A Model for Supply-Chain Decisions for Resource Sharing with an Application to Ventilator Allocation to Combat COVID-19

Sanjay Mehrotra; Hamed Rahimian; Masoud Barah; Fengqiao Luo; Karolina Schantz

We now explain the model in detail. In the first stage, the central agency makes the "here-and-now" decisions x before the stochastic parametersd are realized. As captured in (1a), the goal of the central agency is to minimize the expected total shortage of ventilators over all time periods t ∈ T and all regions n ∈ N . The objective also includes a cost, parameterized by θ of allocating a ventilator by the central agency to a state at a given time. This cost can be set to zero, or set to a small value. In our computations we set θ = 0.01. In the second stage, once the stochastic parametersd are realized, the "wait-and-see" decisions z n,t , y n,t , s t , n ∈ N and t ∈ T , are made. These decisions are scenario-specific, and are indicated by the superscript ω ∈ Ω, in the extensive formulation given in (3) . Constraints (2b) and (2c) ensure the conservation of ventilators for the regions and the central agency at each time period, respectively. Constraint (2d) enforces that at each time period, a region is not sending out any ventilator to the central agency if its in-hand inventory is lower than its safety stock, where the safety stock is determined as ρ ndn,t , for t ∈ T and n ∈ N . Constraint (2e) ensures that at each time period, the total number of outgoing ventilators from the central agency to the regions cannot be larger than the available inventory, after incorporating the newly produced ventilators and the incoming ones from other regions. Constraints (2f) and (2g) set the initial inventory at the regions and central agency, respectively. The remaining constraints ensure the non-negativity of decision variables.