Joint grey time windows and two-dimensional loading design for modelling and solving multi-objective vehicle routing problems

Abstract

This study defines grey time windows as periods during which customers expect deliveries at a specific time, with an allowance for a certain amount of advance or delay. For model solving, the grey time window is first whitened using the whitening triangle function based on different satisfaction thresholds and corresponding delivery costs. A tightness function is then designed for two-dimensional loading, which measures the effectiveness of different key points and loading postures. Finally, the Iori dataset was utilised to verify the effectiveness of the proposed solution algorithm. To the best of our knowledge, this is the first time the joint design concept of grey time windows and the whitening method have been introduced. Additionally, a grey opportunity-constrained programming model was constructed, with the model transformation and solution methods provided. Simulation results demonstrate that the proposed two-dimensional loading algorithm is both simple and efficient, and that the constructed grey chance-constrained programming model is solvable and has practical application value. Ultimately, this study provides insights for logistics companies on optimising delivery plans under various time window concepts.