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On a discrete-in-time deterministic inventory model for deteriorating items with time proportional demand. (English) Zbl 0575.90019

The paper considers the economic order quantity inventory system for deteriorating items with time proportional demand. The model is continuous in the quantity units and discrete in time. The system operates only for a prescribed period of H units of time. The total demand for the item is D units of quantity. The demand rate R(t) is equal to \(2Dt/H(h+1)\), \(t=1,2,...,H\). Let m be the number of the replenishments to be made during the period H and let \(T_ i=Hi/m\), \(i=0,1,...,m\). Then, determine the replenishment cycles where \(\theta\) is a constant expressing the fraction of the on-hand inventory which deteriorates per time unit.
The inventory process during the i-th cycle is described by the equation \(\Delta Q_ i(t)+\theta Q_ i(t)+2Dt/H(H+1)=0\); \(t=T_{i-1}+1\), \(T_{i-1}+2\),..., \(T_ i\) \((i=1,2,...,m)\), \(Q_ i(T_ i+1)=0\) where \(Q_ i(t)=Q_ i(t+1)-Q_ i(t)\). The orders are made when the stock level falls to zero. At the beginning of the i-th cycle (at the moment \(T_{i-1}+1)\) the order level \(S_ i\) is given by \(S_ i=Q_ i(T_{i-1}+1)\). The cost per unit of the item and the inventory holding cost per unit time and the replenishment cost per order are known and constant during the period H.
The problem is to find the number of the replenishments \(m=m_ 0\) so as to minimize the average total cost per time unit of the system during the period H. A necessary condition for the cost to be minimum at \(m_ 0\) is formulated. A numerical example followed by the sensitivity analysis is given to illustrate the derived results. The more general problem when shortages are allowed and are completely backlogged was discussed by R. S. Sachan [J. Oper. Res. Soc. 35, 1013-1019 (1984; Zbl 0563.90035)].
Reviewer: R.Rempala

MSC:

90B05 Inventory, storage, reservoirs

Citations:

Zbl 0563.90035
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References:

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