#### Inventory Problems with Partially Observed
Demands and Lost Sales

**Abstract:** This paper considers the case of partially
observed demand in the context of a multi-period inventory problem with lost
sales. Demand in a period is observed if it is less than the inventory level in
that period and the leftover inventory is carried over to the next period.
Otherwise, only the event that it is larger than or equal to the inventory level
is observed. These observations are used to update the demand distributions over
time. The state of the resulting dynamic program consists of the current
inventory level and the current demand distribution, which is infinite
dimensional. The state evolution equation for the demand distribution becomes
linear with the use of unnormalized probabilities. We study two demand cases.
First, the demands evolve according to a Markov chain. Second, the demand
distribution has an unknown parameter which is updated in the Bayesian manner.
In both cases, we prove the existence of an optimal feedback ordering policy.