SUPPLEMENT
Solving Transportation Problems
Reid & Sanders, Operations Management
© Wiley 2002
C
Learning Objectives
• Define the problem & prepare the
transportation tableau
• Obtain an initial feasible solution
• Identify the optimal solution
• Understand special situations
Reid & Sanders, Operations Management
© Wiley 2002
Page 2
Transportation Problems
• Transportation problems determine how
much of the demand at each one of several
destinations is supplied by each one of
several sources
• The goal is to minimize costs
• Terminology:
– Points of demand:
• The destination that products are shipped to
– Points of supply:
• Where products are shipped from
Reid & Sanders, Operations Management
© Wiley 2002
Page 3
LP Notation
M
N
Minimize z   cij xij
i 1 j 1
subject to :
M
x
i 1
ij
N
x
j 1
ij
 Si
 Dj
for every source , i  1,...M
for every destinatio n j  1,...N
where :
xij  the number of units transporte d to destinatio n j from source i
cij  the cost of transporti ng one unit from source i to destinatio n j
Reid & Sanders, Operations Management
© Wiley 2002
Page 4
Preparing the Problem
• To use the stepping stone or modified
distribution (MODI) methods, supply
must equal demand.
– If not, create a dummy source or
destination to make up the difference.
– In the solution, shipments from the dummy
source represent unmet demand &
deliveries to a dummy destination
represent excess supply capacity.
Reid & Sanders, Operations Management
© Wiley 2002
Page 5
Required Information
• Demand values for each destination (blue)
• Capacity level at each source (green)
• Cost of delivering 1 unit to each destination from each source (yellow)
Plant
Destinations
Sources
A
2
4
1
3
300
B
8
2
6
5
300
C
6
1
4
2
200
Demand
200
200
300
100
Reid & Sanders, Operations Management
© Wiley 2002
Page 6
Initial Solutions
• Common heuristics (rules of thumb):
– Select a cell & allocate as large a shipment as
possible without violating capacity or demand
constraints (this eliminates a row or column
constraint)
– Continue selecting new cells until all row & column
constraints are satisfied
• Examples:
– Northwest Corner Method (NWC)
– Vogel’s Approximation Method (VAM)
Reid & Sanders, Operations Management
© Wiley 2002
Page 7
Northwest Corner Method
• Begin in the upper left-hand corner of the
tableau (the NW corner)
• Assign the largest shipment possible
– If the column constraint is satisfied, move to the
column on the right
– If the row constraint is satisfied, move to the row
below
• Continue until all row & column constraints
are satisfied
Reid & Sanders, Operations Management
© Wiley 2002
Page 8
NWC Example: Step 1
Plant
Destinations
Sources
4
1
3
300
B
8
2
6
5
300
6
1
4
2
200
200
200
300
100
C
Demand
Column Satisfied
A
200
2
Reid & Sanders, Operations Management
© Wiley 2002
Page 9
NWC Example: Step 2
Plant
Destinations
Sources
300 – 200 =
A
200
2
100
4
B
8
2
6
5
300
C
6
1
4
2
200
Demand
200
200
300
100
Reid & Sanders, Operations Management
© Wiley 2002
1Row Satisfied
3
100
Page 10
NWC Example: Step 3
Plant
Destinations
Sources
100
4
1
3
300
B
8
1002
6
5
300
C
6
1
4
2
200
300
100
Demand
200
Reid & Sanders, Operations Management
© Wiley 2002
Column Satisfied
A
200
2
200 – 100
= 100
Page 11
NWC Example: Step 4
Plant
Destinations
Sources
200
2
100
4
B
8
100
2
C
6
1
4
2
Demand
200
200
300
100
A
Reid & Sanders, Operations Management
© Wiley 2002
1
3
200
6Row Satisfied
5
300
300 – 100 =
200
200
Page 12
NWC Example: Step 5
Plant
Destinations
A
200
2
100
4
B
8
C
Demand
Sources
3
300
1002
200
6
5
300
6
1
100
4
2
200
200
200
Reid & Sanders, Operations Management
© Wiley 2002
Column Satisfied
1
300 – 200
= 100
100
Page 13
NWC Example: Step 6
Plant
Destinations
Sources
A
200
2
100
4
1
3
300
B
8
1002
200
6
5
300
100
4
100
2
C
6
1
Demand
200
200
Reid & Sanders, Operations Management
© Wiley 2002
300
200 –100 =
100
100
Page 14
NWC Initial Solution
Plant
Destinations
Sources
A
200
2
100
4
1
3
300
B
8
1002
200
6
5
300
C
6
1
100
4
100
2
200
Demand
200
200
Reid & Sanders, Operations Management
© Wiley 2002
300
100
Page 15
Limitations
• NW Corner Method ignores the
objective function coefficients (costs)
• Solution often isn’t very good:
Total cost:
200 units ($2) + 100 units ($4) + 100 units ($2) + 200
units ($6) + 100 units ($4) + 100 units ($2) =
$2800 to transport the 800 units
Reid & Sanders, Operations Management
© Wiley 2002
Page 16
Vogel’s Approximation Method
• Compute penalties for each row & column:
– Compute penalties by subtracting the smallest cij
from the next smallest cij
• Select the row or column with the largest
penalty
• Select the cell with the lowest cij
• Allocate as many units as possible to that cell
• Continue until all constraints are satisfied
Reid & Sanders, Operations Management
© Wiley 2002
Page 17
VAM Example: Step 1
Plant
B
C
Demand
Penalties
200
2
Sources
Penalties
4
1
3
300
1
8
2
6
5
300
3
6
1
4
2
200
1
200
200
300
100
4
1
3
1
Column Satisfied
A
Destinations
Reid & Sanders, Operations Management
© Wiley 2002
Page 18
VAM Example: Step 2
Plant
Destinations
Sources
Penalties
A
200
2
4
1
3
100
2
B
8
2
6
5
300
3
C
6
1
4
2
200
1
Demand
200
200
300
100
Tied
1
3
1
Penalties
Reid & Sanders, Operations Management
© Wiley 2002
Page 19
Arbitrarily Chose 3rd Destination
Plant
Destinations
Sources
Penalties
A
200
2
4
100
1
B
8
2
6
5
300
3
C
6
1
4
2
200
1
Demand
200
200
300
100
1
3
1
Penalties
Reid & Sanders, Operations Management
© Wiley 2002
Row3 Satisfied
100
2
Page 20
VAM Example: Step 3
Plant
Destinations
A
200
2
B
Sources
Penalties
4
100
1
3
300
8
2
6
5
300
3
C
6
1
4
2
200
1
Demand
200
200
200
100
Tied
1
2
3
Penalties
Reid & Sanders, Operations Management
© Wiley 2002
Page 21
Arbitrarily Chose 4th Destination
Destinations
A
200
2
4
100
1
B
8
2
6
Sources
Penalties
300
2
5
300
3
100
2
200
1
3
Column Satisfied
Plant
C
6
1
4
Demand
200
200
200
100
1
2
3
Penalties
Reid & Sanders, Operations Management
© Wiley 2002
Page 22
VAM Example: Step 4
Plant
B
200
2
4
8
200
2
C
6
Demand
200
Penalties
Reid & Sanders, Operations Management
© Wiley 2002
Sources
Penalties
100
1
3
300
2
6
5
300
4
1
4
100
2
100
3
200
200
1
2
Column Satisfied
A
Destinations
100
Page 23
VAM Example: Step 5
(only 1 column left & only one feasible solution)
Plant
A
B
Destinations
200
2
8
4
3
300
200
2
100
6
5
100
3
100
4
100
2
100
1
6
1
Demand
200
200
Reid & Sanders, Operations Management
© Wiley 2002
Penalties
100
1
C
Penalties
Sources
300
100
2
Page 24
VAM Initial Solution
Plant
A
B
Destinations
200
2
8
4
100
1
3
300
200
2
100
6
5
300
100
4
100
2
200
C
6
1
Demand
200
200
Reid & Sanders, Operations Management
© Wiley 2002
Sources
300
100
Page 25
Better Initial Solution
• Total Costs:
200 units ($2) + 100 units ($1) + 200 units
($2) + 100 units ($6) +100 units ($4) + 100
units ($2) =
$2100 to transport the 800 units
• Compared to $2800 using the Northwest
Corner Method
Reid & Sanders, Operations Management
© Wiley 2002
Page 26
Finding the Optimal Solution
• Initial solutions are feasible, but may not
be optimal
• Use the Stepping Stone or Modified
Distribution Method to identify
improvements & confirm optimality
Reid & Sanders, Operations Management
© Wiley 2002
Page 27
The End
Copyright © 2002 John Wiley & Sons, Inc. All rights reserved.
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use of these programs or from the use of the information
contained herein.
Reid & Sanders, Operations Management
© Wiley 2002
Page 28