A practical d-relaxed priority rule for balancing cost and quality in fresh food delivery: Bi-objective optimization with time-dependent travel speed

The growth of fresh product e-commerce raises challenges in managing delivery demands with non-uniform priorities. This study introduces the d-relaxed priority rule into the pickup and delivery problem with time-dependent travel speed, thereby enhancing the flexibility of customer order services. We present a bi-objective optimization model that balances delivery cost and quality loss, and propose a bi-objective adaptive large neighborhood search algorithm based on decomposition idea to solve the model. Numerical experiments validate the proposed model and solution approach. Sensitivity analysis further reveals that increasing priority levels generally increases travel distance and quality loss, while a larger relaxation of priorities offers more routing flexibility, potentially improving both objectives. Excessive priority relaxation, however, may harm service qualities of high-priority orders. The proposed method performs stably in handling instances of different sizes, offering a practical tool to balance operational efficiency and customer satisfaction via careful selections of prioritization and relaxation level. Managerial insights are provided for decision-makers to improve fresh food delivery operation management.