A grey-interval cooperative distribution model with goods exchange and customer reassignment under uncertain travel times

Purpose The purpose of this study is to develop a grey-interval cooperative distribution model for a two-depot same-region distribution system under incomplete travel-time information and to use the model to examine how goods exchange, customer reassignment and the transfer-cost mechanism affect cooperative performance and internal coordination. Design/methodology/approach Travel-time uncertainty is represented by interval grey numbers. Through a center–width decomposition, each grey travel-time interval is transformed into a robust-equivalent arc cost that incorporates the center and grey-risk components. A non-cooperative grey benchmark model and a grey-interval cooperative distribution model are formulated under the same grey-information environment. The cooperative model introduces a fixed goods-exchange point and post-exchange customer reassignment. Two-depot same-region test instances are constructed from classical multi-depot vehicle routing benchmarks, and exact optimization, statistical risk analysis, sensitivity analysis and mechanism analysis are used for evaluation. Strict performance comparisons are based on the instances for which both models reach proven optimality. Findings For the 16 instances in which both models reach proven optimality, cooperation achieves average instance-wise reductions of 32.32% in total cost, 32.69% in center cost and 32.54% in grey-risk cost. The reduction in grey-risk cost is statistically significant, and the relative cooperative advantage remains stable across the tested levels of risk aversion and uncertainty intensity. Transfer-cost analysis further shows that increasing the net transfer-cost coefficient mainly suppresses transfer imbalance while substantial bidirectional exchange and customer reassignment remain active. The scalability analysis also indicates that the cooperative mixed-integer linear programming becomes substantially more difficult to solve as instance size increases. Originality/value The study develops a grey-interval cooperative distribution model that integrates incomplete travel-time information with goods exchange and post-exchange customer reassignment. The center–width representation converts grey travel-time intervals into deterministic center and grey-risk components, enabling implementation with standard exact optimization without requiring a probability distribution for travel times. A dual-mode modelling architecture compares non-cooperative and cooperative operations under the same grey-information environment. Beyond conventional cost comparison, the proposed model reveals how customer reassignment restructures route exposure and how a net-transfer-cost mechanism regulates transfer imbalance, providing a mechanism-oriented interpretation of cooperation under incomplete travel-time information.