## 1 Introduction
It addresses a multi-trip route scheduling problem specific to a particular type of AMR in a stochastic environment characterized by stochastic travel and service times. The study aims to improve service efficiency and minimize the number of required AMRs by considering the battery level, time window, and load capacity during the service process. The primary objectives of this problem are to minimize operational costs while fulfilling all service requests. To achieve cost reduction and improve AMR efficiency, a stochastic mixed-integer programming model is developed to minimize hospital investment costs, encompassing both the fixed and dynamic expenses associated with the AMRs. Although initially motivated by medical services, this problem has wide-ranging applications in other domains, such as city logistics and drone operations within large-scale transportation networks. Given the stochastic nature of the scheduling problem for AMRs, this issue is addressed within the framework of stochastic programming. There are two commonly employed approaches in stochastic programming: the expected value model and the chance-constrained model [14]. For variables with less randomness, the mean values can replace the random variables in the objective function and constraints, allowing for the use of deterministic methods to solve the problem. Conversely, when dealing with highly random variables, the chance-constrained model is typically adopted, although it introduces some additional complexity due to the variable distributions. The chance-constrained model, proposed by [PERSON] and [PERSON] [14], is a powerful tool for modeling stochastic decision systems. It requires that the probability of the constraint condition is no less than a certain confidence level [15]. [PERSON] et al. [16] explored the electric vehicle routing problem with stochastic demands by constructing a probability constraints model. In our study, given the stochastic travel time and service time, violations of request time windows are inevitable. Consequently, we develop a chance-constrained mixed-integer programming model in which the probability of an AMR arriving within the designated time window should be greater than a given threshold [17]. The AMR routing scheduling problem is an NP-hard combinatorial optimization problem [4]. Exact algorithms are only suitable for small- to medium-scale data instances of the routing problem. To tackle complex routing problems, research has primarily focused on developing heuristic algorithms that can generate high-quality solutions within reasonable computational times. The most common heuristic algorithms are the genetic algorithm [18], variable neighborhood search algorithm [19], ant colony algorithm [20], and tabu search algorithm [21; 22]. Among these approaches, the variable neighborhood search heuristic (VNS) has emerged as an effective and competitive method for solving large-scale routing problems [19; 23; 24; 25]. Initially proposed by [PERSON] and [PERSON] [26], VNS offers a framework for constructing heuristics by systematically exploring different neighborhoods while employing local search techniques to reach local optima. The VNS algorithm effectively mitigates the risk of getting trapped in local optima and rapidly identifies optimal solutions. Numerous variants have been derived from this fundamental algorithmic framework and applied to various combinatorial optimization problems such as vehicle routing [27; 28], electric vehicle routing with stochastic demands [29], traveling salesman problems [30; 31], and scheduling problems [32; 33; 34]. The VNS algorithm is particularly advantageous for handling large problem instances where local search procedures are time-consuming, and it can also serve as an effective means of generating initial solutions before decomposition. The algorithm operates by systematically exploring the solution space, using neighborhood operators to drive the search process toward promising regions.