Parameter estimation of 3d model

Suppose we have cost data on some product line, let's say clothing. A simple model of the true cost of the clothing could look something like this:

C$_{i,j,k}$ = c$_i \cdot \alpha_j \cdot \beta_k$,

for some specific article of clothing, $i$, made in some month, $j$, in some region $k$. Then c$_i$ is the true cost, $\alpha_j$ is a factor for the month and $\beta_k$ is a factor for the region.

The question is how could we estimate the parameters of this model? I am having lots of trouble setting something up. I am only somewhat familiar with Maximum Likelihood Estimation, but I have no idea how the Likelihood would look like in the first place. How does this multi-dimensional aspect of the model get taken care of?