Deterministic Economic Order Quantity (EOQ) models have been studied intensively in the literature, where the demand process is described by an ordinary differential equation, and the objective is to obtain an EOQ, which minimizes the total cost per unit time. The total cost per unit time consists of a “discrete” part, the setup cost, which is incurred at the time of ordering, and a “continuous” part, the holding cost, which is continuously accumulated over time. Quite formally, such deterministic EOQ models can be viewed as fluid approximations to the corresponding stochastic EOQ models, where the demand process is taken as a stochastic jump process. Suppose now an EOQ is obtained from a deterministic model. The question is how well does this quantity work in the corresponding stochastic model. In the present paper we justify a translation of EOQs obtained from deterministic models, under which the resulting order quantities are asymptotically optimal for the stochastic models, by showing that the difference between the performance measures and the optimal values converges to zero with respect to a scaling parameter. Moreover, we provide an estimate for the rate of convergence. The same issue regarding specific Economic Production Quantity (EPQ) models is studied, too.

Consider an inventory item which is demanded. So the inventory level gradually decreases and is backed up by ordering new inventories from time to time. There are two costs to consider: a positive inventory level results in a holding cost, and every order induces a setup cost. The objective is to determine an order quantity that minimizes the total cost per unit time (

Arguably the simplest EOQ model (sometimes referred to as the classic EOQ model) is based on the following assumptions: (i) the instantaneous holding cost rate is constant; (ii) the setup cost is constant; (iii) the demand comes in a deterministic and continuous process at a constant rate; (iv) no backlogging is allowed so that at inventory level zero all the arrived demand is rejected; (v) the inventory item is homogeneous and nonperishable so that only the demand reduces the inventory level; (vi) the inventory level is reviewed continuously so that it can be described by an ordinary differential equation; (vii) the replenishment takes place instantaneously after ordering. A rigorous description of the classic EOQ model is shortly given in Section

Suppose now that a fluid EOQ is obtained and the corresponding stochastic EOQ model is appropriately formulated. Then the issue of interest is how to translate the fluid EOQ into an order quantity for the stochastic model, where the expected total cost per unit time (

Therefore, the main contribution of this paper is to provide a refinement of the results obtained in [

The rest of this paper is organized as follows. In Sections

In what follows, the trivial case of an order quantity taking zero is excluded from consideration, and the context should always make it clear when

Suppose some order quantity

In the corresponding stochastic model, the inventory item is measured in small units, so that a scaling parameter

The scaled stochastic model can be linked to the fluid model by taking

We are interested in minimizing the performance measure given by

We say that order quantity

(a) There exist constants

(b) There exist finite intervals

Note that Condition

Under Condition

In particular, Proposition

The same calculations as in [

Suppose that Condition

Under Condition

Under Condition

(a) For any

(b) For any large enough

Corollary

Fix some order quantity for the fluid model

Suppose that Condition

The fluid approximations of the EOQ models are also briefly considered in [

Firstly, the present paper is based on a weaker condition. Indeed, instead of Condition

(a) Condition

(b) Functions

The global Lipschitz property is essential to the corresponding proof in [

Secondly, the approach in [

In EOQ models the inventory is backed up at once by ordering new inventory items from external suppliers. In this section we consider the situation where the inventory is gradually backed up by producing new items. In greater detail, the inventory level decreases gradually to meet the demand, and when it hits zero, the production is switched on and new inventory items are being produced to back up the inventory. The production is switched off as soon as the inventory is backed up to a predetermined level. Here we have to account for the cost incurred from switching on the production as well as from holding the inventory items. The aim is to obtain an Economic Inventory Backup Level (EIBL) that minimizes

Suppose that we fix some real inventory backup level

Suppose that we fix some inventory backup level positive integer

The concept of AFO and AO inventory backup level

(a) There exist some constants

(b) There exist finite intervals

Note that Condition

Under Condition

As in the case of EOQ models, we observe from Proposition

Under Condition

If

If

If

For

In this section we firstly verify our results by considering a specific EPQ model, where

As for EOQ models, one may refer to [

For the stochastic EPQ model described above,

The proof of this proposition is based on solving (quite tediously) the associated Poisson equation for

The corresponding deterministic EPQ model can be solved easily, and we have

Clearly, if we put

The dotted line (resp., dashed line, solid line) corresponds to

The illustrative graph of

Secondly, if

Let us comment on the applicability issues of our main results (Propositions

Although we assume the ordering point to be always zero, our results are still applicable when it is set to be some fixed positive level, because Lemma

The state-dependence given in Conditions

Finally, Propositions

To sum up, in this work we formally justified a general class of inventory level-dependent deterministic EOQ and EPQ models, regarded as the fluid approximations to their stochastic versions, by showing a translation of the fluid EOQ (EIBL) to provide an order quantity (inventory backup level) asymptotically achieving some optimality for the stochastic model. The efficiency of the translation was obtained, as distinguished from the majority of the works on fluid approximations. The class of inventory models are quite broad so that to various extent, the obtained results are directly applicable to the existing works such as [

To aid our proof, firstly, let us consider the following one-dimensional birth-and-death process

(a) There exist constants

(b ) There exist finite intervals

Note that Condition

The following lemma is a slightly stronger version of [

Suppose that Condition

It can be easily checked in the proof of [

For both the fluid model and (scaled) stochastic model let us call the time duration between two consecutive replenishments a cycle and denote them by

Under Condition

(a)

(b)

(a) Let us denote by

(b) Let us denote by

Under Condition

Now let us easily observe that

For any fixed

We consider the case of a convergent sequence

Now consider the case of a divergent sequence

(a) Under the conditions of the statement we have

(b) According to Lemma

Now

Let us call a cycle the time duration between two consecutive moments when the inventory is fully backed up. Arguing similarly as for EOQ models, it suffices to consider the inventory level process

Under Condition

(a)

(b) with nonnegative

(a) Let us concentrate on the inventory level process over the production-on phase. In the fluid model, it appears convenient to reflect the trajectory

Absolutely similar arguments are applicable to the (scaled) stochastic model. Consequently, we can consider the inventory level process during a production-on phase as a birth-and-death process

(b) The production-off phase has already been covered when analyzing EOQ models. Therefore, one can directly refer to Lemma

Lemma

(a) Suppose that the statement does not hold. That is, for some subsequence

(b) The proof of this part coincides with the one of Corollary

(c1) Let us notice first of all that under the conditions of the statement, we have that the expression

Now let us prove (c1) of the corollary. Observe that under the conditions of the statement, Proposition

(c2) Let us notice that

Now with the help of (c1) and Proposition

The authors are grateful to the reviewers for their valuable comments. This research is partially supported by the Alliance: Franco-British Research Partnership Programme, project “Impulsive Control with Delays and Application to Traffic Control in the Internet” (PN08.021). Y. Zhang thanks Mr. Daniel S. Morrison for his comments which improve the English presentation of this paper.