Exact algorithm for a half-open multi-depot multi-commodity unpaired pickup and delivery problem with redistribution in omnichannel retailing

This study investigates a half-open multi-depot multi-commodity unpaired pickup and delivery problem with redistribution in an integrated omnichannel retail transportation system. In this setting, a retailer operates multiple warehouses and physical stores, served by a fleet of vehicles that fulfills online customer orders while replenishing and recovering inventory for stores. Two types of products are considered: saleable items, which can be redistributed among stores and customers, and unsaleable items, which must be collected and returned to warehouses. The problem is formulated as a mixed-integer programming model that jointly optimizes routing, pickup-and-delivery demand pairing, and warehouse outbound quantities to minimize the total cost. To solve the model, we develop a branch-and-price-and-cut algorithm that incorporates a bidirectional labeling scheme for the pricing subproblem, acceleration strategies, and subset-row inequalities to strengthen the linear programming relaxation. Extensive computational experiments demonstrate the effectiveness of the proposed method and provide managerial insights on the impacts of redistribution strategies, demand heterogeneity, half-open routes, and online-offline integration.