The LP formulation of the TP problem is:

Let,

X_ij = Quantity of Product (transport from source i to destination j)

C_ij = Per unit transporting cost from sources i to destination j

S_i be the row i total supply (where i= 1, 2, 3,....)

D_j be the column j total demand (where j= 1, 2, 3.....)

For this type of problem all units are available

m Warehouse and n Stores

No. of Variables is (mXn)

No. of Constraints is (m+m) (Constraints are for warehouses capacity and stores demand)

To solve the transportation problem by its special purpose algorithm, it is required that the sum of the supplies at the warehouses equal the sum of the demands at the stores. 

 ∑S_{i(i=1,2,3....)} = ∑D_{j(j=1,2,3...)} = units

LP Formulation

The linear programming formulation in terms of the amounts shipped from the origins to the destinations, X_ij, can be written as:

Objective function:

Minimize Z = X₁₁ +X₂₁ +X₃₁ + X₂₁ + X₂₂ + X₂₃ + X₃₁ +X₃₂ +7X_{33....................}   

Subject to the constraints:

X₁₁+X₂₁+X₃₁ > .....

X₁₂+X₂₂+X₃₂ > .....

X₁₃+X₂₃+X₃₃ > ....

X₁₁+X₁₂+X₁₃ < ....

X₂₁+X₂₂+X₂₃ < ......

X₃₁+X₃₂+X₃₃ < .....

With non negativity condition: X₁₁,X₁₂,X₁₃,X₂₁,X₂₂,X₂₃,X₃₁,X₃₂,X_33....... > 0

APPROCH AND METHODOLOGY

The transportation problem is solved in two phases: 

Phase I — obtaining an initial feasible solution

Phase II — moving toward optimality

In Phase I, the Minimum-Cost Procedure can be used to establish an initial basic feasible solution without doing numerous iterations of the Simplex Method.

There are three different ways:

·         Northwest corner method

·         The Minimum cell cost method

·         Vogel’s approximation method (VAM)

SENSITIVITY ANALYSIS

Sensitivity Analysis investigates the change in the optimum solution resulting from making changes in parameters of the linear programming of transportation problem, So the changes in coefficients of (C_ij) Cost Factors.