I have a set of sources and a set of sinks.
Each source $s$ can produce a set of different products $P_s$.
The transportation inter-sources and inter-sinks are allowed.
The sources do not necessarily produce the same products, but they may have products in common.
The same thing for sinks, a sink $t$ can absorb more than a single type of product (i.e., each sink has a set $D_t$ of demanded products), two sinks do not necessarily absorb exactly the same type of products, but they may have products in common though.
Transportation costs on arcs
Production costs on source nodes
Maximal production capacity on each node
Sinks have demands.
N.B. There are no maximal capacities on edges.
The decision is 1) which quantity to produce on each source for each product and 2) on which arcs to send the produced quantities? The objective is to reduce the total cost while satisfying the demand for each sink for each product. N.B. The transportation costs have an influence on the decision of how much quantity we produce in each source.
How do we call this problem in the literature? As far as I know, it's not a multi-commodity flow since in a multi-commodity flow we assume that each source(respectively sink) produce(respectively absorb) a different and unique commodity $k$, i.e, we have different pairs $(s_1,t_1), ..,(s_k,t_k)$ (thus the quantity produced in each source is simply equal to the quantity demanded by the sinks for the commodity $k$ but in my problem, it's part of the decision). Is there a paper that model/solve a similar problem? How to model it? I am also curious about the different approaches to solve it(exactly or not). Feel free to suggest any approach if you have an idea.
EDIT : The answer of @marco shows how to do some tricks in order to reduce the problem into a multi-commodity flow. (actually, a mincost flow, as the different commodities do not have a capacity on a common arc)
What if the transportation costs defined on arcs and the production costs defined on source nodes are of the type fixed_cost + cost/unit instead of a cost/unit as in case 1. Is this still a multi-commodity flow problem? I think that the transformations described by @marco are still valid since only the costs change but the problem still have the same structure. But how this change can affect the hardness of the problem?