Optimal stationary policies in general state space Markov decision chains with finite action sets
Mathematics of Operations Research
Average optimality in dynamic programming with general state space
Mathematics of Operations Research
A note on the Ross-Taylor theorem
Applied Mathematics and Computation
Dynamic Programming and Optimal Control
Dynamic Programming and Optimal Control
Dynamic Programming and Optimal Control, Two Volume Set
Dynamic Programming and Optimal Control, Two Volume Set
Markov Decision Processes: Discrete Stochastic Dynamic Programming
Markov Decision Processes: Discrete Stochastic Dynamic Programming
Introduction to Stochastic Dynamic Programming: Probability and Mathematical
Introduction to Stochastic Dynamic Programming: Probability and Mathematical
Stochastic Optimal Control: The Discrete-Time Case
Stochastic Optimal Control: The Discrete-Time Case
A Periodic Review Inventory System with Emergency Replenishments
Management Science
A new approach for the stochastic cash balance problem with fixed costs
Probability in the Engineering and Informational Sciences
Coordinated Inventory Planning For New And Old Products Under Warranty
Probability in the Engineering and Informational Sciences
A new approach for the stochastic cash balance problem with fixed costs
Probability in the Engineering and Informational Sciences
Average Cost Markov Decision Processes with Weakly Continuous Transition Probabilities
Mathematics of Operations Research
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Consider a single-commodity inventory system in which the demand is modeled by a sequence of independent and identically distributed random variables that can take negative values. Such problems have been studied in the literature under the name cash management and relate to the variations of the on-hand cash balances of financial institutions. The possibility of a negative demand also models product returns in inventory systems. This article studies a model in which, in addition to standard ordering and scrapping decisions seen in the cash management models, the decision-maker can borrow and store some inventory for one period of time. For problems with back orders, zero setup costs, and linear ordering, scrapping, borrowing, and storage costs, we show that an optimal policy has a simple four-threshold structure. These thresholds, in a nondecreasing order, are order-up-to, borrow-up-to, store-down-to, and scrap-down-to levels; that is, if the inventory position is too low, an optimal policy is to order up to a certain level and then borrow up to a higher level. Analogously, if the inventory position is too high, the optimal decision is to reduce the inventory to a certain point, after which one should store some of the inventory down to a lower threshold. This structure holds for the finite and infinite horizon discounted expected cost criteria and for the average cost per unit time criterion. We also provide sufficient conditions when the borrowing and storage options should not be used. In order to prove our results for average costs per unit time, we establish sufficient conditions when the optimality equations hold for a Markov decision process with an uncountable state space, noncompact action sets, and unbounded costs.