1
Transportation,
Transshipment,
and Assignment (TTA)
Chapter 6
2
Chapter Outline
■
To examine special types of LP problems:
■
Transportation Problem
■
Transshipment Problem
■
Assignment Problem
3
Transportation Model: Characteristics (1 of 2)
■
Consider the transportation of a product from supply
source
S
i
to demand destination
D
j
4
Transportation Model: Characteristics (2 of 2)
■
A product is transported from a number of sources to a number of
destinations with the possible
objective
:
■
Minimization of total transportation cost
, or
■
Maximization of total profit
, or
■
Minimization of total distance traveled
■
Each
source is able to supply a fixed number
of units of the
product, and
each destination has a fixed demand
for the
product with
constraints
:
■
Satisfy the
demand requirement
at each destination
■
Satisfy the
supply capacity
available at each source
■
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
■
Assumption
: Transportation cost is proportional to number of
units transported
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5
Transport Cost from Grain Elevator to Mill ($/ton)
Grain Elevator
A. Chicago
B. St. Louis
C. Cincinnati
1. Kansas City
2. Omaha
3. Des Moines
$ 6
7
4
$ 8
11
5
$ 10
11
12
Transportation Model Example
Problem Definition and Data [Supply=Demand] (1 of 9)
How many tons of wheat to transport from each grain
elevator to each mill on a monthly basis in order to
minimize the total cost of transportation?
Grain Elevator
Supply
Mill
Demand
1. Kansas City
150
A. Chicago
220
2. Omaha
175
B. St. Louis
100
3. Des Moines
275
C. Cincinnati
300
Total
600 tons
Total
600 tons
6
Transportation Model Example
Transportation Network Routes (2 of 9)
Figure 6.1
Network of Transportation Routes for Wheat Shipments
7
The 3 mills demand 600 total tons, which is the exact amount that can
be supplied by the 3 grain elevators (300 total tons)
=> balanced transportation model (supply exactly equals demand)
Let x
ij
= tons of wheat from each grain elevator i (i = 1, 2, 3)
to each mill j (j = A,B,C)
Minimize total transportation cost, Z = 6x
1A
+ 8x
1B
+ 10x
1C
+ 7x
2A
+ 11x
2B
+ 11x
2C
+ 4x
3A
+ 5x
3B
+ 12x
3C
subject to:
x
1A
+ x
1B
+ x
1C
= 150
x
2A
+ x
2B
+ x
2C
= 175
x
3A
+ x
3B
+ x
3C
= 275
x
1A
+ x
2A
+ x
3A
= 200
x
1B
+ x
2B
+ x
3B
= 100
x
1C
+ x
2C
+ x
3C
= 300
x
ij
≥
0
Transportation Model Example
Model Formulation [Supply=Demand] (3 of 9)
Supply Capacity
Demand Requirement
8
Transport Cost from Grain Elevator to Mill ($/ton)
Grain Elevator
A. Chicago
B. St. Louis
C. Cincinnati
1. Kansas City
2. Omaha
3. Des Moines
$ 6
7
4
$ 8
11
5
$ 10
11
12
Transportation Model Example
Problem Definition and Data [Supply<Demand] (4 of 9)
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 Spring '10
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 Supply And Demand, Equals sign, grain elevator, Transshipment

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