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

Authors: Alexey Piunovskiy; Yi Zhang
Journal: Advances in Operations Research
 (2011)

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## 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.


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## Body

## 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 Section2.1. 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 Sections2 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 quantityy>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(torder+0)=y, where x(t) reaches state zero at torder. Here, μ(x)>0 is the demand rate, and the impulsive jump of x(t) at torder reflects the instantaneous replenishment assumption. Let tcycle 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
(2.1)tcu⁡(y)≜lim⁡T→∞1T{∫0Tg(x(t))dt+K[Ttcycle⁡]}.
Let us call y* the EOQ for the fluid model, so that tcu(y*)=infy>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 parametern=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 {nXt,t≥0} is modelled as a continuous-time Markov chain with the state space {0,1,2,…,Yn} and the transition rates {nAi,j}i,j=0nY given by
(2.2)nA0,j={nκ,ifj=Yn,-nκ,ifj=0,0,otherwise,
and in the case ofi∈{1,2,3,…,Yn}(2.3)nAi,j={nμ(in),ifj=i-1,-nμ(in),ifj=i,0,otherwise,
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 takingnY≜[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(2.4)nTCU⁡(nY)≜lim⁡T→∞1TEnY[∫0∞{g(nXtn)+Knμ(1n)I{nXt=1}}dt],
where EnY denotes the expectation operator with the initial inventory level (immediately after the replenishment) nX0=Yn. Let us call nY* the EOQ for the (scaled) stochastic model, so that nTCU(nY*)=infnY=1,2,…TCUn(nY).We say that order quantity[ny] for the stochastic model is AFO if (2.5)lim⁡n→∞|nTCU⁡([ny])-tcu⁡(y*)|=0
and AO if (2.6)lim⁡n→∞|nTCU⁡([ny])-TCU⁡n(nY*)|=0.
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 constantsd1≥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 limj→∞xj=∞ such that on each of them g(x) and μ(x) are Lipschitz continuous with a common Lipschitz constant dg and dμ, respectively.Note that Condition1 implies that for any fixed y>0, there exist L (possibly y-dependent) finite intervals (0,x1),(x1,x2),…,(xL,3[y+1]) such that on each interval 1/μ(x) and g(x)/μ(x) are Lipschitz continuous with Lipschitz constants dμ/δ2 and (d1dμ+k1dg)/δ2, respectively. For simplicity, we define d2≜max{dμ/δ2,(d1dμ+k1dg)/δ2}.Proposition 2.2.
Under Condition1, for any fixed order quantity in the fluid model y>0(2.7)|nTCU⁡([ny])-tcu⁡(y)|≤(1+d1)k12κδ([ny]κ+k1){k1d23[y+1]δ+(1+k1δ)3d1Lδ+max⁡{g(0)κ,1κ}}.In particular, Proposition2.2 implies that limn→∞|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 Condition1 is satisfied and g(x)≥0, then
(2.8)y*≥δ2Kk1d1,
and nY* satisfies nY*/n≥δ2K/k1d1-δ/nκ. In particular, lim̲n→∞(Yn*/n)>0.Corollary 2.4.
Under Condition1, if lim̲n→∞(Yn*/n)>0 (see Lemma 2.3), then one has limn→∞|nY*/n-[ny*]/n|=0, and limn→∞|nTCU([ny*])-TCUn(nY*)|=0, that is, [ny*] is AO.Corollary 2.5.
Under Condition1, suppose in addition that L is y-independent and g(x)≥0 on [0,∞). Then the following statements hold.
(a) For anyy≥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,(2.9)|nTCU⁡([ny])-tcu⁡(y)|≤3(1+d1)k13d2δ21n{1-κk1d1/(nκδ2K-δk1d1)}+(1+d1)k12nδ{δ2K/k1d1-δ/nκ-1/n}×{3k1d2δ+(1+k1δ)3d1Lδ+max⁡{g(0)κ,1κ}}≜E(n).
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 limn→∞(E(n)/(1/n))=C with some (nonnegative) constant C.
(b) For any large enoughn≥N,(2.10)|nTCU⁡(nY*)-TCU⁡n([ny*])|≤2E(n)=O(1n).
Here we recall that nY* and y* are the EOQ for the (scaled) stochastic model and fluid model, respectively.Corollary2.5 refines Corollary 2.4 for certain cases by providing an estimate for the rate of convergence.Fix some order quantity for the fluid modely and scaling parameter n, and let nΠ(i),i=0,1,…,[ny] be the stationary distribution of the inventory process {nXt,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 Condition1 is satisfied and some order quantity for the fluid model y>0 is fixed. Then one has(2.11)|nΠ(0)-π(0)|≤k1k1+κ[ny]
and for i=1,2,…,[ny](2.12)|nΠ(i)-∫(i-1)/ni/nπ(x)dx|≤2yk12κ/δ2+(k12κ/nδ)(3k1d2(y+1)/δ+(1+k1/δ)3d1L/δ+1/κ)([ny]+1)y.


### 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 Condition1, the following stronger condition is assumed in [14].Condition 2.
(a) Condition1(a) holds.
(b) Functionsg(·) 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].


## 2.1. Description of Mathematical Models

(a) Fluid Model
Suppose some order quantityy>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(torder+0)=y, where x(t) reaches state zero at torder. Here, μ(x)>0 is the demand rate, and the impulsive jump of x(t) at torder reflects the instantaneous replenishment assumption. Let tcycle 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
(2.1)tcu⁡(y)≜lim⁡T→∞1T{∫0Tg(x(t))dt+K[Ttcycle⁡]}.
Let us call y* the EOQ for the fluid model, so that tcu(y*)=infy>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 parametern=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 {nXt,t≥0} is modelled as a continuous-time Markov chain with the state space {0,1,2,…,Yn} and the transition rates {nAi,j}i,j=0nY given by
(2.2)nA0,j={nκ,ifj=Yn,-nκ,ifj=0,0,otherwise,
and in the case ofi∈{1,2,3,…,Yn}(2.3)nAi,j={nμ(in),ifj=i-1,-nμ(in),ifj=i,0,otherwise,
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 takingnY≜[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(2.4)nTCU⁡(nY)≜lim⁡T→∞1TEnY[∫0∞{g(nXtn)+Knμ(1n)I{nXt=1}}dt],
where EnY denotes the expectation operator with the initial inventory level (immediately after the replenishment) nX0=Yn. Let us call nY* the EOQ for the (scaled) stochastic model, so that nTCU(nY*)=infnY=1,2,…TCUn(nY).We say that order quantity[ny] for the stochastic model is AFO if (2.5)lim⁡n→∞|nTCU⁡([ny])-tcu⁡(y*)|=0
and AO if (2.6)lim⁡n→∞|nTCU⁡([ny])-TCU⁡n(nY*)|=0.
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 constantsd1≥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 limj→∞xj=∞ such that on each of them g(x) and μ(x) are Lipschitz continuous with a common Lipschitz constant dg and dμ, respectively.Note that Condition1 implies that for any fixed y>0, there exist L (possibly y-dependent) finite intervals (0,x1),(x1,x2),…,(xL,3[y+1]) such that on each interval 1/μ(x) and g(x)/μ(x) are Lipschitz continuous with Lipschitz constants dμ/δ2 and (d1dμ+k1dg)/δ2, respectively. For simplicity, we define d2≜max{dμ/δ2,(d1dμ+k1dg)/δ2}.Proposition 2.2.
Under Condition1, for any fixed order quantity in the fluid model y>0(2.7)|nTCU⁡([ny])-tcu⁡(y)|≤(1+d1)k12κδ([ny]κ+k1){k1d23[y+1]δ+(1+k1δ)3d1Lδ+max⁡{g(0)κ,1κ}}.In particular, Proposition2.2 implies that limn→∞|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 Condition1 is satisfied and g(x)≥0, then
(2.8)y*≥δ2Kk1d1,
and nY* satisfies nY*/n≥δ2K/k1d1-δ/nκ. In particular, lim̲n→∞(Yn*/n)>0.Corollary 2.4.
Under Condition1, if lim̲n→∞(Yn*/n)>0 (see Lemma 2.3), then one has limn→∞|nY*/n-[ny*]/n|=0, and limn→∞|nTCU([ny*])-TCUn(nY*)|=0, that is, [ny*] is AO.Corollary 2.5.
Under Condition1, suppose in addition that L is y-independent and g(x)≥0 on [0,∞). Then the following statements hold.
(a) For anyy≥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,(2.9)|nTCU⁡([ny])-tcu⁡(y)|≤3(1+d1)k13d2δ21n{1-κk1d1/(nκδ2K-δk1d1)}+(1+d1)k12nδ{δ2K/k1d1-δ/nκ-1/n}×{3k1d2δ+(1+k1δ)3d1Lδ+max⁡{g(0)κ,1κ}}≜E(n).
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 limn→∞(E(n)/(1/n))=C with some (nonnegative) constant C.
(b) For any large enoughn≥N,(2.10)|nTCU⁡(nY*)-TCU⁡n([ny*])|≤2E(n)=O(1n).
Here we recall that nY* and y* are the EOQ for the (scaled) stochastic model and fluid model, respectively.Corollary2.5 refines Corollary 2.4 for certain cases by providing an estimate for the rate of convergence.Fix some order quantity for the fluid modely and scaling parameter n, and let nΠ(i),i=0,1,…,[ny] be the stationary distribution of the inventory process {nXt,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 Condition1 is satisfied and some order quantity for the fluid model y>0 is fixed. Then one has(2.11)|nΠ(0)-π(0)|≤k1k1+κ[ny]
and for i=1,2,…,[ny](2.12)|nΠ(i)-∫(i-1)/ni/nπ(x)dx|≤2yk12κ/δ2+(k12κ/nδ)(3k1d2(y+1)/δ+(1+k1/δ)3d1L/δ+1/κ)([ny]+1)y.


## 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 Condition1, the following stronger condition is assumed in [14].Condition 2.
(a) Condition1(a) holds.
(b) Functionsg(·) 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 minimizesTCU, 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 levely>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
(3.1)dxdt={-μ(x)(production-offphase),λ(x)-μ(x)(production-onphase),
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 tcycle the time duration between two consecutive production switching-offs. So
(3.2)tcu⁡(y)≜lim⁡T→∞1T{∫0Tg(x(t))dt+K[Ttcycle⁡]}.
Let us denote by y* the EIBL for the fluid model, so that tcu(y*)=infy>0tcu(y).(b) (Scaled) Stochastic Model
Suppose that we fix some inventory backup level positive integernY, meaning that the production is always on until the inventory reaches level nY. Let {nXt,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
(3.3)nA(0,off⁡),(0,on⁡)=nκ,A(0,off⁡),(0,off⁡)n=-nκ,nA(0,on⁡),(1,on⁡)=nλ(0),A(0,on⁡),(0,on⁡)n=-nλ(0),nA(nY-1,on⁡),(nY,off⁡)=nλ(nY-1n),A(nY-1,on⁡),(nY-2,on⁡)n=nμ(nY-1n),nA(nY-1,on⁡),(nY-1,on⁡)=-nλ(nY-1n)-nμ(nY-1n),
for alli=1,…,nY:(3.4)nA(i,off⁡),(j,off⁡)={nμ(in),ifj=i-1,-nμ(in),ifj=i,
and finally for all i=1,…,Yn-2:
(3.5)nA(i,on⁡),(j,on⁡)={nμ(in),ifj=i-1;nλ(in),ifj=i+1;-nμ(in)-nλ(in),ifj=i,
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
(3.6)nTCU⁡(nY)≜lim⁡T→∞1TEnY[∫0T{g(nXtn)+Knμ(1n)I{nXt=1}}dt],
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*)=infnY=1,2,…TCUn(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 constantsd1≥0,  k1>0,δ>0,  δμλ>0,  η̃>1 and measurable functions  μ(x),λ(x), and  g(x) defined on  [0,∞) such that  δ≤μ(x)≤μ(x)+δμλ≤λ(x)≤k1,  λ(x)+μ(x)≤k1,  |g(x)|≤d1, and  infx>0(λ(x)/μ(x))=η̃.
(b) There exist finite intervals(0,x1),(x1,x2),… with limj→∞xj=∞ such that on each of them λ(x),μ(x), and g(x) are Lipschitz continuous with Lipschitz constants dλ,dμ, and dg, respectively.Note that Condition3 implies that for any fixed y>0, there exists an integer L (possibly y-dependent) and L+1 intervals (0,x1), (x1,x2),…,(xL,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 dμ/δ2, (k1dg+d1dμ)/δ2, and (dλ+dμ)/δμλ2, 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 (k1dg+d1dμ)/δ2 and (dλ+dμ)/δμλ2, respectively. Let us now denote the common Lipschitz constant by d2≜max{dμ/δ2,(k1dg+d1dμ)/δ2,(dλ+dμ)/δμλ2}.Proposition 3.1.
Under Condition3, for any fixed (fluid) inventory backup level y>0,
(3.7)|nTCU⁡([ny])-tcu⁡(y)|≤2δ¯κk1(1+d1)δ̲{B1+B2nη̃-2n[y+1]+max⁡{g(0)/2κ,1/2κ}2[ny]κ+k1},
where one puts δ¯≜max{δ,δμκ} and δ̲≜min{δ,δμκ} 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 Proposition3.1 that [ny*] is AFO.Corollary 3.2.
Under Condition3, the following statements hold.(a)
Ifg(x)≥0, then lim̲n→∞(Y*n/n)>0.(b)
Iflim̲n→∞(Y*n/n)>0, then limn→∞|nY*/n-[ny*]/n|=0 and limn→∞|TCUn([ny*])-TCUn(nY*)|=0, that is, [ny*] is AO.(c)
IfL is y-independent and g(x)≥0, then the following two substatements hold.
(c1)
Fory≥Kδ̲2/4k1d1 and big enough n so that n>max{4k1d1/Kδ̲2,k1/2κ-1},
(3.8)|nTCU⁡([ny])-tcu⁡(y)|≤2δ¯κk1(1+d1)δ̲{B̂1+B̂2nη̃-2n(Kδ̲2/4k1d1+1)+max⁡{g(0)/2κ,1/2κ}2(n(Kδ̲2/4k1d1)-1)κ+k1}≜F(n)=O(1n).
Here F(n) is y-independent, and
(3.9)B̂1=k1d2(η̃+1)3(Kδ̲2/4k1d1+1)δ(η̃-1)+(1+k1η̃δ(η̃-1))3d1L(η̃+1)δ(η̃-1),B̂2=(1+6k1(Kδ̲2/4k1d1+1)η̃δ(η̃-1))d1η̃2(η̃+1)δ(η̃-1)2.One has (c2)
|nTCU(nY*)-TCUn([ny*])|≤2F(n)=O(1/n).


## 3.1. Description of Mathematical Models

(a) Fluid Model
Suppose that we fix some real inventory backup levely>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
(3.1)dxdt={-μ(x)(production-offphase),λ(x)-μ(x)(production-onphase),
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 tcycle the time duration between two consecutive production switching-offs. So
(3.2)tcu⁡(y)≜lim⁡T→∞1T{∫0Tg(x(t))dt+K[Ttcycle⁡]}.
Let us denote by y* the EIBL for the fluid model, so that tcu(y*)=infy>0tcu(y).(b) (Scaled) Stochastic Model
Suppose that we fix some inventory backup level positive integernY, meaning that the production is always on until the inventory reaches level nY. Let {nXt,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
(3.3)nA(0,off⁡),(0,on⁡)=nκ,A(0,off⁡),(0,off⁡)n=-nκ,nA(0,on⁡),(1,on⁡)=nλ(0),A(0,on⁡),(0,on⁡)n=-nλ(0),nA(nY-1,on⁡),(nY,off⁡)=nλ(nY-1n),A(nY-1,on⁡),(nY-2,on⁡)n=nμ(nY-1n),nA(nY-1,on⁡),(nY-1,on⁡)=-nλ(nY-1n)-nμ(nY-1n),
for alli=1,…,nY:(3.4)nA(i,off⁡),(j,off⁡)={nμ(in),ifj=i-1,-nμ(in),ifj=i,
and finally for all i=1,…,Yn-2:
(3.5)nA(i,on⁡),(j,on⁡)={nμ(in),ifj=i-1;nλ(in),ifj=i+1;-nμ(in)-nλ(in),ifj=i,
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
(3.6)nTCU⁡(nY)≜lim⁡T→∞1TEnY[∫0T{g(nXtn)+Knμ(1n)I{nXt=1}}dt],
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*)=infnY=1,2,…TCUn(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 constantsd1≥0,  k1>0,δ>0,  δμλ>0,  η̃>1 and measurable functions  μ(x),λ(x), and  g(x) defined on  [0,∞) such that  δ≤μ(x)≤μ(x)+δμλ≤λ(x)≤k1,  λ(x)+μ(x)≤k1,  |g(x)|≤d1, and  infx>0(λ(x)/μ(x))=η̃.
(b) There exist finite intervals(0,x1),(x1,x2),… with limj→∞xj=∞ such that on each of them λ(x),μ(x), and g(x) are Lipschitz continuous with Lipschitz constants dλ,dμ, and dg, respectively.Note that Condition3 implies that for any fixed y>0, there exists an integer L (possibly y-dependent) and L+1 intervals (0,x1), (x1,x2),…,(xL,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 dμ/δ2, (k1dg+d1dμ)/δ2, and (dλ+dμ)/δμλ2, 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 (k1dg+d1dμ)/δ2 and (dλ+dμ)/δμλ2, respectively. Let us now denote the common Lipschitz constant by d2≜max{dμ/δ2,(k1dg+d1dμ)/δ2,(dλ+dμ)/δμλ2}.Proposition 3.1.
Under Condition3, for any fixed (fluid) inventory backup level y>0,
(3.7)|nTCU⁡([ny])-tcu⁡(y)|≤2δ¯κk1(1+d1)δ̲{B1+B2nη̃-2n[y+1]+max⁡{g(0)/2κ,1/2κ}2[ny]κ+k1},
where one puts δ¯≜max{δ,δμκ} and δ̲≜min{δ,δμκ} 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 Proposition3.1 that [ny*] is AFO.Corollary 3.2.
Under Condition3, the following statements hold.(a)
Ifg(x)≥0, then lim̲n→∞(Y*n/n)>0.(b)
Iflim̲n→∞(Y*n/n)>0, then limn→∞|nY*/n-[ny*]/n|=0 and limn→∞|TCUn([ny*])-TCUn(nY*)|=0, that is, [ny*] is AO.(c)
IfL is y-independent and g(x)≥0, then the following two substatements hold.
(c1)
Fory≥Kδ̲2/4k1d1 and big enough n so that n>max{4k1d1/Kδ̲2,k1/2κ-1},
(3.8)|nTCU⁡([ny])-tcu⁡(y)|≤2δ¯κk1(1+d1)δ̲{B̂1+B̂2nη̃-2n(Kδ̲2/4k1d1+1)+max⁡{g(0)/2κ,1/2κ}2(n(Kδ̲2/4k1d1)-1)κ+k1}≜F(n)=O(1n).
Here F(n) is y-independent, and
(3.9)B̂1=k1d2(η̃+1)3(Kδ̲2/4k1d1+1)δ(η̃-1)+(1+k1η̃δ(η̃-1))3d1L(η̃+1)δ(η̃-1),B̂2=(1+6k1(Kδ̲2/4k1d1+1)η̃δ(η̃-1))d1η̃2(η̃+1)δ(η̃-1)2.One has (c2)
|nTCU(nY*)-TCUn([ny*])|≤2F(n)=O(1/n).


## 4. Example and Comments

In this section we firstly verify our results by considering a specific EPQ model, wheretcu(·) 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 {nXt,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,(4.1)nTCU⁡(Z)={RhZ2(D-R)2+2Kn2D(R-D)3+ZhR(3D-R)(D-R)+2RhD2(1-(DR)Z)}×12n(R-D){ZR2-D2+D2(D/R)Z-ZRD}.The proof of this proposition is based on solving (quite tediously) the associated Poisson equation fornTCU(Z) and is omitted here.The corresponding deterministic EPQ model can be solved easily, and we have(4.2)tcu⁡(y)=KD(R-D)yR+hy2.Clearly, if we putZ=[ny] in the expression for nTCU(Z), where y is the inventory backup level for the fluid model, then one can easily see that (4.3)lim⁡n→∞nTCU⁡([ny])=KD(R-D)yR+hy2=tcu⁡(y),|tcu⁡(y)-TCU⁡n([ny])|=O(1n),
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 to1TCU([y]) (resp., 5TCU([5y]), tcu(y)).Figure 2
The illustrative graph ofonx̂(t).Secondly, ifn 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 lim̲n→∞(Y*n/n)>0 in Corollary 3.2 is satisfied.Comments
Let us comment on the applicability issues of our main results (Propositions2.2 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 LemmaA.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)=αxβ, 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 Conditions1 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, Propositions2.2 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)=αxβ, 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].



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*Source: 301205-2011-06-18.xml*