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Cost formulas for continuous review inventory models with fixed delivery lags. (English) Zbl 0543.60088

The authors consider an inventory situation where demand is characterized by a renewal reward process, \((X_ i,Y_ i)\), \(i=1,2,...\), where \(X_ i\) represents the interarrival times of the customers, \(Y_ i=-D_ i\), where \(D_ i\) is the demand of the i-th customer, and the \((X_ i,Y_ i)\) are independent and identically distributed. Backlogging is allowed and future costs are discounted.
An ordering decision is made each time a customer demands goods. Delivery of additional inventory requires a fixed time delay T. If \(S_ n\) represents the time of the n-th order, then, because of the time delay, an order placed at time \(S_ n\) does not affect inventory costs between \(S_ n\) and \(S_ n+T\), but only costs from \(S_ n+T\) to \(S_{n+1}+T.\)
If \(C_ n(y)\) represents the (discounted) inventory costs from \(S_ n+T\) to \(S_{n+1}+T\) when the inventory level at \(S_ n\) is y, then the \(C_ n\) are independent and identically distributed. Hence, one can restrict attention to \(C_ 0(y)\) (where \(S_ 0=0).\)
A general formula is provided for \(c(y)=E (C(y))\), which depends upon \(\beta\), the discount rate, h(y), the holding (shortage) cost when inventory on hand is y (y demand is backlogged), and the distributions of the \(X_ i\) and the \(Y_ i\). Specific cases where the \(X_ i\) have an exponential or an Erlang distribution are studied in more detail, as well as the case where a fixed penalty is assessed when any backlogging is required.
Reviewer: E.Boylan

MSC:

60K15 Markov renewal processes, semi-Markov processes
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