Is this a contextual bandit problem? I have an operations research problem that I would like to solve but I am unsure what the correct framework for analyzing it would be. The closest I've come across so far is contextual bandits I believe but I don't know enough to know for sure.
Here is the problem:
How to efficiently get rid of inventory when selling across several sales channels. For instance, you have a lot of shirts, you want to divide how many you sell in store A, store B, store C, etc. Where each store has it's own probability distribution of being sold and is subject to individual features and each time a decision is made where to place the inventory for sale, the features change.
Is contextual bandits the best place to start? Any recommended readings on this subject?
 A: There is a huge literature on stochastic inventory control.  You will probably want  references for stochastic inventory control rather than, for example, the APICS literature that will describe MRP and bar coding... Axsater is an excellent intro reference at about a graduate level Intro to Production and Operations Management class.  Sherbrooke is more specialized but the approach he takes to multi-echelon problems will likely be quite applicable to your problem, even though it might seem out of context at first as he deals with repairable parts systems.
As a necessarily brief intro, typically you start with a site x item specific cost function where there is a penalty for expected lost sales (stockout cost $p$) and a penalty for expected leftover inventory (holding cost $h$).  I'll be a little simplistic and assume we are dealing with a single item and that $p$ and $h$ don't vary across sites, but that hardly matters, it just saves subscripts.  The expected cost of a policy that stocks $S_i$ units at site $i$ which has a probability distribution of demand $F_i$ is therefore:
$$\text{E}C(S_i) = h\int_0^{S_i}(S_i-x)\text{d}F_i(x) + p\int_{S_i}^{\infty}(x-S_i)\text{d}F_i(x)$$
where I'm using integrals instead of sums for technical reasons, but you can see how it would have to work for sums.
Writing $S$ as the joint policy $S_1, \dots, S_N$ and adding a constraint on total inventory results in the following convex constrained optimization problem:
$$\min_S \text{E}C(S) = \sum_{i=1}^N h\int_0^{S_i}(S_i-x)\text{d}F_i(x) + \sum_{i=1}^N p\int_{S_i}^{\infty}(x-S_i)\text{d}F_i(x)$$
subject to:
$$\sum_{i=1}^N S_i = S^*$$
The unconstrained problem is a sum of "Newsvendor" problems and has a closed-form solution, $S_i^{opt} = F_i^{-1}(p/(p+h))$.  You will see different formulations of this solution, depending on the details of the cost function, but don't worry about it.  Unfortunately, the site-specific problems are linked by the constraint, but we deal with that via Lagrange multipliers:
$$\min_S \text{E}C(S;\lambda) = \sum_{i=1}^N h\int_0^{S_i}(S_i-x)\text{d}F_i(x) + \sum_{i=1}^N p\int_{S_i}^{\infty}(x-S_i)\text{d}F_i(x) + \lambda\sum_{i=1}^NS_i $$
A moderate amount of algebra will reveal that the optimum solution for each site $i$ is:
$$S_i(\lambda) = F^{-1}_i\left({p-\lambda\over p+h}\right)$$
where $\lambda$ is found by solving the one-dimensional root-finding problem:
$$\sum_{i=1}^NS_i(\lambda) - S^* = 0$$
This gives you a general idea of how I, and others, might approach your problem, although of course I'm only guessing at a reasonable cost function, and yours might not have such a nice analytic solution.  The general approach of using a Lagrange multiplier on the constraint and thereby creating $N$ separate univariate minimization problems is the way to go, if you can do it.  Inventory control solutions are generally quite robust against errors in the cost function and parameter estimates.  Multi-period problems have some extra features but can often be solved in exactly the same way.  But Axsater will help, and Sherbrooke will expand on the Lagrange multiplier approach and relate it to marginal analysis, so those references would be where I would start.
