The growing adoption of drones in last-mile logistics has raised urgent demands for controlling energy consumption costs while ensuring delivery efficiency. To address this, we formulate the Two-Echelon Vehicle Routing Problem with Load-dependent Drones (2E-VRPLD) within a depot-satellite-customer framework, which reflects realistic logistics conditions where drones experience continuously changing payloads as they pick up and deliver parcels. These load fluctuations cause discrete shifts in power consumption, making their actual flight endurance highly uncertain under a fixed energy budget and thereby significantly increasing the complexity of planning feasible routes and minimizing overall transportation costs. To solve this complex VRP variant, we develop a novel Constraint Relaxation Co-evolutionary Algorithm (CRCEA) that integrates a customer-satellite matching mechanism to minimize drone endurance redundancy and a minimal impact assessment strategy for route segment merging. Experiments show that CRCEA attains nearly negligible optimality gaps relative to Gurobi on small-scale instances, significantly outperforms seven representative heuristics on large-scale instances, and consistently delivers superior solutions in real-world delivery scenarios from leading express carriers JD and SF Express in Changsha, China. Furthermore, ablation studies identify key factors affecting algorithmic performance, and sensitivity analyses illustrate how drone performance parameters influence the efficiency of solving the 2E-VRPLD. This work provides a robust theoretical and practical approach for solving the two-echelon pickup-and-delivery routing problem with load-dependent drones, offering important insights for advancing smart logistics and improving enterprise operational efficiency.

Multimodal freight transportation is essential for enhancing logistics efficiency and reducing costs, with route optimization as its core component. However, uncertainties are prevalent in multimodal systems, posing challenges for the construction and validation of simulation models. Furthermore, the involvement of multiple stakeholders introduces conflicting optimization objectives. Effectively balancing these objectives to maximize overall benefits has become a critical issue that requires resolution. For the multimodal transport route optimization problem in hybrid uncertainty environments (e.g., there are uncertainties in transportation demand, transportation time, and transfer time), this study constructs a multiobjective optimization model based on fuzzy numbers. The complexity of the model is reduced by introducing the chance-constrained programming theory. To solve the model, a data-driven multi-objective evolutionary algorithm is designed, integrating Monte Carlo simulation with surrogate models to effectively reduce the computational cost of uncertainty estimation. Furthermore, a constraint prioritization strategy is developed to handle multiple conflict objectives, and complex constraints efficiently. Simulation results demonstrate that the proposed algorithm exhibits excellent performance across networks of varying scales, providing robust decision support for multimodal transportation decision-making.

Multi-objective optimization problems (MOPs) in science and engineering frequently involve intricate multi-constraints. This paper extends the application of the Pareto dominance in MOPs on addressing complex multi-constraints and enhancing algorithmic conflicting multi-performance, such as convergence, diversity, and feasibility. The approach begins by identifying non-dominated constraints that closest approximate the actual constrained Pareto Front (CPF) through Pareto non-dominated sorting of every single constrained Pareto Front (SCPF). Subsequently, a Pareto non-dominated sorting multi-performance methodology is employed under the determined non-dominated constraints, considering convergence, diversity, and feasibility as competing objectives. Building upon extending the Pareto dominance approach for constrained multi-objective optimization (EPDCMO), this paper introduces a dual-population multi-archive optimization mechanism to optimize multiple constraints and performance simultaneously. The effectiveness of the proposed approach is validated through the evaluation of 23 constrained multi-objective problems (CMOPs) and practical applications in the domain of CMOPs. The results demonstrate the algorithm's capability to generate competitive solutions for MOPs characterized by multi-constraints.

The constrained-domination principle (CDP) is one of the most popular constraint-handling techniques (CHTs), since it is simple, nonparametric, and easily embedded in unconstrained multiobjective evolutionary algorithms (MOEAs). However, the CDP overly emphasizes the importance of feasibility, which may lead to the search getting stuck in some locally feasible regions or locally optimal, especially when encountering problems with discontinuous and/or narrow feasible regions. This article seeks to capitalize on the strengths of the CDP while overcoming its weaknesses. Accordingly, we propose a novel constrained MOEA (called MOEA/D-LCDP), in which the CDP is applied in a local manner. Unlike most CHTs that emphasize feasibility, which use the feasibility rule in the whole search space, the proposed localized CDP only adopts the CDP within the niche. That is, to maintain the diversity of the population, only solutions within the niche are compared based on the localized CDP. The niche radius is determined a priori by the acute angle between the current subproblem and its nearest subproblem. Additionally, a population-based status detection strategy is developed to allocate computing resources more rationally, and a diversity-enhanced CDP is designed to enhance the exploitation of the search. Comprehensive experiments conducted on four benchmark test suites with a total of 34 problems and three real-world applications demonstrate that MOEA/D-LCDP is very competitive with representative algorithms.

Balancing constraints and objective functions in constrained evolutionary multiobjective optimization is not an easy task. Overemphasis on constraints satisfaction may easily lead to the search to get stuck in local optimal regions, and overemphasis on objectives optimization may lead to substantial search resources wasted on infeasible regions. This article proposes a constrained multiobjective optimization algorithm, called CMOEA-SDE, aiming to achieve a good balance between the above two issues. To do so, CMOEA-SDE presents a strictly constrained dominance relation and a constrained shift-based density estimation strategy. Specifically, the former defines a new dominance relation that considers both constraint satisfaction and the objective functions. It favors good feasible solutions but still leaves room for infeasible solutions to be selected. Unlike most density estimation methods, which only consider the diversity of solutions, our shift-based density estimator covers both the feasibility and the diversity of solutions. That is, our estimator shifts the solutions’ positions based on the extent of the constraints they violate so that solutions violating constraints more severely are shifted to crowded areas, thus being eliminated early. Systematic experiments were conducted on four benchmark test suites and six real-world constrained multiobjective optimization problems. The experimental results suggest that the proposed algorithm can achieve very competitive performance against state-of-the-art constrained multiobjective evolutionary algorithms.