End-of-period vs. continuous accounting of inventory related costs
Nils Rudi • Harry Groenevelt • Taylor R. Randall
INSEAD, Boulevard de Constance, 77305 Fontainebleau, France
Simon School of Business, University of Rochester, Rochester, NY 14627
David Eccles School of Business, University of Utah, Salt Lake City, UT 84112
[email protected] • [email protected] • [email protected]
March 2004, revised January 2005, January 2006, September 2006
Abstract
This paper investigates the effect of using an end-of-period accounting scheme for inventory related costs
when costs actually accrue in continuous time. Using a simple model, we show that (i) the end-of-period
scheme results in higher than optimal order-up-to levels and inventory cost if the cost and demand parameters
are unchanged, and (ii) it is possible to replicate both the optimal base stock level and its cost by selecting
the values of the cost or demand parameters judiciously. The cost adjustments often require extreme values
and no systematic cost parameter adjustment scheme is robust. However, we find a systematic adjustment to
the demand parameters which serves as a good approximation and is robust. We therefore conclude that endof-period cost accounting without parameter adjustments is in general inappropriate when costs are incurred
continuously, but there are adjustments that can make it work well.
1
Introduction
Stochastic models used to manage inventory are often classified as continuous or periodic review (Nahmias 1997,
p. 267). Continuous review models naturally lend themselves to account for inventory-related costs in continuous
time. The continuous accounting scheme reflects the empirical reality that most inventory-related costs accrue
continuously. However, for reasons of mathematical tractability and historic convention, the majority of periodic
review models account for inventory-related costs at the end of each review period. The convention of accounting
for costs at the end of a review period began with the seminal papers of Arrow et al. (1951) and Bellman et al.
(1955). With few exceptions,1 this convention has become standard practice in the analysis of periodic review
inventory models. The end-of-period accounting scheme may inaccurately reflect costs and inventory levels,
particularly when stock-outs result in backorders. For example, while the cost consequences of stock-outs under
backlogging realistically depend on the duration and quantity of the stock-out, the end-of-period scheme only
considers the quantity. Taking snapshots of inventory at the end of a review period tends to reflect a condition
in which the company is least exposed to inventory holding costs and most exposed to stock-outs. The purpose
of this paper is to examine the implications of using an end-of-period accounting scheme with periodic review
models, and to investigate whether one can adjust cost or demand parameters such that the end-of-period cost
accounting method can be used.
To facilitate comparison in this investigation, we formulate a continuous accounting scheme for periodic review
models that serves as a benchmark. We here seek to strip the model to its bare minimum while still reflecting
1 Hadley
and Whitin (1963) formulate a periodic review model with a continuous accounting scheme, but to our knowledge, the
first analytical characterizations of it appear, independently of each other, in two recent papers: in the context of setting length of
review period (Rao 2003) and in studying the combination of a slow and a more expensive but fast freight mode (Groenevelt and
Rudi 2002), where Groenevelt and Rudi’s solution makes the link to the fill rate service measure.
1
the issue of focus. Note here that only the accounting schemes are different – all models in this paper employ the
same periodic review mechanism. An important contrast between these two schemes is seen in the optimality
conditions. While the optimality condition of the periodic accounting scheme prescribes a service level in terms
of in-stock probability (also often referred to as the probability of not stocking out in a review cycle), the
optimality condition of the continuous accounting scheme prescribes a service level in terms of the fill rate. This
distinction is important, as Nahmias (1997, p. 289) notes: “...[the in-stock probability] service is not how service
is interpreted in most applications... fill rate is generally what managers mean by service.” So not only is the
continuous accounting scheme more appropriate, it also prescribes a service measure which corresponds to the
managerial interpretation of service. To this end, the paper has four key objectives:
1. Identify how the end-of-period accounting scheme affects inventory levels and resulting costs.
2. Identify how the end-of-period accounting scheme performs in predicting inventory-related costs.
3. Demonstrate how discrepancies caused by the end-of-period accounting scheme may be alleviated by judiciously selecting values for the cost or demand distribution parameters to replicate the inventory policy
and cost prediction of the continuous accounting scheme.
4. Examine the robustness of using such adjustments in setting inventory policies. Specifically, firms often
keep a very large number of stock keeping units (SKUs). When practitioners estimate model parameters,
they need an approach which is (I) simple to apply to large numbers of SKUs, (II) easy to interpret and
accept, and (III) accurate for SKUs with different characteristics. We consider an estimation scheme which
satisfies (I)-(III) as robust for our purposes.
We find that when compared to the continuous accounting scheme, the end-of-period accounting scheme using
continuous time cost parameters will always result in higher than optimal inventory levels. Also, the end-of-period
scheme then tends to under-predict costs when compared to the continuous accounting scheme. The magnitude
of discrepancies between the two accounting schemes can be surprisingly large. One can select values for model
parameters to replicate the levels of inventory and the prediction of inventory-related costs that result with the
continuous accounting scheme. While we are not able to identify a robust adjustment of the cost parameters,
for the adjustment of demand parameters we are able to derive a systematic approximation that is quite robust.
2
Models
We consider a simple setting to put the issue of end-of-period vs. continuous accounting of inventory-related
costs in focus. Specifically, we normalize the review period to be equal to one time unit, assume zero lead time,
proportional procurement costs and no discounting, and restrict our attention to models with infinite horizon
and where all demands not met from on-hand inventory are backlogged. Let h denote the per-unit holding cost
per time unit and p denote the per-unit backlog penalty cost per time unit. A Brownian motion with drift µ
and infinitesimal variance σ 2 is used to approximate the demand process, since this results in smooth functions
and also results in the demand in a time interval following a Normal distribution, which appears to be the most
2
commonly applied distribution in inventory theory. We also define
+
+
G (y, d) = h (y − d) + p (d − y) ,
where the operator superscript + is defined as x+ = max (0, x). In the expression for G, one can think of y
as the supply (at the beginning of the review period) and d as the cumulative demand (so far in the review
period). Then G is the cost rate, consisting of h times the on-hand inventory and p times the backlog. Finally,
we define S as the order-up-to level, D(t1 ,t2 ] as the demand in time interval (t1 , t2 ], t1 < t2 , and Ft (·) as the
cumulative distribution of demand during a time interval of length t, which is Normal with mean tµ and variance
tσ 2 . (It can easily be shown that the cost functions for each of the accounting schemes are convex, and, hence,
an order-up-to policy is optimal.)
End-of-period cost accounting
We use subscript e (mnemonic for end-of-period cost accounting) to indicate
when a notation is for the end-of-period cost accounting scheme. The per-period expected cost under the endof-period accounting scheme is given by
+
+
Ce(he ,pe ,µe ,σe ) (S) = he S − D(µe ,σe )
+ pe D(µe ,σe ) − S .
(h,p,µ,σ)
Note that Ce (S) = Ce
(S) = EG S, D(0,1] . The corresponding optimal order-up-to level Se satisfies
(µe ,σe )
F1
(Se ) =
pe
.
pe + he
(1)
This corresponds to the well-known newsvendor solution, whose left-hand side is the service measure of in-stock
probability, where the optimal order-up-to level is such that this service measure equals the right-hand side
(he ,pe ,µe ,σe )
fractile. Let the solution to (1) for arbitrary (he , pe , µe , σe ) be denoted by Se
, where we will omit the
superscripts (he , pe ) and/or (µe , σe ) whenever (he , pe ) = (h, p) and/or (µe , σe ) = (µ, σ).
Continuous cost accounting The per-period expected cost under the continuous accounting scheme is given
by
Z
C (S) = E
1
G S, D(0,t] dt.
0
Taking the derivative gives
Z 1
Z 1
Z 1
d
d
C 0 (S) =
E
G S, D(0,t] dt = E
G S, D(0,t] dt =
[hFt (S) − p (1 − Ft (S))] dt,
dS
0
0 dS
0
where the interchange of the derivative and expectation operators is justified by the dominated convergence
theorem. The expected cost function can easily be established to be convex in S. Equating the derivative to
zero and rearranging, we characterize the optimal solution
Z 1
Ft (S) dt =
0
p
,
p+h
(2)
which has a close resemblance to the solution when employing end-of-period cost accounting: the right-hand side
is the same, while the left-hand side is the expected fraction of time with positive inventory. This service measure
(i.e., the left-hand side) is equivalent to the expected fraction of demand satisfied from on-hand inventory and
is typically referred to as the fill rate, and the optimal order-up-to level S ∗ is such that the fill rate equals the
right-hand side of (2).
3
3
Investigation
We investigate the relationship between the two cost accounting schemes in correspondence to the four objectives
laid out in the introduction. We begin with considering the case when the continuous accounting scheme param(h,p,µ,σ)
eters are naı̈vely used in the end-of-period accounting scheme. Then the cost function is Ce (S) = Ce
and the order-up-to level as prescribed by (1) is Se =
(h,p,µ,σ)
Se
,
(S)
when using the continuous accounting scheme
parameters.
Objective 1
The following lemma gives some properties of the relationship between the two models, addressing
Objective 1:
LEMMA 1 Se > S ∗ and C (Se ) > C (S ∗ ).
Proof: It is sufficient to note that, for an arbitrary S, F1 (S) <
R1
0
Ft (S) dt.
Lemma 1 shows that the naı̈ve use of end-of-period cost accounting as a basis for making ordering decisions
always leads to overstocking of inventory and results in higher than optimal cost. The intuition behind this
is that the end-of-period cost accounting scheme considers the point in time when one is the least exposed to
positive inventory and the most exposed to stock-outs. This effect will be larger when the review period is longer.
3.1
Adjusting the cost parameters of the end-of-period accounting scheme
We here consider adjustments in the cost parameters (he , pe ) of the periodic accounting scheme.
o
n
(h ,p )
S
DEFINITION 1 (a) (he , pe ) = (he , pe ) : Se e e = S ∗ ,
o
n
(h ,p )
(h ,p )
C
(b) (he , pe ) = (he , pe ) : Ce e e Se e e = C (S ∗ ) , and
∗
S
C
(c) (he , pe ) = (he , pe ) ∩ (he , pe ) .
These sets of cost parameters make the end-of-period accounting scheme replicate (a) the order-up-to level S ∗ ,
(b) the cost prediction C (S ∗ ), (c) S ∗ and C (S ∗ ).
Objectives 1, 2 and 3 The following lemma gives some properties of the relationships between the two
models, addressing Objective 3:
S
C
LEMMA 2 The sets (he , pe ) and (he , pe )
∗
contain continua of elements, while their intersection (he , pe ) con-
tains exactly one element .
C
Proof: Using standard arguments it is not hard to show that a plot of the set (he , pe )
has a continuous,
C
convex, and strictly decreasing shape with the axes as asymptotes. Similarly, a plot of the set (he , pe )
is an
upward sloping half-line starting in the origin. It follows that the intersection consists of a single point (see
Figure 1b).
Lemma 2 guarantees that there exist unique values of he and pe that make the model with the end-of-period
accounting scheme replicate both the optimal order-up-to level and the corresponding cost prediction of the
4
S
C
Figure 1: (a) C(S), Ce (S), and CeC∗ (S), (b) (he , pe ) , (he , pe )
∗
and (he , pe ) .
∗
continuous accounting scheme model. We denote the single element of (he , pe ) by (h∗e , p∗e ), and write CeC∗ (S) =
∗
(h∗
e ,pe )
Ce
(S).
For our numerical investigations, we consider the following base example: µ = 100, σ = 20, h = 1 and p = 9,
and where appropriate, we fix h + p = 10.2 Figure 1(a) illustrates the cost curves C(S), Ce (S) and CeC∗ (S).
The prescribed order-up-to levels are S ∗ = 94.3 and Se = 125.6, and hence the end-of-period cost accounting
model overestimates the optimal order-up-to level by 33%. Using Se in the continuous accounting cost function
yields C (Se ) = 76.3, while the optimal cost is C (S ∗ ) = 57.7, resulting in cost being 32% above optimal. We
then look at the effect of using the end-of-period cost function Ce to predict the actual cost C. If we employ
the end-of-period optimal cost function, we get Ce (Se ) = 35.1, which underestimates the actual cost when using
the suboptimal Se by 54%. (Alternatively, if one could somehow estimate the true optimal order-up-to quantity
S ∗ but use this in the inaccurate cost function Ce , this would give Ce (S ∗ ) = 105.8, or an overestimation of
cost by 83%.) For the base case example, (h∗e , p∗e ) = (4.6, 2.9), i.e., relative to the continuous accounting scheme
parameters, he is adjusted up by 360% and pe is adjusted down by 68%. This adjustment suggests that the
end-of-period accounting scheme puts too little weight on on-hand inventory and too much weight on stock-outs.
S
The resulting cost function CeC∗ (S) touches C (S) at its minimum. Figure 1(b) illustrates the sets (he , pe ) ,
C
(he , pe )
and (h∗e , p∗e ).
Objective 4
In Table 1, we vary (h, p) and σ in search of a robust method to find good (defined as close to
satisfying Definition 1(c) – or at least (a)) values for (he , pe ) to employ in the end-of-period accounting scheme.
Table 1 provides evidence that: (i) Se deviates more from S ∗ as σ increases and when h is large relative to p, with
median overestimate of 33.42% and average overestimate of 265.67%, (ii) except for h being large relative to p,
the resulting cost increase by using Se instead of S ∗ tends to increase in σ, with median cost increase of 29.47%
and average cost increase of 135.98%, (iii) h∗e is highly variable for different parameters (the median adjustments
∗
from h is 807.94% up) , and p∗e less so (the median adjustment from p is 63.79% down), and (iv) (he , pe ) depends
on the parameters in a highly non-linear way. Finally (not presented in Table 1), Ce (Se ) underestimates the true
cost C (Se ) of using Se , but less so as σ increases (with average under-prediction of about 65.56%). We conclude
that we are not able to come up with a robust way to set the cost parameters (he , pe ) as laid out by conditions
2A
web page for interactive numerical experiments is available at www.nilsrudi.com under RESEARCH.
5
(I)-(III) of Objective 4. In fact, the order of magnitude of the necessary adjustments can be so extreme that it
completely defies interpretation.
(h, p)
(0.1,9.9)
(0.5,9.5)
(1,9)
(5,5)
(9,1)
σ
5
10
20
40
5
10
20
40
5
10
20
40
5
10
20
40
5
10
20
40
(h∗e , p∗e )
(0.98,2.09)
(0.45,1.83)
(0.25,1.77)
(0.16,1.87)
(15.70,3.39
(4.16,2.84)
(1.72,2.56)
(0.96,2.55)
(172.66,3.86)
(16.00,3.31)
(4.61,2.92)
(2.19,2.84)
(4.19 · 1023 ,2.48)
(1.10 · 107 ,2.44)
(492.53,2.32)
(24.72,2.22)
(8.05 · 1071 ,0.50)
(6.95 · 1018 ,0.50)
(2.33 · 105 ,0.49)
(69.07,0.50)
S∗
102.34
108.54
123.27
156.06
95.38
97.63
105.00
124.07
89.92
90.51
94.31
106.57
49.88
49.50
48.04
44.48
9.88
9.50
8.00
2.21
Se
S∗
· 100%
109.08
113.56
118.87
123.70
113.46
119.28
126.57
133.63
118.34
124.65
133.21
141.93
200.50
202.02
208.16
224.81
947.77
917.73
929.61
2206.41
Se
111.63
123.26
146.53
193.05
108.22
116.45
132.90
165.79
106.41
112.82
125.63
151.26
100.00
100.00
100.00
100.00
93.59
87.18
74.37
48.74
C (S ∗ )
5.50
6.32
8.17
12.15
24.83
27.16
33.09
46.83
46.12
49.09
57.72
79.12
125.63
127.51
135.17
164.10
45.13
45.51
47.20
55.92
C (Se )
6.17
7.34
9.68
14.41
29.13
33.29
41.69
58.70
56.45
62.99
76.26
103.36
250.60
252.31
258.59
279.94
395.50
347.03
266.42
159.84
C(Se )
C(S ∗ )
· 100%
112.10
116.05
118.58
118.57
117.30
122.58
125.99
125.35
122.41
128.31
132.12
130.64
199.48
197.87
191.30
170.60
876.44
762.50
564.45
285.83
Table 1: The effects of different combinations of (h, p) and σ.
3.2
Adjusting the demand distribution of the end-of-period accounting scheme
An alternative to using an adjusted set of cost parameters to replicate the optimal base stock level and/or its
cost, as laid out by Definition 1, is to make adjustments to the demand distribution parameters while keeping
the cost parameters unchanged. The following definitions formalize this:
n
o
(µ ,σ )
S
DEFINITION 2 (a) (µe , σe ) = (µe , σe ) : Se e e = S ∗ ,
n
o
(µ ,σ )
(µ ,σ )
C
(b) (µe , σe ) = (µe , σe ) : Ce e e Se e e = C (S ∗ ) ,
∗
S
C
(c) (µe , σe ) = (µe , σe ) ∩ (µe , σe ) .
Objectives 1, 2 and 3 The following lemma gives some properties of the relationships between the two
models, addressing Objective 3:
S
C
p
LEMMA 3 (µe , σe ) is characterized by the linear relationship µe = S ∗ − σe Φ−1 p+h
; σe ≥ 0, and (he , pe )
.h
i
p
is characterized by the relationship σe = C (S ∗ ) (p + h) φ Φ−1 p+h
and µe arbitrary; both sets contain
∗
a continuum of elements. The intersection (µe , σe ) contains exactly one element .
(µe ,σe )
S
Proof: The characterization of (µe , σe ) is found by rewriting equation (1) as Se
= µe + σe Φ−1
p
p+h
,
(µ ,σ )
Se e e
substituting S ∗ for
and solving for µe . The cost at optimality in the end-of-period scheme can be rewritten
(µe ,σe )
(µe ,σe )
p
as Ce
Se
= σe (p + h) φ Φ−1 p+h
. Equating this expression to C (S ∗ ) gives the stated result.
This confirms that it is also possible to replicate the order-up-to level and/or the cost prediction of the continuous
∗
scheme by making adjustments to the demand parameters. We denote the single element of (µe , σe ) by (µ∗e , σe∗ ),
∗
(µ∗
e ,σe )
and write CeD∗ (S) = Ce
(S). A natural follow-up question to this is: can we find a demand distribution
6
which, when used instead of the Normal distribution in the end-of-period scheme, actually replicates the true
cost function C for every value of S? The answer is yes, as shown in the following lemma. Let DF denote a
random variable with arbitrary CDF F.
LEMMA 4 Define F̃ (x) =
µ̃ = E [DF̃ ] =
1
2µ
R1
0
Ft (x) dt. Then F̃ is the unique CDF such that EG (S, DF̃ ) = C (S) for all S, and
and σ̃ = V ar (DF̃ ) = 12 σ 2 +
2
Proof: We have
Z ∞
Z
G (S, x) dF̃ (x) =
−∞
∞
Z
1 2
12 µ .
1
Z
−∞
Z 1
0
Z
∞
Z
1
Z
Z
∞
G (S, x) dFt (x) dt
G (S, x) ft (x) dxdt =
0
EG S, D(0,t] dt = E
=
1
G (S, x) ft (x) dtdx =
−∞
0
−∞
1
G S, D(0,t] dt = C (S) .
0
0
As Rao (2003) points out, F̃ is the CDF of the random variable D̃ = D(0,U ] , where U is a random variable with
a Uniform distribution on [0, 1], independent of the demand process D. The mean and variance follow readily:
h i
h h ii Z 1
1
µ̃ = E D̃ = EU E D̃ U =
µu du = µ,
2
0
Z
1
h h
ii 1
1
1
1
σ̃ 2 = V ar D̃ = E D̃2 − E 2 D̃ = EU E D̃2 U − µ2 =
σ 2 u + µ2 u2 du − µ2 = σ 2 + µ2 .
4
4
2
12
0
To show that F̃ is unique, assume that F̄ satisfies CeF̄ (S) = C (S) for every S. Taking the derivative with
R1
respect to S on both sides and simplifying gives F̄ (S) = 0 Ft (S) dt = F̃ (S) for all S.
S
C
Figure 2: (a) C (S), Ce (S), CeD∗ (S), and C̃e (S), (b) (µe , σe ) , (µe , σe )
∗
and (µe , σe ) .
Lemma 4 gives rise to the following simple approximation: use a Normal distribution with the first two moments
(µ̃,σ̃)
of F̃ in the end-of-period scheme. Let S̃e = Se
(µ̃,σ̃)
and C̃e (S) = Ce
(S) denote the solution to (1) and the
end-of-period cost function when using this approximation, respectively.
Similarly to Figure 1, Figure 2 illustrates Lemmas 3 and 4. Note here that the cost function CeD∗ (S) when
adjusting the demand parameters is much closer to C (S) over all values of S than the cost function CeC∗ (S)
obtained by adjusting the cost parameters. In contrast to (µ∗e , σe∗ ), the approximation using (µ̃, σ̃) does not
depend on the cost parameters (h, p), and it is also very simple to use. Hence, it is more robust in terms of
criterion (I) of Objective 4. Figure 3 illustrates, for different σ’s, the probability density functions (PDFs) and
cumulative distribution functions (CDFs) of the distribution F̃ that makes the end-of-period scheme replicate
7
Figure 3: The CDFs and PDFs of N (µ∗e , σe∗ ), N (µ̃, σ̃), and F̃, for σ ∈ {5, 10, 20, 40}.
the continuous scheme (its PDF is denoted f˜), its Normal approximation N (µ̃, σ̃), and the Normal distribution
N (µ∗e , σe∗ ) which replicate the order-up-to levels and cost predictions of the continuous scheme. From these
graphs, we make the following observations: (i) for low σ, F̃ is close to Uniform (for the deterministic case, i.e.,
σ = 0, it actually is Uniform), (ii) for larger σ’s, the PDF of F̃ increases rapidly for very low values and then
decreases more gradually, (iii) for each σ, the CDF of N (µ̃, σ̃) tends to be close to the CDF of F̃, which indicates
that it is a good approximation (and particularly so for p > h, which one would expect to occur more often in
practice).
Figure 4: Varying p while keeping h + p fixed, (a) S ∗ , Se , and S̃e , (b) C (S ∗ ), C (Se ), Ce (Se ), C S̃e , and
C̃e S̃e .
Objective 4 The quality of the approximation is illustrated in Figures 4 (actual values) and 5 (deviations
from S ∗ and C (S ∗ ) for different values of σ). Its order-up-to level S̃e tends to be smaller than S ∗ for most
practical values of p (i.e., p > h) except for very large ones (Figures 4(a) and 5(a)). The cost prediction using
the approximation tends to be an underestimate of actual cost when p is not too different from h, and to be an
overestimate for very large values of p (except for very large σ). As also noted in Section 3.1, the naı̈ve use of
8
the end-of-period scheme performs poorly.
Figure 5: Varying p while keeping h + p fixed, and for σ ∈ {5, 10, 20, 40} (illustrated by arrows) (a) percentage
deviation of Se and S̃e , both from S ∗ (b) percentage deviation of C (Se ), Ce (Se ), C S̃e , and C̃e S̃e , all from
C (S ∗ ).
Since the error of the approximation is very small compared to other deviations depicted in Figure 5(b), we
graph its deviation from the cost at optimality separately in Figure 6. Considering the base case of σ = 20 in the
interval p ∈ [5, 9.9], which captures the more practical values of p, the average error is 0.22%, and the maximum
error is 0.52%, which is achieved at p = 8.8. While the errors for other values of σ tend to be higher, they are still
very small. We can conclude that the approximation results in costs very close to those of the optimal solutions
across a variety of parameters, and, hence, it satisfies robustness criterion (III) of Objective 4.
Figure 6: Varying p while keeping h + p fixed, and for σ ∈ {5, 10, 20, 40} (illustrated by arrow), percentage
deviation of C S̃e from C (S ∗ ).
In terms of criterion (II) of Objective 4, we concede that the adjustment to the demand parameters in the
approximation is not very intuitive. Hence, the approximation of using the demand distribution N (µ̃, σ̃) in the
end-of-period accounting scheme satisfies criteria (I) and (III) and is “quite” robust, as laid out by Objective 4.
Finally, we note that if costs accrue at multiple (say k) evenly spaced points in time during the review period,
1
Lemma 4 remains valid as long as we calculate E D̃ = 21 1 + k1 µ and V ar D̃ = 21 1 + k1 σ 2 + 12
1 − k1 µ2 .
9
Setting k = 1 corresponds to the standard end-of-period cost model, and letting k → ∞ gives continuous costing.
We conjecture that a 2-moment normal approximation would be very effective for intermediate values of k as
well.
4
Summary and conclusions
This paper demonstrates the effect of using an end-of-period accounting scheme when inventory-related costs
actually accrue continuously. Our approximation approach can readily be applied when lead time is positive
(which would simply change the integral in (2) to being from L to L + 1) and/or when demand is seasonal within
the period, as well as when costs are incurred at multiple discrete points in time during the review period. How
these approximations would perform in these cases is an open question. The results of this work invite further
investigation of the following four questions: How does the length of the review period affect the results of the
investigation? Are there simple schemes that in general perform better than the end-of-period scheme? What
are the effects of other assumptions that go hand in hand with end-of-period accounting, such as discounting cost
in discrete time and lead time being in multiples of the review period? How will the results change for the case
of lost sales, where the cost of stockouts predominantly depends on the quantity stocked out, not the duration,
while inventory holding costs still accrue continuously?
Acknowledgment
We thank the Associate Editor and three anonymous referees for suggestions which significantly improved this
paper. In particular, one of the referees and the Associate Editor provided feedback which helped facilitate
Section 3.2.
References
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