Consider the problem:

$$\begin{array}{ll} \text { Minimize } & \sum_{i=1}^{m} \sum_{j=1}^{n} c_{i j} x_{i j} \\ \text { subject to } & \sum_{j=1}^{n} x_{i j}=a_{i}, \quad i=1, \ldots, m, \\ & \sum_{i=1}^{m} x_{i j}=b_{j}, \quad j=1, \ldots, n \\ & x_{i j} \geqslant 0, \quad \text { for all } i, j \end{array}$$

where $a_{i} \geqslant 0, b_{j} \geqslant 0$ satisfy $\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j}$.

Formulate the dual of this problem and state necessary and sufficient conditions for optimality.