### The Transportation problem

• Brief history

 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

• Transportation problem: description

• 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

• 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) !!!

• Linear Programming formulation of a Transportation problem

• 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

• 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 >= 0 ```

Output:

 ``` 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)