Multi-product profit maximization
At the end of the day, the firm seeks to maximize profits, which is a one-dimensional object. So the firm simply solves 
\begin{equation}
\max_{\mathbf x}\;\mathbf{p}\cdot\mathbf f(\mathbf x)-c(\mathbf x)
\end{equation}
where $\mathbf x\in\mathbb R_+^\ell$ is a vector of $\ell$ inputs, $\mathbf p\in\mathbb R_+^n$ is a vector of $n$ prices for the final output, $\mathbf f:\mathbb R_+^\ell\to\mathbb R_+^n$ is the production function mapping $\ell$ inputs to $n$ outputs, and $c:\mathbb R_+^\ell\to \mathbb R_+$ is the cost function. This is a standard unconstrained optimization exercise that any first year grad student in economics should know how to solve. The standard textbooks --- MWG, Jehle and Reny, Simon and Blume --- have sections that cover the techniques.