A heuristic and lower bound for a multi-depot routing problem

The multi-depot routing problem considered here deals with a situation where a fleet of τ trucks is used to transport π different raw materials available at σ sources to π plants. A heuristic is presented for solving this problem. An initial feasible solution is found by determining the least costly way of supplying each plant with the material demanded, one plant at a time. Then routes are exchanged for each truck if a net cost savings can be realized, while maintaining feasibility. The heuristic is compared to a lower bound obtained from a relaxed binary formulation. Both methods are applied for four sets of randomly generated problems with different values for τ, σ, π, and for the maximum number of trips allowed. The problems ranged in size from 52 to 609 nodes in the lower bound formulation. The results from the heuristic exceed the bound by 2–4% for problem sets with 52–303 nodes. For the larger problems this percentage reaches 8%. The gap is due partly to the lower bound having a smaller value than the optimal solution and partly to the heuristic being non-optimal.