- The well-known
transportation problem
is sometimes called
the Hitchcock problem
- Frank Lauren Hitchcock (1875-1957)
was an American mathematician and physicist
notable
for vector analysis.
He formulated the transportation problem in 1941.
Wikipedia: click here
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- Transporting goods to warehouses:
- A manufacturer has a number of
factories,
each of which produces good at a fixed
output rate
- The manufacturer also has a number of
warehouses,
each of which has a fixed
storage capacity
- The manufactured goods are
transported from
the factories to
the warehouses for
(temporal) storage.
- There is a cost
to transport goods from
a factory to
a warehouse
- Find the transportation of good from factory → warehouse that has the lowest possible cost
- A manufacturer has a number of
factories,
each of which produces good at a fixed
output rate
- Example:
- Factories:
- A1 makes 5 units
- A2 makes 4 units
- A3 makes 6 units
- Warehouses:
- b1 can store 5 units
- b2 can store 3 units
- b3 can store 5 units
- b3 can store 2 units
- Transportation costs:
| b1 b2 b3 b4 ----+--------------------- A1 | 5 4 7 6 A2 | 2 5 3 2 A3 | 6 3 4 4
- A possible (not min. cost) solution:
Note:
- One factory can transport
product to
multiple warehouses
- One warehouse can receive
product from
multiple factoriess
- The solution of a transportation problem is not a matching (in a bi-partite graph) !!!
- One factory can transport
product to
multiple warehouses
- Factories:
- The Transportation problem
is closely related to the
assignment problem
(it is in fact easier than the
assignment problem)
- The Transportation problem
can be formulated as
a ordinary
linear constrained optimization problem
(i.e.: LP)
- Example:
Cost Matrix: warehouses -----------------> 1 2 3 4 +- -+ 1 | 5 4 7 6 | C = 2 | 2 5 3 2 | 3 | 6 3 4 4 | +- -+
Define:
- Cost matrix
C = [cij]
where
cij =
cost of
man i working on
job j
- Variable xij = amount of product shipped from factory i to warehouse j
- Cost matrix
C = [cij]
where
cij =
cost of
man i working on
job j
- Linear program for the above Transportation problem:
min: 5x11 + 4x12 + 7x13 + 6x14 + 2x21 + 5x22 + 3x23 + 2x24 + 6x31 + 3x32 + 4x33 + 4x34 s.t.: x11 + x12 + x13 + x14 = 5 // Total shipment from factory 1 = 5 units x21 + x22 + x23 + x24 = 4 x31 + x32 + x33 + x34 = 6 x11 + x21 + x31 <= 5 // Total shipment to warehouse 1 <= 5 x12 + x22 + x32 <= 3 x13 + x23 + x33 <= 5 x14 + x24 + x34 <= 2 xij >= 0Output:
Value of objective function: 54.00 Actual values of the variables: x11 2 x12 3 x13 0 x14 0 x21 3 x22 0 x23 0 x24 1 x31 0 x32 0 x33 5 x34 1
- Example Program:
(LP input file of problem above)
- LP Prog file: click here