Scheduling Diagnostic Testing Kit Deliveries with the Mothership and Drone Routing Problem

Abstract

A critical component in the public health response to pandemics is the ability to determine the spread of diseases via diagnostic testing kits. Currently, diagnostic testing kits, treatments, and vaccines for the COVID-19 pandemic have been developed and are being distributed to communities worldwide, but the spread of the disease persists. In conjunction, a strong level of social distancing has been established as one of the most basic and reliable ways to mitigate disease spread. If home testing kits are safely and quickly delivered to a patient, this has the potential to significantly reduce human contact and reduce disease spread before, during, and after diagnosis. This paper proposes a diagnostic testing kit delivery scheduling approach using the Mothership and Drone Routing Problem (MDRP) with one truck and multiple drones. Due to the complexity of solving the MDRP, the problem is decomposed into 1) truck scheduling to carry the drones and 2) drone scheduling for actual delivery. The truck schedule (TS) is optimized first to minimize the total travel distance to cover patients. Then, the drone flight schedule is optimized to minimize the total delivery time. These two steps are repeated until it reaches a solution minimizing the total delivery time for all patients. Heuristic algorithms are developed to further improve the computational time of the proposed model. Experiments are made to show the benefits of the proposed approach compared to the commonly performed face-to-face diagnosis via the drive-through testing sites. The proposed solution method significantly reduced the computation time for solving the optimization model (less than 50 minutes) compared to the exact solution method that took more than 10 hours to reach a 20% optimality gap. A modified basic reproduction rate (i.e., mR₀) is used to compare the performance of the drone-based testing kit delivery method to the face-to-face diagnostic method in reducing disease spread. The results show that our proposed method (mR₀= 0.002) outperformed the face-to-face diagnostic method (mR₀= 0.0153) by reducing mR₀ by 7.5 times.

Appendix: Launch-Return Location Optimization Model

The LRO chooses an optimal set of launch and return nodes for a current sets of patients to minimize the MDRP objective function. Unlike MDRP, P and SP are pre-determined in LRO. The primary purpose of this model is to check the feasibility of an initial solution provided by TSP_BFS.

$$ \begin{array}{@{}rcl@{}} &&\underset{f,w,h}{\min} \text{LRO} = {\sum}_{(l,i,r)\in LFR}t^{i}_{l,r}f^{i}_{l,r}+{\sum}_{i\in P}w_{i}+{\sum}_{i\in N} h_{i} \end{array} $$

(1)

$$ \begin{array}{@{}rcl@{}} && (7),(8),(32) \end{array} $$

(2)

$$ \begin{array}{@{}rcl@{}} && T_{p_{r}}-T_{p_{l}}+{\sum}_{k=l+1}^{r-1}w_{p_{k}}-M(1-f^{j}_{p_{l},p_{r}})\leq {t}^{j}_{p_{l},p_{r}}+{\sum}_{i \in {S}_j}h_{i} \forall (p_{l},j,p_{r})\in FLR \end{array} $$

(3)

$$ \begin{array}{@{}rcl@{}} && {t^j_{p_l,p_r}+{\sum}_{i\in S_j}h_i-M(1-f^j_{p_l,p_r})}{\leq T_{p_r}-T_{p_l}+{\sum}_{k=l}^{r}w_{p_k} \ \ \ \forall (p_l,j,p_r)\in FLR} \end{array} $$

(4)

$$ \begin{array}{@{}rcl@{}} && {D_{p_{i-1}}-D_{p_{i}}}{={\sum}_{(l,j,p_{i})\in LFR}f^j_{l,p_i}-{\sum}_{(p_i,j,r)\in LFR}f^j_{p_i,r} \ , i=2,...,|P|} \end{array} $$

(5)

$$ \begin{array}{@{}rcl@{}} && {D_{C}+{\sum}_{(C,j,r)\in LFR}f^j_{C,r}}{=n } \end{array} $$

(6)