# Literature on fixed-charge network flow problems with inventory levels

I'm looking for optimization problem classes with inventory control constraints. Given a directed graph $$G(V,A)$$ where vertex set $$V$$ consists of production nodes and demand nodes. The arc set $$A$$ contains an outgoing connection from every production node to every other node. Demand nodes only have incoming arcs. We also have a set of commodities $$K$$, and a set of time periods $$T=0,1,2,\dots,H$$. With each arc $$(i,j)\in A$$ we associate a transfer lead time $$\tau_{ij}$$ (measured in time periods) and a per unit cost $$c_{ijk}$$. Let parameter $$s_{ik}^t$$ be the amount of commodity $$k$$ that is produced at production facility $$i$$ in period $$t$$. We define two types of variables, e.g. flow variables $$x_{ijk}^t$$ representing the quantity of commodity $$k$$ transferred from $$i$$ to $$j$$ in period $$t$$, as well as inventory level variables $$I_{ik}^t$$ representing the inventory level of production facility $$i$$ of commodity $$k$$ in period $$t$$. The inventory levels can be tracked through the following type of constraints:

\begin{align} &I_{ik}^t=I_{ik}^{t-1}-\sum_{(i,j)\in A}x_{ijk}^t+\sum_{(j,i)\in A}x_{jik}^{t-\tau_{ij}}+s_{ik}^t & \forall i\in V, k\in K, t=1,2,\dots,H\\ &I_{ik}^0=s_{ik}^0 &\forall i\in V, k\in K \end{align} In words, the first constraint states that the inventory level of facility $$i$$ at time $$t$$ for commodity $$k$$ equals the inventory level in the previous time period $$(t-1)$$ minus the volume that was transferred to other nodes plus the volume received from other facilities plus the amount $$s_{ik}^t$$ that was produced. The second constraint sets the initial inventory levels at the start of the time horizon.

I am looking for references to papers and problem classes that contain constraints with this kind of structure. I'm particularly interested in optimization problems that minimize total transfer costs plus some fixed charge component (e.g. a costs for using an arc $$(i,j)$$). Although I can think of a variety of real-life applications that exhibit this general structure, I can't seem to find many relevant references? Perhaps this is a well-studied class of problems and I'm just searching for the wrong kind of literature. There's various kinds of fixed-charge network flow problems, but none of the variants I found track inventory levels. Any references on problem classes with inventory level constraints (with or without a fixed charge cost component) are appreciated.