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This paper considers inventory models with advance demand information and flexible delivery. Customers place their orders in advance, and delivery is flexible in the sense that early shipment is allowed. Specifically, an order placed at time t by a customer with demand leadtime T should be fulfilled by period t+T; failure to fulfill it within the time window [t, t+T] is penalized. We consider two situations: (1) customer demand leadtimes are homogeneous and demand arriving in period t is a scalar dt to be satisfied within T periods. We show that state-dependent (s,S) policies are optimal, where the state represents advance demands outside the supply leadtime horizon. We find that increasing the demand leadtime is more beneficial than decreasing the supply leadtime. (2) Customers are heterogeneous in their demand leadtimes. In this case, demands are vectors and may exhibit crossover, necessitating an allocation decision in addition to the ordering decision. We develop a lower-bound approximation based on an allocation assumption, and propose protection level heuristics that yield upper bounds on the optimal cost. Numerical analysis quantifies the optimality gaps of the heuristics (2 % on average for the best heuristic) and the benefit of delivery flexibility (14 % on average using the best heuristic), and provides insights into when the heuristics perform the best and when flexibility is most beneficial