Method for (binary) optimization under constraints

Question

I would like to know if there is a method to solve the Problem.

Problem:
Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \cdots & p_{m,i} \end{bmatrix}*\begin{bmatrix}e_1 \\ e_2 \\ \vdots \\ e_m \end{bmatrix} * c_i$$

where we know the values of each $e_j$ ($j \in \{1, \dotsc, m\}$) and $c_i$.
The values of the $p_{j,i}$ should be either 1 or 0 and we have the following constraints:

1. For all $j \in \{1,\dotsc,m\}$: $\sum_{i=1}^n p_{j,i} \le 1$.
2. For all $i \in \{1,\dotsc,n\}$: $\sum_{j=1}^m p_{j,i} = t_i$, where $t_i$ is a known value.

I would be grateful for every hint, method or solution.

Answer 1

This is the transportation problem in a bipartite network, with a supply of $1$ at each $j$ node and a demand of $t_i$ at demand node $i$. The problem can be solved via linear programming, a minimum-cost network flow algorithm, or a specialized algorithm. Because of total unimodularity, you can relax the binary $\{0,1\}$ restriction to $\ge 0$ and still obtain an optimal binary solution.