Q.Give mathematical form of Assignment problem.

Assignment problem: minimise $Z=\sum\sum c_{ij}x_{ij}$ with each row-sum and column-sum $=1$ and $x_{ij}\in\{0,1\}$.

In the Tamil Nadu HSC Business Maths & Statistics syllabus (Operations Research), an assignment problem assigns $n$ jobs to $n$ facilities on a one-to-one basis so that the total cost (or time) is minimum. Let $c_{ij}$ be the cost of assigning the $i$-th person to the $j$-th job, and let

$x_{ij}=\begin{cases}1,&\text{if the }i\text{-th person is assigned the }j\text{-th job}\\[2pt]0,&\text{otherwise.}\end{cases}$

The mathematical form is:

$\text{Minimise } Z=\sum_{i=1}^{n}\sum_{j=1}^{n}c_{ij}\,x_{ij}$

subject to …