22
27
0
Baltimore
2
28
50 48
20
Demand
41
34 2
35 28
60 4
16
186 \ 186
cross out 4 and cross out the cell costs for Column D. Revise the Row Baltimore total to
16. The result is shown below.
Chapter 08S – The Transportation Model
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
18
16
22
27
0
Baltimore
2
28
4
50 48
20 16
Demand
41
34 2
35 28
60 4
16
186 \ 186
(i) Assign the extra 16 units in Row Baltimore to Cell Baltimore-Dummy. The initial solution
is shown below:
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
18
16
22
27
0
Baltimore
2
28
4
16
50
Demand
41
34
35
60
16
186 \ 186
Total cost = (41 x 14) + (7 x 18) + (56 x 16) + (32 x 16) + (2 x 16) + (28 x 22) + (4 x 27)
= $2,864.
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.
Chapter 08S – The Transportation Model
8S–123
(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 = 14)
B
(Index = 12)
C
(Index = 18)
D
(Index = 23)
Dummy
(Index = –4)
Supply
14
24
18
28
0
1 (Index = 0)
41
7
48
17
18
25
16
0
2 (Index = –7)
56
56
30
16
22
30
0
3 (Index = 4)
32
32
18
16
22
27
0
Baltimore
(Index = 4)
2
28
4
16
50
Demand
41
34
35
60
16
186 \ 186
Row 1 Index + Column A Index = 14
0 + Column A Index = 14
Column A Index = 14 – 0 = 14.
Shift to Cell 1-C:
Row 1 Index + Column C Index = 18
Row Baltimore Index + 18 = 22
Row Baltimore Index = 22 – 18 = 4.
Shift to Cell Baltimore-D:
Row Baltimore Index + Column D Index = 27
4 + Column D Index = 27
Shift to Cell Baltimore-B:
Row Baltimore Index + Column B Index = 16
4 + Column B Index = 16
Column B Index = 16 – 4 = 12.
8S–124
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
Row 3 Index + 12 = 16
Row 3 Index = 16 – 12 = 4.
(5) There are no other occupied cells in Row 3, so shift from Cell Baltimore-D to Row 2,
Cell 2-D:
Row 2 Index = 16 – 23 = -7.
(c) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-B
24 – (0 + 12) = 12
1-D
28 – (0 + 23) = 5
1-Dummy
0 – (0 + –4) = 4
2-A
17 – (-7 + 14) = 10
2-B
18 – (-7 + 12) = 13
2-C
25 – (-7 + 18) = 14
2-Dummy
0 – (-7 + –4) = 11
3-A
30 – (4 + 14) = 12
3-C
22 – (4 + 18) = 0
3-D
30 – (4 + 23) = 3
3-Dummy
0 – (4 + –4) = 0
Baltimore-A
18 – (4 + 14) = 0
and Baltimore-A are all zero indicates that at least one other equivalent alternative exists.
Chapter 08S – The Transportation Model
8S–125
Baltimore Option
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
18
16
22
27
0
Baltimore
2
28
4
16
50
Demand
41
34
35
60
16
186 \ 186
Chapter 08S – The Transportation Model
8S–126
Philadelphia Option
From:
To:
A
B
C
D
Supply
14
24
18
28
1
48
17
18
25
16
2
56
30
16
22
30
3
32
31
25
19
20
Philadelphia
50
Demand
41
34
35
60
170 \ 186
(a) Check to see if supply and demand are equal. They are not equal—supply exceeds
demand. We need a dummy warehouse with demand of 16 units (see below). Note: We
make assignments to dummy cells last.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
48
17
18
25
16
0
2
56
30
16
22
30
0
3
32
31
25
19
20
0
Philadelphia
50
Demand
41
34
35
60
16
186 \ 186
(b) Find the cell in the table above that has the lowest unit transportation cost. Cell 1-A has
Chapter 08S – The Transportation Model
8S–127
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
48 7
17
18
25
16
0
2
56
30
16
22
30
0
3
32
31
25
19
20
0
Philadelphia
50
Demand
41
34
35
60
16
186 \ 186
(c) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 2-D and Cell 3-B are tied for next lowest cost ($16). Break the tie
arbitrarily by assigning as many units as possible to Cell 3-B: minimum of 32 & 34 = 32.
This exhausts the Row 3 total, so cross out 32, and cross out the cell costs for Row 3.
Revise the Column B total to 2. The result is shown below.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
48 7
17
18
25
16
0
2
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
50
Demand
41
34 2
35
60
16
186 \ 186
out 56, and cross out the cell costs for Row 2. Revise the Column D total to 4. The result
is shown below.
Chapter 08S – The Transportation Model
8S–128
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
50
Demand
41
34 2
35
60 4
16
186 \ 186
(e) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 1-C has the next lowest cost ($18). Assign as many units as
possible to this cell: minimum of 7 & 35 = 7. This exhausts the Row 1 total, so cross out
7, and cross out the cell costs for Row 1. Revise the Column C total to 28. The result is
shown below.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
50
Demand
41
34 2
35 28
60 4
16
186 \ 186
units as possible to this cell: minimum of 50 & 28 = 28. This exhausts the Column C total,
so cross out 28, and cross out the cell costs for Column C. Revise the Row Philadelphia
total to 22. The result is shown below.
Chapter 08S – The Transportation Model
8S–129
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
28
50 22
Demand
41
34 2
35 28
60 4
16
186 \ 186
so cross out 4, and cross out the cell costs for Column D. Revise the Row Philadelphia
total to 18. The result is shown below.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
28
4
50 22 18
Demand
41
34 2
35 28
60 4
16
186 \ 186
(h) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell Philadelphia-B has the next lowest cost ($25). Assign as many
units as possible to this cell: minimum of 18 & 2 = 2. This exhausts the Column B total, so
cross out 2, and cross out the cell costs for Column B. Revise the Row Philadelphia total
to 16. The result is shown below.
Chapter 08S – The Transportation Model
8S–130
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
2
28
4
50 22 18 16
Demand
41
34 2
35 28
60 4
16
186 \ 186
(i) Assign the extra 16 units in Row Philadelphia to Cell Philadelphia-Dummy. The initial
solution is shown below:
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48
17
18
25
16
0
2
56
56
30
16
22
30
0
3
32
32
31
25
19
20
0
Philadelphia
2
28
4
16
50
Demand
41
34
35
60
16
186 \ 186
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.