## 2 Problem Description  
The motivation for this study arises from the hospital's intention to acquire a fleet of AMRs to assist medical personnel in providing certain medical services. There are some daily medical requests to be completed by some homogeneous AMRs. The primary task of the AMRs involves transporting medicines from depots (e.g., pharmacies) to different demand points (e.g., wards) and returning to the depots upon completing the service. However, considering the diverse demands associated with each request, the service time for the AMRs is uncertain. Furthermore, during the driving process, multiple unpredictable factors influence travel time, such as obstacle avoidance or waiting for the elevator. As a result, both service time and travel time are treated as random variables. Each medical request is assigned a specific time window within which it must be completed, and the AMRs are responsible for delivering a specified quantity of demands for each request. It is important to note that each request is fulfilled by one and only one AMR. Assuming that the capacity of each AMR is denoted by \(Q\), the AMRs consume electricity during their operations. To improve the efficiency of AMRs and minimize the hospital's daily costs, it becomes imperative to plan the optimal number of participating AMRs and devise the most efficient AMR service routes. The following characteristics of the AMR service process should be taken into consideration:  
1. Stochastic travel time: due to unforeseeable obstacles, the travel time between any two requests is subject to randomness. Additionally, the AMR waiting time for elevators is also stochastic. 2. Stochastic service time: the duration of AMR service varies depending on the service object or environment. 3. Time window: medical requests are associated with specific time windows. Failure to deliver medicine to the patient within the designated timeframe can result in severe consequences. 4. Battery constraint: low battery levels require immediate recharging at the charging station. 5. Capacity constraint: the AMR's load capacity sets an upper bound on the items it can carry. To facilitate the analysis of the routing problem of the medical AMRs, a complete directed graph \(G=(V,A)\) is defined, where \(V=D_{1}\cup R\cup C\cup D_{2}\), \(D_{1}\) is the set of starting points of the AMRs, \(D_{2}\) is the set of endpoints, \(C\) is the set of charging stations, and \(R=\left\{1,2,3,\ldots,n\right\}\) is the set of medical requests issued by the hospital. Let \(A=\left\{\left(i,j\right)\middle|i,j\in V,i\
eq j\right\}\) be the set of arcs. If an AMR needs to charge during operation, it selects the nearest charging point to complete the charge. Without loss of generality, let \(D_{1}=D_{2}=\left\{d\right\}\). Suppose that the time window of request \(i\) is \(\left[e_{i},h_{i}\right]\). If an AMR arrives at the request point \(i\) earlier than the lower bound of its time window \(e_{i}\), it waits until \(e_{i}\). To ensure service quality, the chance constraint is adopted. Let the probability that AMRs serve requests in their specific time window be greater than a given threshold \(1-\varepsilon\). The service time of the request \(i\) is a random variable \(S_{i}\). The AMR needs to transport the corresponding demand \(q_{i}\) to the request point \(i\), \(i\in R\). The distance between any two medical requests is \(d_{ij}\), and their floor difference is \(\left|f_{ij}\right|\), \(\left(i,j\right)\in A\). In the stochastic environment there are various uncertainties, such as avoiding obstacles or taking elevators during the travel of AMRs. The travel time \(T_{ij}\) between request \(i\) and request \(j\) is a random variable. Figure 1 illustrates a simple example. It involves three AMRs, twelve medical requests, a charging point, and a depot (pharmacy). All AMRs depart from the depot and eventually return to the depot. The two elements in vector \(\left(l_{i}\text{ }level,\ q_{i}\text{ }kg\right)\) represent the floor and the demand of the request point \(i\), respectively.