Optimal Inventory Under Stochastic Demand with Two Supply Options

The problem of determining the optimal ordering policies under stochastic demand is examined when two supply options, air and surface, are available, with different costs and different delivery times. Assuming mild conditions on the holding-penalty cost functions, linear ordering costs and backlogging, some sufficient conditions are obtained to show when it is optimal to order nothing by air and others to show when it is optimal to order nothing by surface. Explicit formulae are derived for the optimal orders in the case when air delivery time is $\kappa $ periods and surface delivery time is $\kappa + 1$ periods, respectively, under more general conditions than before.