Cross-docking and milk run logistics in a consolidation network: A hybrid of harmony search and simulated annealing approach

•A novel integer programming model to transport goods in a consolidation network via three transportation systems; direct shipment, cross-docking, and milk run logistics.•A novel solution approach which is a new combination of harmony search and simulated annealing meta-heuristics.•Convergence of the proposed solution approach to reasonable solutions in a sensible amount of CPU time.

In this paper, we develop a novel integer programming model for the transportation problem of a consolidation network where a set of vehicles are used to transport goods from suppliers to their corresponding customers via three transportation systems: direct shipment, shipment through cross-dock (indirect shipment) and milk run. Since the proposed problem formulation is NP-hard, we offer a hybrid of harmony search (HS) and simulated annealing (SA) based heuristics (HS-SA algorithm) in order to solve the problem. The objective of this problem is to minimize the total shipping cost in the network, so it is tried to reduce the number of required vehicles using an efficient vehicle routing strategy in the algorithm. Solving several numerical examples demonstrates that our solving approach performs much better than GAMS/CPLEX in reducing both the shipping cost in the network and computational time requirement, especially for large size problem instances.