An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging

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Abstract

Wee (International Journal of Production Economics 59 (1999) 511), in his interesting paper presented a deterministic inventory model with the following characteristics. Quantity discount schemes for the unit cost, partial backlogging at a fixed rate, deterioration of stock in time and demand rate being a linear function of the selling price. In this article we generalize the work of Wee (1999). More specifically, we consider a model where the demand rate is a convex decreasing function of the selling price and the backlogging rate is a time-dependent function, which ensures that the rate of backlogged demand increases as the waiting time to the following replenishment point decreases.

Introduction

The deterioration of inventory in stock during the storage period constitutes an important factor, which has attracted the attention of researchers. Deterioration, in general, may be considered as the result of various effects on the stock, some of which are damage, spoilage, obsolescence, decay, decreasing usefulness, and many more. The first inventory model where the factor of deterioration was considered as significant seems to be that proposed by Ghare and Shrader (1963). These authors studied a model, having a constant rate of deterioration and a constant rate of demand over a finite planning horizon. The model of Ghare and Schrader was extended by Covert and Philip (1973) by introducing variable rate of deterioration. A further generalization to the above models was proposed by Shah (1977) by considering a model allowing complete backlogging of the unsatisfied demand.

There is an extended literature concerning deteriorating inventories. A common characteristic to the most of these models is that they allow shortages, while the unsatisfied demand is completely backlogging. The consideration of partial backlogging for non-perishable product has been undertaken by Montgomery et al. (1973), Rosenberg (1979) and others. The study of models allowing for partial backlogging of unsatisfied demand and deterioration of inventory in stock has been pursued by Wee (1995), Abad (1996), and Chang and Dye (1999).

All these models, and many more, treated the demand rate as constant or time-dependent function. However, in the real-world application the unit selling price is effected the demand, as low selling price exposures demand, while high selling prices decline demand records to zero. This reality has led researchers to consider and study models where demand is taken as a function of the selling price, the so-called optimal pricing policies.

Eilon and Mallaya (1996), were the first authors who studied the pricing policy. They considered perishable items with maximum shelf life and no deterioration before the expiration date. Deterioration is a rather continuously going on phenomenon affected inventory stock. The model by Cohen (1977), takes into account this pattern of deterioration by introducing an exponentially decaying scheme for the inventory in stock and treats demand rate as a linear function of selling price. The simultaneous optimal pricing and ordering levels are determined. Research continues with Kang and Kim (1983), Aggarwall and Jaggi (1989), Abad (1988). Shiue (1990) developed an inventory model over a prescribed scheduling period, taking general rate of deterioration, constant demand rate, partial backlogging. Further, he used three purchasing price, scheme i.e. all units quantity discount, incremental quantity discount and time discount. In a most recent article, Wee (1999) studied an inventory model considering, joint pricing and replenishment policy, where stock deteriorates with time following the Weibull rate of deterioration. The purchase cost for the product follows the all units quantity discount scheme and demand rate is a decreasing linear function of the selling price. The model allows for shortages, which are partially backlogged at a constant rate.

In this article we generalize the work of Wee (1999) as follows. The demand rate is described by any convex decreasing function of the selling price and instead of a constant rate of partial backlogging we consider a variable backlogging rate, as proposed by Abad (1996). For this model we found the optimal solution and we compare it, via examples, to the approximated results produced by Wee (1999). Moreover, we propose a correction for the revenue function considered in the model of Wee.

Section snippets

Notation and assumptions

The following notation is used throughout the paper:

Tcycle length
T1inventory cycle interval with positive stock (decision variable)
qorder quantity (units/cycle)
I(t)the inventory level at time t
Immaximum starting inventory level of the cycle (units)
Ibthe per cycle amount of shortages backlogged (units)
Ilthe per cycle lost sales quantity (units)
sunit selling price (decision variable)
d(s)deterministic demand rate for the product, a function of the selling price s (units/unit time)
mithe ith price

Mathematical formulation of the model

The fluctuation of the inventory level, in this model during a cycle of length T is given in Fig. 1. The differential equation, which describes the variation of inventory level, I(t), with respect to time, t, during the interval [0,T1] isdI(t)dt=−αβtβ−1I(t)−d(s),0⩽t⩽T1,with boundary condition I(T1)=0.

Solving (1) we obtainI(t)=d(s)e−αtβtT1eαuβdu,0⩽t⩽T1,Im=I(0)=d(s)0T1eαuβdu.From (2) we obtain the inventory carried, CI, during the cycle:CI=0T1d(s)e−αtβtT1eαuβdudt.The amount of deteriorated

The solution procedure

For convenience let us setg(T1,T)=uiT1TB1+γ(T−u)du+0T1eatβdt+c20T1e−αtβtT1eauβdudt+c3BT1TT−u1+γ(T−u)du+c4T1T1−B+γ(T−u)1+γ(T−u)du,NP(T1,s,T)=R(T1,s,T)−1Td(s)g(T1,T)−c1T.Note that g(T1,T) is only a function of T1, so its partial derivative w.r.t. s is zero and w.r.t. T1 is∂g(T1,T)∂T1=uieαT1βB1+γ(T−T1)+c2eαT1β0T1e−αtβdt−c3BT−T11+γ(T−T1)c41−B+γ(T−T1)1+γ(T−T1).Taking the partial derivatives we have the following necessary conditions for an extreme point:NP∂T1=d(s)Ts−sB1+γ(T−T1)∂g(T1,T)∂T1

Numerical examples

To illustrate the preceding theory we consider the following examples.

Example 1

Taken from Wee (1999) and adapted to our model

ui(q)=u1=$9.00/gallonfor0<q⩽60,u2=$8.00/gallonfor60<q⩽80,u3=$7.00/gallonfor80<q,c1=$250.00/replenishment,c2=$(0.35+Iui(q))/gallon/week,I=2%,c3=$2.00/gallon/weekbacklogged,c4=$15.00/gallon,d(s)=25–0.5s,α=0.05,β=1.5.This example was solved for many sets of values for the parameters B,γ and T. The optimal values for T1,s and the results of testing the second-order conditions for a maximum are given in Table 1. In this example

Concluding remarks

  • 1.

    The demand rate function used in this model is quite general. So it gives some flexibility to cover many demand scenarios. Some examples are the following family of functions, a−bs,ae−bs,a/(s+b).

  • 2.

    The model studied in this paper gives as special cases the following previous work:

    • (a)

      For γ=0 and d(s)=abs we obtain the model studied by Wee (1999).

    • (b)

      For γ=0, d(s)=d, c4=0ui=(1−p)Ci with p representing the discount percentage of the unit cost Ci and keep the Weibull distribution for the time that an item

Acknowledgements

We would like to thank the referees for their constructive suggestions to improve the paper.

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