Abstract
The existence of an optimal $(s,S)$-policy is shown for the single product continuous review inventory model with infinite planning horizon. The demand process is assumed to be a superposition of a compound Poisson process and a continuous deterministic (state-dependent) process. The inventory position is continuously monitored and at any epoch an order of any size can be placed. An order placed at time t is delivered at time $t + T$, where T is a known nonnegative real number. If $T > 0$ we assume that excess demand is backlogged. When $T = 0$ the backlog and lost sale case can be treated simultaneously. The cost structure consists of fixed set-up costs for each order, a linear purchase cost and a holding and penalty cost function. Both the discounted and average cost criterion are considered.
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