Probabilistic multi-item inventory model with varying order cost under two restrictions: A geometric programming approach
Introduction
Most of the literature dealing with probabilistic inventory models assumes that the demand rate is probabilistic since the probability distribution of the future demand rate rather than the exact value of demand rate itself, is known. Most of the probabilistic inventory models assume that the units of cost are constant and independent of the number of periods. Unconstrained probabilistic inventory models with constant unit of costs have been treated by Gupta and Hira (1994), Hadley and Whitin (1963), and Taha (1997).
Fabrycky and Banks (1967) studied the probabilistic single-item, single-source (SISS) inventory system with zero lead time, using the classical optimization. Recently, Abou-El-Ata and Mousa (1998) studied the deterministic multi-item inventory model with varying order cost under two restrictions. Also, Fergany (1999) discussed the multi-item inventory system with both demand-dependent unit costs and varying leading time using the Lagrangian multiplier.
In this research, we investigate a probabilistic multi-item, single-source (MISS) inventory model with varying order cost under two restrictions, one of them is on the expected order cost and the other on the expected holding cost. The optimal number of periods Nr* and the optimal expected total cost are obtained. Also, some special cases are deduced and an illustrative numerical example is added with some graphs.
Section snippets
Model development
The following notations are adopted for developing our model:the purchase cost of the rth item the varying order cost of the rth item per cycle the holding cost of the rth item per period a random variable demand rate of the rth item per period the probability density function of the demand rate the expected value of the demand rate the expected order quantity of the rth item the maximum inventory level of the rth item the number of periods of the rth item (a decision
Special cases
We deduce four special cases of our model as follows:
Case (1): Let K2→∞ (this cancels the linear constraint) ⇒W4r*=0, W1r*=1+(β−1)W3r*/2−β and W2r*=((β−1)(1+W3r*)/β−2). This is a probabilistic multi-item inventory model with varying order cost under a non-linear order cost constraint. Then , , becomeand
An illustrative example
Let us find the optimal expected number of periods and the minimum expected total cost from Table 1.
Also assuming that E(Dr)=1.8, K1=1500, K2=1200, 0⩽β<1 and b=3.
Solution (Table 2)
From the data given in Table 2, we can calculate , at different values of β and draw the graphs of both Nr* and against β as in Fig. 2, Fig. 3.
Conclusion
We have evaluated the optimal expected number of periods ; then we deduced the minimum expected total cost of the considered probabilistic multi-item inventory model. We draw the curves Nr* and against β, which indicate the values of Nr* and β that give minimum value of the expected total cost of our numerical example.
Acknowledgements
The authors are very grateful to both referees for their valuable comments, which clarify our work to the researchers.
References (7)
- et al.
Multi-item inventory model with varying order cost under two restrictionsA geometric programming approach
Journal of the Egyptian Mathematical Society
(1998) - Duffin, R.J., Peterson, E.L., 1974. Constrained minima treated by geometric means. Westinghouse Scientific paper...
- et al.
Procurement and Inventory SystemsTheory and Analysis
(1967)