An adaptive algorithm balances cost and Mahalanobis workload equity in multicommodity vehicle routing with simultaneous pickup and delivery

Balancing operational cost with route-level workload equity is challenging in multicommodity vehicle routing with simultaneous pickup and delivery, where bidirectional flows continuously alter vehicle loads and service requirements. A mixed-integer programming model is formulated for the multicommodity vehicle routing problem with simultaneous pickup and delivery. Route-level workload is represented by total operational time and the number of served customer nodes, and a Mahalanobis-distance penalty is introduced to account for differences in scale and covariance between these indicators. To solve the resulting discrete cost-equity routing problem, an elite archive-based hybrid Red-billed Blue Magpie optimization algorithm (EA-HRBMO) is developed by integrating Clarke-Wright savings initialization, an elite archive, adaptive exploration–exploitation control and variable-neighborhood local search. Experiments on nine extended Solomon instances from clustered, random and mixed customer distributions were conducted over 20 independent runs per algorithm-instance combination. EA-HRBMO obtained the lowest best fitness in six instances and had a lower mean running time than adaptive large neighborhood search (ALNS) in eight instances. At 100 nodes, its mean running time ranged from 584.2 to 697.9 s, compared with 4315.1 to 5767.4 s for ALNS. ALNS achieved lower best fitness in the remaining three instances, indicating that relative performance depends on instance structure. Overall, the results show that EA-HRBMO provides an effective and computationally efficient approach for static day-ahead multicommodity pickup-and-delivery planning under the adopted bivariate workload-equity definition.