I read that the dual of the assignment problem is
$$ \begin{align} \text{maximize } & \sum_{i=1}^n \lambda_i + \sum_{i=1}^n \beta_i \\ & \lambda_i + \beta_j \leq c_{ij} \end{align} $$
where the primal problem is
$$ \begin{align} \text{minimize } & \sum\limits_{i=i}^n\sum\limits_{j=1}^n c_{ij}x_{ij} \\ &\sum_{i=1}^n x_{ij} = 1\; \forall j \\ &\sum_{j=1}^n x_{ij} = 1\; \forall i \\ &x_{i,j} ~\in (0,1) \end{align} $$
But unfortunately, I am not able to derive dual from primal using Lagrangian. Anyone can help ?