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.
