  We design an Inventory Allocation System (INVALS) that, for each item-store
combination, plans the quantity to be allocated from a warehouse replenishing
multiple stores using trailers, while respecting the typical supply-chain
constraints. We formulate a linear objective function which when maximised
computes the allocation by considering not only the immediate store needs, but
also its future expected demand. Such forward-looking allocation significantly
improves the labour and trailer utilisation at the warehouse. To reduce
overstocking, we adapt from our objective to prioritise allocating those items
in excess which are sold faster at the stores, keeping the days of supply (DOS)
to a minimum. For the proposed formulation, which is an instance of Mixed
Integer Linear Programming (MILP), we present a scalable algorithm using the
concepts of submodularity and optimal transport theory by: (i) sequentially
adding trailers to stores based on maximum incremental gain, (ii) transforming
the resultant linear program (LP) instance to an instance of capacity
constrained optimal transport (COT), solvable using double entropic
regularization and incurring the same computational complexity as the Sinkhorn
algorithm. When compared against the planning engine that does the allocation
only for immediate store needs, INVALS increases on an average the labour
utilization by 34.70 and item occupancy in trailers by 37.08. The DOS
distribution is also skewed to the left indicating that higher demand items are
allocated in excess, reducing the days they are stocked. We empirically
observed that for ~ 90% of replenishment cycles, the allocation results from
INVALS are identical to the globally optimal MILP solution.
