Optimal replenishment rate for inventory systems with...

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Annals of Operations Research ISSN 0254-5330 Ann Oper ResDOI 10.1007/s10479-015-1998-y

Optimal replenishment rate for inventorysystems with compound Poisson demandsand lost sales: a direct treatment of time-average cost

Michael N. Katehakis, BenjaminMelamed & Jim Junmin Shi

Ann Oper ResDOI 10.1007/s10479-015-1998-y

Optimal replenishment rate for inventory systemswith compound Poisson demands and lost sales: a directtreatment of time-average cost

Michael N. Katehakis1 · Benjamin Melamed2 ·Jim Junmin Shi3

© Springer Science+Business Media New York 2015

Abstract Supply contracts are designed to minimize inventory costs or to hedge againstundesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular,replenishment terms stipulated by supply contracts need to be optimized with respect tooverall costs, profits, service levels, etc. In this paper, we shall be primarily interested inminimizing an inventory cost functionwith respect to a constant replenishment rate. Considera single-product inventory system under continuous review with constant replenishment andcompound Poisson demands subject to lost-sales. The system incurs inventory carrying costsand lost-sales penalties, where the carrying cost is a linear function of on-hand inventory anda lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We firstderive an integro-differential equation for the expected cumulative cost until and includingthe first lost-sale occurrence. From this equation, we obtain a closed form expression for thetime-average inventory cost, and provide an algorithm for a numerical computation of theoptimal replenishment rate that minimizes the aforementioned time-average cost function.In particular, we consider two special cases of lost-sales penalty functions: constant penaltyand loss-proportional penalty. We further consider special demand size distributions, such asconstant, uniform andGamma, and take advantage of their functional form to further simplifythe optimization algorithm. In particular, for the special case of exponential demand sizes,

B Jim Junmin [email protected]

Michael N. [email protected]

Benjamin [email protected]

1 Department of Management Science and Information Systems, Rutgers Business School – Newarkand New Brunswick, 100 Rockafeller Road, Piscataway, NJ 08854, USA

2 Department of Supply Chain Management, Rutgers Business School – Newark and NewBrunswick, 100 Rockafeller Rd., Piscataway, NJ 08554, USA

3 School of Management, New Jersey Institute of Technology, University Heights,Newark, NJ 07102, USA

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we exhibit a closed form expression for the optimal replenishment rate and its correspondingcost. Finally, a numerical study is carried out to illustrate the results.

Keywords Compound Poisson ·Continuous replenishment · Integro-differential equation ·Production-inventory systems · Lost sales · Replenishment rate · Time-average cost

1 Introduction

Supply contracts are designed to minimize inventory costs or to hedge against undesirableevents (e.g., shortages) in the face of demand or supply uncertainty [Simchi-Levi et al.(2008)]. In particular, replenishment terms stipulated by supply contracts need to be optimizedwith respect to overall costs, profits, service levels, etc. In this paper, we shall be primarilyinterested in deriving and minimizing an inventory cost function with respect to a constantreplenishment rate. The corresponding replenishment regimewill be referred to as continuousreplenishment.

Production-inventory systems with continuous replenishment are common in both manu-facturing and service organizations. For example, blood banks and food banks are typicallyreplenished at a deterministic rate and face random demands. Moreover, the pharmaceuticalmanufacturing and chemical industries set up production lines to satisfy incoming randomdemands from customers; the attendant setup times and costs are quite high, which in turnrender modifications of the production line prohibitively costly. An appropriate productionrate trades off carrying costs (if the production rate is too high, thereby incurring excessivestocks) for penalty costs (if the production rate is too low, thereby incurring stockouts). Thus,selecting a production rate (e.g., the number of production lines) is a key strategic decisionduring production planning, as is the selection of a service rate in service organizations.

Consider a continuous-review single-product inventory systemwith continuous replenish-ment, compound Poisson demands and lost sales. In this system, unsatisfied demand can bepartially satisfied from on-hand inventory (if any) and excess demand (shortage) is lost. Theexcess demand is referred to as the lost-sale size. Replenishment is continuous at a constant(deterministic) rate, which can also be the production rate of a replenishing production facil-ity. The system incurs two types of costs: carrying cost and lost-sales cost. A carrying costis incurred continuously as a function of the inventory on hand. A lost-sales cost is incurreddiscretely at each loss occurrence. The objective of this paper is to formulate and optimizethe time-average inventory cost with respect to the replenishment rate decision variable.

There is a large body of literature on managing inventory systems with compound Poissondemands, where demand arrivals follow a Poisson process, and the corresponding demandsizes are independent and identically distributed (iid), independent of arrivals. Early paperswhich study the inventory process include Richards (1975), Thompstone and Silver (1975),Archibald and Silver (1978), and Feldman (1978). Tijms (1972), Sahin (1979, 1983), andFedergruen and Schechner (1983) generalize the compound Poisson assumption to a generalcompound renewal processes, in which both the demand inter-arrival times and demand sizeshave arbitrary distributions. For a detailed literature review, see Presman and Sethi (2006)and references therein. The aforementioned papers assume various replenishment policies,but do not treat continuous replenishment.

Production-inventory systems with continuous replenishment and various demandprocesses have been previously studied in the literature. Graves and Keilson (1981) considersa one-product, one-machine production-inventory problem. The demand process is governed

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by a compound Poisson process with exponential demand sizes. The system is subject to an(r, R) policy with a constant replenishment rate. The paper analyses the cost optimizationproblem as a constrained Markov process using the compensation method. The optimal pol-icy is then obtained via a search over the policy space. Graves (1982) presents twomodels forinventory systemswith continuous production and perishable items. For each of thesemodels,the paper uses a queuing-theoretic approach to derive analytical expressions for the steady-state distribution of the inventory level. The steady-state results are then used to evaluatevarious performance metrics. Gullu and Jackson (1993) considers a one-product production-inventory system with a constant production rate and a demand process with stationaryand independent (i.e., time-homogeneous and additive) increments, and the produce-up-to-Sreplenishment policy. The paper derives the steady-state distribution of the inventory levelby extending existing results for dam systems, and then optimizes the time-average costof the system by exhibiting a closed form formula for the optimal policy. Recently, Shiet al. (2013) studies a production-inventory system with a constant replenishment rate and acompound renewal demand stream, subject to underage and overage penalties. By construct-ing auxiliary martingale processes in term of the inventory process, explicit expressions ofpenalty functions for underage and overage are obtained for the case inwhich demand arrivalsfollow a Poisson process. Shi et al. (2014) studies a similar production-inventory problembut focusing on minimizing the long-run expected discounted cost. In contrast, the currentstudy treats the time-average cost and attempts to derive more results and insights into theproduction/inventory system.

A number of papers study the derivation of optimal or near-optimal inventory replen-ishment policies that minimize the time-average cost. Springael and Nieuwenhuyse (2005)studies a lost-sales inventory model with a compound Poisson demand process, in whichreplenishment lead times are negligible. On-hand inventory is managed according to a (0,B∗) policy, namely, when the on-hand inventory drops to 0, the retailer instantaneously getsa fixed amount B∗ of units from the central stockroom as replenishment. The paper ana-lyzes the time-average cost of the system and provides a steepest-descent-based algorithmto calculate the optimal B∗ parameter. In a similar vein, Minner and Silver (2007) studies aninventory system with compound Poisson demands and negligible replenishment lead times.The paper formulates the optimization problem as a Markov decision problem, applicableto inventory systems with a small number of products. For a larger number of products, thepaper proposes several heuristics for the optimal reorder points and reorder quantities.

Several scholars have noted similarities between the mathematical formulations of queue-ing models and inventory models fairly early. An early paper on the linkage between thosetwo areas is Prabhu (1965). However, from a managerial point of view, inventory modelsdiffer from the classical G/M/1 queue. For example, under the lost-sales policy, service timesand waiting times (for customer demand) are immaterial, since there is no inventory notionof waiting queue of customers. From a revenue management perspective, the basic objectiveof inventory management is to minimize the (discounted or time-average) inventory cost.For more recent related work in the broader area of service systems, the reader is referred toAdan et al. (2005), Li and Glazebrook (2010), Perry and Stadje (2003).

A recent comprehensive survey of related inventory models with Markovian demand maybe found in Beyer et al. (2010) and references therein. However, we are not aware of anyprevious work on stochastic models with continuous replenishment, except Shi et al. (2013,2014). Accordingly, this paper makes a number of contributions to the field of inventorysystems with compound Poisson demands and continuous replenishment, subject to the lost-sales policy. The main contributions of this paper are summarized as follows:

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(i) A closed formexpression for the time-average inventory cost of the systemunder study isobtained in Theorem 2 for arbitrarily distributed demand and a general penalty function.

(ii) A characterization of the optimal constant replenishment rate that minimizes the time-average inventory cost is derived in Theorem 3 for general demand distributions andgeneral penalty functions.

(iii) An algorithm to numerically calculate the optimal replenishment rate is presented inEq. (5.3).

(iv) Closed form expressions for the optimal replenishment rates and their attendant costsare obtained for the case of exponential demand for both constant penalty and loss-proportional penalty functions.

(v) We further consider special demand-size distributions, such as constant, uniform andGamma, and take advantage of their functional form to further simplify the optimizationalgorithm.

The rest of this paper is organized as follows. Section 2 introduces notation and formulatesthe inventory model under study. Section 3 uses a renewal argument to derive an integro-differential equation for the cost function until and including the first lost-sale occurrence.From this equation, we obtain a closed form formula for the time-average cost, expressedas a ratio of the conditional expected total cost until the first lost-sale occurrence divided bythe conditional expected time to the first lost-sale occurrence, given zero initial inventorylevel. It also derives a closed form expression for the time-average cost function. Section 4investigates the existence and uniqueness of the optimal replenishment rate which minimizesthe aforementioned cost function aswell as asymptotic costs. Section 5 treats the time-averagecost optimization problem with general demand distribution and studies the special cases ofconstant penalty and loss-proportional penalty. Section 6 contains three numerical studiesthat illustrate our results. Finally, Section 7 concludes this paper.

2 Model formulation

Throughout this paper, we use the following notational conventions and terminology. Let Rdenote the set of real numbers. For any real number x ∈ R, x+ = max{x, 0}. The indicatorfunction 1A(x) is 1 if x belongs to A and 0 otherwise. For a real function f (x), f (x+)

denotes the right limit of f (x) at x. For a random variable X , let f X (x), FX (x) andFX (x) = 1−FX (x) denote respectively its probability density function (pdf), its cumulativedistribution function (cdf) and its complementary cdf. The Laplace transform of a functionf (x) is defined by

f (z) = L [ f ] (z) =∫ ∞

0e−zx f (x)dx.

If z is sufficiently small, the above integral might not exists, in which case we denote f (z) =∞. If real functions f (x) and g(x) are defined on [0,∞), then the convolution function off (x) and g(x) is given by

⟨ f ∗g⟩ (u) =∫ u

0f (u − x)g(x)dx.

Also, we assume that the continuous compound interest rate, r ≥ 0, is a given constant, sothat the present value of one unit of cash flow at time t is e−r t .

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Finally, we shall make repeated use of the relation

˜FD(z) =∫ ∞

0e−zx FD(x)dx = 1

z

[1+

∫ ∞

0e−zx d FD(x)

]= 1

z

[1 − f D(z)

], (2.1)

where the second equality follows from integration by parts.

2.1 The inventory model

Let the demand arrival stream {Ai : i ≥ 0} follow a compound Poisson process withrate λ, where time A0 = 0 by convention. Thus, the corresponding sequence of demandinterarrival times, {Ti : i ≥ 1}, where Ti = Ai − Ai−1, have iid exponential distributions.The corresponding demand sizes form an iid sequence {Di : i ≥ 1} with a common densityfunction fD(x)and common mean µD = E[D] < ∞, where demand Di arrives at time Ai .Continuous replenishment occurs at a constant (deterministic) rate ρ > 0. Throughout thispaper, we assume the stability condition,

ρ < λµD. (2.2)

By Prabhu (1965), this stability condition implies that the inventory level will not blowup over an infinite time horizon, namely, lim

t→∞I(t) < ∞ and there exists a lost sale with

probability 1, where {I(t) : t ≥ 0} denotes the right-continuous inventory process, given by

I(t) = I(0)+ ρ t −!N(t)

i=1[Di − L(Ai )], (2.3)

N(t) is the number of demands arriving over the interval (0, t] and

L(t) ={ [Di − I(Ai−)]+, if t = Ai , i = 1, 2, ...0, otherwise

(2.4)

is the lost-sale size (excess demand that cannot be satisfied from on-hand inventory underthe lost-sales rule) associated with the i-th demand arrival, and zero, otherwise. Denote byI the infinite-horizon time-average of inventory, namely,

I = limt→∞

1t

∫ t

0I(z)dz. (2.5)

Let {τi : i ≥ 0} be the sequence of lost-sale occurrence times, given by

τi = inf{t > τ i−1 : L(t) > 0}, (2.6)

where τ 0 = 0 by convention. Let {Jk : k = 1, 2, . . .} be the sequence of random arrivalindices at which loss occurs, namely, τk = AJk . Figure 1 illustrates the evolution of theinventory process under study. A similar dynamic production-inventory process has beenstudied by Shi et al. (2013, 2014) but with different focus.

Figure 2 depicts the detailed evolution of a sample path of the inventory process over theinterval [0, τ 1].

2.2 Inventory costs

Inventory cost is consisted comprised of two types: carrying costs and lost-sales penalties.These cost components are described below.

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Fig. 1 A sample path of the inventory level process, {I(t)}

Fig. 2 A sample path of the inventory level process over the interval [0, τ1]

• Carrying costs. While there is inventory on hand, a carrying cost is incurred at rate hper unit time and per inventory unit. Accordingly, the cumulative carrying cost processHρ = {Hρ(t) : t ≥ 0} is given by

Hρ(t) = h∫ t

0I(z)dz (2.7)

• Lost-sales penalties. Whenever a customer’s demand cannot be completely satisfiedfrom on-hand inventory, a penalty w(x) is incurred as a non-decreasing function of thelost-sale size, x, with the proviso thatw(0) = 0. In particular, we shall consider a constantlost-sales penalty function and a loss-proportional penalty function in Section 5 as specialcases. Accordingly, the penalty process Wρ = {Wρ(t) : t ≥ 0} is given by

Wρ(t) =!N(t)

i=1w(L(Ai )). (2.8)

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The inventory cost process Cρ = {Cρ(t) : t ≥ 0} is given by

Cρ(t) = Hρ(t)+ Wρ(t) = h∫

0

tI(z)dz +

!N(t)

i=1w(L(Ai )). (2.9)

For a given initial inventory level, u, the infinite-horizon time-average inventory cost isdefined by

cρ = limt→∞

E[Cρ(t)|I(0) = u]t

. (2.10)

In a similar vein, the infinite-horizon time-average carrying cost is defined by

hρ = limt→∞

E[Hρ(t)|I(0) = u]t

, (2.11)

and the infinite-horizon time-average penalty is defined by

wρ = limt→∞

E[Wρ(t)|I(0) = u]t

. (2.12)

Thus, we have cρ = hρ + wρ .

3 Solutions for the time-average cost function

In this section we derive closed form expressions for the time-average cost functions. Specif-ically, we first derive an integro-differential equation for the conditional expected discountedcost function until and including the first lost-sale occurrence, and then use it to solve for thetime-average cost cρ . The main result of this section is the closed form expression for thetime average cost, cρ , which is given by Theorem 2 in Sect. 3.4.

3.1 The function cρ

To derive the time-average cost function cρ we first consider the inventory cost until andincluding the first lost-sale occurrence. It is given by

Cρ(τ 1) = h∫ τ 1

0I(z)dz + w(L(τ 1)). (3.1)

Its expected value, conditional on I(0) = u, is denoted by

cρ(u) = E[Cρ(τ 1)|I(0) = u]. (3.2)

Note that the inventory process over intervals of the form (τi , τ i+1] is a renewal process andthe corresponding cost process can be regarded as a renewal reward process. Consequently,by Theorem 3.6.1 in Ross (1996), with probability 1, the time-average cost in Eq. (2.10) isindependent of the initial inventory level, I(0) = u, and given by

cρ = cρ(0)E[τ 1|I(0) = 0] . (3.3)

Our goal is to obtain a formula for cρ , by deriving cρ(0) and E[τ1|I(0) = 0]. To this end,we shall make use of the total discounted inventory cost until and including the first lost-saleoccurrence, given by

Cρ,r(τ1) = h∫ τ 1

0I(z)e−r zd z + w(L(τ1))e−rτ 1 (3.4)

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and its associated conditional expected discounted cost function, given by

cρ,r(u) = E[Cρ,r(τ1)|I(0) = u]. (3.5)

We shall utilize the discounted quantities above as follows. To compute cρ(0), we shall usethe fact that cρ(u) can be obtained from cρ,r(u) in Eq. (3.5) by setting there r = 0, so thatcρ(u) = cρ,0(u). Next, define

dρ,r(u) = E[e−rτ1 |I(0) = u]. (3.6)

Note that setting h = 0 and w(x) = 1(0,∞)(x) in Eq. (3.4) implies

dρ,r(u) = cρ,r(u). (3.7)

3.2 The function cρ,r(u)

For any given initial inventory level u ≥ 0 and time interval (0, s], where s > 0 is small, con-sider the following disjoint events and the corresponding discounted cost function, cρ,r(u).

(1) On the event {A1 > s}, the corresponding cost is

E[Cρ,r1{A1>s}|I(0) = u]

=∫ ∞

sλe−λt

[h∫ s

0(u + ρ z)e−r zd z + cρ,r(u + ρs)e−rs

]dt

= e−λs[h∫ s


](3.8)

where the first term in the sums above is the discounted carrying cost over (0, s], and thesecond is the discounted residual cost over (s, τ 1], since {A1 > s} ⊂ {s ≤ τ 1}.

(2) On the event {A1 ≤ s}, the corresponding cost is

E[Cρ,r1{A1≤s}|I(0) = u] =∫ s

0λe−λt M(u, t)dt (3.9)

where M(u, t) = E[Cρ,r |A1 = t, I(0) = u] is given by

M(u, t) = h∫ t

0(u + ρ z)e−r zd z

+ e−r t∫ u+ρ t

0fD(x)cρ,r(u + ρ t − x)dx

+ e−r t∫ ∞

u+ρ tfD(x)w(x − (u + ρ t))dx (3.10)

It follows that

M(u, 0) =⟨fD ∗ cρ,r

⟩(u)+

"

u

∞fD(x)w(x − u)dx. (3.11)

Thus, adding Eqs. (3.8) and (3.9) yields

cρ,r(u) = e−λs[h∫ s


]+∫ s

0λe−λt M(u, t)dt

(3.12)

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Next, differentiating Eq. (3.12) with respect to s, and setting s = 0, we have

0 = hu − (λ + r)cρ,r(u)+ ρ∂

∂ucρ,r(u)+ λM(u, 0). (3.13)

Finally, substituting Eq. (3.11) into Eq. (3.13) yields after rearranging terms

ρ∂

∂ucρ,r(u) − (λ + r)cρ,r(u)+ λ

⟨fD∗cρ,r

⟩(u) = −g(u), (3.14)

where

g(u) = hu + λ

∫ ∞

ufD(x)w(x − u)dx. (3.15)

It is convenient to decompose the function above into g(u) = g1(u)+ g2(u), where

g1(u) = hu, (3.16)

g2(u) = λ

∫ ∞

ufD(x)w(x − u)dx. (3.17)

Thus, g1(u) corresponds to the carrying cost component, while g2(u) corresponds to thelost-sales penalty component.Next, we proceed to solve Eq. (3.14) for cρ,r(u). To this end, we take the Laplace transformon both sides of that equation to get

ρ[zcρ,r(z) − cρ,r(0)] − (λ + r)cρ,r(z)+ λ f D(z)cρ,r(z) = − g(z), z > 0 (3.18)

Rearranging and simplifying the above equation yields

[λ f D(z)+ ρ z − λ − r]cρ,r(z) − ρcρ,r(0) = − g(z), z > 0, (3.19)

Denoting the first factor in Eq. (3.19) as the auxiliary function

ψr(z) = λ f D(z)+ ρ z − λ − r, (3.20)

Equation (3.19) can now be written as

ψr (z)cρ,r(z) − ρcρ,r(0) = − g(z), z > 0, (3.21)

For ease of exposition, denote the first root of the equationψr (z) = 0 by η(r) and the secondroot by θ(r), where η(r) < θ(r), provided they exist.

The following lemma summarizes some key properties of the equation ψr (z) = 0.

Lemma 1 For any r ≥ 0, the equation ψr (z) = 0 has two distinct roots, η(r) and θ(r)satisfying the following relations:

(a) If r = 0, then η(0) = 0 and θ(0) > 0.(b) If r > 0, then η(r) < 0 and θ(r) > 0.

Proof We first prove that the function ψr(z) is convex by computing its first and secondderivatives,

∂

∂ zψr (z) = ρ − λ

∫ ∞

0xe−zx fD(x)dx, (3.22)

∂2

∂ z2ψr (z) = λ

∫ ∞

0x2e−zx fD(x)dx. (3.23)

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Fig. 3 Illustration of the function ψr (z) and its root structure

Since the case of zero demand with probability 1 is precluded, it follows from Eq. (3.23) that

∂2

∂ z2ψr(z) > 0. (3.24)

Toprove part (a) for r = 0,wehaveψ0(0) = 0, namely, zero is a root ofψ0(z) = 0. It remainsto show the existence of exactly one more positive root. First, note that ∂

∂ zψ0(z)|z=0 < 0by Eqs. (3.22) and (2.2). Therefore, there exists z′ > 0 such that ψ0(z

′) < 0. But sinceψ0(∞) = ∞, there must be a positive root. Next, we prove by contradiction that therecannot be more than two roots. Otherwise, by Rolle’s Theorem, there must be more than onez∗ such that ∂

∂ zψ0(z∗) = 0. This contradicts the fact that there is at most one z∗ such that

∂∂ zψ0(z

∗) = 0 by Eq. (3.24), thereby establishing part (a).To prove part (b) note that for r > 0, one hasψr(0) < 0,ψr (∞) = ∞ andψr(−∞) = ∞.

Consequently, there must be at least one positive root and one negative root of ψr (z) = 0.An argument similar to that in part (a) establishes that there cannot be more than two rootsas required. ⊓-

Figure 3 illustrates the key features of the function ψr(z) and the root structure for theequation ψr(z) = 0.

In particular, for r = 0, we denote

ξ = θ(0).

In view of Lemma 1, we can write,

λ f D(ξ)+ ρξ − λ = 0, (3.25)

and generally,λ fD(θ)+ ρθ − λ = r. (3.26)

where θ = θ(r) is the unique positive root of Eq. (3.26), ξ = θ(0) is the unique positive rootof Eq. (3.25), and it follows that lim

r→0θ(r) = ξ , as illustrated in Fig. 3.

Lemma 2 (a) For ρ ≥ 0 and ξ = ξ(ρ),

ρ = λ ˜FD(ξ). (3.27)

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(b) For ρ ≥ 0, the mapping ρ .→ ξ(ρ) is strictly monotone decreasing.

Proof To prove part (a), note first that by Eq. (3.25),

ρ =λ[1 − f D(ξ)

]

ξ, (3.28)

Equation (3.27) now follows by Eqs. (2.1) and (3.28).To prove part (b), differentiate Eq. (3.27) with respect to ρ, yielding

1 = −λξ ′(ρ)∫ ∞

0xe−xξ(ρ) FD(x)dx.

The equation above implies ξ ′(ρ) < 0 since the integral on the right hand side is strictlypositive for all ρ ≥ 0, which in turn implies the result. ⊓-

Corollary 1limρ→0

ρξ = λ. (3.29)

where ξ = ξ(ρ).

Proof Sending ρ ↓ 0 on both sides of Eq. (3.27) implies limρ→0

˜FD(ξ(ρ)) = 0, which in turn

implieslimρ→0

ξ(ρ) = ∞. (3.30)

Furthermore, Eq. (3.25) can be written as

ρξ = λ − λ fD(ξ). (3.31)

The proof immediately follows by sending ρ ↓ 0 in Eq. (3.31) and the fact that limρ→0

f D(ξ) =0. ⊓-

We are now in a position to derive a closed form formula for cρ,r(0).

Lemma 3

cρ,r(0) =g(θ(r))

ρ. (3.32)

Proof The result follows by setting z = θ(r) in Eq. (3.21) and noting that the first termvanishes by Lemma 1. ⊓-

Corollary 2

cρ(0) =g(ξ)ρ

. (3.33)

Proof Follows from Eq. (3.32) and the fact that limr→0

θ(r) = ξ . ⊓-

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3.3 The function dρ,r(u)

Consider the special case h = 0 and w(x) = 1(0,∞)(x). Then, Eq. (3.15) becomes

g(u) = λ

∫ ∞

ufD(x)dx = λFD(u)

and consequently, by Eq. (3.14), dρ,r(u) satisfies the following integro-differential equation,

ρ∂

∂udρ,r(u) − (λ + r)dρ,r(u)+ λ

⟨fD ∗ dρ,r

⟩(u) = −λFD(u). (3.34)

The following lemma provides a closed form formula for dρ,r(0).

Lemma 4dρ,r(0) = 1 − r

ρθ(r). (3.35)

Proof Set h = 0 and w(x) = 1(0,∞)(x) in Eq. (3.32), which becomes

dρ,r(0) =λ

ρ˜FD(θ), (3.36)

in view of Eq. (3.7). Using Eq. (2.1), we can rewrite Eq. (3.36) as

dρ,r(0) =λ − λ fD(θ)

ρθ(3.37)

Furthermore, by Eq. (3.26), the numerator of Eq. (3.37) can be written as

λ − λ fD(θ) = ρθ − r. (3.38)

The result now follows by substituting Eq. (3.38) into Eq. (3.37). ⊓-

Theorem 1E[τ 1|I(0) = 0] = 1

ρξ. (3.39)

Proof In view of Eq. (3.35), we have

E[τ 1|I(0) = 0] = − ∂

∂ rdρ,r(0)|r=0 = − lim

r→0

∂

∂ r

[1 − r

ρθ(r)

]

= 1ρ

limr→0

{1

θ(r)− r

[θ(r)]2 θ ′(r)}= 1

ρξ

Here, the first equality holds by Eq. (3.6); the second equality holds by Eq. (3.35); the fourthequality is due to the fact that lim

r→0r

[θ(r)]2 θ′(r) = 0, and it remains to show that it suffices to

prove that limr→0

1[θ(r)]2 θ

′(r) exists and is finite. To see that, we first note that limr→0

[θ(r)]2 = ξ2.

Secondly, since r #→ θ(r) is a one-to-one mapping by Lemma 1 and Eq. (3.26), one haslimr→0

θ ′(r) = 1limθ→ξ

r ′(θ) . Furthermore, by Eqs. (3.20) and (3.26), one has

r ′(θ) = ∂

∂ zψr(z) |z=θ .

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Therefore, by continuity of ∂∂ zψr (z) at z = ξ [cf. the proof of Lemma 1], we have

limθ→ξ

r ′(θ) = ∂

∂ zψ0(z)

∣∣z=ξ > 0, (3.40)

again by the proof for Lemma 1. We conclude that limr→0

1[θ(r)]2 θ

′(r) is finite as claimed. ⊓-

We point out that for ρ = 0,E[τ 1|I(0) = 0] = λ−1 clearly holds (that is, in a systemwithout replenishment, started with zero initial inventory, each demand arrival results in a lostsale, and consequently, the mean time between losses is λ−1). Indeed, this result is obtainedby sending ρ ↓ 0 on both sides of Eq. (3.39) and substituting Eq. (3.29) into Eq. (3.39).

3.4 Closed form expression for cρ

The following theorem provides computable representations for the infinite-horizon time-average total cost and its components (carrying cost and lost-sales penalty).

Theorem 2

cρ = ξ g(ξ), (3.41)

hρ = ξ g1(ξ), (3.42)

wρ = ξ g2(ξ) (3.43)

where ξ = ξ(ρ).

Proof To prove Eq. (3.41), substitute Eqs. (3.33) and (3.39) into Eq. (3.3). Equations (3.42)and (3.43) readily follow by noting that g = g1 + g2 implies g = g1 + g2. ⊓-

We mention that Eq. (3.41) has been derived by Shi et al. (2014), but in a different way.

Corollary 3

hρ = hξ, (3.44)

I = 1ξ. (3.45)

Proof Note that g1(z) = hz2 by Eq. (3.16). Equation (3.44) now readily follows by substitut-

ing g1(ξ) into Eq. (3.42). Finally, Eq. (3.45) follows immediately from Eq. (3.44) by notingthat the inventory time average is equivalent to the time-average carrying cost with h=1. ⊓-

Since our model does not stipulate a base-stock level, excursions of the inventory processcan generally reach high levels. The following lemma provides a bound on the probabilityof the inventory level exceeding a given value.

Lemma 5 Forany given replenishment rate,ρ, consider the corresponding inventory processin the steady state regime. Then, for s > 0,

P{I(t) ≥ s} ≤ 1sξ

, t ≥ 0. (3.46)

Proof By the Markov inequality [cf. Karr (1993)],

P{I(t) ≥ s} ≤ E[I(t)]s

= Is

The result now follows from Eq. (3.45). ⊓-

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4 Cost function properties

In this section, we study properties of the cost function cρ , given by Eq. (3.41), and itscomponents, hρ and wρ . To this end, we first provide some asymptotic results for these costfunctions, and then demonstrate the existence and uniqueness of the minima of cρ .

We first rewrite Eq. (3.41) as

cρ = L[g′] (ξ)+ g(0+) = L

[g′] (ξ)+ λE[w(D)], (4.1)

where the first equality holds by a property of the Laplace transform [cf. Widder (1959)],and the second equality follows from Eq. (3.15) and the representation

g′(u) = h + g′2(u) = h − λ

[∫ ∞

ufD(x)w′(x − u)dx + fD(u)w(0+)

]. (4.2)

Lemma 6 (a) hρ is monotone increasing and convex in ρ ≥ 0, and has the followingasymptotes

limρ→0

hρ = 0; (4.3)

limρ→∞ hρ = ∞. (4.4)

(b) wρ is monotone decreasing and concave in ρ ≥ 0, and has the following asymptotes

limρ→0

wρ = λE[w(D)]; (4.5)

limρ→∞ wρ = 0. (4.6)

(c) cρ has the following asymptotes

limρ→0

cρ = λE[w(D)]; (4.7)

limρ→∞ cρ = ∞. (4.8)

Proof Part (a) readily follows from Eq. (3.44). To prove part (b), we first prove Eq. (4.5) bywriting

limρ→0

wρ = limξ→∞

ξ g2(ξ) = limu→0

g2(u) = λE[w(D)].

Here, the first equality follows fromEq. (3.43) and themonotone decreasing relation betweenρ and ξ exhibited in Eq. (3.27); the second equality holds by the initial value theorem of theLaplace transform [cf. Widder (1959)]; and the third equality holds by Eq. (3.17).

Next, to prove Eq. (4.6) we write

limρ→∞ wρ = lim

ξ→0ξ g2(ξ) = lim

u→∞ g2(u) = ∞.

Here, the first equality holds by Eq. (3.43) and the decreasing monotone relation betweenρ and ξ exhibited in Eq. (3.27); the second equality holds by the final value theorem of theLaplace transform [cf. Widder (1959)]; and the last equality holds by Eq. (3.17).

We next show that the monotonicity and concavity of wρ follow from its first and secondderivatives, respectively. To this end, we write

wρ = ξ g2(ξ) = L[g′

2](ξ)+ g2(0+) =

∫ ∞

0e−ξ x g′

2(x)dx + λE[w(D)], (4.9)

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where the first equality holds by Eq. (3.43); the second equality holds by a property of theLaplace transform in Widder (1959); the first term in the third equality holds by definition;and the second term in the third equality holds by Eq. (3.17). Differentiating Eq. (4.9) nowyields

∂

∂ξwρ = −

∫ ∞

0xe−ξ x g′

2(x)dx ≥ 0 (4.10)

∂2

∂ξ2wρ =

∫ ∞

0x2e−ξ x g′

2(x)dx ≤ 0 (4.11)

Here, we use the fact that Eq. (3.17) implies

g′2(u) = −λ

[∫ ∞

ufD(x)w′(x − u)dx + fD(u)w(0+)

]≤ 0, (4.12)

since the equality holds by the generalized Leibniz integral rule, and the inequality holds inview of fD(u) ≥ 0 and the fact that the inequalities w(0+),w′(x) ≥ 0 hold by assumption.This completes the proof for part (b).

Finally, Eqs. (4.7) and (4.8) follow by adding Eq. (4.3) to Eq. (4.5), and adding Eq. (4.4)to Eq. (4.6), respectively. ⊓-

We are now in a position to study the existence and uniqueness of the minima of cρ . Wemention that it is straightforward to prove the existence of these minima; however the proofof uniqueness is much more challenging. Still, we can prove uniqueness for some importantcost functions. To this end, we shall make use of the following result (see “Appendix” forit’s proof).

Proposition 1 Let f (x) be a continuous function, not identically zero, satisfying∫ ∞

0f (x)dx = 0, (4.13)

and assume that there exists a constant x0 > 0 such that f (x) ≤ 0 for 0 ≤ x ≤ x0, andf (x) ≥ 0 for x > x0. Then, f (z) = 0 if and only if z = 0. ⊓-

The following lemma provides results for the case w(0+) = 0.

Lemma 7 For w(0+) = 0,

(a) if h = 0, then cρ attains a unique minimum at ρ∗ = λE[D], where ξ∗ = 0;(b) if

0 < h < λE[w′(D)] < ∞, (4.14)

then cρ has a unique and finite minimum at ρ∗ = λ ˜FD(ξ∗), where ξ∗ > 0;

(c) if h ≥ λE[w′(D)] > 0, then cρ attains a unique minimum at ρ∗ = 0, where ξ∗ = ∞.

Proof If w(0+) = 0, then Eq. (4.2) implies

g′(u) = h − λ

∫ ∞

ufD(x)w′(x − u)dx = R(u), (4.15)

where R(u) is an increasing function of u. Furthermore, Eqs. (4.1) and (4.15) jointly imply

cρ = R(ξ)+ λE[w(D)]. (4.16)

Equation (4.16) shows that minimizing cρ in ρ is equivalent to minimizing R(ξ) in ξ .

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To prove part (a), observe that h = 0 implies that R(u) < 0 because w(x) is a non-decreasing function (of the loss) by assumption (see Section 3), and consequently, R(ξ) isstrictly increasing. Part (a) now follows since R(ξ) attains a unique minimum at ξ∗ = 0.

To prove part (b), note that the existence of the minimum follows from the continuity ofcρ and part (c) in Lemma 6. It remains to prove the uniqueness of the minimum. To this end,differentiate Eq. (4.16) with respect to ξ , and set the derivative to zero, yielding

∂

∂ξcρ(ξ) = −

∫ ∞

0xe−ξ x R(x)dx =

∫ ∞

0fξ (x)dx = 0, (4.17)

where fξ (x) = −xe−ξ x R(x).Next, Eq. (4.15) implies

limu→∞ R(u) = h > 0. (4.18)

Furthermore, the assumption h < λE[w′(D)] < ∞ and Eq. (4.15) imply R(0+) < 0. Usingthe two bounds above and the continuity and monotonicity of R(u), it follows that thereexists a constant u0 > 0, such that R(u) ≤ 0 for 0 ≤ u ≤ u0, while R(u) ≥ 0 for u ≥ u0.Consequently, we conclude that for any ξ ≥ 0, one has fξ (x) ≥ 0 for 0 ≤ u ≤ u0, whilefξ (x) ≤ 0 for u ≥ u0.Letting ξ∗ denote a solution of Eq. (4.17), we next prove its uniqueness by contradiction.

Suppose there exists another solution ξ ′ of Eq. (4.17), such that without loss of generality,ξ∗ < ξ ′. Then, by Eq. (4.17),

∂

∂ξcρ(ξ ′) =

∫ ∞

0fξ ′(x)dx =

∫ ∞

0e−(ξ ′−ξ∗)x f ξ∗(x)dx = 0.

In view of Proposition 1, wemust have ξ ′−ξ∗ = 0which contradicts the assumption ξ∗ < ξ ′,thereby completing the proof for part (b).

Finally, to prove part (c), note that if h ≥ λE[w′(D)], then R(u) ≥ 0 by Eq. (4.15). Itfollows that R(ξ) is non-increasing, which completes the proof of part (c). ⊓-

Figure 4 illustrates a typical cρ as function of the original domain variable (the replen-ishment rate, ρ) and a Laplace domain variable (the positive root, ξ); recall that ρ and ξ arerelated by Eq. (3.27).

Fig. 4 A typical cρ as function of ρ (left) and ξ (right)

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5 Optimal replenishment rate

We are now in a position to derive the optimal replenishment rate. More specifically, we willproceed to optimize the time-average cost of Eq. (2.10) with respect to the replenishmentrate, ρ. To this end, we first obtain a general structural result for optimal replenishment rates,ρ∗, and then study some special cases.

Note that for ξ = 0, ˜FD(ξ) = µD. Hence, by the one-to-one mapping between ρ and ξ ,established by Eq. (3.27), the stability condition given by Eq. (2.2) is equivalent to ξ > 0.In this case, we shall restrict cost optimization to ξ > 0. We shall admit the possibility ofmultiple optimal replenishment rates.

Theorem 3 The optimal replenishment rates for Eq. (2.10) are given by

ρ∗ = λ ˜FD(ξ∗), (5.1)

whereξ∗ = argmin

ξ>0{ξ g(ξ)}. (5.2)

Proof In view of Eq. (3.41), minimizing cρ = ξ(ρ) g(ξ(ρ)) with respect to ρ is equivalentto minimizing cρ = ξ g(ξ) with respect to the nonnegative variable ξ . To this end, we firstcompute Eq. (5.2), namely, perform optimization on cρ = ξ g(ξ) in the Laplace domain tofind the optimal values ξ∗.

Finally, recall that by Lemma 2(b), the mapping ρ .→ ξ(ρ) is one-to-one. Consequently,we can invert each ξ∗ = ξ(ρ∗) via Eq. (5.1) to obtain the corresponding optimal replenish-ment rate, ρ∗. ⊓-

The minimum values, ξ∗, given in Eq. (5.2), can be computed in several ways. A straight-forward but relatively time consuming method is global search. However, when ξ∗ is unique,the availability of derivatives of cρ(ξ) with respect to ξ allows us to apply the relativelyfast Newton’s Method, where successive approximations of the minimum are given by theiterative scheme,

ξn+1 = ξn −∂∂ξ cρ(ξn)∂2

∂ξ2cρ(ξn)

, n = 0, 1, . . . . (5.3)

We next proceed to study production-inventory systems with specialized lost-sales penaltystructures, and specifically the constant lost-sales penalty and the loss-proportional penalty.Under each penalty structure,we study the optimal average costs, subject to particular demanddistributions, such as constant, uniform, Exponential and Gamma distributions.

5.1 Constant lost-sales penalty

In this case, w(x) = K 0 for x ≥ 0 where K 0 > 0 is a constant. Then, Eq. (3.15) becomes

g(u) = hu + λK 0

∫ ∞

ufD(x)dx = hu + λK 0 FD(u), (5.4)

and the corresponding Laplace transform is given by

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Table 1 Optimal quantities for production-inventory systems subject to constant penalty and various demanddistributions

D ξ∗ ρ∗ cρ∗

D = d argminξ>0

{hξ

− λK0e−ξd}

λξ∗[1 − e−ξ∗d

]hξ∗ + K0ρ

∗ξ∗

d > 0

D ∼Exp(β) β√h√

λβK0−√h

λβ −

√λh

β√

βK02√

hλK0β − h

β

β > 0

βλK0 > h

D ∼ U(a, b) argminξ>0

{hξ

− λK0e−aξ −e−bξ

(b−a)ξ

}λξ∗

[1 − e−aξ∗−e−bξ∗

(b−a)ξ∗

]hξ∗ + K0ρ

∗ξ∗

0 ≤ a < b

D ∼ Γ (α,β) argminξ>0

{hξ

− λK0

(1+ ξ

β

)−α}

λξ∗

[1 −

(1+ ξ∗

β

)−α]

hξ∗ + K0ρ

∗ξ∗

α,β > 0

g(z) = hz2

+ λK 0˜FD(z) =

hz2

+ λK 01 − f D(z)

z,

where the second equality holds by Eq. (2.1). In view of Eq. (3.41), we now have

cρ = ξ g(ξ) = hξ+ λK 0

[1 − f D(ξ)

]= h

ξ+ ρK 0ξ , (5.5)

where the last equality holds by Eq. (3.28). By Eq. (5.2), the optimal ξ∗ is given by

ξ∗ = argminξ>0

{hξ

− λK 0 f D(ξ)}. (5.6)

Wemention that ξ∗ is amonotonically decreasing function of λK0h . To see that, Eq. (5.6) can be

rewritten as ξ∗ = argminξ>0

{1ξ − λK0

h f D(ξ)}, so that the derivative of the rewritten objective

function with respect to λK0h is − f D(ξ) < 0, which implies the result. It follows that ρ∗

is a monotonically increasing function of λK0h , because ξ∗ is a monotonically decreasing

function of λK0h , while ξ = ξ(ρ) is monotonically decreasing in ρ.

Table 1 displays expressions for ξ∗, ρ∗and cρ∗ for selected demand distribution withdetailed derivations given in the “Appendix”.

5.2 Loss-proportional penalty

In this case, w(x) = K 1x for x ≥ 0 where K 1 > 0 is constant. Then, Eq. (3.15) becomes

g(u) = hu + λK 1

∫ ∞

u(x − u) fD(x)dx, (5.7)

and the corresponding Laplace transform is given by

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g(z) = hz2

+ λK 1

[µD

z− 1 − f D(z)

z2

]

. (5.8)

In view of Eq. (3.41), we now have

cρ = hξ+ λK 1

[

µD − 1 − f D(ξ)ξ

]

, (5.9)

where µD = E[D]. Note that since λξ−1[1 − f D(ξ)

]= ρ by Eq. (3.25), it follows that

cρ = hξ

− K 1ρ + λK 1µD (5.10)

Consequently, by Eq. (5.9), the optimal ξ∗ is given by

ξ∗ = argminξ>0

{hξ

− λK 11 − f D(ξ)

ξ

}

. (5.11)

We mention that ξ∗ is a monotonically decreasing function of λK1h . To see that, note that by

Eq. (5.9), we have ξ∗ = argminξ>0

{1ξ + λK1

h

[µD − 1− f D(ξ)

ξ

]}, so that the derivative of the

rewritten objective function with respect to λK1h is

µD − ξ−1[1 − f D(ξ)

]= µD − ˜FD(ξ)

= µD −∫ ∞

0e−ξ x FD(x)dx < µD −

∫ ∞

0FD(x)dx = 0,

which implies the result. It follows that ρ∗ is a monotonically increasing function of λK1h ,

because ξ∗ is a monotonically decreasing function of λK1h , while ξ = ξ(ρ) is monotonically

decreasing in ρ.Table 2 displays expressions for ξ∗, ρ∗ and cρ∗ for selected demand distribution with

detailed derivations given in the “Appendix”.

5.3 Exponential demand: relationship between the optimal and cost-balancedrates

In this section, we assume that demand is exponential, and under this assumption we relatethe optimal replenishment rate, ρ∗, and the corresponding cost-balanced replenishment rate,ρ, defined as the replenishment rate satisfying

hρ = wρ . (5.12)

Let β > 0 be the rate parameter of an exponential demand distribution, so

fD(x) = βe−βx, x ≥ 0. (5.13)

and

f D(z) =β

β + z. (5.14)

Accordingly, φ0(z) becomes

ψ0(z) = ρ z − λ + λβ

β + z,

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Table 2 Optimal quantities for production-inventory systems subject to loss-proportional penalty and variousdemand distributions

D ξ∗ ρ∗ cρ∗

D = d argminξ>0

{hξ

− λK11−e−ξd

ξ

}λξ∗ [1 − e−ξ∗d ] h

ξ∗ − kρ∗ + λK1d

d > 0

D ∼Exp(β) β√h√

λK1−√h

λβ − 1

β

√hλK1

2β

√hλK1 − h

β

β > 0

λK1 > h

D ∼ U(a, b) argminξ>0

{hξ

− λK1ξ

[1 − e−aξ −e−bξ

(b−a)ξ

]}λξ∗

[1 − e−aξ∗−e−bξ∗

(b−a)ξ∗

]hξ∗ − K1ρ

∗ + λK1[b−a]2

0 ≤ a < b

D ∼ Γ (α,β) argminξ>0

⎧⎨

⎩hξ

− λK11−(1+ ξ

β

)−α

ξ

⎫⎬

⎭λξ∗

[1 −

(1+ ξ∗

β

)−α]

hξ∗ − K1ρ

∗ + λK1µD

α,β > 0

and the equation ψ0(z) = 0 can be written as(z − λ

ρ+ β

)z = 0. (5.15)

Hence, the positive root of Eq. (5.15) is given by

ξ = λ

ρ− β. (5.16)

We then have the following result (see the “Appendix” for a proof).

Proposition 2 Let the demand distribution be exponential, and assume that the penaltyfunction is of the form w(x) = K 0 or w(x) = K 1x for x ≥ 0. Then, for any ρ > 0,

ρ > ρ∗; (5.17)

hρ∗ ≤ wρ∗ . (5.18)

A numerical study illustrating the relations in Eqs. (5.17) and (5.18) appears in Sect. 6.

6 Numerical study

In this section, we study three special cases subject to the constant lost-sales penalty with λ =0.5, h = 1 and K 0 = 100 in all cases. As a check on the accuracy of the approximation givenby Eq. (5.3), we performed paired evaluations of the requisite cost functions: by analyticalformulas developed earlier and by simulation.Accordingly, in the figures below, time-averagecost-function curves are paired as follows: curveswith circles correspond to analytical results,while curves with asterisks correspond to their simulation counterparts.

In the first case, we study the time-average total cost, cρ , as function of the replenishmentrate, ρ, under three demand distributions: constant, exponential and uniform. To ensure that

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0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.9515

20

25

30

35

40

45

ρ

Aver

age

Tota

l Cos

tConstant Demand (Simulation)Constant Demand (Analytical)Exponential Demand (Simulation)Exponential Demand (Analytical)Uniformal Demand (Simulation)Uniformal Demand (Analytical)

Constant

Uniform

Exponential

Fig. 5 Time-average costs for inventory systems with various demand distributions as functions of the replen-ishment rate

these systems are comparable, we letµD = 2 be the commonmean of all the aforementioneddemand distributions.

Figure 5 depicts cρ as a function of ρ for each demand distribution. Here, curve stylescorrespond to demand distributions: solid curves to the constant distribution, dashed curvesto the exponential distribution and dotted curves to the uniform distribution. Figure 5 showsa good agreement between all pairs of analytical and simulation results. Furthermore, thesystemwith constant demand has the largest optimal replenishment rate, while its exponentialcounterpart has the smallest one.

In the second case, we study the time-average total cost, cρ , and its components (time-average carrying cost, hρ , and time-average penalty, wρ) as functions of the replenishmentrate, ρ, under an exponential demand distribution with rate parameter, β = 0.5.

Figure 6 depicts cρ, hρ and wρ as functions of ρ. Here, curve styles correspond to costtypes: solid curves to the time-average total costs, dashed curves to time-average carryingcosts and dotted curves to time-average penalties. Figure 6 shows a good agreement betweenall pairs of analytical and simulation results. Furthermore, the optimal solution P1 (withreplenishment rate ρ∗) for the time-average total costs differs from its cost-balanced coun-terpart, P2 (with replenishment rate ρ), such that ρ > ρ∗, in agreement with Eq. (5.17).

In the third case, we study analytically-computed quantities associated with the optimalsolution, (ρ∗, cρ∗), under various demand distributions: constant, exponential, uniform andGamma.

Table 3 displays ρ∗ and ξ∗ as functions of the mean demand, β−1, with the four afore-mentioned demand distributions. From Table 3 it can be seen that the respective optimalreplenishment rates increase in this order of distributions: constant, uniform, Gamma and

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0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950

5

10

15

20

25

30

35

40

ρ

Aver

age

Cos

tAverage Total Cost (Analytical)Average Total Cost (Simulation)Average Holding Cost (Analytical)Average Holding Cost (Simulation)Average Penalty Cost (Analytical)Average Penalty Cost (Simulation)

P1

P2

Total

Penalty

Carrying

Fig. 6 Time-average total costs, carrying costs and penalties as functions of the replenishment rate underexponential demand

exponential. Note that as the average demand decreases (i.e., β gets larger), the optimalreplenishment rate approaches zero for all demand distributions, as it should, since the opti-mal replenishment must be zero in the absence of demand.

7 Concluding remarks

This paper investigated a single-product production-inventory system under continuousreview with constant replenishment and compound Poisson demands subject to the lost-sales policy. The total cost function of the system is defined as the sum of carrying costsand lost-sales penalties. For arbitrary demand-size distributions, we developed an integro-differential equation in terms of the expected discounted total cost function, conditioned onthe initial inventory level, and then derived a closed-form formula for the time-average totalcosts in terms of Laplace transforms. It was shown that this cost function can be readilyoptimized with respect to the replenishment rate by simple search or some computationally-efficient search algorithms (e.g., Newton’ Method) in the transform domain; however, forthe special case of exponential demand size, we derived a closed-form optimal solution.Finally, we studied optimal solutions using analytical and simulation evaluations for variousdemand-size distributions.

For the cost optimization problem treated in Sect. 5, we only considered the time-averageinventory cost. Another component of system cost, the production cost p(ρ), can be assessedcontinuously over time, where p(ρ) is typically increasing in ρ. By Eq. (3.27), the productioncost can be furtherwritten as p(ρ) = p

(λ ˜FD(ξ)

). In this case, tominimize the time-average

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Table 3 Analytically-computed optimal quantities under various demand distributions

β D = 1β Exp (β) U

(0, 2

β

)Γ(4, 1

4β

)

ξ∗ ρ∗ ξ∗ ρ∗ ξ∗ ρ∗ ξ∗ ρ∗

0.1 0.060 3.798 0.080 2.806 0.068 3.367 0.065 3.515

0.6 0.121 0.762 0.133 0.689 0.125 0.736 0.124 0.743

1.1 0.159 0.427 0.170 0.398 0.163 0.417 0.162 0.420

2.1 0.215 0.229 0.226 0.217 0.218 0.225 0.217 0.226

3.1 0.258 0.156 0.269 0.150 0.262 0.154 0.261 0.155

4.1 0.295 0.119 0.306 0.115 0.299 0.117 0.298 0.118

5.1 0.328 0.096 0.339 0.093 0.332 0.095 0.331 0.095

6.1 0.358 0.080 0.369 0.078 0.361 0.080 0.361 0.080

7.1 0.385 0.069 0.396 0.067 0.389 0.069 0.388 0.069

8.1 0.411 0.061 0.421 0.059 0.414 0.060 0.413 0.060

9.1 0.435 0.054 0.445 0.053 0.438 0.054 0.437 0.054

10.1 0.457 0.049 0.468 0.048 0.461 0.049 0.460 0.049

11.1 0.479 0.045 0.490 0.044 0.483 0.044 0.482 0.044

12.1 0.500 0.041 0.510 0.040 0.503 0.041 0.502 0.041

13.1 0.520 0.038 0.530 0.037 0.523 0.038 0.522 0.038

14.1 0.539 0.035 0.549 0.034 0.542 0.035 0.541 0.035

system cost, we can first find ξ∗ by

ξ∗ = argminξ>0

{ξ g(ξ)+ p

(λ ˜FD(ξ)

)}.

Consequently, the optimal production rate ρ∗ can be obtained via Eq. (5.1).In future work one may consider production economies-of-scale in the form of cost sav-

ings; cf. Katehakis and Smit (2012) and references therein.The methodology for computing time-average costs is developed in the discounted-costs

framework. Some of the obtained results can be used to derive more detailed analyses ofthe system; cf. Shi et al. (2013, 2014). Given that the time-average cost does not depend onthe initial inventory level, one may derive the cost by other methodologies, e.g., steady stateanalysis; cf. Perry and Stadje (2002, 2003).

The research of this paper suggests future work in several directions. First, for generalsystems (with general cost functions and general demand distributions) it is highly likelythat optimal replenishment rates are unique under fairly general conditions. The extent ofconditions that ensure such uniqueness is a future research problem. Secondly, it is morepractical to consider a continuous-replenishment inventory with base-stock capacity level(where replenishment is suspended when the inventory level reaches or is at capacity) ratherthan one with unlimited capacity. Finally, it is of interest to investigate costs in a make-to-stock inventory with discrete replenishment sizes, where replenishment orders are triggeredby demand arrivals that drop the inventory level below the base-stock level.

Acknowledgments The first author would like to acknowledge the support for this project from the NationalScience Foundation (NSF grant CMMI-14-50743). The third author was supported in part by Research SeedGrant and Startup Grant at NJIT and Seed Grant by Leir Charitable Foundation.

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8 Appendix

Proof of Proposition 1 The proof of the necessary condition is trivial. To prove the sufficientcondition, we first write Eq. (4.13) as

∫ ∞

x0f (x)dx = −

∫ x0

0f (x)dx. (8.1)

Next, for all z > 0,

f (z) =∫ x0

0e−zx f (x)dx +

∫ ∞

x0e−zx f (x)dx

≤∫ x0

0e−zx f (x)dx + e−zx0

∫ ∞

x0f (x)dx

=∫ x0

0e−zx f (x)dx − e−zx0

∫ x0

0f (x)dx

=∫ x0

0

[e−zx − e−zx0

]f (x)dx < 0

Here, the first inequality holds since∫∞x0

e−zx f (x)dx ≤ e−zx0∫∞x0

f (x)dx since e−zx ≤e−zx0 for x ≥ x0; the second equality holds by Eq. (8.1), and the last inequality holds by therelations e−zx > e−zx0 , whence e−zx − e−zx0 > 0 for 0 ≤ x < x0 and f (x) ≤ 0 but notidentically zero for 0 ≤ x < x0. This completes the proof. ⊓-

Proofs of Table 1 Formulas (1) Constant demand sizeConsider the first distribution row of Table 1, where D = d > 0 is a constant, so that

f D(z) = exp{−zd}. (8.2)

The corresponding ξ∗ follows by substituting Eq. (8.2) into Eq. (5.6), the correspondingρ∗ follows by substituting this ξ∗ and Eq. (8.2) into Eq. (3.28), and the corresponding cρ∗

follows by substituting ξ∗ and ρ∗ into Eq. (5.5).

(2) Exponentially-distributed demand sizeConsider the second distribution row of Table 1, where D ∼Exp(β). Substituting Eq. (5.14)into Eq. (5.5) yields

cρ = λK 0 +hξ

− λK 0β

β + ξ. (8.3)

The corresponding ξ∗ is obtained by straightforward minimization of Eq. (8.3) in ξ , thecorresponding ρ∗ follows by substituting this ξ∗ into Eq. (5.16), and the corresponding cρ∗follows by substituting this ξ∗ into Eq. (8.3).

(3) Uniformly-distributed demand sizeConsider the third distribution row of Table 1, where D ∼U(a, b), so that

f D(z) =e−az − e−bz

(b − a)z. (8.4)


follows by substituting ξ∗ and ρ∗ into Eq. (5.5).

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Ann Oper Res

(4) Gamma-distributed demand sizeConsider the fourth distribution row of Table 1, where D ∼ Γ (α,β), so that

f D(z) =(1+ z

β

)−α

. (8.5)


follows by substituting ξ∗ and ρ∗ into Eq. (5.5). ⊓-

Proofs of Table 2 Formulas (1) Constant demand size

Consider the first distribution row of Table 2, where D = d. Then, the corresponding ξ∗

follows by substituting Eq. (8.2) into Eq. (5.11), the corresponding ρ∗ follows by substitutingthis ξ∗ and Eq. (8.2) into Eq. (3.28), and the corresponding cρ∗ follows by substituting ξ∗

and ρ∗ into Eq. (5.10).

(2) Exponentially-distributed demand sizeConsider the second distribution row of Table 2, where D∼Exp(β). Substituting Eq. (5.14)into Eq. (5.9) yields,

cρ = hξ

− λK 1

β + ξ+ λK 1µD. (8.6)

The corresponding ξ∗ is obtained by straightforward minimization of Eq. (8.6), the corre-sponding ρ∗ follows by substituting this ξ∗ into Eq. (5.16), and the corresponding cρ∗ followsby substituting this ξ∗ into Eq. (8.6).

(3) Uniformly-distributed demand sizeConsider the third distribution row of Table 2, where D ∼ U(a, b). Then, the correspondingξ∗ follows by substituting Eq. (8.4) into Eq. (5.11), the corresponding ρ∗ follows by substi-tuting this ξ∗ and Eq. (8.4) into Eq. (3.28), and the corresponding cρ∗ follows by substitutingξ∗ and ρ∗ into Eq. (5.10).

(4) Gamma-distributed demand sizeConsider the fourth distribution rowof Table 2,where D ∼ Γ (α,β). Then, the correspondingξ∗ follows by substituting Eq. (8.5) into Eq. (5.11), the corresponding ρ∗ follows by substi-tuting this ξ∗ and Eq. (8.5) into Eq. (3.28), and the corresponding cρ∗ follows by substitutingξ∗ and ρ∗ into Eq. (5.10). ⊓-

Proof of Proposition 2 Assume first that the lost-sales penalty is of the form w(x) =1(0,∞)(x)K 0 (see Sect. 6.1). Substituting Eq. (5.16) into Eq. (5.5), we have

cρ = ρhλ − ρβ

+ K 0[λ − ρβ] = hρ + wρ, (8.7)

where the time-average carrying cost is

hρ = ρhλ − ρβ

, (8.8)

and the time-average lost-sales penalty is

wρ = K 0[λ − ρβ]. (8.9)

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Ann Oper Res

Next, equate Eqs. (8.8) and (8.9) and solve for ρ, yielding

ρ = λ

β−√(

h

2K 0β2

)2+ λh

K 0β3 + h

2K 0β2 . (8.10)

Letting a = h2K0β

2 > 0 and b =√

λhK0β

3 > 0 above, and noting that√a2 + b2 < a+ b, we

get √(h

2K 0β2

)2+ λh

K 0β3 − h

2K 0β2 <

√λh

K 0β3 . (8.11)

Equations (8.10) and (8.11) readily imply

ρ >λ

β−√

λh

K 0β3 = ρ∗, (8.12)

where the equality in Eq. (8.12) follows from the exponential case in Table 1. This completesthe proof of Eq. (5.17).

To prove Eq. (5.18), note first that hρ is an increasing function of ρ by part (a) of Lemma 6,while wρ is a decreasing function of ρ by part (b) of Lemma 6. Second, the aforementionedmonotonicities of hρ and wρ in conjunction with Eq. (8.12) imply hρ ≥ hρ∗ and wρ ≤ wρ∗ .Eq. (5.18) now follows from the last two inequalities together with Eq. (5.12).

Finally, the corresponding proofs for the case w(x) = K 1x are readily seen to be analo-gous to the proofs above for w(x) = K 0, but with K 0 replaced by K1

β . ⊓-

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