Decision Sciences
Volume 37 Number 2
May 2006
C 2006, The Author
C 2006, Decision Sciences Institute
Journal compilation TECHNICAL NOTE
Recursive Behavior of Safety Stock
Reduction: The Effect of Lead-Time
Uncertainty
Ping Wang
Fisher College of Business, The Ohio State University, Columbus, OH 43210,
e-mail: wang.607@osu.edu
James A. Hill†
Fisher College of Business, The Ohio State University, Columbus, OH 43210,
e-mail: hill.249@osu.edu
ABSTRACT
Motivated by a recent paper on the effect of lead-time variability reduction on safety
stocks, we provide evidence of the recursive nature of safety stock changes. When lead
times follow a gamma distribution we demonstrate that, for cycle service levels between
.60 and .70, the reduction of lead-time variability will first increase safety stock and then
either recursively decrease safety stock or make it remain constant. We also numerically
show the existence of the recursive effect. A two-by-two matrix is introduced to assist
managers in making decisions regarding safety stock policy.
Subject Areas: Inventory, Lead-Time Variability, Probability Models, and
Safety Stock.
INTRODUCTION
In a recent paper, Chopra, Reinhardt, and Dada (2004; hereafter referred to as
CRD) argued that many firms in practice operate at cycle service levels (CSLs) in
the 50–70% range. CRD proved that, for CSLs above 50% but below a threshold,
reducing the lead-time variability increases the reorder point and safety stock. They
concluded, “In this range of CSLs, a manager who wants to decrease inventories
should focus on decreasing lead times rather than lead-time variability.” This result contradicts the prescriptions drawn from using the normal approximation of
demand during lead time. However, their results did not articulate where the threshold point is in the 50–70% range. We demonstrate in this article, numerically, that
CRD’s results most likely hold at CSLs in the 50–60% range. CRD demonstrate
that at a CSL of 60%, reducing the standard deviation of lead time by 20% (from 5
to 4) increases the safety stock. What they did not show is what happens at different
levels of lead-time variability and, more important, at CSLs between 60% and 70%.
† Corresponding
author.
285
Recursive Behavior of Safety Stock Reduction
286
Table 1: Safety stocks for gamma lead time at different lead-time variability and
service levels (mean lead time = 10).
CSL
.95
.75
.70
.65
.60
.55
.51
ρ t = .1 ρ t = .2 ρ t = .3 ρ t = .4 ρ t = .5 ρ t = .6 ρ t = .7 ρ t = .8 ρ t = .9
98
45
36
29
22
15
10
119
50
40
31
22
14
8
148
57
44
33
22
12
4
181
65
49
35
22
9
0
218
72
52
35
20
5
–6
256
78
55
35
17
0
−12
296
83
56
33
13
–6
–20
337
86
56
30
7
–14
–29
379
89
55
26
1
–22
–39
After replicating their calculation of safety stocks at different lead-time variability
and CSLs (see Table 1), we find that at a CSL of 60%, any further reduction of the
standard deviation of lead time (from 4 to 3 and so on) does not increase safety
stock and that at a CSL of 65% there exists a recursive effect of the reduction of
lead-time variability on safety stock, where safety stock first increases and then
decreases recursively. We also find that this threshold value of CSL is dynamic in
that it changes with a reduction of lead-time variability.
We offer the following proposition when lead time follows a gamma
distribution:
For a fixed CSL above 60% but below a threshold CSL, the reduction of leadtime variability will first increase the safety stock, until the lead-time variability
goes below some threshold lead-time variability; any further reduction will
either have no effect on the reduction of safety stock, or recursively decreases
the safety stock.
In the remainder of this article we refer to this phenomenon as the recursive effect.
The proof of the existence of such an effect is given in the Appendix.
NUMERICAL ANALYSIS AND BOUNDARY CHARACTERISTICS
OF THE RECURSIVE ZONE
To further explore the recursive effect, we replicated CRD’s numerical experiment
(CRD, 2004, p. 8). Parameters we used for the replication are summarized in
Table 2.
Our computational results reveal that the recursive effect is more pronounced
within the range of CSL from .60 to .70, defined as the recursive zone. See Hill
and Wang (2005) for detailed computational results. From our results we observe
that CSL = .70 behaves as an upper bound of the recursive zone and CSL = .60
behaves as its lower bound. We call these bounds as CSL bounds of the recursive
zone. The lower bound of the recursive zone partitions the CSL range of 50–
70% into two subranges, where in the 50–60% range, hereafter referred to as the
counter zone, the effect observed by CRD occurs, and in the 60–70% range the
recursive effect occurs. At CSL above 70%, hereafter referred to as the conventional
zone, the effect of reductions of lead-time variability on safety stocks follows the
conventional prescription of normal approximation.
Wang and Hill
287
Table 2: Parameters.
Variables
Default Value
Viable Values
Description
μd
σd
μt
ρt
20
10
10
.1, .2, . . ..8, .9
10, 30
5, 15
5, 15
Mean demand
SD of demand
Mean lead time
Coefficient of variance
of lead time
Cycle service level
CSL
.95, .90, .85, .80,
.75, .70, .65, .60,
.55, .51
Table 3: Boundary behaviors of recursive zone as reducing the lead-time variability.
High Lead-time Variability
(ρt ≥ .60)
Low Lead-time Variability
( ρt < .60)
Upper bound
Has no significant impact on Decreases reorder point significantly
(CSL close to .70) reorder point (stable zone)
(conventional zone)
Lower bound
Increases reorder point
Has no significant impact on
(CSL close to .60) significantly (counter zone) reorder point (stable zone)
The boundary characteristics of the recursive zone are summarized in
Table 3, where the columns represent the lead-time variability. High lead-time
variability ranges from .60 to .90 (inclusive), while low lead-time variability represents the region below .60. The entire table constitutes the recursive zone.
To simplify, we refer to the two extreme quadrants, “Low ρt /upper bound”
and “High ρt /lower bound,” as conventional zone and counter zone, respectively.
The other two zones, that is, “High ρt /upper bound” and “Low ρt /lower bound,” are
referred to as stable zones. When operating in the conventional zone, the reorderpoint behavior follows the conventional effect. On the other hand, when operating
in the counter zone, the reorder point behaves as described by CRD. The stable
zones mean that in those areas the reduction of lead-time variability has either no
impact or only a slight impact on reducing the reorder point. In-depth observations
reveal that the recursive effect is more pronounced at CSL = .65 than at other
levels.
Figure 1 shows how CSL = .60 partitions the CSL range of 50–70% and how
the recursive zone functions as a transition area between the conventional zone and
the counter zone.
Finding the Recursive Zone
We now analytically show the existence of the recursive zone. Figure 2 is a
schematic illustration of three CDFs with lead-time variability at different levels:
Recursive Behavior of Safety Stock Reduction
288
Figure 1: Recursive effects when lead time is gamma.
Figure 2: Finding the recursive zone.
CSL
1
α
α2
γ
α1
β
cdf for y
cdf for y+2
cdf for y+1
ROP
Wang and Hill
289
(y + 2), (y + 1), and y. ᾱ2 is the crossover between cdf for (y + 2) and cdf for
(y + 1), while ᾱ1 is the crossover between cdf for (y + 1) and cdf for y. The change
from ᾱ2 to ᾱ1 shows the dynamic feature of the threshold CSL, which is neglected
in CRD. For a CSL at α, which is higher than both ᾱ2 and ᾱ1 , any reduction of
lead-time variability from (y + 2) to (y + 1 ) or from (y +1) to y will decrease
the reorder point, as indicated by the conventional effect. For a CSL at β, which
is lower than both ᾱ2 and ᾱ1 , any reduction of lead-time variability from (y + 2)
to (y + 1) or from (y + 1) to y will increase the reorder point, as indicated by the
counter effect. For a CSL at γ , which is in the range between ᾱ1 and ᾱ2 , a reduction
of lead-time variability from (y + 2) to (y + 1) will first increase the reorder point,
but a further reduction from (y + 1) to y will recursively decrease the reorder point,
as indicated by the recursive effect. Thus the CSL range between ᾱ1 and ᾱ2 is the
recursive zone.
CONCLUSION
We demonstrated in this article that the results of CRD did not fully explain the
behavior of the reduction of lead-time variability on the direction of reorder-point
changes. The reason for that is the neglect of the dynamic feature of the threshold
CSL. We prove theoretically and demonstrate numerically that a recursive zone
exists in the CSL range of .60–.70. Within the recursive zone a reduction of leadtime variability might first increase the reorder point and then recursively decrease
the reorder point or make it remain constant. A more robust model of lead-time
variability and demand variability remains an open and challenging problem. [Received: October 2005. Accepted: April 2006.]
REFERENCES
Chopra, S., Reinhardt, G., & Dada, M. (2004). The effect of lead-time uncertainty
on safety stocks. Decision Sciences, 35(1), 1–24.
Hill, J. A., & Wang, P. (2005). Demand uncertainty and lead-time uncertainty:
Implications on the boundary behavior of the recursive zone. Working Paper.
APPENDIX
We are going to prove the existence of the recursive effect by following the logic
of CRD’s proof of Theorem 3, Part 2.
Theorem 1. Given a CSL α, .5 < α < 1, there exists y ∗ > 0 such that (i)
R y ∗ (α) < R0 (α) and (ii) R y ∗ (α) < R y ∗ +1 (α), where y ∗ is a threshold lead-time
variability.
Proof. First, following CRD’s logic in their proof of Theorem 3, Part 2, we
can have R y (α) < R0 (α), for a small y > 0. However, we find a condition,
, which is weaker than k < (4γ +1 4γ 2 ) , the one identified by CRD.
k = 2cY 2 < (4γ4 ++ 3γ
4γ 2 )
To find the weaker condition, let the LHS of (A1) be B(γ , β):
Recursive Behavior of Safety Stock Reduction
290
4γβ + 2γ 2 β + 2β 2
B(γ , β) ≡ exp −k
1 − β2
−
2+γ +β
≥ 0.
(1 + β)3/2
2+γ −β
(1 − β)3/2
(A1)
Now, to make sure that B(γ , β) is initially a nonincreasing function at β = 0,
we take a partial derivative of B(γ , β) with respect to β and set it bigger than zero,
that is,
∂ B(γ , β) = −k(4γ + 4γ 2 ) + 4 + 3γ > 0.
∂β β=0
.
This will lead to k = 2cY 2 < (4γ4 ++ 3γ
4γ 2 )
Next, we need to illustrate that there exists a y ∗ , such that for y > y ∗ , such as
y = y ∗ + 1, R y ∗ (α) < R y ∗ +1 (α). It is equivalent to showing that after increasing y
beyond a certain value y ∗ , any further increase in y will make B(γ , β) negative.
Without loss of generality, we choose β → 1− . After simple algebra we have
4γβ + 2γ 2 β + 2β 2
2+γ −β
2+γ +β
lim
exp −k
−
β → 1−
1 − β2
(1 − β)3/2
(1 + β)3/2
=0−
2+γ +β
2+γ +β
=−
< 0.
3/2
2
23/2
By continuity we know that there exists a lead-time variability y∗ , y ∗ ∈ (0, Y ),
such that A changes from a positive value to a negative value (refer A to CRD,
p.19). Following CRD’s logic we conclude that, when the lead time is uniformly
, a small increase in lead-time variability but
distributed, for k = 2cY 2 < (4γ4 ++ 3γ
4γ 2 )
below a threshold lead-time variability y∗ will make the CSL initially increase
and the associated reorder point decrease, that is, R y ∗ (α) < R0 (α) holds; once the
lead-time variability increases above y ∗ , the CSL will recursively decrease and the
associated reorder point will increase, that is, R y ∗ (α) < R y ∗ +1 (α) holds.
Ping Wang is a PhD candidate at The Ohio State University. He received his
Master’s of engineering in logistics from MIT and his Master’s of engineering in
computer science from the Chinese Academy of Science. His research interests
include lead-time uncertainty in supply chains and supplier relationship management.
James A. Hill is an assistant professor at The Ohio State University. He
received his PhD in operations management from The Ohio State University.
He conducts research on master production scheduling in process industries,
lead-time uncertainty in supply chains, and the value of cross-docking in supply
chains. His research has appeared in the International Journal of Production
Research and Interfaces.