Chapter 08S – The Transportation Model
8S–81
Because no cell evaluations are negative, we have found the optimal solution (repeated
below):
Optimal Solution
From:
To:
A
B
C
Supply
3
4
2
1
15
25
40
5
1
7
2
15
45
60
8
7
4
3
50
50
Demand
30
45
75
150 \ 150
8S–82
6. a. Obtain the optimal distribution plan. Develop the initial solution using the intuitive
lowest-cost approach. Use MODI for cell evaluations.
Given:
From:
To:
A
B
C
Supply
3
6
2
1
40
3
1
3
2
50
7
6
4
3
65
Demand
55
55
45
155 \ 155
(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 2–B
for Row 2. Revise the Column B total to 5. The result is shown below.
From:
To:
A
B
C
Supply
3
6
2
1
40
3
1
3
2
50
50
7
6
4
3
65
Demand
55
55 5
45
155 \ 155
(c) 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 ($2). Assign as many units as
possible to this cell: minimum of 40 & 45 = 40. This exhausts the Row 1 total, so
cross out 40, and cross out the cell costs for Row 1. Revise the Column C total to 5.
The result is shown below.
Chapter 08S – The Transportation Model
From:
To:
A
B
C
Supply
3
6
2
1
40
40
3
1
3
2
50
50
7
6
4
3
65
Demand
55
55 5
45 5
155 \ 155
The result is shown below.
From:
To:
A
B
C
Supply
3
6
2
1
40
40
3
1
3
2
50
50
7
6
4
3
5
65 60
Demand
55
55 5
45 5
155 \ 155
cross out 5, and cross out the cell costs for Column B. Revise the Row 3 total to 55.
The result is shown below.
Chapter 08S – The Transportation Model
From:
To:
A
B
C
Supply
3
6
2
1
40
40
3
1
3
2
50
50
7
6
4
3
5
5
65 60 55
Demand
55
55 5
45 5
155 \ 155
possible to this cell: minimum of 55 & 55 = 55. This exhausts the Row 3 and Column
A totals, so we are finished. The initial solution is shown below.
From:
To:
A
B
C
Supply
3
6
2
1
40
40
3
1
3
2
50
50
7
6
4
3
55
5
5
65
Demand
55
55
45
155 \ 155
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
(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 = 5)
B (Index = 4)
C (Index = 2)
Supply
3
6
2
1 (Index = 0)
40
40
3
1
3
2 (Index = –3)
50
50
7
6
4
3 (Index = 2)
55
5
5
65
Demand
55
55
45
155 \ 155
(1) Row 1 Index = 0.
(2) Shift to Cell 1-C:
Row 1 Index + Column C Index = 2
0 + Column C Index = 2
Column C Index = 2 – 0 = 2.
(3) There are no other occupied cells in Row 1, so we shift to Row 3, Cell 3-C:
Row 3 Index + Column C Index = 4
Row 3 Index + 2 = 4
Row 3 Index = 4 – 2 = 2.
(4) We use the Row 3 Index to determine the Column A Index & the Column B
Index.
Row 3 Index + Column B Index = 6
2 + Column B Index = 6
Column B Index = 6 – 2 = 4.
(5) There are no other occupied cells in Row 3, so we shift to Row 2, Cell 2-B:
Row 2 Index + Column B Index = 1
Chapter 08S – The Transportation Model
8S–86
(c) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-A
3 – (0 + 5) = –2
1-B
6 – (0 + 4) = 2
2-A
3 – (-3 + 5) = 1
2-C
3 – (-3 + 2) = 4
Cell 1-A Stepping Stone Path
From:
To:
A (Index = 5)
B (Index = 4)
C (Index = 2)
Supply
(+)
3
6
(-)
2
1 (Index = 0)
40
40
3
1
3
2 (Index = –3)
50
50
(-)
7
6
(+)
4
3 (Index = 2)
55
5
5
65
Demand
55
55
45
155 \ 155
The quantities in the cells that have – signs are potential candidates for shifting units.
Cell 1-C has 40 units and Cell 3-A has 55 units. Therefore, 40 units can be shifted.
The result is shown below.
From:
To:
A
B
C
Supply
3
6
2
1
40
40
3
1
3
2
50
50
7
6
4
3
15
5
45
65
Demand
55
55
45
155 \ 155
Total cost = (40 x 3) + (50 x 1) + (15 x 7) + (5 x 6) + (45 x 4) = $485.
8S–87
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
R + C – 1 = 3 + 3 – 1 = 5.
The solution above is not degenerate.
(e) 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 = 3)
B (Index = 2)
C (Index = 0)
Supply
3
6
2
1 (Index = 0)
40
40
3
1
3
2 (Index = –1)
50
50
7
6
4
3 (Index = 4)
15
5
45
65
Demand
55
55
45
155 \ 155
(1) Row 1 Index = 0.
Row 3 Index + Column A Index = 7
Row 3 Index + 3 = 7
Row 3 Index = 7 – 3 = 4.
(4) We use the Row 3 Index to determine the Column C Index and the Column B
Index:
Row 3 Index + Column B Index = 6
4 + Column B Index = 6
Column B Index = 6 – 4 = 2.
(5) There are no other occupied cells in Row 3, so shift to Row 2, Cell 2-B:
Row 2 Index + Column B Index = 1
Chapter 08S – The Transportation Model
8S–88
(f) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-B
6 – (0 + 2) = 4
1-C
2 – (0 + 0) = 2
2-A
3 – (-1 + 3) = 1
2-C
3 – (-1 + 0) = 4
Because no cell evaluations are negative, we have found the optimal solution
(repeated below):
From:
To:
A
B
C
Supply
3
6
2
1
40
40
3
1
3
2
50
50
7
6
4
3
15
5
45
65
Demand
55
55
45
155 \ 155
b. Total cost = (40 x 3) + (50 x 1) + (15 x 7) + (5 x 6) + (45 x 4) = $485.
c. There is not an alternate optimal solution. No cell evaluations = 0.
8S–89
7. a. Develop the initial solution using the intuitive lowest-cost approach.
Given:
From:
To:
A
B
C
D
Supply
18
12
14
16
1
40
23
24
27
33
2
80
42
34
31
26
3
130
Demand
90
80
30
50
250 \ 250
(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-B
has the lowest cost ($12). Assign as many units as possible to this cell: minimum of
From:
To:
A
B
C
D
Supply
18
12
14
16
1
40
40
23
24
27
33
2
80
42
34
31
26
3
130
Demand
90
80 40
30
50
250 \ 250
The result is shown below.
Chapter 08S – The Transportation Model
8S–90
From:
To:
A
B
C
D
Supply
18
12
14
16
1
40
40
23
24
27
33
2
80
80
42
34
31
26
3
130
Demand
90 10
80 40
30
50
250 \ 250
(d) Find the cell (that is not crossed out) in the table above that has the next lowest unit
From:
To:
A
B
C
D
Supply
18
12
14
16
1
40
40
23
24
27
33
2
80
80
42
34
31
26
3
50
130 80
Demand
90 10
80 40
30
50
250 \ 250
possible to this cell: minimum of 80 & 30 = 30. This exhausts the Column C total, so
cross out 30, and cross out the cell costs for Column C. Revise the Row 3 total to 50.
The result is shown below.