Question

Explain Vogel’s approximation method by obtaining an initial basic feasible solution to the following transportation problem.

Answer 1

Total supply (a_i ) = 6+1 + 10=17 

Total demand (b_j ) = 7 + 5 + 3 + 2 = 17 

Σa_i = Σb_j . So given transportation problem is a balanced problem and hence we can find a basic feasible solution by VAM.

 First allocation:

The largest penalty is 6. So allot min (2, 1) to the cell (O₂ , D₄ ) which has the least cost in column D₄ . 

Second allocation:

The largest penalty is 5. So allot min (5, 6) to the cell (O₁ , D₂ ) which has the least cost in column D₂ . 

Third allocation:

The largest penalty is 5. So allot min (7, 1) to the cell (O₁ , D₁ ) which has the least cost in row O₁ . 

Fourth allocation:

Largest penalty = 4. So allocate min (6, 10) to the cell (O₃ , D₁ ) which has the least cost in row O₃ 

Fifth allocation:

The largest penalty is 6. So allot min (1, 4) to the cell (O₃ , D₄ ) which has the least cost in row O₃ . The balance 3 units is allotted to the cell (O₃ , D₃ ). We get the final allocation as given below.

Transportation schedule: 

O₁ → D₁ , O₁ → D₂ , O₂ → D₄ , O₂ → D₁ , O₃ → D₃ , O₃ → D₄ 

(i.e) x₁₁ = 1, x₁₂ = 5, x₂₄ = 1, x₃₁ = 6, x₃₃ = 3, x₃₄ = 1 

Total cost is given by = (1 × 2) + (5 × 3) + (1 × 1) + (6 × 5) + (3 × 15) + (1 × 9) 

= 2 + 15 + 1 + 30 + 45 + 9 

= 102 

Thus the optimal (minimal) cost of the given transportation problem by Vogel’s approximation method is Rs. 102.