OPTIMAL INVENTORY POLICIES UNDER STOCHASTIC PRODUCTION …vkulkarn/papers/EOQ.pdf · Optimal Stochastic Inventory Policies 175 set-up cost and the holding cost. In this paper, we

Stochastic Models, 24:173–190, 2008Copyright © Taylor & Francis Group, LLCISSN: 1532-6349 print/1532-4214 onlineDOI: 10.1080/15326340802016977

OPTIMAL INVENTORY POLICIES UNDER STOCHASTICPRODUCTION AND DEMAND RATES

K. Yan1 and V. Kulkarni2

1SAS Institute Inc., Cary, North Carolina, USA2Department of Statistics and Operations Research, University of North Carolina,Chapel Hill, North Carolina, USA

� We consider a single stage production-inventory system whose production and demand ratesare modulated by an environment process modeled as a finite state Continuous Time MarkovChain (CTMC). When the inventory level reaches zero, an order is placed from an externalsupplier, and it arrives instantaneously. We derive an Economic Order Quantity (EOQ ) policythat minimizes the long-run average cost if one replaces the deterministic demand rate by theexpected demand – production rate in steady state. Finally, we extend the model to allowbacklogging.

Keywords Backlogging; Economic order quantity; Inventory; Stochastic demand;Stochastic production.

Mathematics Subject Classification Primary 90B05; Secondary 60J25.

1. INTRODUCTION

In this paper, we study a production-inventory model operating ina stochastic environment that is modulated by a finite state ContinuousTime Markov Chain (CTMC). A production rate and a demand rate areassociated with each state of the CTMC. The inventory increases whenthe production rate exceeds the demand rate, and decreases when thedemand rate exceeds the production rate. The inventory level undercontinuous review thus can be viewed as a fluid process that fluctuatesaccording to the evolution of the underlying background environment.When the inventory level drops to 0 a replenishment order of size qis placed. We assume that the lead time is zero, i.e., the replenishment

Received October 2006; Accepted December 2007Address correspondence to K. Yan, SAS Institute Inc., Cary, NC 27513, USA; E-mail: Keqi.Yan@

sas.com

174 Yan and Kulkarni

order is delivered instantaneously, thus the inventory level jumps from 0to q instantaneously. Figure 1 illustrates a sample path of the environmentprocess and inventory level process.

This model reflects situations in which the production and demandrates undergo recurring changes in a stochastic fashion, and may bemodeled as Markovian. For example, production rates and demandrates change due to weather, economy, competition, seasonal promotion,customer status, and forecasting, etc. Some other examples are asdescribed in Mitra[10] where the author studies a producer and consumerproblem in a machine shop where the production rate changes accordingto the number of working machines. We assume the order size is qregardless of the state of the CTMC when the inventory level reaches zero.This is an appropriate model when we are forced to base our inventoryreplenishment decisions only on the inventory level and not on the stateof the CTMC. This may be because the knowledge of the state of thebackground CTMC is unavailable, or to simplify the ordering policies.There are costs to hold products in inventory, to purchase and to produce.There is also a fixed set-up cost every time an order is placed with anexternal supplier. The objective is to find the replenishment order size qthat minimizes the long-run average cost.

In a deterministic setting with constant demand rate, the classicalEconomic-Order-Quantity (EOQ) formula describes the trade-off between the

FIGURE 1 A sample path of the (environment–inventory) process.

Optimal Stochastic Inventory Policies 175

set-up cost and the holding cost. In this paper, we establish the stochasticEOQ theorem that shows in a CTMC environment the standard deterministicEOQ formula remains optimal if we replace deterministic demand rate bythe expected net demand rate in steady state.

For stochastic models, in the literature, dynamic control of stochasticinventory systems have been classified as periodic review models andcontinuous review models. The continuous review model can be furtherclassified according to whether the demand and production are discreteor continuous. In the case of discrete demands and production,Poisson demands in the continuous review models are studied in depth,(Scarf [11]). Song and Zipkin[12] considered the case of Markov modulatedPoisson demands. In the case of continuous fluid models, Berman andPerry[5] studied a fluid model where the production and demand ratesdepend on the inventory level. Browne and Zipkin[6] studied a modelwith continuous demand driven by a Markov process. To our knowledge,although there are papers studying similar problems, none includes theresult about the explicit Stochastic EOQ theorem presented in this paper.

In addition to the ordering policy, we also consider the optimalproduction policy: choose the optimal production rates that achievethe best combination of outsourcing and inhouse-production. We dothis mainly with numerical results for a specific example. Interestingly,the optimal policy does not suggest that one should depend entirelyon inhouse-production, even if the production cost is less than theoutsourcing price.

The rest of this paper is organized as follows. In Section 2, we describethe model mathematically. In Section 3 we summarize the results ofKulkarni and Yan[9] to make this paper self contained. In Section 4, wepresent the optimal order size q∗ that minimizes the total cost per unittime. Furthermore, in Section 5 we extend the results to a more generalcase that allows backorders, where we investigate the optimal reorder-point/order-quantity (r , q) policy. We also prove that for a given orderquantity q , the optimal reorder point r ∗(q) can be given explicitly by thewell-known news-vendor solution. In a special case when the productionrate is always less than the demand rate, we show that the deterministicEOQ formula with backlogging is still optimal in the Markovian settingwith a similar modification as in the no backlogging case. In Section 6 wepropose several extensions worthy of study.

2. THE MODEL

Consider a production-inventory system that is modulated by astochastic process �Z (t), t ≥ 0� on state space � = �1, 2, � � � ,n�. We assumethat �Z (t), t ≥ 0� is an irreducible CTMC on �1, 2, � � � ,n� with rate matrixQ = [qij ]. When Z (t) is in state i , the production occurs continuously at


rate ri , and there is a demand at rate di . The net production rate is thusRi = ri − di . Note that Ri may be negative or positive. Let X (t) be theinventory level at time t . Then as long as Z (t) = i , �X (t), t ≥ 0� changesat rate Ri . When X (t) reaches zero we place an order of size q > 0 froman external supplier who delivers it instantaneously. Note that X (t) is acontinuous random variable taking values in [0,∞).

Let

� = [�1, �2, � � � , �n] (1)

be the limiting distribution of the CTMC, i.e., it is the unique solution to

�Q = 0, � · e = 1,

where e = [1, � � � , 1]t is an n-dimensional column vector of ones. Thesystem is stable if the expected net input rate

∑i∈�

�iRi < 0� (2)

(Kulkarni[8].) We assume that this stability condition holds for the rest ofthis paper.

Next we consider costs to operate the system. The total cost consistsof four parts: holding cost, ordering cost, production cost and externalprocurement cost. Let h be the cost to hold one item in inventory for oneunit of time, and k be the fixed ordering cost incurred whenever an orderis placed from an external supplier, p1 be the cost of producing one itemand p2 be the cost of procuring one item from the external supplier.

We are interested in computing the optimal order size q∗ thatminimizes the long-run total cost per unit of time. In order to do this wefirst need to compute the cumulative distribution function of the inventorylevel in steady state. We do this in the next section.

3. LIMITING BEHAVIOR OF THE INVENTORY LEVEL PROCESS

In Kulkarni and Yan[9], the authors consider a bivariate fluid processthat behaves like the process described in the previous section, except thatif the fluid level reaches zero while the background environment is in statei , the fluid level jumps instantaneously to q and the environment changessimultaneously to j with probability �ij . The jump probabilities form thematrix A = [�ij ]. The bivariate process �(X (t),Z (t)), t ≥ 0� described inthe previous section can be obtained as a special case of this fluid modelwith jumps by setting A = I , the identity matrix.

In this section we summarize the results of Kulkarni and Yan[9] asthey apply to our case in order to set the notation and to make this


paper self-contained. Assume the stability condition (2) holds so that�(X (t),Z (t)), t ≥ 0� has a limiting distribution. Let

(X q ,Z q)d= lim

t→∞(X (t),Z (t)),

where d= represents equality in distribution. Note that the limiting caseq = 0 corresponds to the standard fluid model without jumps where oncethe fluid level reaches zero it stays there until the environment changes toa state with a positive drift.

Now let U (0, q) be a random variable distributed uniformly over (0, q).The decomposition result in Theorem 6.1 of Ref.[9] can be stated as

X q d= U (0, q) + X 0 (3)

where U (0, q) and X 0 are independent random variables. Thus, in steadystate the fluid level is a sum of two independent random variables: auniform random variable over (0, q), and the limiting fluid level in astandard fluid model with no jumps. In particular when R ≤ 0, we haveX 0 = 0 with probability one, and hence, in steady state the inventory levelis uniformly distributed on (0, q), and is independent of Z q . The abovedecomposition result greatly simplifies the cost calculations as shown in thenext section.

Using the above decomposition we show how to compute the limitingdistribution of (X q ,Z q). Let

Gqj (x) = P �X q > x ,Z q = j�, x ≥ 0, t ≥ 0, j ∈ �, (4)

and

Gq(x) = [Gq

1 (x), � � � ,Gqn (x)

]� (5)

Equation (3) implies that

Gq(x) = U q ∗ G 0(x), (6)

where U q represents the cdf of a U (0, q) random variable, and ∗ representsthe convolution operator. For completeness, we recapitulate below theresults about the computation of G 0(x) from Kulkarni[8]. First, G 0(x)satisfies the following differential equations

ddx

G 0(x)R = G 0(x)Q , x > 0, (7)


with boundary conditions

G 0j (0) = 0, ∀j : Rj > 0, (8)

G 0(0)e = 1� (9)

Next we describe the solution to the above equations for completeness.Let (�i ,�i), i = 1, 2, � � � ,n, be the n (eigenvalue, eigenvector) pairs thatsatisfy

�iQ = �i�iR � (10)

When all the eigenvalues are distinct, the solution to the differentialEquation (7) is given by

G 0(x) =∑

i:Re(�i )<0

aie �i x�i , x ≥ 0� (11)

where the unknowns ai are computed by using the boundary conditionsin Equations (8) and (9). This, along with Equation (6) gives a completemethod of computing Gq(x).

What we have presented here is called the spectral method of analysisof the fluid queue. Recently several authors have studied various aspectsof fluid queues with matrix geometric methods, see, for example Ahnand Ramaswami[1] and Latouche and Soares[7] for steady state analysis,Bean et al.[4] for return probabilities, Ahn and Ramaswami[2,3], for transientanalysis.

4. OPTIMAL ORDER SIZE

In this section, we demonstrate the primary result of this paper. Weshow that the total cost rate c(q) is a convex function of q and establish theequivalence of the classical deterministic EOQ formula in this stochasticenvironment. The main result is given in the following Theorem. Let be the expected net demand rate (i.e., demand rate–production rate) insteady state, given by

= −∑i

�iRi � (12)

Theorem 4.1. Suppose > 0. Then the optimal order size q∗ that minimizes thetotal cost rate c(q) is given by

q∗ =√2kh

� (13)


Proof. Let c1(q), c2(q), c3(q) and c4(q) be the holding cost, ordering cost,procurement cost and production costs per unit time in steady state. Thenthe total cost rate c(q) is given by

c(q) = c1(q) + c2(q) + c3(q) + c4(q)� (14)

First we calculate c1(q). From Equation (3) we get

c1(q) = hE(X q) = h(E(U q) + E(X 0)) = hq/2 + hE(X 0)� (15)

Note that hE(X 0) is independent of q . Next we calculate c2(q). Let T bethe time between two consecutive order placements in steady state. Fromthe results on renewal reward processes we get

c2(q) = k/E(T )� (16)

In steady state, the average net demand during the time T between twoconsecutive orders has to be equal to the amount of the external supply.Hence we have

E(T ) = q � (17)

Thus

c2(q) = k/q � (18)

Next, a procurement cost of p1q is incurred every time an order is placed,hence

c3(q) = p1qE(T )

= p1� (19)

Finally, the production rate in steady state is given by∑

i∈� �i ri , which isindependent of q . Hence,

c4(q) = p2∑i∈�

�i ri � (20)

This is also independent of q . Thus the total cost rate is

c(q) = hq2

+ kq

+ K , (21)

where K = hE(X 0) + p1 + p2∑

i∈� �i ri .Clearly, C(q) is a convex function of q , and it is minimized at q∗ given

by Equation (13).


Remark 4.1. The optimal order quantity q∗ of Equation (13) is theclassical EOQ formula with the deterministic demand rate replaced by thesteady-state expected net demand rate.

A Machine Shop Example. Consider a machine shop that has nindependent and identical machines. Each machine produces itemscontinuously at rate u when it is up, and it fails after an exp() amount oftime. If it is down, there is no production, and it takes exp(�) amountof time to fix it. Each machine has its own repair person. The machineis as good as new after repairs complete. Let Z (t) be the number ofworking machines at time t . Thus the CTMC �Z (t), t ≥ 0� has n + 1 states,i.e., � = �0, 1, � � � ,n�. Suppose the demand rate is directly proportional tothe number of machines. Thus we have di = n and ri = i · u for all i ∈ �.Whenever the inventory reaches zero, an external supply of amount q isordered and arrives instantaneously. Next we investigate the effect of theproduction rate on the optimal order size q∗. Consider a system with � = 1, = 2, h = 10, k = 2, p1 = 8 and p2 = 5. We plot the optimal values of q∗

in Figure 2 for 1 ≤ n ≤ 5 and u varying in (0, 3).Note that for a fixed n, the q∗ decreases with u. This makes intuitive

sense because as the production increases the net demand rate decreases.Note that the q∗ reaches zero when u increases to 3. This is because the

FIGURE 2 The optimal order quantity vs. production rate.


system is unstable for u ≥ 3 and hence we do not need to order from theexternal supplier.

It should be noted that there are numerical difficulties when theparameter values make the rate Ri in some states close to zero. We havesimply avoided such regions and used interpolation to produce the abovegraphs. Perhaps the matrix geometric methods studied by Ahn andRamaswami[1] may yield a way around these difficulties. However, authorshave not implemented them.

4.1. Minimum Cost Rate

Given the optimal order quantity q∗, we can compute thecorresponding minimum cost rate by using Equation (21):

c(q∗) = √2kh + hE(X 0) + p1 + p2

∑i∈�

�i ri � (22)

Here hE(X 0) can be calculated by using G 0(·) of Equation (11). In Figure 3we plot the optimal cost rate as a function of production rate for differentvalues of n in the machine shop example of the previous section.

FIGURE 3 The minimum total cost and the optimal production rate.


From this figure we can see that under the optimal ordering policy,the corresponding minimum cost rate seems to be a convex function ofthe production rate u. We have no analytic proof of this and leave it asa conjecture. Assuming this to be true, we see that exists one optimalproduction rate u∗ that achieves the trade-off point between outsourcingand producing. Interestingly, even if the production cost rate p2 is less thanthe outsourcing cost rate p1, the optimal policy does not suggest dependingon inhouse-production too much, in which circumstance the increase inholding cost due to the high production rate annihilates the advantage ofthe cost difference between the outsourcing and inhouse-production.

Also note that although q∗ → 0 as u ↑ 3, the expected total cost doesnot approach zero. This is a consequence of the stochastic model, asopposed to the deterministic one.

5. INVENTORY MODEL WITH BACKLOGGING

In the previous sections we considered a model where we place anorder as soon as the inventory on hand reaches zero. Many businessesfind it practical to operate with planned backlogging. In this section weconsider the same system as in the sections above, but allow backlogging,and assume that unsatisfied demands are fully backlogged. Let X (t) bethe net inventory level at time t (i.e., the inventory on hand at timet -backorders at time t). We always use any inventory on hand to filldemands; backorders accumulate only when we run of stock entirely. Thusif X (t) is positive, it represents the amount of inventory on hand. If it isnegative, it represents the negative of the amount of backorders at time t .

Now besides all the costs occurring in the previous setting, there isalso backlogging cost. We consider a policy under which we place an orderof size q whenever the inventory level decreases to the reorder point r .We assume zero lead times, so the orders arrive instantaneously and theinventory level jumps to q + r . Clearly an optimal policy should have r ≤ 0to reduce unnecessary holding cost. However, unlike in the deterministicset up, we may not have q + r > 0. The inventory level is always in (r ,∞).Figure 4 illustrates a typical sample path of the �X (t), t ≥ 0� process.

Note that under this policy, the stability condition is the same as in (2).Let

Gr ,qj (x) = lim

t→∞P (X (t) > x ,Z (t) = j)� (23)

The next theorem shows how to compute Gr ,q(x)= [Gr ,q1 (x),Gr ,q

2 (x), � � � ,Gr ,q

n (x)].


FIGURE 4 The inventory level process when allowing backlogging.

Theorem 5.1. Let Gq(x)(x ≥ 0) be defined in Equation (5). Then

Gr ,q(x) = Gq(x − r ), x ≥ r � (24)

Proof. Follows from the fact that the sample path of the inventory levelprocess with backorder is identical to that without the backorder shifteddown by r . �

5.1. Cost Rate Calculation of the (r , q) Policy

Now suppose it costs b to backlog one unit of demand for one unit oftime. Let c1(r , q), c2(r , q), c3(r , q), c4(r , q) and c5(r , q) be the steady stateholding, order cost, procurement cost, production cost and backloggingcost rates respectively as functions of the order quantity q and reorderpoint r . The total cost rate c(r , q) is thus given by

c(r , q) = c1(r , q) + c2(r , q) + c3(r , q) + c4(r , q) + c5(r , q)� (25)

The next theorem gives the long-run average costs.

Theorem 5.1.1. Let Gq(x)(x ≥ 0) be defined in Equation (5). Then

c1(r , q) = h∫ ∞

y=−rG q(y)dy · e , (26)

c2(r , q) = kq, (27)

c3(r , q) = p1, (28)

c4(r , q) = p2n∑

i=1

�i ri , (29)

c5(r , q) = −br − b∫ −r

y=0Gq(y)dy · e � (30)


Proof. First, for the holding cost rate, we get

c1(r , q) = hE(X +) = h∫ ∞

0limt→∞

P �X (t) > x�dx

= h∫ ∞

0Gr ,q(x)dx · e = h

∫ ∞

0Gq(x − r )dx · e �

Letting y = x − r , we get

c1(r , q) = h∫ ∞

y=−rG q(y)dy · e �

The backlogging cost rate is given by

c5(r , q) = bE(X −) = b∫ 0

−∞limt→∞

P �X (t) ≤ x�dx

= b∫ 0

r

(Gr ,q(r ) − Gr ,q(x)

)dx · e = −br − b

∫ 0

rG q(x − r )dx · e �

Letting y = x − r , we get

c5(r , q) = −br − b∫ −r

y=0Gq(y)dy · e �

Ordering cost rate, procurement cost rate and production cost rateare all independent of r , and hence can be obtained directly fromEquations (18)–(20).

5.2. Optimal (r , q) Policy

Theorem 5.2. gives the long-run average costs in terms of thedistribution function Gq(·). Thus we can find the optimal (r ∗, q∗) pair thatminimizes the total cost c(r , q).

5.2.1. News-Vendor Solution for the Optimal r for a Given qThe next theorem shows that for a given q , the corresponding optimal

r is given by the well-known news-vendor solution.

Theorem 5.2.1.1. For a given q, the optimal reorder point r = r ∗(q) is givenby the unique solution to

Gq(−r ) · e = bh + b

� (31)


Proof. Since c2(r , q), c3(r , q) and c4(r , q) are independent of r , we have

�

�rc(r , q) = �

�rc1(r , q) + �

�rc5(r , q)�

Substituting c1(r , q) and c5(r , q) by their expressions in Equations (26)and (30), and noticing that Gq(x) is independent of r , we get

�

�rc(r , q) = hGq(−r ) · e − b + bGq(−r ) · e �

We also have

�2

�r 2c(r , q) = −hGq ′

(−r ) · e − bGq ′(−r ) · e ≥ 0�

Hence c(r , q) is a convex function of r for a given q . Thus it is minimizedwhen �

�r c(r , q) = 0. This yields Equation (31).

Unfortunately, proving joint convexity of c(r , q) is hard. Numerically,one can minimize the function of single variable c(r ∗(q), q) to obtain theoptimal order quantity q∗. Then the optimal reorder point is given byr ∗ = r ∗(q∗). There is one special case when q∗ and r ∗ can be obtainedanalytically. We describe it below.

5.2.2. Stochastic EOQ Policy with BackloggingWe consider a special case when Ri < 0 for all i ∈ �. It follows from

Theorems 5.1 and 6.2 in Kulkarni and Yan[9] that the limiting distributionof the �X (t), t ≥ 0� process is uniformly distributed on (r , q + r ), and isindependent of Z . Thus the cost rate is given by

c(r , q) = h∫ q+r

0

xqdx − b

∫ 0

r

xqdx + k

q+ p1 + p2

n∑i=1

�i ri

= hq(q + r )2 + br 2

q+ k

q+ p1 + p2

n∑i=1

�i ri

It can be shown that c(r , q) is jointly convex function of (r , q), and isminimized at

q∗ =√2k(b + h)

hb(32)

r ∗ = −(

hb + h

)q∗� (33)


Equations (32) and (33) are identical to the well-known optimal (r , q)policy (Zipkin[13]) with deterministic demand rate . Here we haveshown that it remains optimal in this stochastic setting if we replace thedeterministic demand rate with the steady-state expected net demand rate.

5.2.3. A Numerical ExampleConsider the machine shop example in Section 4. Suppose the

backlogging cost rate is b = 8. We plot the optimal q∗’s and r ∗’s when thenumber of machines is 1, 2, � � � , 5 in Figures 5 and 6.

From Figures 5 and 6 we can see a few interesting points:

(1) When production rate u < 1, this example falls into the specialcase R < 0. Thus (r ∗, q∗) is given by the deterministic policy Equations (32)and (33) with = n(3 − u)/3.

(2) As u increases to 3, the reorder point r ∗ goes to −∞. Thisis contrary to the deterministic results. Actually, when u ↑ 3, more andmore background states have positive net input rates, and for these states,the input rates are also larger and larger; in addition, for those stateswith negative input rates, the net output rates are smaller and smaller.As a result, even though the expected input rate is still negative, oncethe background changes to a state with positive input rate, the inventory

FIGURE 5 The optimal order quantity vs. production rate.


FIGURE 6 The optimal reorder point vs. production rate.

level increases quickly, and the probability that the background changesto a state with negative input rate decreases. When such a state eventuallyis reached, the inventory level decreases, but slowly. Thus the optimalreorder point decreases in an effort to keep the inventory level close tozero to decrease the holding cost.

(3) The order quantity q∗ is increasing when u ↑ 3. This is alsodifferent from the result in the deterministic model. In this example, aslong as u < 3, there are still states with negative input rates. As u ↑ 3, the r ∗

is further and further below zero. Thus the increasing q∗ is a compensationto bring the inventory level back to zero at the ordering state (which musthave a negative input rate) to reduce the backlogging cost.

(4) As functions of u, q∗ and r ∗ have different characteristics onq∗ + r ∗ < 0 and q∗ + r ∗ > 0. Figure 7 shows the graph of q∗ + r ∗ as afunction of u.

Figure 7 shows that q∗ + r ∗ decreases from a positive quantity to anegative quantity as u ↑ 3. From Equations (26) and (30), we can seethat the holding cost rate and backlogging cost rate both have differentexpressions when q∗ + r ∗ > 0 and q∗ + r ∗ < 0. Hence q∗ and r ∗ also showdifferent qualitative behavior in these two regions.


FIGURE 7 The optimal order-up-to level q∗ + r ∗ vs. production rate.

Minimum Cost Rate. Consider the machine shop example inSection 5.2.3. Given the optimal ordering policy (r ∗, q∗), we calculate thecorresponding minimum cost rate. As in the example without backlogging,in Figure 8 we plot the optimal total cost rate as a function of u.As u ↑ 3 the total cost rate goes to infinity since the holding cost rate andbacklogging cost rate both go to infinity. It is clear from Figure 8 that foreach n, there is an optimal production rate u that minimizes the totalcost rate. This optimal production rate increases with n. Comparing withFigure 3 we can see that the minimum cost is a little lower with the optionof planned backlogging. The difference is slight because we choose arelatively small fixed ordering cost k = 0�5.

6. EXTENSIONS

Several extensions of the system studied in this paper might beinteresting. First, in the current analysis the order size is not allowed todepend on the state of the CTMC when the inventory level hits zero.Clearly, if that information is available, it would lower the costs if the ordersize can be made dependent on that information. Second, in this paperwe have assumed zero lead times. This assumption is reasonable whenlead times are short enough to be neglected. Sometimes it makes more


FIGURE 8 The minimum cost vs. production rate.

sense to model a system with nonzero lead times. Third, in this paper wehave assumed continuous product qualities. It would be interesting to seeif similar conclusions hold in a parallel system with discrete productionand demands. Finally, the results of this paper clearly remain valid if thebackground process is a semi-Markov process with phase-type distributions.This can be shown by constructing an appropriate larger CTMC. Sincephase-type distributions for the sojourn times are dense in the set ofall continuous distributions on [0,∞), it follows that the results holdfor a semi-Markov background process with continuous sojourn times.We believe that the results hold for more general semi-Markov processesas long as the sample paths of the �X (t), t ≥ 0� process are not periodicwith probability one. Rigorous proof of this remains to be shown.

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OPTIMAL INVENTORY POLICIES UNDER STOCHASTIC PRODUCTION …vkulkarn/papers/EOQ.pdf · Optimal Stochastic Inventory Policies 175 set-up cost and the holding cost. In this paper, we