The transportation problem is a well-studied problem in operations research. Given sources $i\in\{1, \ldots, n\}$ and destinations $j\in\{1, \ldots, m\}$, we seek to minimize the total cost of shipping materials from sources to destinations. Each source has supply $s_i$, each destination has demand $d_j$, and the cost of shipping a unit from source $i$ to destination $j$ is $c_{ij}$. This can easily be formulated as a linear program, with decision variables $x_{ij}$ indicating how many units to ship from source $i$ to destination $j$:
\begin{align*} \min & \sum_{i=1}^n\sum_{j=1}^m c_{ij}x_{ij} \\ \text{s.t.} & \sum_{j=1}^m x_{ij}\leq s_i~\forall~i=1,\ldots,n \\ & \sum_{i=1}^n x_{ij}\geq d_j~\forall~j=1,\ldots,m \\ & x_{ij}\geq 0~\forall~i=1,\ldots,n;~j=1,\ldots,m \end{align*}
I am interested in a variant of this problem where there is a cardinality constraint on the suppliers: no more than $k$ suppliers can be used. This can be easily formulated using a binary decision variable $y_i$ for every supplier $i$ indicating whether they are used:
\begin{align*} \min & \sum_{i=1}^n\sum_{j=1}^m c_{ij}x_{ij} \\ \text{s.t.} & \sum_{j=1}^m x_{ij}\leq s_iy_i~\forall~i=1,\ldots,n \\ & \sum_{i=1}^n x_{ij}\geq d_j~\forall~j=1,\ldots,m \\ & \sum_{i=1}^n y_i\leq k \\ & x_{ij}\geq 0~\forall~i=1,\ldots,n;~j=1,\ldots,m \\ & y_i\in\{0, 1\}~\forall~i=1,\ldots,n \end{align*}
Given the popularity of the transportation problem, I would have assumed this simple variant would have been studied, but I can't seem to track down a name or reference for it. The closest I have come is the Transportation Problem with Market Choice (TPMC), which limits the number of destinations served, giving a reward for all destinations selected. However, the variant I'm interested in has no rewards and a fixed cardinality limit $k$, and that limit is on the sources and not the destinations.
Does anybody know what this variant is called, or could you point me to a reference about it?
Your problem has an element of the facility location problem I think. There is a constraint on the number of sources (facilities) you can open. In the general case there will be a cost of opening a facility (activating a source). Literature related to combining facility location with transportation might be helpful. This might be a decent starting point
Melkote, Sanjay, and Mark S. Daskin. "An integrated model of facility location and transportation network design." Transportation Research Part A: Policy and Practice 35.6 (2001): 515-538.