Transportation Problems
• Transportation and assignment problems are important
network structured linear programming problems that have
received a great deal of attention in the literature.
• The assignment problem is itself a special case of the
transportation problem.
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Transportation Problems
• Transportation Problems
 Mathematical Modelling
 Identifying Basic Feasible Solution (BFS)
 The Northwest Corner Rule
 The Least Cost Method
 VOGEL (VAM) Method
 Transportation simplex
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The Transportation Model
Characteristics
A product is transported from a number of sources to a
number of destinations at the minimum possible cost.
Each source is able to supply a fixed number of units of the
product, and each destination has a fixed demand for the
product.
The linear programming model has constraints for supply at
each source and demand at each destination.
All constraints are equalities in a balanced transportation
model where supply equals demand.
Constraints contain inequalities in unbalanced models
where supply does not equal demand.
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The Transportation Model
Characteristics
Transportation modelling is an iterative procedure for solving
problems that involve minimizing the cost of shipping products
from a series of sources to a series of destinations.
 Origin Points (or sources) can be factories, warehouses, or
any other points from which goods are shipped.
 Destinations are any points that receive goods.
Transportation models are useful when considering alternative
facility locations. The choice of a new location depends on
which will yield the minimum cost for the entire system.
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The Transportation Model
Characteristics
To use the transportation model we need to know the following:
 The origin points and the capacity or supply per period at
each
 The destination points and the demand per period at each
 The cost of shipping one unit from each origin to each
destination
 In a transportation problem, we have “m” origins (sources),
with origin “i” processing “ai” items and “n” destinations with
destination “j” requiring “bj” items, and with ai = bj.
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Transportation problem: represented as
a LP model
•
•
•
•
•
m- number of sources (origin)
n- number of destinations
ai- supply at source i
bj – demand at destination j
cij – cost of transportation per unit
from source i to destination j
• Xij – number of units to be
transported from the source i to
destination j
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Transportation problem: represented as
a LP model
m
n
Minimize : Z   cij X ij
i 1 j 1
n
subject to
 X ij  ai i  1,2,...., m
j 1
m
 X ij  b j
i 1
j  1,2,....., n
Supply
constraint
Demand
constraint
X ij  0 for i  1,...m and j  1,..n
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The Assignment Problem
Assumptions
1. The number of assignees and the number of tasks are
the same, and is denoted by n.
2. Each assignee is to be assigned to exactly one task.
3. Each task is to be performed by exactly one assignee.
4. There is a cost cij associated with assignee i performing
task j (i, j = 1, 2, …, n).
5. The objective is to determine how well n assignments
should be made to minimize the total cost.
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Transportation problem: Comparison
Transportation LP model
Assignment LP model
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Format of a Transportation Tableau
Problems solved by hand can use a transportation simplex tableau.
Dimensions:
• The transportation simplex tableau has only m rows and n
columns!
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Transportation Problem Model
The feasible solutions property: a transportation problem has
feasible solution if and only if
• If the supplies represent maximum amounts to be distributed, a
dummy destination can be added.
• Similarly, if the demands represent maximum amounts to be
received, a dummy source can be added.
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Transportation Problem Model
• Balanced transportation problems
m
a
i
i 1

n
b
j 1
j
• Unbalanced transportation problems
m
a
i 1
i

n
b
j 1
j
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The Powerco Example
13
The Powerco Example: Graphical
Representation
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The Powerco Example: Problem
Formulation
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The Powerco Example: Problem
Formulation
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The Powerco Example: Balancing a
Transportation Problem
If total supply exceeds total demand:
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The Powerco Example: Balancing a
Transportation Problem
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The Powerco Example: Balancing a
Transportation Problem
If total supply is less than total demand:
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Solution of Transportation Problems
Two phases:
• 1st phase:
– Find an basic feasible solution (bfs) by using
• Northwest corner method
• Least cost method
• Vogel’s approximation (penalty cost) method
• 2nd phase:
– Transportation simplex
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The Powerco Example: Transportation
Tableau
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The Powerco Example: Balancing the
Transportation Tableau
Dummy
demand
point
Dummy
supply
point
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Finding BFS
•
There are three basic methods:
1.
Northwest Corner Method
2.
Minimum Cost Method
3.
Vogel’s Method
23
Finding BFS
•
There are three basic methods:
1.
Northwest Corner Method
2.
Minimum Cost Method
3.
Vogel’s Method
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The Northwest Corner Method
25
The Powerco Example: Finding a BFS
using the Northwest Corner Method
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The Powerco Example: Finding a BFS
using the Northwest Corner Method
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The Powerco Example: Finding a BFS
using the Northwest Corner Method
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The Powerco Example: Finding a BFS
using the Northwest Corner Method
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The Powerco Example: Finding a BFS
using the Northwest Corner Method
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The Powerco Example: Finding a BFS
using the Northwest Corner Method
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The Powerco Example: Finding a BFS
using the Northwest Corner Method
32
Finding BFS
•
There are three basic methods:
1.
Northwest Corner Method
2.
Minimum Cost Method
3.
Vogel’s Method
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The Minimum Cost Method
 Minimum Cost Starting Procedure
• Step 1: Select the cell with the least cost. Assign to this cell
the minimum of its remaining row supply or remaining
column demand.
• Step 2: Decrease the row and column availabilities by this
amount and remove from consideration all other cells in the
row or column with zero availability/demand. (If both are
simultaneously reduced to 0, assign an allocation of 0 to any
other unoccupied cell in the row or column before deleting
both.) GO TO STEP 1.
The Minimum Cost Method
The Minimum Cost Method
Step 1: Select the cell with minimum cost.
2
3
5
6
5
2
1
3
5
10
3
12
8
8
4
4
6
6
15
The Minimum Cost Method
Step 2: Cross-out column 2
2
3
5
6
5
2
1
3
5
2
8
3
12
8
X
4
4
6
6
15
The Minimum Cost Method
Step 3: Find the new cell with minimum shipping
cost and cross-out row 2
2
3
5
6
5
2
1
3
5
X
2
8
3
10
8
X
4
4
6
6
15
The Minimum Cost Method
Step 4: Find the new cell with minimum shipping
cost and cross-out row 1
2
3
5
6
X
5
2
1
3
5
X
2
8
3
5
8
X
4
4
6
6
15
The Minimum Cost Method
Step 5: Find the new cell with minimum shipping
cost and cross-out column 1
2
3
5
6
X
5
2
1
3
5
X
2
8
3
8
4
6
5
X
X
4
6
10
The Minimum Cost Method
Step 6: Find the new cell with minimum shipping
cost and cross-out column 3
2
3
5
6
X
5
2
1
3
5
X
2
8
3
8
5
4
6
4
X
X
X
6
6
The Minimum Cost Method
Step 7: Finally assign 6 to last cell. The bfs is found
as: X11=5, X21=2, X22=8, X31=5, X33=4 and X34=6
2
3
5
6
X
5
2
1
3
5
X
2
8
3
8
5
4
4
X
X
6
6
X
X
X
Finding BFS
•
There are three basic methods:
1.
Northwest Corner Method
2.
Minimum Cost Method
3.
Vogel’s Method
43
Finding BFS
•
There are three basic methods:
1.
Northwest Corner Method
2.
Minimum Cost Method
3.
Vogel’s Method
44
Vogel’s Method
• Begin with computing each row and column a penalty.
• The penalty will be equal to the difference between the two
smallest shipping costs in the row or column. Identify the
row or column with the largest penalty.
• Find the first basic variable which has the smallest shipping
cost in that row or column.
• Then assign the highest possible value to that variable, and
cross-out the row or column as in the previous methods.
• Compute new penalties and use the same procedure.
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Vogel’s Method
An example for Vogel’s Method
Step 1: Compute the penalties.
6
7
15
Demand
Column Penalty
Supply
Row Penalty
10
7-6=1
15
78-15=63
8
80
78
15
5
5
15-6=9
80-7=73
78-8=70
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Vogel’s Method
Step 2: Identify the largest penalty and assign the
highest possible value to the variable.
6
7
Supply
Row Penalty
5
8-6=2
15
78-15=63
8
5
15
Demand
Column Penalty
80
78
15
X
5
15-6=9
_
78-8=70
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Vogel’s Method
Step 3: Identify the largest penalty and assign the
highest possible value to the variable.
6
7
5
Column Penalty
Row Penalty
0
_
15
_
8
5
15
Demand
Supply
80
78
15
X
X
15-6=9
_
_
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Vogel’s Method
Step 4: Identify the largest penalty and assign the
highest possible value to the variable.
6
0
7
5
Supply
Row Penalty
X
_
15
_
8
5
15
80
78
Demand
15
X
X
Column Penalty
_
_
_
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Vogel’s Method
Step 5: Finally the bfs is found as X11=0, X12=5, X13=5,
and X21=15
6
0
7
5
Supply
Row Penalty
X
_
X
_
8
5
15
80
78
15
Demand
X
X
X
Column Penalty
_
_
_
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Transshipment Problems
• A transportation problem allows only shipments that go
directly from supply points to demand points.
• In many situations, shipments are allowed between
supply points or between demand points.
• Sometimes there may also be points (called
transshipment points) through which goods can be
transshipped on their journey from a supply point to a
demand point.
• Fortunately, the optimal solution to a transshipment
problem can be found by solving a transportation
problem.
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