| Adjust the supply and demand requirements in the respective rows and columns. Then following cases arise: |

Case 1: | If the allocation ${X}_{ij}={a}_{i}$ , i-th row is to be crossed out and ${b}_{j}$ is reduced to ( ${b}_{j}-{a}_{i}$ ). Now complete the allocation along j-th column by making the allocation/allocations in the smallest cost cell/cells continuously. Consider that, j-th column is exhausted for the allocation ${X}_{kj}$ at the cell (k, j). Now, follow the same procedure to complete the allocation along k-th row and continue this process until entire rows and columns are exhausted. Again if the allocation ${X}_{ij}={b}_{j}$ , just reverse the process for ${X}_{ij}={a}_{i}$ . |

Case 2: | If the allocation ${X}_{ij}={a}_{i}={b}_{j}$ , find the next smallest cost cell (i, k) from the rest of the cost cells along i-th row and j-th column. Assign a zero in the cell (i, k) and cross out i-th row and j-th column. After that complete the allocation along k-th row/column following the process described in Case-1 to complete the allocations. |

Step 8: | Compute the total transportation cost using the original transportation cost matrix and allocations obtained in Step 6 and Step 7. |

Step 9: | Finally calculate the total transportation cost from the cost table. This calculation is the sum of the product of cost and corresponding allocated value of the cost table. |