Chapter 08S – The Transportation Model
8S–21
Row N2 Index + 3 = 6
Row N2 Index = 6 – 3 = 3.
(4) Use the Row N2 Index to determine the Column C Index, the Dummy Index, and the
Column A Index:
Row N2 Index + Dummy Index = 0
3 + Dummy Index = 0
Dummy Index = 0 – 3 = -3.
Shift to Cell N2-A:
Row N2 Index + Column A Index = 10
Row 2 Index = 5 – 7 = -2.
(c) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-A
8 – (0 + 7) = 1
1-C
7 – (0 + 1) = 6
1-Dummy
0 – (0 + –3) = 3
2-B
10 – (-2 + 3) = 9
2-C
9 – (-2 + 1) = 10
2-Dummy
0 – (-2 + –3) = 5
Because no cell evaluations are negative, we have found the minimum cost solution for
N2 (repeated below).
Chapter 08S – The Transportation Model
8S–22
Location N2 Option
From:
To:
A
B
C
Supply
8
3
7
0
1
500
500
5
10
9
0
2
400
400
10
6
4
0
N2
100
350
500
Demand
400
600
350
1,400 \ 1,400
Total cost = (500 x 3) + (400 x 5) + (100 x 6) + (350 x 4) = $5,500.
Conclusion: Location N2 would result in a lower transportation cost ($5,500).
The transportation cost for Location N1 = $6,750.
Chapter 08S – The Transportation Model
8S–23
3. A manufacturer wants to open a new manufacturing facility. The new facility will have a
supply capacity of 160 units per week. Two locations are being studied, Toledo and
Cincinnati. Transportation costs are shown below. Select the location for the new
manufacturing facility that will minimize total transportation cost.
Toledo Option
From:
To:
A
B
C
Supply
10
14
10
1
210
12
17
20
2
140
11
11
12
3
150
18
8
13
Toledo
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.
(b) Find the cell in the table above that has the lowest unit transportation cost. Cell Toledo-B
has the lowest cost ($8). Assign as many units as possible to this cell: minimum of 160 &
220 = 160. This exhausts the Row Toledo total, so cross out 160, and cross out the cell
costs for Row Toledo. Revise the Column B total to 60. The result is shown below.
From:
To:
A
B
C
Supply
10
14
10
1
210
12
17
20
2
140
11
11
12
3
150
18
8
13
Toledo
160
160
Demand
220
220 60
220
660 \ 660
Chapter 08S – The Transportation Model
(c) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 1-A and Cell 1-C are tied for the next lowest cost ($10). Break the
tie arbitrarily by assigning as many units as possible to Cell 1-A: minimum of 210 & 220
= 210. This exhausts the Row 1 total, so cross out 210, and cross out the cell costs for
Row 1. Revise the Column 1 total to 10. The result is shown below.
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
140
11
11
12
3
150
18
8
13
Toledo
160
160
Demand
220 10
220 60
220
660 \ 660
(d) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 3-A and Cell 3-B are tied for the next lowest cost ($11). Break the
10. This exhausts the Column A total, so cross out 10, and cross out the cell costs for
Column A. Revise the Row 3 total to 140. The result is shown below.
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
140
11
11
12
3
10
150 140
18
8
13
Toledo
160
160
Demand
220 10
220 60
220
660 \ 660
(e) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 3-B has the next lowest cost ($11). Assign as many units as
Chapter 08S – The Transportation Model
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
140
11
11
12
3
10
60
150 140
80
18
8
13
Toledo
160
160
Demand
220 10
220 60
220
660 \ 660
(f) Find the cell (that is not crossed out) in the table above that has the next lowest unit
Supply
10
14
10
12
17
20
2
140
11
11
12
3
10
60
80
150 140
80
18
8
13
Demand
Chapter 08S – The Transportation Model
8S–26
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
140
140
11
11
12
3
10
60
80
150
18
8
13
Toledo
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.
Row Index + Column Index = Cell Cost
From:
To:
A
(Index = 10)
B
(Index = 10)
C
(Index = 11)
Supply
10
14
10
1 (Index = 0)
210
210
12
17
20
2 (Index = 9)
140
140
11
11
12
3 (Index = 1)
10
60
80
150
18
8
13
Toledo
(Index = –2)
160
160
Demand
220
220
220
660 \ 660
Row 1 Index + Column A Index = 10
0 + Column A Index = 10
Column A Index = 10 – 0 = 10.
(3) There are no other occupied cells in Row 1, so shift to Row 3, Cell 3-A:
Row 3 Index + Column A Index = 11
Column B Index = 11 – 1 = 10.
Shift to Cell 3-C:
Row 3 Index + Column C Index = 12
1 + Column C Index = 12
Column C Index = 12 – 1 = 11.
Cell Toledo-B:
Row Toledo Index + Column B Index = 8
Row Toledo Index + 10 = 8
Row Toledo Index = 8 – 10 = –2.
(c) Evaluate the empty cells using the following formula:
Chapter 08S – The Transportation Model
8S–30
(f) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-B
14 – (0 + 17) = –3
1-C
10 – (0 + 18) = –8
2-B
17 – (2 + 17) = –2
3-A
11 – (-6 + 10) = 7
Toledo-A
18 – (-9 + 10) = 17
Toledo-C
13 – (-9 + 13) = 9
Three cells have negative evaluations: Cell 1-B (-3), Cell 1-C (-8), & Cell 2-B (-2). Cell 1-
C has the largest negative value, so we shift as many units as possible to Cell 1-C. The
stepping stone path for Cell 1-C is shown below.
Cell 1-C Stepping Stone Path
From:
To:
A
B
C
Supply
(-)
10
14
(+)
10
1
210
210
(+)
12
17
(-)
20
2
10
130
140
11
11
12
3
60
90
150
18
8
13
Toledo
160
160
Demand
220
220
220
660 \ 660