Cost minimization in multi−commodity multi−mode generalized networks with time windows

The purpose of this research is to develop a heuristic algorithm to minimize total costs in multi-commodity, multi-mode generalized networks with time windows problems. The proposed mathematical model incorporates features of the congestion of vehicle flows and time restriction of delivering commodities. The heuristic algorithm, HA, has two phases. Phase 1 provides lower and upper bounds based on Lagrangian relaxations with subgradient methods. Phase 2 applies two methods, early due date with overdue-date costs and total transportation costs, to search for an improved upper bound. Two application networks are used to test HA for small and medium-scale problems. A different number of commodities and various lengths of planning time periods are generated. Results show that HA can provide good feasible solutions within the reasonable range of optimal solutions. If optimal solutions are unknown, the average gap between lower and upper bounds is 0.0239. Minimal and maximal gaps are 0.0007 and 0.3330. If optimal solutions are known, the maximal gap between upper bounds and optimal solutions is less than 10% ranges of optimal solutions