# A Multi-Strategy ALNS for the VRP with Flexible Time Windows and Delivery Locations  
[PERSON]  
1 School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China; [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (X.D. ); [PERSON]ADDRESS] (P.L.) 1  
[PERSON]  
1 School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China; [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (X.D. ); [PERSON]ADDRESS] (P.L.) 1  
[PERSON]  
1 School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China; [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (X.D. ); [PERSON]ADDRESS] (P.L.) 1  
[PERSON]  
2 School of Information Engineering, Hubei University of Economics, Wuhan 430205, China 23 Hubei Key Laboratory of Digital Finance Innovation, Hubei University of Economics, Wuhan 430205, China 34 Correspondence: [EMAIL_ADDRESS] 42334234  
market competitiveness of enterprises [2]. In logistics delivery, the last mile of delivery is directly facing the customers [3], which is the key link to improve the service quality. In this process, customer satisfaction, as an important indicator for measuring the service quality of logistics companies, has received widespread attention. In last-mile delivery, customer satisfaction is usually assessed by the timeliness of delivery [4]. Researchers have proposed a variety of descriptions related to delivery time, such as customer waiting time cumulative function [5] and fuzzy time window function [6]. However, customer satisfaction is a subjective metric that is influenced by customers' psychological behavior. The above methods cannot adequately capture customers' psychological behaviors, so the obtained customer satisfaction often deviates from the actual situation [7; 8; 9]. For this reason, this paper improves the traditional way of describing customer satisfaction based on prospect theory. Specifically, when considering the customer's psychological reference point in delivery time, the customer's expectations are met when the delivery vehicle arrives early or on time; while when the vehicle is late, the customer's satisfaction will show a different response based on the customer's psychological behavior. This update better reflects customers' psychological responses in different delivery scenarios, thus providing a more realistic evaluation of customer satisfaction. In addition, due to the division of urban areas (e.g., work, living, and recreational areas) and the differences in the living habits of customers, logistics companies often have to make secondary deliveries due to the absence of the customer when designating the delivery location [10], which leads to a reduction in the efficiency of delivery. Obviously, this not only has a negative impact on transport cost but is also detrimental to the achievement of ecological goals. Current delivery models usually allow customers to choose only one delivery location; however, since the customer's activity location is not fixed, this can easily lead to a mismatch between the customer's actual activity location and the selected delivery location. To address this problem, this paper proposes a new delivery model that allows customers to provide flexible optional delivery locations in a single delivery service, thereby improving delivery flexibility. This improvement is expected to reduce the occurrence of secondary deliveries and enhance logistics efficiency while reducing transport costs and environmental impact. Vehicle Routing Problems (VRPs) represent a core challenge in logistics and operations research, focusing on optimizing delivery routes to minimize costs while meeting various constraints. In today's competitive landscape, particularly in e-commerce driven last-mile delivery, ensuring high levels of customer satisfaction has become as critical as cost efficiency.