$maximize: -\sum^{n}_{i=1}\sum^{m}_{j=1} f_{ij}·p_j+\sum^{n}_{j=1}\tilde{f_i}·\tilde{p_i}$
subject to: $1^{T}c_j=1^T\tilde{c_i}=1\\\tilde{c}^{min}\leqslant\tilde{c_i}\leqslant\tilde{c}^{max}\\\sum^{n}_{i}f_{ij}\leqslant F_j \ (0\leqslant f_{ij},0<F_j)\\\tilde{f_i}\leqslant\tilde{F_i}\\\sum^{m}_{j=1}f_{ij}·c_j=\tilde f_i·\tilde{c_i}$
and the $f_{ij} \ \tilde{f_i} \ \tilde{c_i}$ here are the variables, the rest are known.
I have no idea how to deal with this part.. $\sum^{m}_{j=1}f_{ij}·c_j=\tilde f_i·\tilde{c_i}$
You do not give many sign restrictions, but let me assume that every parameter and variable is nonnegative. You can replace all occurences of $\tilde{c_i}$ with $\left(\sum^{m}_{j=1}f_{ij}·c_j\right) / \tilde f_i$, and then multiply each constraint with the denominator to make them linear again.