## Vogel's Approximation Method (VAM)

Vogel's approximation method is an improved version of the least cost entry method. It gives better starting solution as compared to any other method.

Consider a general transportation problem with $m$ origins and $n$ destinations.

Origin Destination | $D_1$ | $D_2$ | $\cdots$ | $D_j$ | $\cdots$ | $D_n$ | Availability |
---|---|---|---|---|---|---|---|

$O_1$ | $c_{11}$ | $c_{12}$ | $\cdots$ | $c_{1j}$ | $\cdots$ | $c_{1n}$ | $a_1$ |

$O_2$ | $c_{21}$ | $c_{22}$ | $\cdots$ | $c_{2j}$ | $\cdots$ | $c_{2n}$ | $a_2$ |

$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |

$O_i$ | $c_{i1}$ | $c_{i2}$ | $\cdots$ | $c_{ij}$ | $\cdots$ | $c_{in}$ | $a_i$ |

$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |

$O_m$ | $c_{m1}$ | $c_{m2}$ | $\cdots$ | $c_{mj}$ | $\cdots$ | $c_{mn}$ | $a_m$ |

Requirement | $b_1$ | $b_2$ | $\cdots$ | $b_j$ | $\cdots$ | $b_n$ | $\sum_i a_i = \sum_j b_j$ |

If the transportation problem is unbalanced (i.e. the total availability is not equal to the total requirement, $\sum_i a_i \neq \sum_j b_j$) then convert it into a balanced transportation problem by adding a dummy row or dummy column as per the requirement taking zero costs.

## Step by step procedure

The step by step procedure to obtain the initial basic feasible solution to the transportation problem using **Vogel's Approximation method** is as follows:

#### Step 1

For each row (column), determine the penalty measure by subtracting the **smallest unit cost** element in the row (column) from the **next smallest unit cost** element in the same row (column).

#### Step 2

Select the row or column with the **largest** penalty. If a tie occurs, use any arbitrary tie breaking choice.

Let the largest penalty corresponds to $i^{th}$ row and let `$c_{ij}$`

be the smallest cost in the $i^{th}$ row. Allocate as much as possible amount `$x_{ij} = min(a_i, b_j)$`

in the cell $(i,j)$ and cross-out the $i^{th}$ row or $j^{th}$ column in the usual manner.

#### Step 3

Again determine the penalties for rows and column ignoring the costs of cross-out row and column for the reduced transportation table. Then go to Step 2.

#### Step 4

Repeat Step 2 and 3 until all the requirements and availabilities are satisfied.