# A Multi-Strategy ALNS for the VRP with Flexible Time Windows and Delivery Locations
## 6 Conclusions  
A Vehicle Routing Problem considering Customer Satisfaction (VRP-CS) is studied, where customer preferences for delivery time and location are incorporated. Satisfaction is evaluated using prospect theory, and a MSALNS algorithm is developed to address the resulting bi-objective optimization. Numerical experiments yielded several key findings:  
1. The proposed MSALNS algorithm demonstrated superior performance compared to standard ALNS, TSA, and SCSO on the VRP-CS benchmark instances, finding better solutions in the majority of cases. 2. By adopting a prospect theory-based customer satisfaction evaluation method, the VRP-CS achieves an average improvement of 15.32% in customer satisfaction, with only a 5.68% increase in delivery cost. This indicates that service quality and customer satisfaction can be effectively improved with a controllable increase in cost. 3. Utilizing optional delivery locations led to substantial benefits, reducing average total costs by 14.22% and delivery costs by 11.30%, while simultaneously increasing average customer satisfaction by 12.37%. These findings demonstrate the effectiveness of the VRP-CS in achieving a balanced trade-off between delivery cost and customer satisfaction. This paper proposes a mathematical model that considers customer satisfaction and alternative delivery locations. In view of the complexity of the problem, an MSALNS algorithm is designed, introducing some new strategies that jump out of local optimal, adapting some classic and effective removal and insertion operators, and designing some new removal and insertion operators that consider customer satisfaction to reconstruct solutions to expand the search space. In addition, new efficient removal operators are introduced to accelerate the convergence of MSALNS. Numerical experiments are used to verify the efficiency and effectiveness of the MSALNS algorithm and to analyze the impact of customer satisfaction on delivery cost as well as the impact of optional delivery locations on delivery cost and customer satisfaction. This study has several limitations, which also point to directions for future research. The current VRP-CS assumes deterministic travel and service times, ignoring real-world factors such as traffic conditions, which may affect the feasibility of routes and the accuracy of customer satisfaction evaluation. The parameters (\(\alpha\), \(\beta\), \(\lambda\)) in the prospect theory function are adopted from recommended values in the existing literature, without being tailored to individual customer characteristics, which limits the precision of the satisfaction assessment. In addition, computational experiments are conducted on simulated datasets. To fully evaluate the practical significance of the VRP-CS in real-world logistics distribution, it is necessary to validate the approach using large-scale real-world data from logistics companies. To address the limitations of the VRP-CS, future research can focus on several directions. One direction is to develop a stochastic VRP-CS model that explicitly considers uncertainty in travel time or customer service time, thereby enhancing the robustness of the solution. Another is to improve the accuracy of customer satisfaction evaluation by incorporating real customer feedback or historical delivery data, potentially through reinforcement learning or other data-driven methods. Further, additional factors influencing customer satisfaction--such as the demand for delivery consistency--could be included to enrich the satisfaction evaluation framework. Finally, validating the VRP-CS using real-world datasets is of great importance. From an algorithmic perspective, exploring alternative optimization techniques, such as constraint programming combined with LNS (CP-LNS) or reinforcement learning-based methods, could help strike a better balance between solution quality and computational effort. **Author Contributions:** Conceptualization, X.D. and X.Z. ; funding acquisition, X.Z. and P.L. ; investigation, X.D. and P.L. ; methodology, X.D. and J.H.