# Row Minima Method for Transportation Problem

## Row Minima Method

Row minima method is a method of finding initial basic feasible solution for a transportation problem.

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

Step by step procedure of Row Minima method is as follows:

#### Step 1

Select the smallest cost in the first row of the transportation table. Let it be $c_{1j}$. Allocate as much as possible amount $x_{1j} = min_j(a_1, b_j)$ in the cell $(1,j)$, so that either the capacity of origin $O_1$ is exhausted or the requirement at destination $D_j$ is satisfied or both.

#### Step 2

• If $x_{1j} = a_1$, the availability at origin $O_1$ is completely exhausted, cross-out the first row of the table and move down to the second row.
• If $x_{1j}= b_j$, the requirement at destination $D_j$ is satisfied, cross-out the $j^{th}$ column and reconsider the first row with the remaining availability of origin $O_i$.
• If $x_{1j} = a_1= b_j$, the availability at origin $O_1$ and the requirement at destination $D_j$ are completely exhausted. So cross-out $1^{st}$ row and $j^{th}$ column simultaneously. Move down to the second row.

#### Step 3

Repeat Step 1 and Step 2 for the reduced transportation table until all the requirements and availabilities are satisfied.