Asymptotic Optimality of Order-up-to Policies in Lost Sales Inventory Systems Journal: Management Science Manuscript ID: MS-01100-2006.R3 Manuscript Type: Operations and Supply Chain Management Date Submitted by the Author: 09-Jul-2008 Complete List of Authors: Huh, Woonghee; Columbia University, Industrial Engineering and Operations Research Janakiraman, Ganesh; New York University, Stern School of Business Muckstadt, John; Cornell University, School of Operations Research and Information Engineering Rusmevichientong, Paat; Cornell University, School of Operations Research and Information Engineering Keywords: Inventory-Production : Policies, Inventory-production : Stochastic, Inventory- production : Approximations-heuristics, Inventory-Production, Policies, Leadtime ScholarOne, 375 Greenbrier Drive, Charlottesville, VA, 22901 Management Science
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ScholarOne, 375 Greenbrier Drive, Charlottesville, VA, 22901th2113/files/Asymptotic_LS_FinalSubmission.pdfretail and service parts environments. In this paper, we propose simple inventory
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Asymptotic Optimality of Order-up-to Policies in Lost Sales Inventory Systems
Journal: Management Science
Manuscript ID: MS-01100-2006.R3
Manuscript Type: Operations and Supply Chain Management
Date Submitted by the Author: 09-Jul-2008
Complete List of Authors: Huh, Woonghee; Columbia University, Industrial Engineering and Operations
Research
Janakiraman, Ganesh; New York University, Stern School of Business
Muckstadt, John; Cornell University, School of Operations Research and
Information Engineering
Rusmevichientong, Paat; Cornell University, School of Operations Research and
Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 3meet these agreements, the equipment manufacturers frequently expedite service parts to customer
locations when the closest stocking locations do not have the necessary parts. Consider a $100 part
that has to be expedited at an additional cost of $14. These systems are typically reviewed once a
day. Assuming a cost of capital of 25% per year, the cost of holding this part in inventory for one
day is about $0.07. Here, the ratio between the lost sales penalty cost – in this case, the expediting
premium of $14 – and the holding cost is 200.
In the retail example, our reasoning for the high penalty cost to holding cost ratio is more likely to
hold for non-perishable products with long life cycles than for products that are either perishable
or have short life-cycles. Similarly, our discussion in the service parts case holds only under the
assumption that the expiration of the service contract is not in the near future, thus precluding
the possibility of end-of-horizon behavior.
Our main result is that, under mild assumptions on the demand distribution, the class of order-
up-to policies is asymptotically optimal for these systems as the lost sales penalty increases. In
fact, we show asymptotic optimality for a specific order-up-to policy that is computed using the
newsvendor formula with appropriate parameters. For any given cost parameters, we also establish
an upper bound on the increase in the total cost from using this specific order-up-to policy instead
of the optimal policy. Finally, we present several computational results to evaluate the perfor-
mance (relative to the optimal policy) of the best order-up-to policy, and the specific order-up-to
policy mentioned above, for a wide range of demand distributions and cost parameters. Despite
its asymptotic optimality, there is a noticeable difference in the cost performance of this specific
order-up-to policy and the best order-up-to policy in our computational experiments when the
b/h ratio is relatively low. So, we modify this order-up-to level and propose another intuitively
appealing second heuristic order-up-to level which is based on two newsvendor expressions; in our
experiments, this heuristic has an average cost increase of 2.52% relative to the best order-up-to
policy. Moreover, a simple modification of our asymptotic analysis shows the asymptotic optimality
of this new order-up-to policy, also. Thus, this new heuristic satisfies three important criteria: (a)
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Huh et al.: Asymptotic Optimality of Order-up-to Policies4 Article submitted to Management Science; manuscript no. MS-01100-2006.R3
it is easy to compute (it only involves the newsvendor formula), (b) it performs almost as well as
the best order-up-to policy, and, (c) it is asymptotically optimal.
1.1. Notation and Problem Description
To facilitate the discussion of our main results, let us introduce the notation and the problem
description. We will consider both the lost sales and the backorder systems. In both systems, the
lead time between the placement of a replenishment order and its delivery is denoted by τ . (We
assume τ ≥ 1 since the τ = 0 case is nothing but the simple newsvendor problem.) The index for
time periods is t and Dt is the demand in period t. We assume D1,D2, . . . are independent and
identically distributed random variables and we use D to denote a generic random variable with
the same distribution as Dt. Also, let D =∑τ+1
t=1 Dt denote the total demand over τ + 1 periods,
representing the total demand over the lead time including the period when we place the order.
Let F denote the distribution function of D.
At the beginning of period t, the replenishment order placed in period t − τ is received. Let
XLt ∈ [0,∞) denote the inventory on hand at this instant in the lost sales system. For the backorder
system, letXBt ∈ (−∞,∞) denote the net-inventory in period t, that is, the inventory on hand minus
backorders at the instant after receiving the delivery due in period t. After receiving deliveries, a
new replenishment order is placed after which the demand Dt is observed.
For any h≥ 0 and b≥ 0, we denote by L(h, b) the lost sales system and by B(h, b) the backorder
system, which is identical to L(h, b) except that excess demand is backordered. In both the lost
sales L(h, b) and backorder B(h, b) systems, we charge holding costs on inventory on hand at the
end of each period at the rate of $h per unit per period. While we incur a lost sales penalty of
$b per unit of unmet demand in the lost sales model L(h, b), the shortage costs in the backorder
system B(h, b) are charged at the rate of $b per unit of backordered demand per period. We stress
that the meaning of b depends on whether this parameter is used in the backorder system or in
the lost sales system1; it is helpful to use a common notation as we compare these two systems
1 In practice, if some customers are willing to wait (that is, backorder) for a product while others are impatient (thatis, lead to lost sales) for procuring the same product, the lost sales penalty cost will be much larger than the backorder
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Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 5(this practice is consistent with Janakiraman et al. (2007)). It should be noted that we use B(h, b)
primarily as a tool for deriving bounding formulas for L(h, b); that the two penalty cost parameters
here are identical is of no other consequence to our results on lost sales inventory systems.
Given the holding cost h and lost sales penalty b, we denote by CL,S(h, b) and CL∗(h, b) the long
run average cost in the lost sales system L(h, b) under an order-up-to-S policy and under an optimal
policy, respectively. The corresponding quantities CB,S(h, b), and CB∗(h, b) are defined similarly,
with the interpretation of b as the backorder cost per unit per period. We denote by SL∗(h, b) and
SB∗(h, b) the best order-up-to levels in the lost sales system, L(h, b), and the backorder system,
B(h, b), respectively. We note that in the backorder system B(h, b), order-up-to policies are optimal,
and the best order-up-to level is given by the newsvendor formula under the distribution function
F of D.
1.2. An Intuitive Overview of Our Approach
Our main goal is to show that the ratio, minS CL,S(h,b)
CL∗(h,b) , approaches one as b grows infinitely. This
seems quite intuitive; in fact, a simple line of reasoning is as follows. When b is large, the probability
of demand exceeding supply in a period is so small that the issue of whether excess demand is
lost or backordered should not make a big difference. So, the optimal policy for B, which is an
order-up-to policy, should be close to optimal for L. Unfortunately, this intuition does not tranlate
to a proof, and it appears that a formal proof is not quite as straightforward. Next, we provide an
intuitive overview of the key steps in our proof.
The ratio, minS CL,S(h,b)
CL∗(h,b) , is difficult to work with directly because neither the numerator nor the
denominator can be expressed as a simple function of the primitives of the problem. In contrast,
the corresponding quantities for B are both identical since order-up-to policies are optimal in B,
and, they can be expressed using expectations and the newsvendor formula; in fact, CB∗(h, b) is
nothing but the optimal cost for a newsvendor facing a demand distribution of D. This motivates
penalty cost. The latter can be thought of as the margin lost plus a goodwill loss whereas the former can be thoughtof as the interest lost on the revenue by delaying the sale plus a goodwill loss.
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Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 7on the demand distribution, the expected backorder cost converges to a finite limit. Thus, for large
b, the holding cost is the primary determinant of CB∗(h,β · b). Thus, it is intuitive that for large
b, the cost CB∗(h,β · b) is insensitive to β. Therefore, the ratio CB∗(h,b+τ ·h)CB∗(h,b/(τ+1))
approaches one, which
is the desired result. Moreover, we show through an example that this result does not hold for all
demand distributions.
1.3. Contributions and Organization of the Paper
Category Description Results
Backorder Robustness of Optimal For any ν > 0, limb→∞CB∗(h,νb)CB∗(h,b) = 1, and
System Cost and Newsvendor limb→∞CB,Sνb (h,b)CB∗(h,b) = limb→∞
CB,Sb (h,νb)CB∗(h,νb) = 1,
(Section 4) Solution (Theorem 2) where Sb = SB∗(h, b) and Sνb = SB∗(h,νb)Asymptotic Equivalence
Connections of the Optimal Costs limb→∞CB∗(h,b)CL∗(h,b) = 1
Between (Theorem 3)Lost Sales Bounds on the Cost For any S,
and of Any Order-up-to CB,S(h, b/(τ + 1))≤CL,S(h, b)≤CB,S(h, b+ τh)Backorder Policy (Lemma 5)Systems Bounds on the Best
Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 9When the parameter b is large, this result enables us to use the (easily computed) optimal cost of
the backorder system B(h, b) as an approximation for the optimal cost in the corresponding lost
sales system.
In addition to asymptotic equivalence of the optimal costs, the long run average cost of any
order-up-to policy in the lost sales L(h, b) system is bounded above and below by the cost of the
same policy in the backorder systems B(h, b+τh) and B(h, b/(τ +1)), respectively. Lemma 5 shows
that for any order-up-to level S,
CB,S(h, b/(τ + 1))≤CL,S(h, b)≤CB,S(h, b+ τh).
We also develop bounds on the best order-up-to level in the lost sales system, as shown in Theorem
Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 15Proof: The first and second inequalities follow from the fact that CB∗(h, b) ≤ CB,Sνb(h, b) and
CB∗(h,νb)≤CB,Sb(h,νb), respectively. To establish the first equality, note that
CB∗(h,νb)CB,Sνb(h, b)
=νbE [D−Sνb]+ +hE [Sνb−D]+
bE [D−Sνb]+ +hE [Sνb−D]+=
1 +ψ (Sνb;h,νb)1 + (1/ν)ψ (Sνb;h,νb)
,
where the last equality follows from dividing the numerator and denominator by hE [Sνb−D]+.
The proof of the second equality of the lemma is similar.
The bounds in Lemma 1 lead directly to the main asymptotic result of this section, which is
stated in the following theorem.
Theorem 2. Under Assumption 1, the following results hold for any h≥ 0.
(a) The ratio between the expected backorder cost per period to the expected holding cost per
period under the optimal policy converges to zero as the backorder cost b increases, that is,
limb→∞
ψ(SB∗(h, b);h, b
)= 0.
(b) For large values of b, the optimal cost and the optimal policy are robust against changes in
the backorder cost; that is, for any ν > 0,
limb→∞
CB,Sνb(h, b)CB∗(h, b)
= limb→∞
CB∗(h,νb)CB∗(h, b)
= limb→∞
CB,Sb(h,νb)CB∗(h,νb)
= 1,
where Sb = SB∗(h, b) and Sνb = SB∗(h,νb).
Proof: To establish the result in part (a), note that the optimal order-up-to level SB∗(h, b) is given
by the newsvendor formula:
SB∗(h, b) = inf{y :P {D≤ y} ≥ b
b+h
},
which implies that P {D≤ SB∗(h, b)} ≥ b/(b+ h) and P {D>SB∗(h, b)} ≤ h/(b+ h) by the right
continuity of the distribution function. Therefore,
ψ(SB∗(h, b);h, b
)=b · P {D>SB∗(h, b)}E
[D−SB∗(h, b)
∣∣ D>SB∗(h, b)]
h · P {D≤ SB∗(h, b)}E[SB∗(h, b)−D
∣∣ D≤ SB∗(h, b)]
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Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 19Proof: Please see Appendix EC.3 in the electronic companion.
Since the long run average cost is the quantity of interest to us in this paper, Lemma 3 implies
that we can limit our analysis to any specific starting state. In the next lemma, we show that for
a specific starting state we choose, the stationary distribution of the on hand inventory, {XL,St },
exists.
Lemma 4. Assume the starting state (in period 1) is such that there are S/(τ +1) units on hand
and S/(τ + 1) units due to be delivered in each of the periods 2, . . . , τ . Then, the sequence of the
distributions of the random variables {XL,St } converges.
Proof: Please see Appendix EC.4 in the electronic companion.
We will use XL,S∞ to denote a random variable whose distribution is the limiting distribution
from Lemma 4. We can now define CL,S(h, b) mathematically as follows:
CL,S(h, b) = hE[(XL,S∞ −D
)+]+ bE
[(D−XL,S∞
)+],
where the random variable D denotes the demand in a single period.
Corollary 1. The random variable XB,S∞ is stochastically smaller than the random vari-
able XL,S∞ and the random variable LOSTL,S∞ is stochastically smaller than the random variable
BACKB,S∞ , that is, for any z ≥ 0,
P{XB,S∞ > z
}≤P
{XL,S∞ > z
}and P
{LOSTL,S∞ > z
}≤P
{BACKB,S∞ > z
}.
Proof: Notice that we assumed that XL,S∞ represents the limiting distribution of XL,St when the
starting state vector has S/(τ + 1) units in each component. So, this starting state satisfies the
assumption of Lemma 2. The result follows directly from this lemma.
The next result establishes upper and lower bounds on the cost of any order-up-to policy in the
lost sales system in terms of the costs in the backorder system.
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Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 25Our new heuristic order-up-to level is S̃. The motivation for this formula is the following. For
large values of b/h, we know from our theoretical results that the order-up-to SB∗(h, b, τ) policy
is close to optimal (this is true because SB∗(h, b, τ) and SB∗(h, b+ τh, τ) are close to each other
for large b/h values). On the other hand, when b/h is very small, the optimal policy will be such
that the probability of stock-out is close to 1. When stock-outs are guaranteed to occur in every
period, the inventory problem is similar to that of the newsvendor problem for which SB∗(h, b,0)
is the optimal order-up-to level. Our suggested order-up-to level, S̃, is simply a weighted average
of these two order-up-to levels where the weights are determined by the newsvendor ratio. It can
be shown, using arguments similar to those used in Section 6, that the order-up-to S̃ policy is
also asymptotically optimal under Assumption 1 with the added assumption that the demand
distribution has no mass at the upper end of the support, that is, P (D =M) = 0. (See Appendix
EC.7 in the electronic companion for a proof). It should be noted that the order-up-to S̃ policy is
not optimal when demand is deterministic - despite this drawback, our computational investigation
of this heuristic (Tables 2 through 5) reveals that the cost increase from the use of this heuristic
relative to the best order-up-to policy is only 2.52%, on an average. Furthermore, the order-up-to
S̃ policy performs well even with moderate b/h ratios; in our computation, its costs are 4.77%,
3.02%, and 2.80% higher than the best base stock policies when the b/h ratios are 1, 4, and 9,
respectively.
7.2. Representative Problems
In this section, we report the computational results for representative problems considered in
Zipkin (2006b), enabling us to compare the cost of our base stock policies with other replenishment
heuristics. We consider Poisson and Geometric demand distributions, both with mean 5. The lead
time ranges from 1 to 4 periods. Assuming a holding cost of $1, we consider the lost sales penalty
ranging from $1 to $199. We compare the cost of the optimal policy, the best base stock policy, the
order-up-to-SB∗(h, b+ τh) policy, and the order-up-to S̃ policy. Tables 2 and 3 show the costs of
these polices for Poisson and geometric distributions, respectively. (We remark that Zipkin (2006b)
reports the cases where the lost sales penalty is $4, $9 or $19.)
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Huh et al.: Asymptotic Optimality of Order-up-to PoliciesArticle submitted to Management Science; manuscript no. MS-01100-2006.R3 29Karush, W. 1957. A queuing model for an inventory problem. Operations Research 5(5) 693–703.
Levi, R., G. Janakiraman, M. Nagarajan. 2008. A 2-approximation algorithm for stochastic inventory control
models with lost-sales. Mathematics of Operations Research 33(2) 351–374.
Levi, R., Martin Pal, R. O. Roundy, D. B. Shmoys. 2007. Approximation algorithms for stochastic inventory
control models. Mathematics of Operations Research 32(2) 284–302.
Morton, T. E. 1969. Bounds on the solution of the lagged optimal inventory equation with no demand
backlogging and proportional costs. SIAM Review 11(4) 572–596.
Morton, T. E. 1971. The near-myopic nature of the lagged-proportional-cost inventory problem with lost
sales. Operations Research 19(7) 1708–1716.
Nahmias, S. 1979. Simple approximations for a variety of dynamic leadtime lost-sales inventory models.
Operations Research 27(5) 904–924.
Reiman, M. I. 2004. A new and simple policy for the continuous review lost sales inventory model. Working
Paper, Bell Laboratories.
Ross, S. M., J. G. Shanthikumar, Z. Zhu. 2005. On increasing-failure-rate random variables. Journal of
Applied Probability 42(3) 797–809.
Shaked, M., J.G. Shanthikumar. 1994. Stochastic Orders and Their Applications. Academic Press.
Zipkin, P. 2006a. Old and new methods for lost-sales inventory systems. Forthcoming in Operations Research
.
Zipkin, P. 2006b. On the structure of lost-sales inventory models. Forthcoming in Operations Research .
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e-companion to Huh et al.: Asymptotic Optimality of Order-up-to Policies ec1
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e-companion to Huh et al.: Asymptotic Optimality of Order-up-to Policies ec3To prove part (c), we can assume without loss of generality that F (x)< 1 for all x≥ 0. For any
x, let F̄ (x) = 1−F (x). It then follows from the definition of mD(t) that
mD(t)t
=
∫∞tF̄ (u)dutF̄ (t)
.
Since E [D2]<∞, it follows that E [D] =∫∞0F̄ (u)du<∞. Therefore,
limt→∞
∫ ∞t
F̄ (u)du= 0.
Moreover, we have that E [D2] =∫∞0
2uF̄ (u)du. Since D has a finite second moment, it follows that
limt→∞
tF̄ (t) = 0,
implying that both the numerator and the denominator in the expression for mD(t)/t converge
to zero at t increases to infinity. Since D is assumed to be a continuous random variable, we can
apply L’Hospital’s Rule to conclude that
limt→∞
mD(t)t
= limt→∞
−F̄ (t)F̄ (t)− tf(t)
= limt→∞
1t · r(t)− 1
= 0,
which is the desired result.
EC.2. Proof of Proposition 1
Proposition 1. For any θ > 1, b ≥ 0, h ≥ 0, and ν > 0, if D has a distribution function Fθ,
then
limt→∞
mD(t)t
=1
θ− 1and lim
b→∞
CB∗(h,νb)CB∗(h, b)
= ν1/θ .
Proof: It is easy to verify that 1− Fθ(x) = 1/(1 + x)θ. It follows that E [D] =∫∞0
1− Fθ(x)dx =
1/(θ− 1). Then, using the fact that mD(t) =∫∞tP {D> z}dz/P {D> t}, we can also show that
mD(t) = (1 + t)/(θ− 1), which proves the first part of Proposition 1.
To establish the second part, note that by definition SB∗(h, b) = F−1θ (b/(b+ h)), which implies
that SB∗(h, b) =(b+hh
)1/θ− 1. Then, we have that
E[(
D−SB∗(h, b))+]
= P{D>SB∗(h, b)
}·E[D−SB∗(h, b)
∣∣∣ D>SB∗(h, b)]
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e-companion to Huh et al.: Asymptotic Optimality of Order-up-to Policies ec5exists and equals b · (µ−S/(τ + 1)) for all S ≤M · (τ + 1), where µ=E [D]. We will first show that
the limsup of this sequence is bounded above by this quantity, and then the lim inf is bounded
below by this quantity.
Upper Bound. Let IP1 denote the inventory position at the beginning of period 1. Consider
the following policy π whose on-hand inventory process and lost sales process will be denoted
by {XL,πt } and {LOSTL,πt }, respectively. If IP1 > S, π mimics the order-up-to S policy until the
first period in which the inventory position falls below S before the ordering opportunity (and
therefore, reaches S after ordering). Let us call this period as T . It is easy to verify that E[T ]<∞
if E[Dt]> 0. For all t∈ {1, . . . , T}, XL,πt = XL,St . For all t∈ {1, . . . , T −1}, LOSTL,πt = LOSTL,St .
The policy π deviates from the order-up-to S policy in the following sense from period T onwards.
We introduce a standard modification of introducing the sales decision to the inventory system;
in addition to the inventory replenishment decision, the manager can determine the sales quantity,
which is bounded above by, but can be strictly less than, the minimum of the on-hand inventory
level and realized demand. If the manager does not satisfy demand to the maximum extent possible
in a period, then both a lost sales penalty and a holding cost are incurred. Since cost parameters
are stationary over time, it is easy to show that under the order-up-to-S policy, the introduction of
this sales lever does not decrease the T -horizon cost, for any T ≥ 1. Let the policy π order-up-to S
each period, as usual, but we define the sales decision of this policy from period period T onwards
as follows: do not sell any unit in the interval [T ,T + τ −1], and sell exactly S/(τ +1) units in each
period of the interval [T + τ,∞).
(We claim that the above policy π is well-defined. We need to demonstrate that it is possible
to sell exactly S/(τ + 1) units in period T + τ onwards. First, the demand in each period exceeds
M , which, by assumption, exceeds S/(τ + 1). Second, observe that the inventory position at the
beginning of period T is S by definition. By construction, π does not sell any units in the interval
[T ,T + τ − 1]. This implies that XL,πT+τ
= S. Since the demand in every period exceeds S/(τ + 1)
and we have S units on hand at the beginning of period T +τ , it is possible to sell exactly S/(τ+1)
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e-companion to Huh et al.: Asymptotic Optimality of Order-up-to Policies ec7policy, we know that for all t≥ T , the inventory position at the beginning of a period is exactly
S. This means that the maximum number of units that can be sold in the interval [t, t+ τ ] is S.
The expected demand in this interval is µ · (τ + 1). So, b · (µ · (τ + 1)−S) is a lower bound on the
expected lost sales penalty costs incurred in the interval [t, t+ τ ] for any t≥ T . Recall that T has
a finite expectation. Therefore,
limT→∞
inf∑T
t=1E[h · (XL,St −Dt)+ + b · (Dt−XL,St )+]T
≥ b · (µ−S/(τ + 1)) .
Thus, combining the upper and lower bounds, we have shown that
limT→∞
∑T
t=1E[h · (XL,St −Dt)+ + b · (Dt−XL,St )+]T
exists and equals b · (µ−S/(τ + 1)) when S ≤M · (τ + 1).
EC.4. Proof of Lemma 4
Lemma 4. Assume the starting state (in period 1) is such that there are S/(τ +1) units on hand
and S/(τ + 1) units due to be delivered in each of the periods 2, . . . , τ . Then, the sequence of the
distributions of the random variables {XL,St } converges.
Proof: As mentioned earlier, Huh et al. (2006) show the convergence of the stochastic process
{XL,St } for all S >M · (τ + 1) independent of the starting state vector. For any S ≤M · (τ + 1), it
is easy to verify that
XL,St = S/(τ + 1) ∀ t
because Dt ≥M ∀ t. This implies the result with XL,S∞ being the deterministic quantity S/(τ + 1)
for all S ≤M · (τ + 1).
EC.5. Proof of Theorem 4(a)
Theorem 4 (a). For any h ≥ 0 and b ≥ 0, the best order-up-to level in the lost sales system
L(h, b) is bounded above by the best order-up-to level in the backorder system B(h, b+ τh) with a
backorder penalty cost parameter of b+ τh, that is, SL∗(h, b)≤ SB∗(h, b+ τh).
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Asymptotic Optimality ofOrder-up-to Policies in Lost Sales Inventory
Systems
Managerial Relevance
Inventory managers make product stocking decisions by balancing the expected overage and underage costs under demand uncertainty. These decisions are complicated by the fact that there is a lead time between placing a replenishment order and delivery. The stocking decisions are impacted by the consequences of stocking out. While customers sometimes accept reasonable delays in deliveries (leading to back-order models in the inventory literature), in some businesses, unmet demand cannot be postponed. For example, in most retail settings, the inability to meet a customer’s demand results in the loss of that sale to a competitor, and in high-tech service parts industries, the unavailability of a needed part triggers expensive emergency actions (e.g., air-shipment). Both these examples belong to the class of lost sales inventory models. Developing good algorithms to compute stocking levels for the lost sales models has remained a challenging problem since the 1950’s. In this paper, we first argue that, in many environments with lost sales, the cost of a lost sale is significant larger than the holding cost. Then, we show that the class of threshold-based order-up-to policies performs well, enabling the manager to search within this class of simple policies without significant loss of performance. Numerical experiments show that the average cost of the best order-up-to policy is within 1.5% of the optimal cost when the ratio between the lost sales penalty cost and the holding cost is 100. In addition, we propose an order-up-to level that can be easily computed on a spreadsheet. This level corresponds to a newsvendor quantity of a backorder system with a slightly perturbed parameter, and we establish the validity of this order-up-to level both theoretically and numerically.
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