On the Accuracy of Fluid Approximations to a Class of Inventory-Level-Dependent EOQ and EPQ Models

Abstract

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.

1. Introduction

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 (tcu). Such a minimizer is known as an Economic Order Quantity (EOQ), and the model itself is known as an EOQ model.

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 2.1. Amongst the efforts of generalizing the classic EOQ model, a great deal have been made on relaxing assumption (iii). For instance, the EOQ model considered in [1] (see also the references therein) assumes the demand rate to be inventory level dependent, despite the dependence is of a specific form. This is a response to: “At times, the presence of inventory has a motivational effect on the people around it. It is a common belief that large piles of goods displayed in a supermarket will lead the customer to buy more [2].” Generalizations regarding other assumptions include allowing backlogging and periodic reviews in [3], accounting for perishable goods in [4–7], and so on. A comprehensive review of the literature on EOQ models is available in [8]. Note that the aforementioned works mainly focus on EOQ models where the demand process is deterministic. Quite formally these deterministic models can be viewed as the fluid approximations to the corresponding stochastic EOQ models, where the demand comes in a stochastic jump process. (For this reason, in what follows we call the deterministic models also fluid models, and EOQs derived from fluid models are referred to as fluid EOQs.)

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 (TCU) is (nearly) minimal. The fluid approximation would be justified if such a translation is obtained. The formal justification of the fluid approximations to various jump Markov optimization models with local transitions has been addressed by numerous authors, see [9–12], all of which focus on queueing networks. In greater detail the optimization problems considered in [9, 12] are with a discounted criterion over an infinite horizon, the one in [11] is with an expected total cost criterion over a fixed finite time horizon, while the performance measure for the queueing network in [10] is the expected total cost up to the first moment the system gets empty. Here we emphasize that in fluid (resp., stochastic) EOQ models the performance measures are tcus (resp., TCUs), which are long run averages. All of those works compare the optimal value for the fluid model with the performance measure for the (scaled) stochastic model under the translated policy and show that the difference converges to zero as the scaling parameter increases to infinity. However, none of them reveal the rate of convergence, which measures the accuracy of the fluid approximations and the efficiency of the underlying translations. Consequently the more recent development on this topic aims at obtaining the rate of convergence, see [13–15]. In particular, in [14] the author proposes a translation of the fluid EOQ and shows that it results in an asymptotically fluid optimal (AFO) and asymptotically optimal (AO) order quantity for the corresponding stochastic model. (The accurate definitions of “AFO” and “AO” are postponed to Sections 2 and 3.) However, fairly strong conditions on the system parameters are assumed there, restricting the applicability of the obtained results.

Therefore, the main contribution of this paper is to provide a refinement of the results obtained in [14] by relaxing the conditions assumed therein, and thus enlarging the applicability to cover a broader class of EOQ models. In greater detail, by taking the stochastic model as a continuous-time Markov chain, we justify the translation of fluid EOQ proposed in [14] and obtain its efficiency in the form of a rate of convergence. The Markov property of our stochastic model is a result of assuming exponentially distributed interarrival times in the demand process, which is standard in the current literature on inventory systems, see [7, 16–19]. In our models, the demand and holding cost rates are of a rather general inventory-level dependence, and thus with broad applicabilities. In particular, our results are applicable to the important case of discontinuous demand and holding cost rates (compared to that globally Lipschitz continuous rates are assumed in [14]), see more discussions on this in Section 4. Moreover, in this paper results similar to those for EOQ models are derived for Economic Production Quantity (EPQ) models, too.

The rest of this paper is organized as follows. In Sections 2 and 3 we formulate EOQ and EPQ models and state the main results. In Section 4 some comments are given on the issues of possible applicabilities of the obtained results and thus illustrate the contribution of this paper. We finish this paper with conclusions. The proofs of the main statements are postponed to the appendix.

2. Economic Order Quantity Models

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 [⋅] stands for the function taking the largest integer part of its argument.

2.1. Description of Mathematical Models

(a) Fluid Model
Suppose some order quantity 𝑦>0 is fixed. Then let {𝑥(𝑡),𝑡≥0} be the inventory level process subject to the dynamics 𝑑𝑥/𝑑𝑡=−𝜇(𝑥) when 0<𝑥(𝑡)≤𝑦 and 𝑥(𝑡_order+0)=𝑦, where 𝑥(𝑡) reaches state zero at 𝑡_order. Here, 𝜇(𝑥)>0 is the demand rate, and the impulsive jump of 𝑥(𝑡) at 𝑡_order reflects the instantaneous replenishment assumption. Let 𝑡_cycle be the time duration between two consecutive jumps of the inventory level process, 𝑔(𝑥) the instantaneous holding cost rate, and 𝐾>0 the setup cost, incurred immediately whenever an order is made. We are interested in minimizing the long run average cost given by tcu𝑦≜lim_𝑇→∞1𝑇^𝑇₀𝑔(𝑥(𝑡𝑇))𝑑𝑡+𝐾𝑡_cycle. (2.1) Let us call 𝑦^∗ the EOQ for the fluid model, so that tcu(𝑦^∗)=inf_𝑦>0tcu(𝑦). In particular, if 𝜇(𝑥)=𝜇>0 and 𝑔(𝑥)=ℎ𝑥 (i.e., the demand rate and the holding cost rate per unit of inventory are constants 𝜇>0,ℎ>0), then we have the so called classic EOQ model.

(b) (Scaled) Stochastic Model
In the corresponding stochastic model, the inventory item is measured in small units, so that a scaling parameter 𝑛=1,2,… is present to indicate the units. The intuitive meaning of this scaling parameter is explained in Remark 2.1 below. In greater detail, fixing some integer order quantity ^𝑛𝑌>0, the inventory level process {^𝑛𝑋_𝑡,𝑡≥0} is modelled as a continuous-time Markov chain with the state space {0,1,2,…,^𝑛𝑌} and the transition rates {^𝑛𝐴_𝑖,𝑗}^𝑛𝑌_𝑖,𝑗=0 given by ^𝑛𝐴_0,𝑗=𝑛𝜅,if𝑗=^𝑛𝑌,−𝑛𝜅,if𝑗=0,0,otherwise, (2.2) and in the case of𝑖∈{1,2,3,…,^𝑛𝑌}^𝑛𝐴_𝑖,𝑗=𝑖𝑛𝜇𝑛𝑖,if𝑗=𝑖−1,−𝑛𝜇𝑛,if𝑗=𝑖,0,otherwise, (2.3) where 𝑛𝜇(𝑖/𝑛) is the instantaneous demand rate and 𝑛𝜅 is the parameter of the exponentially distributed lead time between the ordering and the corresponding replenishment. In other words, the time between two consecutive demand arrivals is exponentially distributed with mean 1/𝑛𝜇(𝑖/𝑛) when the current inventory level is 𝑖, and new inventories are ordered when the inventory level hits zero.

Remark 2.1. The scaled stochastic model can be linked to the fluid model by taking ^𝑛𝑌≜[𝑛𝑦], where 𝑦>0 is the order quantity for the fluid model (Here [𝑛𝑦]>0 for big enugh 𝑛. So below we assume that [𝑛𝑦]>0, that is, we consider large enough 𝑛.). The above-described scaling is often referred to as a fluid scaling, and its intuitive meaning can be understood as follows. Clearly, when 𝑛=1, the stochastic model is a corresponding version of the fluid model. As we increase 𝑛, the demand comes in smaller units, and inventories are measured more accurately. Take 𝜇(𝑖/𝑛)≜𝑀 as an example. Suppose that 𝑛=1 corresponds to the unit of (kg) so that on average 1⋅𝜇(𝑖/1)=𝑀 (kg) units of demand come per time unit. In the case of 𝑛=1000, the unit will be (g): on average 𝑛𝜇(𝑖/𝑛)=1000𝑀 (g) units of demand come per time unit. Meanwhile, the cost rate is not amplified by multiplying 𝑛 because it costs the same to hold either 1000𝑀 (g) or 𝑀 (kg) of inventories.

We are interested in minimizing the performance measure given by ^𝑛TCU^𝑛𝑌≜lim_𝑇→∞1𝑇𝐸_𝑛𝑌^∞₀𝑔^𝑛𝑋_𝑡𝑛1+𝐾𝑛𝜇𝑛𝐼^𝑛𝑋_𝑡=1𝑑𝑡, (2.4) where 𝐸_𝑛𝑌 denotes the expectation operator with the initial inventory level (immediately after the replenishment) ^𝑛𝑋₀=^𝑛𝑌. Let us call ^𝑛𝑌^∗ the EOQ for the (scaled) stochastic model, so that ^𝑛TCU(^𝑛𝑌^∗)=inf_𝑛^𝑛_𝑌=1,2,…TCU(^𝑛𝑌).

We say that order quantity [𝑛𝑦] for the stochastic model is AFO if lim_𝑛→∞^𝑛TCU𝑛𝑦−tcu𝑦^∗=0 (2.5) and AO if lim_𝑛→∞^𝑛TCU𝑛𝑦−^𝑛TCU^𝑛𝑌^∗=0. (2.6) In what follows, the EOQs for both fluid and stochastic models are assumed to be unique, and the similar assumption applies to the EPQ models.

2.2. Main Results

Condition 1. (a) There exist constants 𝑑₁≥0,𝑘₁>0 and 𝛿>0 such that 𝛿≤𝜇(𝑥)≤𝑘₁ and |𝑔(𝑥)|≤𝑑₁; here functions 𝜇(⋅) and 𝑔(⋅) are measurable and both defined on [0,∞).
(b) There exist finite intervals (0,𝑥₁),(𝑥₁,𝑥₂),… with lim_𝑗→∞𝑥_𝑗=∞ such that on each of them 𝑔(𝑥) and 𝜇(𝑥) are Lipschitz continuous with a common Lipschitz constant 𝑑_𝑔 and 𝑑_𝜇, respectively.

Note that Condition 1 implies that for any fixed 𝑦>0, there exist 𝐿 (possibly 𝑦-dependent) finite intervals (0,𝑥₁),(𝑥₁,𝑥₂),…,(𝑥_𝐿,3[𝑦+1]) such that on each interval 1/𝜇(𝑥) and 𝑔(𝑥)/𝜇(𝑥) are Lipschitz continuous with Lipschitz constants 𝑑_𝜇/𝛿² and (𝑑₁𝑑_𝜇+𝑘₁𝑑_𝑔)/𝛿², respectively. For simplicity, we define 𝑑₂≜max{𝑑_𝜇/𝛿²,(𝑑₁𝑑_𝜇+𝑘₁𝑑_𝑔)/𝛿²}.

Proposition 2.2. Under Condition 1, for any fixed order quantity in the fluid model 𝑦>0^𝑛𝑦≤TCU𝑛𝑦−tcu1+𝑑₁𝑘²₁𝜅𝛿𝑛𝑦𝜅+𝑘₁𝑘₁𝑑₂3𝑦+1𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅. (2.7)

In particular, Proposition 2.2 implies that lim_𝑛→∞|^𝑛TCU([𝑛𝑦^∗])−tcu(𝑦^∗)|=0, that is, [𝑛𝑦^∗] is AFO.

The same calculations as in [14, page 406] result in the next lemma.

Lemma 2.3. Suppose that Condition 1 is satisfied and 𝑔(𝑥)≥0, then 𝑦^∗≥𝛿²𝐾𝑘₁𝑑₁, (2.8) and ^𝑛𝑌^∗ satisfies ^𝑛𝑌^∗/𝑛≥𝛿²𝐾/𝑘₁𝑑₁−𝛿/𝑛𝜅. In particular, lim_𝑛→∞(^𝑛𝑌^∗/𝑛)>0.

Corollary 2.4. Under Condition 1, if lim_𝑛→∞(^𝑛𝑌^∗/𝑛)>0 (see Lemma 2.3), then one has lim_𝑛→∞|^𝑛𝑌^∗/𝑛−[𝑛𝑦^∗]/𝑛|=0, and lim_𝑛→∞|^𝑛TCU([𝑛𝑦^∗])−^𝑛TCU(^𝑛𝑌^∗)|=0, that is, [𝑛𝑦^∗] is AO.

Corollary 2.5. Under Condition 1, suppose in addition that 𝐿 is 𝑦-independent and 𝑔(𝑥)≥0 on [0,∞). Then the following statements hold.
(a) For any 𝑦≥𝛿²𝐾/𝑘₁𝑑₁−𝛿/𝑛𝜅 (see (2.8)) and any large enough 𝑛≥𝑁, where 𝑁 is a positive integer number satisfying 𝑁>(2+𝛿/𝜅)𝑘₁𝑑₁/𝛿²𝐾,^𝑛𝑦≤3TCU𝑛𝑦−tcu1+𝑑₁𝑘³₁𝑑₂𝛿²1𝑛1−𝜅𝑘₁𝑑₁/𝑛𝜅𝛿²𝐾−𝛿𝑘₁𝑑₁+1+𝑑₁𝑘²₁𝑛𝛿𝛿²𝐾/𝑘₁𝑑₁×−𝛿/𝑛𝜅−1/𝑛3𝑘₁𝑑₂𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅≜𝐸(𝑛). (2.9) Here the 𝑦-independent 𝐸(𝑛) goes to zero as fast as 1/𝑛 in the sense of 𝐸(𝑛)=𝑂(1/𝑛), whose meaning is that lim_𝑛→∞(𝐸(𝑛)/(1/𝑛))=𝐶 with some (nonnegative) constant 𝐶.
(b) For any large enough 𝑛≥𝑁,^𝑛TCU^𝑛𝑌^∗−^𝑛TCU𝑛𝑦^∗(𝑛)1≤2𝐸=𝑂𝑛. (2.10) Here we recall that ^𝑛𝑌^∗ and 𝑦^∗ are the EOQ for the (scaled) stochastic model and fluid model, respectively.

Corollary 2.5 refines Corollary 2.4 for certain cases by providing an estimate for the rate of convergence.

Fix some order quantity for the fluid model 𝑦 and scaling parameter 𝑛, and let ^𝑛Π(𝑖),𝑖=0,1,…,[𝑛𝑦] be the stationary distribution of the inventory process {^𝑛𝑋_𝑡,𝑡≥0} and 𝜋(𝑥) the invariant density in the fluid model of the underlying dynamics of 𝑥(𝑡). Then the following proposition shows that the fluid model can also be used to provide approximations to stationary distributions of the inventory level process in the (scaled) stochastic model.

Proposition 2.6. Suppose that Condition 1 is satisfied and some order quantity for the fluid model 𝑦>0 is fixed. Then one has^𝑛Π(0)(0≤𝑘−𝜋)¹𝑘₁+𝜅𝑛𝑦 (2.11) and for 𝑖=1,2,…,[𝑛𝑦]^𝑛Π(𝑖)−₍₎^𝑖/𝑛_{𝑖−1/𝑛}𝜋(𝑥)𝑑𝑥≤2𝑦𝑘²₁𝜅/𝛿²+𝑘²₁𝜅/𝑛𝛿3𝑘₁𝑑₂𝑦+1/𝛿+1+𝑘₁/𝛿3𝑑₁𝐿/𝛿+1/𝜅𝑦.𝑛𝑦+1 (2.12)

2.3. A Comparison with Section 4 of [14]

The fluid approximations of the EOQ models are also briefly considered in [14, Section 4]. Therefore, we mention the main difference of this paper from that one in this subsection.

Firstly, the present paper is based on a weaker condition. Indeed, instead of Condition 1, the following stronger condition is assumed in [14].

Condition 2. (a) Condition 1(a) holds.
(b) Functions 𝑔(⋅) and 𝜇(⋅) are globally Lipschitz continuous.

The global Lipschitz property is essential to the corresponding proof in [14].

Secondly, the approach in [14] is based on the explicit expression of ^𝑛TCU(^𝑛𝑌) obtained by solving the associated Poisson equations. Instead of doing that, the present paper employs the recent results of [15]. One advantage of this approach lies in the weaker condition required (only piecewise Lipschitz continuity is needed). Another advantage is that it allows one to study EPQ models, because the Poisson equations for EPQ models are much more difficult to solve compared to those for EOQ models. That is why EPQ models are not considered in [14].

3. Economic Production Quantity Models

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 TCU, and the resulting model is called an EPQ model. Similar to the previous section, below we justify the fluid approximations to stochastic EPQ models, whose rigorous description is shortly given.

3.1. Description of Mathematical Models

(a) Fluid Model
Suppose that we fix some real inventory backup level 𝑦>0, meaning that the production is always on until the inventory reaches the level 𝑦. Let {𝑥(𝑡),𝑡≥0} represent the inventory level process in the fluid model with state space [0,𝑦], and instantaneous demand and production rates 𝜇(𝑥)>0 and 𝜆(𝑥)>0, respectively. Then the inventory level process is subject to the dynamics 𝑑𝑥=(𝑥),𝑑𝑡𝜆(𝑥)(𝑥,−𝜇production-offphase−𝜇)production-onphase (3.1) where the production-off phase and production-on phase, superseding each other, are triggered by 𝑥(𝑡)=𝑦 and 𝑥(𝑡)=0, respectively. In words, without any delay, once the inventory level reaches zero, production is switched on till it reaches the inventory backup level 𝑦. Let 𝑔(𝑥) be the holding cost rate, 𝐾>0 the setup cost incurred with switching on the production, and 𝑡_cycle the time duration between two consecutive production switching-offs. So tcu𝑦≜lim_𝑇→∞1𝑇^𝑇₀𝑔(𝑥(𝑡𝑇))𝑑𝑡+𝐾𝑡_cycle. (3.2) Let us denote by 𝑦^∗ the EIBL for the fluid model, so that tcu(𝑦^∗)=inf_𝑦>0tcu(𝑦).

(b) (Scaled) Stochastic Model
Suppose that we fix some inventory backup level positive integer ^𝑛𝑌, meaning that the production is always on until the inventory reaches level ^𝑛𝑌. Let {^𝑛𝑋_𝑡,𝑡≥0} represent the inventory level process. We model it as a continuous-time Markov chain with the state space {(^𝑛𝑌,off),(^𝑛𝑌−1,off),…,(0,off),(0,on),…,(^𝑛𝑌−1,on)}, where (𝑖,off) indicates that the inventory level is 𝑖 and the production is off and the denotation of (𝑖,on) can be understood in the same way. Its transition rates are given by ^𝑛𝐴_{(0,off),(0,on)}=𝑛𝜅,^𝑛𝐴_{(0,off),(0,off)}=−𝑛𝜅,_𝑛𝐴_{(0,on),(1,on)}=𝑛𝜆(0),^𝑛𝐴_{(0,on),(0,on)}=−𝑛𝜆_𝑛(0),𝐴_{(𝑛𝑌−1,on),(𝑛𝑌,off)}=𝑛𝜆^𝑛𝑌−1𝑛,^𝑛𝐴_{(𝑛𝑌−1,on),(𝑛𝑌−2,on)}=𝑛𝜇^𝑛𝑌−1𝑛_𝑛,𝐴_{(𝑛𝑌−1,on),(𝑛𝑌−1,on)}=−𝑛𝜆^𝑛𝑌−1𝑛−𝑛𝜇^𝑛𝑌−1𝑛, (3.3) for all𝑖=1,…,^𝑛𝑌∶^𝑛𝐴_{(𝑖,off),(𝑗,off)}=𝑖𝑛𝜇𝑛𝑖,if𝑗=𝑖−1,−𝑛𝜇𝑛,if𝑗=𝑖, (3.4) and finally for all 𝑖=1,…,^𝑛𝑌−2: ^𝑛𝐴_{(𝑖,on),(𝑗,on)}=𝑖𝑛𝜇𝑛𝑖,if𝑗=𝑖−1;𝑛𝜆𝑛𝑖,if𝑗=𝑖+1;−𝑛𝜇𝑛𝑖−𝑛𝜆𝑛,if𝑗=𝑖, (3.5) where 𝑛𝜆(𝑖/𝑛) and 𝑛𝜇(𝑖/𝑛) stand for the instantaneous production and demand rates, 𝑛𝜅 is the parameter of the exponentially distributed lead time between the switching and the actual production-on, and we have ignored all the cases when the transition rates take zero. So we have ^𝑛TCU^𝑛𝑌≜lim_𝑇→∞1𝑇𝐸_𝑛𝑌^𝑇₀𝑔^𝑛𝑋_𝑡𝑛1+𝐾𝑛𝜇𝑛𝐼^𝑛𝑋_𝑡=1𝑑𝑡, (3.6) with the holding cost 𝑔(𝑖/𝑛) and setup cost 𝐾>0. Let us denote by ^𝑛𝑌^∗ the EIBL for the (scaled) stochastic model, so that ^𝑛TCU(^𝑛𝑌^∗)=inf_𝑛^𝑛_𝑌=1,2,…TCU(^𝑛𝑌).
The concept of AFO and AO inventory backup level [𝑛𝑦] can be understood in the same manner as introduced at the end of Section 2.1.

3.2. Main Results

Condition 3. (a) There exist some constants 𝑑₁≥0, 𝑘₁>0,𝛿>0, 𝛿_𝜇𝜆>0, 𝜂>1 and measurable functions 𝜇(𝑥),𝜆(𝑥), and 𝑔(𝑥) defined on [0,∞) such that 𝛿≤𝜇(𝑥)≤𝜇(𝑥)+𝛿_𝜇𝜆≤𝜆(𝑥)≤𝑘₁, 𝜆(𝑥)+𝜇(𝑥)≤𝑘₁, |𝑔(𝑥)|≤𝑑₁, and inf_𝑥>0(𝜆(𝑥)/𝜇(𝑥))=𝜂.
(b) There exist finite intervals (0,𝑥₁),(𝑥₁,𝑥₂),… with lim_𝑗→∞𝑥_𝑗=∞ such that on each of them 𝜆(𝑥),𝜇(𝑥), and 𝑔(𝑥) are Lipschitz continuous with Lipschitz constants 𝑑_𝜆,𝑑_𝜇, and 𝑑_g, respectively.

Note that Condition 3 implies that for any fixed 𝑦>0, there exists an integer 𝐿 (possibly 𝑦-dependent) and 𝐿+1 intervals (0,𝑥₁), (𝑥₁,𝑥₂),…,(𝑥_𝐿,3[𝑦+1]) such that on each interval 1/𝜇(𝑥), 𝑔(𝑥)/𝜇(𝑥), and 1/(𝜆(𝑥)−𝜇(𝑥)) are Lipschitz continuous with Lipschitz constant 𝑑_𝜇/𝛿², (𝑘₁𝑑_𝑔+𝑑₁𝑑_𝜇)/𝛿², and (𝑑_𝜆+𝑑_𝜇)/𝛿²_𝜇𝜆, respectively, and at the same time on each of these intervals belonging to (0,𝑦), functions (with respect to 𝑥) 𝑔(𝑦−𝑥)/(𝜆(𝑦−𝑥)−𝜇(𝑦−𝑥)) and 1/(𝜆(𝑦−𝑥)−𝜇(𝑦−𝑥)) are Lipschitz continuous with Lipschitz constants (𝑘₁𝑑_𝑔+𝑑₁𝑑_𝜇)/𝛿² and (𝑑_𝜆+𝑑_𝜇)/𝛿²_𝜇𝜆, respectively. Let us now denote the common Lipschitz constant by 𝑑₂≜max{𝑑_𝜇/𝛿²,(𝑘₁𝑑_𝑔+𝑑₁𝑑_𝜇)/𝛿²,(𝑑_𝜆+𝑑_𝜇)/𝛿²_𝜇𝜆}.

Proposition 3.1. Under Condition 3, for any fixed (fluid) inventory backup level 𝑦>0, ^𝑛𝑦≤2TCU𝑛𝑦−tcu𝛿𝜅𝑘₁1+𝑑₁𝛿𝐵₁+𝐵₂𝑛𝜂^{−2𝑛[𝑦+1]}+max𝑔(0)/2𝜅,1/2𝜅2𝑛𝑦𝜅+𝑘₁, (3.7) where one puts 𝛿≜max{𝛿,𝛿_𝜇𝜅} and 𝛿≜min{𝛿,𝛿_𝜇𝜅} and the 𝑛-independent terms 𝐵₁,𝐵₂ are given by (A.2) in the appendix.

As in the case of EOQ models, we observe from Proposition 3.1 that [𝑛𝑦^∗] is AFO.

Corollary 3.2. Under Condition 3, the following statements hold.(a)If 𝑔(𝑥)≥0, then lim_𝑛→∞(^𝑛𝑌^∗/𝑛)>0. (b)If lim_𝑛→∞(^𝑛𝑌^∗/𝑛)>0, then lim_𝑛→∞|^𝑛𝑌^∗/𝑛−[𝑛𝑦^∗]/𝑛|=0 and lim_𝑛→∞|^𝑛TCU([𝑛𝑦^∗])−^𝑛TCU(^𝑛𝑌^∗)|=0, that is, [𝑛𝑦^∗] is AO. (c)If 𝐿 is 𝑦-independent and 𝑔(𝑥)≥0, then the following two substatements hold.(c1)For 𝑦≥𝐾𝛿²/4𝑘₁𝑑₁ and big enough 𝑛 so that 𝑛>max{4𝑘₁𝑑₁/𝐾𝛿²,𝑘₁/2𝜅−1}, ^𝑛𝑦≤2TCU𝑛𝑦−tcu𝛿𝜅𝑘₁1+𝑑₁𝛿𝐵₁+𝐵₂𝑛𝜂^{−2𝑛(𝐾𝛿2/4𝑘1𝑑1+1)}+max𝑔(0)/2𝜅,1/2𝜅2𝑛𝐾𝛿²/4𝑘₁𝑑₁−1𝜅+𝑘₁(𝑛)1≜𝐹=𝑂𝑛. (3.8) Here 𝐹(𝑛) is 𝑦-independent, and 𝐵₁=𝑘₁𝑑₂𝜂+13𝐾𝛿²/4𝑘₁𝑑₁+1𝛿+𝑘𝜂−11+¹𝜂𝛿𝜂−13𝑑₁𝐿𝜂+1𝛿,𝐵𝜂−1₂=1+6𝑘₁𝐾𝛿²/4𝑘₁𝑑₁+1𝜂𝛿𝑑𝜂−1¹𝜂²𝜂+1𝛿𝜂−1². (3.9)One has (c2)|^𝑛TCU(^𝑛𝑌^∗)−^𝑛TCU([𝑛𝑦^∗])|≤2𝐹(𝑛)=𝑂(1/𝑛).

4. Example and Comments

In this section we firstly verify our results by considering a specific EPQ model, where tcu(⋅) and ^𝑛TCU(⋅) can be analytically computed. Then we comment on the applications of our results.

Example 4.1. As for EOQ models, one may refer to [14] for an example. Hence we study the following classic setting for the (scaled) stochastic EPQ model: assume constant demand and production rates 𝜇(𝑥)≜𝐷>0 and 𝜆(𝑥)≜𝑅>0, linear holding cost 𝑔(𝑥)=ℎ𝑥 with a constant ℎ>0, constant setup cost 𝐾>0, and finally no lead time between “switching” and “actual production-on,” corresponding to if we take 𝜅→∞. Therefore, if we consider the underlying continuous time Markov chain {^𝑛𝑋_𝑡,𝑡≥0}, state (0,off) will be excluded. The transition rates are modified accordingly. Clearly, Condition 3 is satisfied with this classic setting. The following lemma gives the explicit formula for ^𝑛TCU(𝑍), where for simplicity we have put 𝑍 instead of ^𝑛𝑌 for the inventory backup level.

Proposition 4.2. For the stochastic EPQ model described above, ^𝑛(𝑍)=TCU𝑅ℎ𝑍²(𝐷−𝑅)²+2𝐾𝑛²𝐷(𝑅−𝐷)³+𝑍ℎ𝑅(3𝐷−𝑅)(𝐷−𝑅)+2𝑅ℎ𝐷²𝐷1−𝑅^𝑍×12𝑛(𝑅−𝐷)𝑍𝑅²−𝐷²+𝐷²(𝐷/𝑅)^𝑍.−𝑍𝑅𝐷 (4.1)

The proof of this proposition is based on solving (quite tediously) the associated Poisson equation for ^𝑛TCU(𝑍) and is omitted here.

The corresponding deterministic EPQ model can be solved easily, and we have 𝑦=()tcu𝐾𝐷𝑅−𝐷+𝑦𝑅ℎ𝑦2. (4.2)

Clearly, if we put 𝑍=[𝑛𝑦] in the expression for ^𝑛TCU(𝑍), where 𝑦 is the inventory backup level for the fluid model, then one can easily see that lim^𝑛_𝑛→∞=()TCU𝑛𝑦𝐾𝐷𝑅−𝐷+𝑦𝑅ℎ𝑦2𝑦,𝑦−=tcutcu^𝑛1TCU𝑛𝑦=𝑂𝑛, (4.3) which agree with Proposition 3.1. For the means of illustration, we put 𝐾=5, ℎ=1, 𝐷=5, 𝑅=10. Then the graphs of tcu(⋅) and ^𝑛TCU(⋅) are plotted in Figure 1. Note that when 𝑛=100, the curves of tcu(𝑦) and ^𝑛TCU([𝑛𝑦]) nearly coincide.

Figure 1 

The dotted line (resp., dashed line, solid line) corresponds to 1TCU([𝑦]) (resp., 5TCU([5𝑦]), tcu(𝑦)).

Figure 2 

The illustrative graph of 𝑥̂on(𝑡).

Secondly, if 𝑛 increases, then by inspecting the numerator (especially the first two terms) and the denominator of the expression for ^𝑛TCU(𝑍), we see that if 𝑍 does not increase as fast as 𝑛, ^𝑛TCU(𝑍) will blow up to ∞ (it can be easily checked that the expression 𝑍𝑅²−𝐷²+𝐷²(𝐷/𝑅)^𝑍−𝑍𝑅𝐷>0). Therefore, the condition of lim_𝑛→∞(^𝑛𝑌^∗/𝑛)>0 in Corollary 3.2 is satisfied.

Comments
Let us comment on the applicability issues of our main results (Propositions 2.2 and 3.1). We mainly focus on EOQ models, as absolutely similar comments can be made on EPQ models in the same manner.
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 A.1 holds if we put another absorbing state instead of zero. In particular, if one allows the state taking negative values, by putting some negative state absorbing, our results also impound the case of backlogging. This flexibility regarding the ordering point together with the fact that 𝑔(⋅) is unrestricted in signs enriches the applications of our results, in that although we require 𝜇 to be separated from zero, when profit rather than solely operational cost is counted, the ordering point is most likely positive, meaning in cases of 𝜇(𝑥)=𝛼𝑥^𝛽, 𝛼>0, 0<𝛽<1 as in [20] and 𝜇(𝑥)=𝛼𝑥^−𝛽, 𝛼>0,𝛽≥1 as in [1], 𝜇 will be essentially separated from zero, validating our results.
The state-dependence given in Conditions 1 and 3 is fairly general. In particular, that functions 𝜆(𝑥), 𝜇(𝑥) and 𝑔(𝑥) being bounded is not restrictive, because once some EOQ for the fluid model 𝑦>0 is fixed, to validate Propositions 2.2 and 3.1, they are only required to be bounded on bounded intervals. Note that in addition to the demand rate, some authors such as those of [5, 6] also include a state-dependent deteriorating rate, to indicate that the underlying goods are perishable. Our results are also applicable to such cases: one only needs explain 𝜇(⋅) as the total reduction rate of the inventory level.
Finally, Propositions 2.2 and 3.1 are significant extensions of the relevant results in [14], where the author only focuses on EOQ models and requires global Lipschitz continuity of 𝜇(⋅) and 𝑔(⋅). However, from the modelling point of view, the case of discontinuous functions is interesting and important as demonstrated by [1, 3, 5, 21, 22], where [1] considers a piecewise constant function 𝜇(⋅) and the others consider discontinuous 𝜇(⋅) taking either a constant value or according to 𝜇(𝑥)=𝛼𝑥^𝛽, 𝛼>0,0<𝛽<1. The results in [14] are derived based on the closed form of the solution to a Poisson equation satisfied by ^𝑛TCU(⋅), which is tremendously difficult to get explicitly in the case of (stochastic) EPQ models.

5. Conclusions

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 [1, 5, 6, 8, 20–22]. The present work is a significant extension of the relevant results in [14].

Appendix

To aid our proof, firstly, let us consider the following one-dimensional birth-and-death process {^𝑛𝑍_𝑡,𝑡≥0} with state space {0,1,…} and birth and death rates 𝑛𝛼(𝑖/𝑛) and 𝑛𝛽(𝑖/𝑛), respectively, where nonnegative measurable functions 𝛼 and 𝛽 are defined on [0,∞) and 𝑖 indicates the current state of the process. In addition, 𝛼(0)=𝛽(0)=0, where the equality holds only at 0, meaning that state zero is absorbing. Let 𝐸_𝑖 denote the expectation of any underlying functional of the process with the initial state ^𝑛𝑍₀=𝑖. Let a real measurable function 𝛾(⋅) defined on [0,∞) be fixed with 𝛾(0)=0. Now we are in the position to state the following condition.

Condition 1. (a) There exist constants 𝜂>1, 𝛿>0, 𝑑₁>0, and 𝑘₁<∞ such that inf_𝑧>0(𝛽(𝑧)/𝛼(𝑧))>𝜂, 𝛽(𝑧)≥𝛿, 𝛼(𝑧)+𝛽(𝑧)≤𝑘₁, |𝛾(𝑧)|≤𝑑₁. Here if 𝛼(𝑧)≡0, then 𝜂 can be arbitrary.
(b ) There exist finite intervals (𝑧₀≜0,𝑧₁), (𝑧₁,𝑧₂),… with lim_𝑗→∞𝑧_𝑗=∞ such that on each of them, 𝛾(𝑧)/(𝛽(𝑧)−𝛼(𝑧)) is a Lipschitz continuous function with a common Lipschitz constant 𝑑₂.

Note that Condition 1(b) implies that for any fixed 𝑦>0 there exists an integer 𝐿 (possibly 𝑧-dependent) and 𝐿+1 finite intervals (0,𝑧₁),(𝑧₁,𝑧₂),…,(𝑧_𝐿,3[𝑦+1]) such that on each interval, function 𝛾(⋅)/(𝛽−𝛼) is Lipschitz continuous with a common Lipschitz constant 𝑑₂.

The following lemma is a slightly stronger version of [15, Theorem 2] and will play an important role in our proof.

Lemma A.1. Suppose that Condition 1 is satisfied. Then for each 𝑦>0sup_{0≤𝑖≤𝑛y+1}𝐸_𝑖^∞₀𝛾^𝑛𝑍_𝑠𝑛𝑑𝑠−^∞₀𝛾(𝑧(𝑠≤𝐵))𝑑𝑠¹𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}, (A.1) where regarding the second integral the underlying dynamics is given by 𝑑𝑧/𝑑𝑠=𝛼(𝑧)−𝛽(𝑧), 𝑧(0)=𝑖/𝑛 and 𝐵₁ and 𝐵₂ are given by 𝐵₁=𝑘₁𝑑₂𝜂+13𝑦+1𝛿+𝑘𝜂−11+¹𝜂𝛿𝜂−13𝑑₁𝐿𝜂+1𝛿,𝐵𝜂−1₂=1+6𝑘₁𝑦+1𝜂𝛿𝑑𝜂−1¹𝜂²𝜂+1𝛿𝜂−1². (A.2)

Proof. It can be easily checked in the proof of [14, Theorem 2] that our Condition 1, weaker than the original conditions imposed therein, is sufficient for the statement. See also [23].

Proofs of Proposition 2.2, Corollaries 2.4 and 2.5, and Proposition 2.6
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 𝑡_cycle and ^𝑛𝑇_cycle, respectively. Here for simplicity, we do not explicitly indicate the 𝑦-dependence (resp., ^𝑛𝑌-dependence) of 𝑡_cycle (resp., ^𝑛𝑇_cycle). Clearly {^𝑛𝑋_𝑡,𝑡≥0} is a regenerative process [24, page 425] in that it probabilistically repeats itself from one cycle to the next. It then follows from [25, Theorem 1.1, Proposition 131] (see also [24, Proposition 7.3]) that as far as the long-run average ^𝑛TCU([𝑛𝑦]) is concerned, it suffices to consider the inventory level process and the cost incurred with it over only one cycle. For simplicity, we always consider the cycle starting at time 𝑡=0 with the initial position ^𝑛𝑋₀=[𝑛𝑦]. Let us denote by ^𝑛TC and tc the total cost incurred over the cycle in the stochastic and fluid model, respectively. Then the following lemma indicates that the difference between 𝐸_[𝑛𝑦][^𝑛𝑇_cycle] and 𝑡_cycle and the one between 𝐸_[𝑛𝑦][^𝑛TC] and tc cannot be too big.

Lemma A.2. Under Condition 1 the following two inequalities hold with nonnegative 𝐵₁ and 𝐵₂ given by (A.2):
(a)|𝐸_[𝑛𝑦][^𝑛𝑇_cycle]−𝑡_cycle|≤𝐵₁/𝑛+𝐵₂𝜂^{−𝑛[𝑦+1]}+1/𝑛𝜅;
(b)|𝐸_[𝑛𝑦][^𝑛TC]−tc|≤𝐵₁/𝑛+𝐵₂𝜂^{−𝑛[𝑦+1]}+𝑔(0)/𝑛𝜅.

Proof. (a) Let us denote by ^𝑛𝑇_absorbing the time duration from the starting point 𝑡=0 up to the point when ^𝑛𝑋_𝑡 firstly reaches state zero. Obviously we have𝐸_[𝑛𝑦]^𝑛𝑇_absorbing=𝐸_[𝑛𝑦]^∞₀𝐼^𝑛𝑋_𝑡>0𝑑𝑡. (A.3) Then 𝐸_[𝑛𝑦][^𝑛𝑇_cycle]=𝐸_[𝑛𝑦][^𝑛𝑇_absorbing]+1/𝑛𝜅, where the second term on the right hand side is the expected lead time. Now observe firstly that Condition 1 is a specific version of Condition 1: one can take 𝜂>1 to be arbitrary, and put functions (Here it does no matter to put 𝜇(0)=0.) 𝛼(𝑥)≡0, 𝛽(𝑥)=𝜇(𝑥) and 𝐼{𝑥>0}=𝑔(𝑥)=𝛾(𝑥); and secondly that the inventory level process from 𝑡=0 up to ^𝑛𝑇_absorbing is a pure death process. Therefore, one can refer to Lemma A.1 for |𝐸_[𝑛𝑦][^𝑛𝑇_cycle]−𝑡_cycle|≤|𝐸_[𝑛𝑦][^𝑛𝑇_absorbing]−𝑡_cycle|+1/𝑛𝜅≤𝐵₁/𝑛+𝐵₂𝜂^{−𝑛[𝑦+1]}+1/𝑛𝜅.
(b) Let us denote by ^𝑛TC_absorbing the cost incurred during the interval [0,^𝑛𝑇_absorbing], so that 𝐸_[𝑛𝑦][^𝑛TC]=𝐸_[𝑛𝑦][^𝑛TC_absorbing]+𝑔(0)/𝑛𝜅, where the second term on the right hand side corresponds to the cost incurred over the lead time. In the same way as in part (a), comparing 𝐸_[𝑛𝑦][^𝑛TC_absorbing] with tc first, and then adding 𝑔(0)/𝑛𝜅 results in the statement. Remember that the setup cost cancels out.

Proof of Proposition 2.2. Under Condition 1 we have^𝑛𝑦=𝐸TCU𝑛𝑦−tcu^{[𝑛𝑦]𝑛}TC𝐸_[𝑛𝑦]^𝑛𝑇_cycle−tc𝑡_cycle=𝐸_[𝑛𝑦]^𝑛TC𝑡_cycle−tc𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝑡_cycle=𝐸_[𝑛𝑦]^𝑛TC𝑡_cycle−tc𝑡_cycle+tc𝑡_cycle−tc𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝑡_cycle=𝑡_cycle𝐸_[𝑛𝑦]^𝑛TC−tc+tc𝑡_cycle−𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝑡_cycle≤𝑡_cycle𝐸_[𝑛𝑦]^𝑛T𝐶−tc+tc𝑡_cycle−𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝐸_[𝑛𝑦]^𝑛𝑇_cycle𝑡_cycle≤𝑦/𝛿+𝑑₁𝑦/𝛿𝑦/𝑘₁𝑛𝑦/𝑛𝑘₁𝐵+1/𝑛𝜅𝑛¹+𝐵₂𝜂^{−2𝑛[𝑦+1]}1+max,𝑔(0)𝑛𝜅=𝑛𝜅1+𝑑₁𝑛𝑘²₁𝜅𝛿𝑛𝑦𝜅+𝑘₁𝐵₁𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}1+max,𝑔(0)𝑛𝜅,𝑛𝜅 (A.4) where the last inequality follows from the facts that 𝑦/𝑘₁≤𝑡_cycle≤𝑦/𝛿, tc≤𝑑₁𝑦/𝛿, 𝐸_{[𝑛𝑦][𝑛𝑇cycle]}≥[𝑛𝑦]/𝑛𝑘₁+1/𝑛𝜅 and Lemma A.2.
Now let us easily observe that (𝜂+1)/(𝜂−1) and 𝜂/(𝜂−1) both decrease with 𝜂∈(1,∞). It follows that 𝐵₁/𝑛, 𝐵₂𝜂^{−2𝑛[𝑦+1]}, and thus the above-derived expression all decrease with 𝜂, where we recall that 𝜂 can be an arbitrary number on the interval (1,∞), see Condition 1. This implies that^𝑛𝑦𝜂TCU𝑛𝑦−tcu≤lim_→∞1+𝑑₁𝑘²₁𝑛𝜅𝛿𝑛𝑦𝜅+𝑘₁𝐵₁𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}1+max,𝑔(0)𝑛𝜅=𝑛𝜅1+𝑑₁𝑘²₁𝜅𝛿𝑛𝑦𝜅+𝑘₁𝑘₁𝑑₂3𝑦+1𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅. (A.5)

Proof of Corollary 2.4. For any fixed 𝑛, let us denote ^𝑛𝑌^∗=𝑛𝑦(𝑛). We do the proof in two parts.Part 1. We consider the case of a convergent sequence 𝑦(𝑛). Suppose now that as 𝑛→∞, 𝑦(𝑛) does not go to 𝑦^∗ but lim_𝑛→∞𝑦(𝑛)=𝑦>0; here we allow 𝑦 to be from the extended real line. In particular, for big enough 𝑛, 𝑦(𝑛) is separated from zero. According to Proposition 2.2, we have ^𝑛TCU(𝑛𝑦(𝑛))→tcu(𝑦(𝑛)). But we also have tcu(𝑦(𝑛))→tcu(𝑦), since tcu(𝑦)=(∫^𝑦₀(𝑔(𝑥)/𝜇(𝑥)𝑑𝑥)+𝐾)/∫^𝑦₀(𝑑𝑥/𝜇(𝑥)) is continuous in 𝑦. This gives ^𝑛TCU(𝑛𝑦(𝑛))→tcu(𝑦)>tcu(𝑦^∗). However, it follows from Proposition 2.2 that ^𝑛TCU([𝑛𝑦^∗])→tcu(𝑦^∗). This indicates that at least for big enough 𝑛, ^𝑛TCU([𝑛𝑦^∗])<^𝑛TCU(𝑛𝑦(𝑛))=^𝑛TCU(^𝑛𝑌^∗), which is a desired contradiction. Hence lim_𝑛→∞𝑦(𝑛)=𝑦^∗, and consequently, lim_𝑛→∞|^𝑛TCU([𝑛𝑦^∗])−^𝑛TCU(^𝑛𝑌^∗)|=0, as required.Part 2. Now consider the case of a divergent sequence 𝑦(𝑛). One only needs consider the following two situations: either it has a bounded subsequence, which by Bolzano-Weierstrass theorem further has a convergent subsequence; or it does not have a bounded subsequence, which means that it has a subsequence blowing up to ∞. However, by taking the corresponding subsequences, we find that both situations have been essentially covered in Part 1. Part 2 is thus proved.

Proof of Corollary 2.5. (a) Under the conditions of the statement we have()≤RHSof2.71+𝑑₁𝑘²₁𝛿𝑘𝑛𝑦¹𝑑₂3𝑦+3𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅=1+𝑑₁𝑘³₁3𝑑₂𝑦𝛿²+𝑛𝑦1+𝑑₁𝑘²₁𝛿𝑛𝑦3𝑘₁𝑑₂𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅≤31+𝑑₁𝑘³₁𝑑₂𝛿²𝑦+𝑛𝑦−11+𝑑₁𝑘²₁𝛿𝑛𝑦−13𝑘₁𝑑₂𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅≤3Recall,here𝑛𝑦−1>01+𝑑₁𝑘³₁𝑑₂𝛿²1𝑛1−𝜅𝑘₁𝑑₁/𝑛𝜅𝛿²𝐾−𝛿𝑘₁𝑑₁+1+𝑑₁𝑘²₁𝑛𝛿𝛿²𝐾/𝑘₁𝑑₁×−𝛿/𝑛𝜅−1/𝑛3𝑘₁𝑑₂𝛿+𝑘1+¹𝛿3𝑑₁𝐿𝛿𝑔(0)+max𝜅,1𝜅. (A.6) (Here we use the fact that𝑦/(𝑛𝑦−1)decreases with𝑦and𝑦≥𝛿²𝐾/𝑘₁𝑑₁−𝛿/𝑛𝜅.) (a) is now clear.
(b) According to Lemma 2.3, 𝑦^∗≥𝛿²𝐾/𝑘₁𝑑₁, and ^𝑛𝑌^∗ satisfies ^𝑛𝑌^∗/𝑛≥𝛿²𝐾/𝑘₁𝑑₁−𝛿/𝑛𝜅. Therefore, according to part (a), for 𝑛≥𝑁 we have^𝑛TCU𝑛𝑦^∗≤tcu𝑦^∗+𝐸(𝑛)≤tcu^𝑛𝑌^∗𝑛+𝐸(𝑛)≤^𝑛TCU^𝑛𝑌^∗+2𝐸(𝑛) (A.7) in one direction and ^𝑛TCU𝑛𝑦^∗≥^𝑛TCU^𝑛𝑌^∗≥tcu^𝑛𝑌^∗𝑛−𝐸(𝑛)≥tcu𝑦^∗−2𝐸(𝑛) (A.8) in the other direction. Combining both directions results in the statement.

Proof of Proposition 2.6. Now ^𝑛Π(𝑖) and 𝜋(𝑥) can be easily computed as done in [14]. So we haveΠ(0)=1/𝑛𝜅∑^[𝑛𝑦]_𝑗=1(𝑖)=()1/𝑛𝜇𝑗/𝑛+1/𝑛𝜅,Π1/𝑛𝜇𝑖/𝑛∑^[𝑛𝑦]_𝑗=1,;𝜋(0)(𝑥)=11/𝑛𝜇𝑗/𝑛+1/𝑛𝜅𝑖=1,2,…,𝑛𝑦=0;𝜋𝜇(𝑥)𝑡_cycle,0<𝑥≤𝑛𝑦. (A.9) Here we put 𝜋(0)=0 for convenience. Then |^𝑛Π(0)−𝜋(0)|≤1/𝑛𝜅(1/𝑛𝜅+[𝑛𝑦]/𝑛𝑘₁)≤𝑘₁/(𝑘₁+𝜅[𝑛𝑦]), and ^𝑛Π(𝑖)−₍₎^𝑖/𝑛_{𝑖−1/𝑛}𝜋(𝑥)=()𝑑𝑥1/𝑛𝜇𝑖/𝑛𝐸_[𝑛𝑦]^𝑛𝑇_cycle−∫₍₎^𝑖/𝑛_{𝑖−1/𝑛}1/𝜇(𝑥)𝑑𝑥𝑡_cycle=1()𝑡𝑛𝜇𝑖/𝑛^cycle−₍₎^𝑖/𝑛_{𝑖−1/𝑛}1𝜇(𝑥)𝑑𝑥𝑡_cycle+₍₎^𝑖/𝑛_{𝑖−1/𝑛}1𝜇(𝑥)𝑑𝑥𝑡_cycle−₍₎^𝑖/𝑛_{𝑖−1/𝑛}1𝜇(𝑥)𝑑𝑥𝐸_[𝑛𝑦]^𝑛𝑇_cycle1𝐸^𝑛_[𝑛𝑦][𝑇_{cycle]𝑡cycle}≤𝑡_cycle1/𝑛𝜇(𝑖/𝑛)−∫₍₎^𝑖/𝑛_{𝑖−1/𝑛}𝑑𝑥/𝜇(𝑥)+∫₍₎^𝑖/𝑛_{𝑖−1/𝑛}𝑑𝑥/𝜇(𝑥)𝐸_[𝑛𝑦]^𝑛𝑇_cycle−𝑡_cycle𝐸_{[𝑛𝑦][𝑛𝑇cycle𝑡cycle]}≤2𝑦/𝑛𝛿²+(1/𝑛𝛿)𝐵₁/𝑛+𝐵₂𝜂^{−𝑛2[𝑦+1]}+1/𝑛𝜅𝑛𝑦/𝑛𝑘₁+1/𝑛𝜅𝑦/𝑘₁. (A.10) (Here we recall (a) of Lemma A.2.) Recall that in the above derived expression, 𝜂 can be any number from (1,∞). After passing to the limit 𝜂→∞, we eventually end up with ^𝑛Π(𝑖)−₍₎^𝑖/𝑛_{𝑖−1/𝑛}𝜋(𝑥)≤𝑑𝑥2𝑦𝑘²₁𝜅/𝛿²+𝑘²₁𝜅/𝑛𝛿3𝑘₁𝑑₂𝑦+1/𝛿+1+𝑘₁/𝛿3𝑑₁𝐿/𝛿+1/𝜅𝑦,𝑛𝑦+1 (A.11) as required.

Proofs of Proposition 3.1 and Corollary 3.2
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 {^𝑛𝑋_𝑡,𝑡≥0} and the cost incurred over one complete cycle, for which we put the starting time of 𝑡=0. Let us denote by 𝑡_cycle, ^𝑛𝑇_cycle and tc, ^𝑛TC the duration of a cycle and the cost incurred over a cycle in the fluid and scaled stochastic model, respectively. Notice additionally that a cycle is always constituted to by two phases corresponding to the on and off of the production. This raises another set of denotations: let 𝑡_on, ^𝑛𝑇_on (𝑡_off, ^𝑛𝑇_off), and tc_on,^𝑛TC_on (tc_off, ^𝑛TC_off) be the total cost incurred during the production-on (off) phases in the fluid and (scaled) stochastic model, respectively. We agree on that in both fluid and (scaled) stochastic model the setup cost is accounted for in tc_off and ^𝑛TC_off. Then obviously we have 𝑡_cycle=𝑡_on+𝑡_off, 𝐸[^𝑛𝑇_cycle]=𝐸[^𝑛𝑇_on]+𝐸[^𝑛𝑇_off] and tc=tc_on+tc_off, 𝐸[^𝑛TC]=𝐸[^𝑛TC_on]+𝐸[^𝑛TC_off]. Here and below, for convenience we omit the subscript of the expectation operator.

Lemma A.3. Under Condition 3 the following inequalities hold:
(a)|𝐸[^𝑛TC_on]−tc_on|≤𝐵₁/𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}, |𝐸[^𝑛𝑇_on]−𝑡_on|≤𝐵₁/𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]};
(b) with nonnegative 𝐵₁ and 𝐵₂ given by (A.2),𝐸^𝑛TC_off−tc_off≤𝐵₁𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}+𝑔(0),𝐸𝑛𝜅_𝑛𝑇_off−𝑡_off≤𝐵₁𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}+1.𝑛𝜅 (A.12)

Proof. (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 {𝑥(𝑡),𝑡∈(𝑡_on,𝑡_cycle)} (corresponding to the solid curve in Figure 2) about the horizontal 𝑡-axis first, and then shift the resulting trajectory (corresponding to the curve of crosses in Figure 2) upwards by 𝑦 units, and finally further shift the resulting trajectory to the left by shifting the time by 𝑡_off units to the right to get {_on𝑥(𝑡),𝑡∈[0,𝑡_on]} (corresponding to the curve of solid boxes in Figure 2). Note now, for {_on𝑥(𝑡),𝑡∈[0,𝑡_on]} with _on𝑥(0)=𝑦 the roles of production and demand have switched over: each produced unit reduces _on𝑥 by one unit, and each demanded unit increases _on𝑥 by one unit. More precisely, let us define the following functions:_on𝜇𝑦=𝜆(0),_on𝜇(0)=0,_on𝜇(𝑥)=𝜆𝑦−𝑥,𝑥∈0,𝑦_on,𝜆𝑦=𝜇(0)=0,_on𝜆(0)=0,_on𝜆=𝜇𝑦−𝑥,𝑥∈0,𝑦_on,𝑔𝑦=𝑔(0),_on𝑔(0)=0,_on𝑔(𝑥)=𝑔𝑦−𝑥,𝑥∈0,𝑦. (A.13) Then the dynamics of _on𝑥(0)=𝑦 with 𝑑_on𝑥(𝑡)/𝑑𝑡|_on𝑥_(𝑡)=𝑥=_on𝜆(𝑥)−_on𝜇(𝑥) for 𝑥∈(0,𝑦] is of our interest, because we can write tc_on=∫^∞₀𝑔(_on𝑥(𝑡))𝑑𝑡.
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 {^𝑛_on𝑋_𝑡,𝑡≥0} with initial condition ^𝑛_on𝑋₀=[𝑛𝑦], state space {0,1,…}, birth and death rates given by 𝑛𝛼(𝑖/𝑛)≜𝑛_on𝜆(𝑖/𝑛) and 𝑛𝛽(𝑖/𝑛)=^Δ𝑛_on𝜇(𝑖/𝑛) when the current state is 𝑖, and the cost rate given by 𝛾(𝑖/𝑛)≜_on𝑔(𝑖/𝑛). By recognizing 𝐸_[𝑛𝑦][∫^∞_0on𝑔(^𝑛_on𝑋_𝑡)𝑑𝑡]=𝐸[^𝑛TC_on] and that Condition 3 is a specific version of Condition 1, we can refer to Lemma A.1 for |^𝑛𝐸[^𝑛TC_on]−tc_on|≤𝐵₁/𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}. Arguing similarly as above (see also the proof of Lemma A.2), we have |^𝑛𝐸[^𝑛𝑇_on]−𝑡_on|≤𝐵₁/𝑛+𝐵₂𝜂^{−2𝑛[𝑦+1]}. (a) is now clear.
(b) The production-off phase has already been covered when analyzing EOQ models. Therefore, one can directly refer to Lemma A.2 for the statement.

Proof of Proposition 3.1. Lemma A.3 implies that𝐸^𝑛TC−tc≤𝐸^𝑛TC_on−tc_on+𝐸^𝑛TC_off−tc_off≤2𝑛𝐵₁+𝐵₂𝑛𝜂^{−2𝑛[𝑦+1]}+𝑔(0)2𝜅 (A.14) and similarly |𝐸[^𝑛𝑇_cycle]−𝑡_cycle|≤(2/𝑛)(𝐵₁+𝐵₂𝑛𝜂^{−2𝑛[𝑦+1]}+1/2𝜅). Then according to (A.4) and the facts of 𝐸[^𝑛𝑇_cycle]≥2[𝑛𝑦]/𝑛𝑘₁+1/𝑛𝜅, 2𝑦/𝛿≥𝑡_cycle≥2𝑦/𝛿 and tc≤2𝑦𝑑₁/𝛿 (recall 𝛿=^Δmax{𝛿,𝛿_𝜇𝜆} and 𝛿=^Δmin{𝛿,𝛿_𝜇𝜆}), we have TCU𝑛𝑦−tcu𝑦^𝑛≤2𝑦/𝛿(2/𝑛)𝐵₁+𝐵₂𝑛𝜂^{−2𝑛[𝑦+1]}+max𝑔(0)/2𝜅,1/2𝜅1+𝑑₁2𝑛𝑦/𝑛𝑘₁+1/𝑛𝜅2𝑦/𝛿=2𝛿𝜅𝑘₁1+𝑑₁𝛿𝐵₁+𝐵₂𝑛𝜂^{−2𝑛[𝑦+1]}+max𝑔(0)/2𝜅,1/2𝜅2𝑛𝑦𝜅+𝑘₁. (A.15)

Proof of Corollary 3.2. (a) Suppose that the statement does not hold. That is, for some subsequence {𝑛_𝑗,𝑗=1,2,…} with 𝑛_𝑗→∞ as 𝑗→∞,^𝑛𝑗𝑌^∗=𝑜(𝑛_𝑗) in that lim_𝑗→∞(^𝑛𝑗𝑌^∗/𝑛_𝑗)=0. Under ^𝑛𝑗𝑌^∗ we have𝐸^𝑛𝑗𝑇_cycle=1𝑛_𝑗𝜅+^{𝑛𝑗𝑌∗}_𝑖=11𝑛_𝑗𝜇𝑖/𝑛_𝑗+^{𝑛𝑗𝑌∗−1}_𝑖=01𝑛_𝑗𝜆𝑖/𝑛_𝑗+^𝑖−1_𝑘=0𝜇𝑖/𝑛_𝑗𝜇(𝑖−1)/𝑛_𝑗⋯𝜇(𝑖−𝑘)/𝑛_𝑗𝑛_𝑗𝜆𝑖/𝑛_𝑗𝜆(𝑖−1)/𝑛_𝑗⋯𝜆(𝑖−𝑘−1)/𝑛_𝑗, (A.16) where by [26, Theorem 1, page 175] the term inside the first curry bracket corresponds to 𝐸[^𝑛𝑗𝑇_off] and the second (last) sum corresponds to 𝐸[^𝑛𝑗𝑇_on]. Here we agree on that when 𝑖=0, the term in the second curry bracket reduces to 1/𝑛_𝑗𝜆(0). This gives ^𝑛𝑗TCU^𝑛𝑗𝑌^∗=𝐸^𝑛𝑗TC1/𝑛_𝑗𝜅+∑^{𝑛𝑗𝑌∗}_𝑖=11/𝑛_𝑗𝜇𝑖/𝑛_𝑗+∑^{𝑛𝑗𝑌∗−1}_𝑖=01/𝑛_𝑗𝜆𝑖/𝑛_𝑗≥𝐾+𝒜1/𝑛_𝑗𝜅+∑^{𝑛𝑗𝑌∗}_𝑖=11/𝑛_𝑗𝜇𝑖/𝑛_𝑗+∑^{𝑛𝑗𝑌∗−1}_𝑖=01/𝑛_𝑗𝜆𝑖/𝑛_𝑗,+𝒜 (A.17) where 𝒜=∑^𝑖−1_𝑘=0(𝜇(𝑖/𝑛_𝑗)𝜇((𝑖−1)/𝑛_𝑗)⋯𝜇((𝑖−𝑘)/𝑛_𝑗)/𝑛_𝑗𝜆(𝑖/𝑛_𝑗)𝜆((𝑖−1)/𝑛_𝑗)⋯𝜆((𝑖−𝑘−1)/𝑛_𝑗)), where the last inequality follows from the fact 𝑔(𝑥)≥0. Clearly, the right hand side expression of the above inequality goes to infinity as 𝑛_𝑗→∞, because 𝜆(𝑥) and 𝜇(𝑥) are both bounded and separated from zero, and ^𝑛𝑗𝑌^∗=𝑜(𝑛_𝑗) by supposition. On the other hand, obviously there exists some 𝑦_∗>0 with tcu(𝑦_∗)<∞, which according to Proposition 3.1 leads to that at least for big enough 𝑛_𝑗, [𝑛_𝑗𝑦_∗] outperforms ^𝑛𝑗𝑌^∗, which is a desired contradiction. Part (a) is thus proved.
(b) The proof of this part coincides with the one of Corollary 2.4, and is thus omitted.
(c1) Let us notice first of all that under the conditions of the statement, we have that the expression (𝐵₁+𝐵₂𝑛𝜂^{−2𝑛(𝑦+1)}+max{𝑔(0)/2𝜅,1/2𝜅})/(2(𝑛𝑦−1)𝜅+𝑘₁) is positive and decreases with 𝑦. Here 𝐵₁ and 𝐵₂ come from replacing [𝑦+1] by 𝑦+1 in 𝐵₁ and 𝐵₂. Indeed, as for the positivity part, one only needs to see the denominator 2(𝑛𝑦−1)𝜅+𝑘₁>0 if 𝑛 is subject to the given condition. The decreasing (with respect to 𝑦) part follows from 𝑑[(𝑦+1)/(2𝜅(𝑛𝑦−1)+𝑘₁)]/𝑑𝑦=(𝑘₂−2𝜅−2𝜅𝑛)/[2𝜅(𝑛𝑦−1)+𝑘₁]²<0 whenever 𝑛>𝑘₁/2𝜅−1. Remember that 𝐵₁ and 𝐵₂ are both 𝑦-dependent, and 𝐿 is 𝑦-independent.
Now let us prove (c1) of the corollary. Observe that under the conditions of the statement, Proposition 3.1 implies that𝐸^𝑛𝑦≤2TCU𝑛𝑦−tcu𝛿𝜅𝑘₁1+𝑑₁𝛿𝐵₁+𝐵₂𝑛𝜂^{−2𝑛(𝑦+1)}+max𝑔(0)/2𝜅,1/2𝜅2𝑛𝑦−1𝜅+𝑘₁. (A.18) For 𝑦≥𝐾𝛿²/4𝑘₁𝑑₁, one can bound from the above the right hand side expression in (A.18) by substituting 𝑦=𝐾𝛿²/4𝑘₁𝑑₁ in it, which leads to (c1).
(c2) Let us notice that 𝑦^∗≥𝐾𝛿²/4𝑘₁𝑑₁, and for big enough 𝑛, ^𝑛𝑌^∗/𝑛≥𝐾𝛿²/4𝑘₁𝑑₁. Indeed, due to (b), to justify the second inequality, we only need verify 𝑦^∗≥𝐾𝛿²/4𝑘₁𝑑₁, which is done as follows. For the fluid model, clearly we have𝑦=∫tcu^𝑦0𝑔(𝑥)/𝜇(𝑥)𝑑𝑥+∫^𝑦₀𝑔(𝑥)/𝜆(𝑥)−𝜇(𝑥)+𝐾∫^𝑦₀𝑑𝑥/𝜇(𝑥)+∫^𝑦₀𝑑𝑥/𝜆(𝑥)−𝜇(𝑥), (A.19) where the numerator corresponds to tc and the denominator corresponds to 𝑡_cycle, and 𝑑tcu=𝑔𝑦𝑦𝑦/𝜆𝑦𝑦∫𝑑𝑦/𝜇+𝑔−𝜇^𝑦₀𝑑𝑥/𝜇(𝑥)+∫^𝑦₀𝑑𝑥/𝜆(𝑥)−𝜇(𝑥)∫^𝑦₀𝑑𝑥/𝜇(𝑥)+∫^𝑦₀𝑑𝑥/𝜆(𝑥)−𝜇(𝑥)²−∫^𝑦₀𝑔(𝑥)/𝜇(𝑥)𝑑𝑥+∫^𝑦₀𝑔(𝑥)/𝜆(𝑥)−𝜇(𝑥)+𝐾1/𝜇𝑦+1/𝜆𝑦−𝜇𝑦∫^𝑦₀𝑑𝑥/𝜇(𝑥)+∫^𝑦₀𝑑𝑥/𝜆(𝑥)−𝜇(𝑥)², (A.20) which is negative if (𝑔(𝑦)/𝜇(𝑦)+𝑔(𝑦)/(𝜆(𝑦)−𝜇(𝑦)))(∫^𝑦₀(𝑑𝑥/𝜇(𝑥))+∫^𝑦₀(𝑑𝑥/(𝜆(𝑥)−𝜇(𝑥))))<𝐾/𝜇(𝑡). But the latter inequality holds if 𝑦<𝐾𝛿²/4𝑘₁𝑑₁, because 𝑔(𝑦)/𝜇(𝑦)+𝑔(𝑦)/(𝜆(𝑦)−𝜇(𝑦))≤2𝑑₁/𝛿 and ∫^𝑦₀(𝑑𝑥/𝜇(𝑥))+∫^𝑦₀(𝑑𝑥/(𝜆(𝑥)−𝜇(𝑥)))≤2𝑦/𝛿. This means 𝑦^∗≥𝐾𝛿²/4𝑘₁𝑑₁.
Now with the help of (c1) and Proposition 3.1, (c2) can be proved in the same way as for (b) of Corollary 2.5.

Acknowledgments

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.