Chapter 08S – The Transportation Model
8S–41
Cell 2-A Stepping Stone Path
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
The quantities in the cells that have – signs are potential candidates for shifting units. Cell
1-A has 60 units and Cell 2-C has 70 units. Therefore, 60 units can be shifted. The result
is shown below.
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
60
70
10
140
11
11
12
3
150
150
7
17
13
Cinci
160
160
Demand
220
220
220
660 \ 660
Chapter 08S – The Transportation Model
(d) 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 = 2)
B
(Index = 7)
C
(Index = 10)
Supply
10
14
10
1 (Index = 0)
210
210
12
17
20
2 (Index = 10)
60
70
10
140
11
11
12
3 (Index = 4)
150
150
7
17
13
Cinci
(Index = 5)
160
160
Demand
220
220
220
660 \ 660
(1) Row 1 Index = 0.
Row 2 Index + Column C Index = 20
Row 2 Index + 10 = 20
Row 2 Index = 20 – 10 = 10.
Shift to Cell 2-B:
Row 2 Index + Column B Index = 17
Column A Index = 12 – 10 = 2.
(4) There are no other occupied cells in Row 2, so shift to Row Cinci, Cell Cinci-A:
Row Cinci Index + Column A Index = 7
Row Cinci Index + 2 = 7
Row Cinci Index = 7 – 2 = 5.
Chapter 08S – The Transportation Model
(5) There are no other occupied cells in Row Cinci, so we shift from Cell 2-B to Row 3,
Cell 3-B:
Row 3 Index + Column B Index = 11
Row 3 Index + 7 = 11
Row 3 Index = 11 – 7 = 4.
(f) Evaluate the empty cells using the following formula:
Chapter 08S – The Transportation Model
8S–44
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
60
80
140
11
11
12
3
140
10
150
7
17
13
Cinci
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.
R + C – 1 = 4 + 3 – 1 = 6.
The solution above is not degenerate.
(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 = 4)
B
(Index = 9)
C
(Index = 10)
Supply
10
14
10
1 (Index = 0)
210
210
12
17
20
2 (Index = 8)
60
80
140
11
11
12
3 (Index = 2)
140
10
150
7
17
13
Cinci
(Index = 3)
160
160
Demand
220
220
220
660 \ 660
Chapter 08S – The Transportation Model
8S–45
(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:
Row 3 Index + Column C Index = 12
Row 3 Index + 10 =12
Row 2 Index = 17 – 9 = 8.
Shift to Cell 2-A:
Row 2 Index + Column A Index = 12
8 + Column A Index = 12
Column A Index = 12 – 8 = 4.
(5) There are no other occupied cells in Row 2, so shift to Row Cinci, Cell Cinci-A:
Cell
Evaluation
1-A
10 – (0 + 4) = 6
1-B
14 – (0 + 9) = 5
2-C
20 – (8 + 10) = 2
3-A
11 – (2 + 4) = 5
Cinci-B
17 – (3 + 9) = 5
Cinci-C
13 – (3 + 10) = 0
Because no cell evaluations are negative, we have found the minimum cost solution for
Cincinnati (repeated below). The fact that the evaluation for Cell Cinci-C is zero indicates
that at least one other equivalent alternative exists.
Chapter 08S – The Transportation Model
Cincinnati Option
From:
To:
A
B
C
Supply
10
14
10
1
210
210
12
17
20
2
60
80
140
11
11
12
3
140
10
150
7
17
13
Cinci
160
160
Demand
220
220
220
660 \ 660
Toledo Option (Optimal)
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
Total cost = (210 x 10) + (140 x 12) + (80 x 11) + (60 x 11) + (10 x 12) + (160 x 8) =
$6,720.
To:
15
9
4
1
10
7
11
2
340
14
18
5
4. A large retailer is planning to open a new store in one of three locations: SCP, FI, and LH.
Each of the store locations has a demand potential of 300 units per week. Determine the store
location that would minimize transportation costs for the system.
SCP Option
From:
To:
A
B
SCP
Supply
15
9
4
1
660
10
7
11
2
340
14
18
5
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.
Chapter 08S – The Transportation Model
From:
To:
A
B
SCP
Supply
15
9
4
1
300
660 360
10
7
11
2
340
340
14
18
5
3
200
Demand
400
500 160
300
1200 \ 1200
The result is shown below.
From:
To:
A
B
SCP
Supply
15
9
4
1
160
300
660 360
200
10
7
11
2
340
340
14
18
5
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
possible to this cell: minimum of 200 & 400 = 200. This exhausts the Row 3 total, so
cross out 200, and cross out the cell costs for Row 3. Revise the Column 1 total to 200.
8S–49
(f) Find the cell (that is not crossed out) in the table above that has the next lowest unit
Column A totals. The initial solution is shown below.
From:
To:
A
B
SCP
Supply
15
9
4
1
200
160
300
660
10
7
11
2
340
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200
Total cost = (200 x 15) + (160 x 9) + (300 x 4) + (340 x 7) + (200 x 14) = $10,820.
Step 2: Evaluate empty cells with the MODI method:
(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)
SCP
(Index = 4)
Supply
15
9
4
1 (Index = 0)
200
160
300
660
10
7
11
2 (Index = –2)
340
340
14
18
5
3 (Index = –1)
200
200
Demand
400
500
300
1200 \ 1200
8S–50
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 = 15
0 + Column A Index = 15
Column A Index = 15 – 0 = 15.
Shift to Cell 1-SCP:
Row 1 Index + Column SCP Index = 4
0 + Column SCP Index = 4
(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 2 Index + 9 = 7
Row 2 Index = 7 – 9 = -2.
(4) There are no other occupied cells in Row 2, so we shift from Cell 1-A to Row 3, Cell
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
2-A
10 – (-2 + 15) = –3
2-SCP
11 – (-2 + 4) = 9
3-B
18 – (-1 + 9) = 10
3-SCP
5 – (-1 + 4) = 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
SCP
Supply
(-)
15
(+)
9
4
1
200
160
300
660
(+)
10
(-)
7
11
2
340
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200