# Vogels Approximation Method (VAM)

Vogel's Approximation Method (VAM)

## 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.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.