Chapter 08S – The Transportation Model
8S–51
The quantities in the cells that have – signs are potential candidates for shifting units. Cell
1-A has 200 units and Cell 2-B has 340 units. Therefore, 200 units can be shifted. The
result is shown below.
From:
To:
A
B
SCP
Supply
15
9
4
1
360
300
660
10
7
11
2
200
140
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200
(d) Test for degeneracy in the table above.
The number of occupied cells must equal R + C – 1.
(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 = 12)
B
(Index = 9)
SCP
(Index = 4)
Supply
15
9
4
1 (Index = 0)
360
300
660
10
7
11
2 (Index = –2)
200
140
340
14
18
5
3 (Index = 2)
200
200
Demand
400
500
300
1200 \ 1200
Chapter 08S – The Transportation Model
8S–52
(1) Row 1 Index = 0.
(2) Shift to Cell 1-SCP:
Row 1 Index + Column SCP Index = 4
0 + Column SCP Index = 4
Column SCP Index = 4 – 0 = 4.
Shift to Cell 1-B:
Row 1 Index + Column B Index = 9
(4) There are no other occupied cells in Row 2, so shift to Row 3, Cell 3-A:
Row 3 Index + Column A Index = 14
Row 3 Index + 12 = 14
Row 3 Index = 14 – 12 = 2.
(f) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-A
15 – (0 + 12) = 3
2-SCP
11 – (-2 + 4) = 9
3-B
18 – (2 + 9) = 7
3-SCP
5 – (2 + 4) = –1
One cell has a negative evaluation: Cell 3-SCP (-1). Shift as many units as possible to Cell
3-SCP. The stepping stone path for Cell 3-SCP is shown below.
Cell 3-SCP Stepping Stone Path
From:
To:
A
B
SCP
Supply
15
(+)
9
(-)
4
1
360
300
660
(+)
10
(-)
7
11
2
200
140
340
(-)
14
18
(+)
5
3
200
200
Demand
400
500
300
1200 \ 1200
8S–53
The quantities in the cells that have – signs are potential candidates for shifting units. Cell
1-SCP has 300 units, Cell 2-B has 140 units, and Cell 3-A has 200 units. Therefore, 140
units can be shifted. The result is shown below.
From:
To:
A
B
SCP
Supply
15
9
4
1
500
160
660
10
7
11
2
340
340
14
18
5
3
60
140
200
Demand
400
500
300
1200 \ 1200
Total cost = (500 x 9) + (160 x 4) + (340 x 10) + (60 x 14) + (140 x 5) = $10,080.
(g) 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 = 13)
B
(Index = 9)
SCP
(Index = 4)
Supply
15
9
4
1 (Index = 0)
500
160
660
10
7
11
2 (Index = –3)
340
340
14
18
5
3 (Index = 1)
60
140
200
Demand
400
500
300
1200 \ 1200
Chapter 08S – The Transportation Model
(1) Row 1 Index = 0.
(2) Shift to Cell 1-B:
Row 1 Index + Column B Index = 9
0 + Column B Index = 9
Column B Index = 9 – 0 = 9.
Shift to Cell 1-SCP:
Row 1 Index + Column SCP Index = 4
0 + Column SCP Index = 4
Column SCP Index = 4 – 0 = 4.
(3) There are no other occupied cells in Row 1, so shift to Row 3, Cell 3-SCP:
1 + Column A Index = 14
Column A Index = 14 – 1 = 13.
(4) There are no other occupied cells in Row 3, so shift to Row 2, Cell 2-A:
Row 2 Index + Column A Index = 10
Row 2 Index + 13 = 10
8S–55
FI Option
From:
To:
A
B
FI
Supply
15
9
7
1
660
10
7
6
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.
(b) Find the cell in the table above that has the lowest unit transportation cost. Cell 3-FI has
the lowest cost ($5). Assign as many units as possible to this cell: minimum of 200 & 300
= 200. This exhausts the Row 3 total, so cross out 200, and cross out the cell costs for
From:
To:
A
B
FI
Supply
15
9
7
1
660
10
7
6
2
340
14
18
5
3
200
200
Demand
400
500
300 100
1200 \ 1200
Chapter 08S – The Transportation Model
From:
To:
A
B
FI
Supply
15
9
7
1
660
10
7
6
2
100
340 240
14
18
5
3
200
200
Demand
400
500
300 100
1200 \ 1200
(d) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost. Cell 2-B has the next lowest cost ($7). Assign as many units as
15
9
7
10
7
6
14
18
5
3
200
200
Chapter 08S – The Transportation Model
From:
To:
A
B
FI
Supply
15
9
7
1
260
660 400
10
7
6
2
240
100
340 240
14
18
5
3
200
200
Demand
400
500 260
300 100
1200 \ 1200
(f) Find the cell (that is not crossed out) in the table above that has the next lowest unit
possible to this cell: minimum of 400 & 400 = 400. This exhausts the Row 1 and the
Column A totals. The initial solution is shown below.
From:
To:
A
B
FI
Supply
15
9
7
1
400
260
660
10
7
6
2
240
100
340
14
18
5
3
200
200
Demand
400
500
300
1200 \ 1200
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.
The number of occupied cells = 5.
R + C – 1 = 3 + 3 – 1 = 5.
The solution above is not degenerate.
Chapter 08S – The Transportation Model
8S–58
From:
To:
A
(Index = 15)
B
(Index = 9)
FI
(Index = 8)
Supply
15
9
7
1 (Index = 0)
400
260
660
10
7
6
2 (Index = –2)
240
100
340
14
18
5
3 (Index = –3)
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
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 2 Index + 9 = 7
Row 2 Index = 7 – 9 = -2.
Row 3 Index + Column FI Index = 5
Row 3 Index + 8 = 5
Row 3 Index = 5 – 8 = -3.
(c) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-FI
7 – (0 + 8) = –1
2-A
10 – (-2 + 15) = –3
3-A
14 – (-3 + 15) = 2
3-B
18 – (-3 + 9) = 12
Two cells have negative evaluations: Cell 1-FI (-1) & 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.
Chapter 08S – The Transportation Model
Cell 2-A Stepping Stone Path
From:
To:
A
B
FI
Supply
(-)
15
(+)
9
7
1
400
260
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 400 units and Cell 2-B has 240 units. Therefore, 240 units can be shifted. The
result is shown below.
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
(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.
The solution above is not degenerate.
(e) Obtain an index number of each row and column. Do this using only occupied cells. Index
Chapter 08S – The Transportation Model
8S–60
From:
To:
A
(Index = 15)
B
(Index = 9)
FI
(Index = 11)
Supply
15
9
7
1 (Index = 0)
160
500
660
10
7
6
2 (Index = –5)
240
100
340
14
18
5
3 (Index = –6)
200
200
Demand
400
500
300
1200 \ 1200
(1) Row 1 Index = 0.
(2) Shift to Cell 1-B:
Row 1 Index + Column B Index = 9
0 + Column B Index = 9
(3) There are no other occupied cells in Row 1, so shift to Row 2, Cell 2-A:
Row 2 Index + Column A Index = 10
Row 2 Index + 15 = 10
Row 2 Index = 10 – 15 = -5.
Shift to Cell 2-FI:
Row 3 Index + 11 = 5
Row 3 Index = 5 – 11 = -6.
(f) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-FI
7 – (0 + 11) = –4
2-B
7 – (-5 + 9) = 3
3-A
14 – (-6 + 15) = 5
3-B
18 – (-6 + 9) = 15
One cell has a negative evaluation: Cell 1-FI (-4). Shift as many units as possible to Cell
1-FI. The stepping stone path for Cell 1-FI is shown below.