Probabilistic multi-item inventory model with varying order cost under two restrictions: A geometric programming approach

https://doi.org/10.1016/S0925-5273(02)00327-4Get rights and content

Abstract

A probabilistic multi-item inventory model with varying order cost and zero lead time under two restrictions is treated in this paper under the following assumptions: (1) the maximum inventory level of each item is a constant multiple of the average quantity ordered; (2) the order cost is a continuous increasing function of the replenishment quantity, which itself is proportional to some number of periods covered by the replenishment quantity. The constant of proportionality is the average demand per period. The expected total cost of inventory management is composed of three components: the average purchase cost, which is a constant that does not enter into the optimization consideration; the expected ordering cost, and the expected holding cost. No shortages are to be allowed. An analytical solution of the optimal number of periods Nr* (rounded integer) and the optimal maximum inventory level is derived using a geometric programming approach. There are four special cases corresponding to the three possible relaxations of the constraints plus the case of the classical probabilistic model of constant procurement cost combined with the absence of the constraints. Also, an illustrative numerical example is added with some graphs.

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 minE(TC) 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:

Cprthe purchase cost of the rth item
Cor(Nr)the varying order cost of the rth item per cycle
Chrthe holding cost of the rth item per period
Dra random variable demand rate of the rth item per period
f(Dr)the probability density function of the demand rate
E(Dr)the expected value of the demand rate
E(Qr)the expected order quantity of the rth item
Qmrthe maximum inventory level of the rth item
Nrthe 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 , , becomeNr*=ChrE(Dr)b(1+(β−1)W3r*)2Cor(1−β)(1+W3r*)1/β−2,Qmr*=aChr(E(Dr))β−1b(1+(β−1)W3r*)2Cor(1−β)(1+W3r*)1/β−2andminE(TC)=r=1nCprE(Dr)+ChrCor1/1−βE(Dr)b(1+(β−1)W3r*)2(1−β)(1+W3r*)β−1/β−2+ChrE(Dr)b2ChrE(D

An illustrative example

Let us find the optimal expected number of periods and the minimum expected total cost minE(TC) 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 Cor(Nr*),r=1,2,3, at different values of β and draw the graphs of both Nr* and minE(TC) against β as in Fig. 2, Fig. 3.

Conclusion

We have evaluated the optimal expected number of periods Nr*,r=1,2,...,n; then we deduced the minimum expected total cost minE(TC)of the considered probabilistic multi-item inventory model. We draw the curves Nr* and minE(TC) 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)

  • M.O. Abou-El-Ata 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...
  • W.J. Fabrycky et al.

    Procurement and Inventory SystemsTheory and Analysis

    (1967)
There are more references available in the full text version of this article.

Cited by (0)

View full text