# Optimization of Inventory Management to Prevent Drug Shortages in the Hospital Supply Chain
## 3 System Model
### Problem Formulation  
We aim to design an online solution that dynamically determines when to refill drugs over the entire running span \(T\) while satisfying all requirements with a minimum refilling cost. Specifically, we aim to avoid a shortage situation in the system, which could lead to a domino effect. There is a general trade-off between cost optimization and any shortages due to unknown patient demand: by refilling more drugs, the shortage situation may be avoided, but the cost will be increased, and vice versa. In this work, we combine these dual goals in the objective function by using weighted parameters to weight the priority of each aspect. We design a main variable \(\mathbf{x}=\{x_{i}(t)\}_{\forall i\in\mathcal{I},t=1,2,\ldots,T}\) to represent the amount of drug \(i\) to refill at time \(t\). The refilling cost model in this work contains the following parts:  
* **Medicine cost:** We consider the first term as the medicine cost depending on the amount of drugs being ordered: \(\sum_{t=1}^{T}\sum_{i\in\mathcal{I}}a_{i}p_{i}(t)x_{i}(t)\), where \(a_{i}\in(0,1]\) is the weight parameter of drug \(i\) and \(p_{i}(t)\) is the amount of money required to purchase the required \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbols** & **Description** \\ \hline \(\mathcal{I}\) & Set of drugs \\ \(i\) & Drug index. \\ \(t\) & Time index. \\ \(T\) & Spanning time. \\ \(e_{i}(t)\) & An amount of expired drug \(i\) at time \(t\). \\ \(p_{i}(t)\) & The refilling cost of drug \(i\) at time \(t\). \\ \(\psi(.)\) & The penalty function of drug \(i\). \\ \(\underline{p}_{i}(t)\) & The base price of drug \(i\) at time \(t\). \\ \(\bar{\beta}_{i}(t)\) & The penalty price of drug \(i\) with an emergency demand. \\ \(B_{i}\) & The budget of drug \(i\). \\ \(\mathbb{B}\) & The total budget. \\ \(\alpha_{i}\) & The weight parameters of drug \(i\). \\ \(R_{i}(.)\) & The storage cost function of drug \(i\). \\ \(r_{i}(t)\) & The storage cost of drug \(i\) at time \(t\). \\ \(\lambda_{i}(t)\) & The amount of demand of drug \(i\) at time \(t\). \\ \(e_{i}(t)\) & The amount expired at time \(t\). \\ \(v_{i}(t)\) & The remaining volume of drug \(i\) at time \(t\). \\ \(C_{i}(t)\) & The storage capacity of drug \(i\). \\ \(\mathbb{C}_{i}(t)\) & The lowest volume requirement of drug \(i\). \\ \(\underline{\rho}_{i}(t)\) and \(\bar{\rho}_{i}(t)\) & The upper and lower bound refilling of drug \(i\) at time \(t\). \\ \hline
**Variables** & \\ \hline \(\mathbf{x}\) & The decision refilling volume variable. \\ \(x_{i}(t)\) & The refilling volume of drug \(i\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. amount of drug \(i\). Depending on the drugs' priorities, \(\alpha_{i}\) is set with a high or low value. * **Storage cost:** We design the second term as the storage cost incurred when the hospital stores drugs. Hospitals may incur a variety of costs associated with safely storing medicine, and some must utilize third-party storage. In this work, we generally model the storage cost as a function \(R_{i}(v_{i}(t),r_{i}(t))\) depending on \(v_{i}(t)\), the amount of drugs, and \(r_{i}(t)\), the storage cost. The volume of drug \(i\) at time \(t\) can be obtained by: \[v_{i}(t)=v_{i}(t-1)-\lambda_{i}(t)-e_{i}(t)+x_{i}(t),\] (2) where \(v_{i}(t-1)\) is the remaining volume of drug \(i\) at timeslot \(t-1\), \(\lambda_{i}(t)\) is the amount of the demand, and \(e_{i}(t)\) is the amount of the drug available before expiration at time \(t\). * **Penalty cost:** A penalty cost \(\psi(v_{i}(t))\) is assigned to prevent a shortage situation in the system. This cost depends on the volume and the minimum requirement for the drugs at time \(t\). Without loss of generality, we refer this cost to a quadratic function to formulate our model. As shown in Figure 1, the quadratic penalty cost increases when the volume of drug \(i\) reaches the bounded values \(\underline{C}_{i}\) and \(\bar{C}_{i}\). In other words, the penalty cost function aims to prevent both the shortage and overstock problems.