Optimal Policies and Approximations for A Serial Multi-echelon Inventory System...
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Optimal Policies and Approximations for A Serial Multi-echelon Inventory System with Time-correlated Demand Lingxiu Dong John M. Olin School of Business Washington University St. Louis, MO 63130 [email protected] Hau L. Lee Graduate School of Business Stanford University Stanford, CA 94305 [email protected] August 2000 Revision: November 2001 Subject classification: Inventory/production: approximations, multi-echelon, stochastic.
Transcript of Optimal Policies and Approximations for A Serial Multi-echelon Inventory System...
Optimal Policies and Approximations for A Serial
Multi-echelon Inventory System with
Time-correlated Demand
Lingxiu Dong
John M. Olin School of Business
Washington University
St. Louis, MO 63130
Hau L. Lee
Graduate School of Business
Stanford University
Stanford, CA 94305
August 2000
Revision: November 2001
Subject classification: Inventory/production: approximations,
multi-echelon, stochastic.
Abstract
Since Clark and Scarf’s pioneering work, most advances in multi-echelon inventory
systems have been based on demand processes that are time-independent. This paper
revisits the serial multi-echelon inventory system of Clark and Scarf, and develops
three key results. First, we provide a simple lower bound approximation to the
optimal echelon inventory levels and an upper bound to the total system cost for the
basic Clark and Scarf’s model. Second, we show that the structure of the optimal
stocking policy of Clark and Scarf holds under time-correlated demand processes
using Martingale model of forecast evolution. Third, we extend the approximation
to the time-correlated demand process, and study in particular for an auto-regressive
demand model the impact of leadtimes and auto-correlation on the performance of
the serial inventory system.
1 Introduction
In a seminal paper, Clark and Scarf (1960) study a serial multi-echelon inventory sys-
tem where demand occurs at the lowest echelon, and show that, when there is no fixed
ordering cost in the serial system, the optimal inventory control of the overall system
follows a base-stock policy for every echelon. The base-stock levels at each echelon
can be solved from a series of single location inventory problems with appropriately
defined penalty functions for not having enough inventory to bring the downstream
site to its target base-stock level.
Extensions of the Clark and Scarf approach have been developed for other
production/inventory systems. Schmidt and Nahmias (1985) use the Clark and Scarf
decomposition technique to characterize the optimal policy for an assembly system
with two components. Rosling (1989) shows, under appropriate assumptions, that a
general assembly system is equivalent to a serial system and thus Clark and Scarf’s
results apply. Axäter and Rosling (1993) establish the condition under which the
echelon and installation based policies are equivalent. Chen (2000) generalizes the
Clark and Scarf result to the serial systems and assembly systems where the material
flow from one level to another has to be in batches (e.g., truck loads). Iida(2000)
studies error bounds of near myopic policies in a multi-echelon system. Lee and
Whang (1999) show that a performance measurement scheme based on Clark and
Scarf’s penalty function exists that enables a decentralized serial inventory system to
achieve the optimal performance of a centralized system.
Although the optimal inventory policy follows a simple form of a base-stock
inventory policy, the computation of the optimal base-stock levels can be quite com-
plicated due to the complexity of the induced penalty cost functions. Federgruen and
Zipkin (1984) extend Clark and Scarf’s results to the infinite horizon case, and show
that the computation can be much simplified if one minimizes discounted cost or
long-term average cost. For a two-echelon system with Normal demand distribution,
they provide a closed-form equation for solving the optimal inventory levels and a
closed-form expression for the corresponding optimal cost. Similar calculations can
be carried over to a system with more than two echelons, but the complexity of the
computation increases considerably. Moreover, even with a closed-form expression,
the analysis of the system inventory performance is still quite tedious and often has
to rely on numerical methods. Gallego and Zipkin (1999) develop several heuristic
methods and conduct numerical studies on system performance sensitivity to the
stock positioning under different assumptions of inventory holding costs. Shang and
Song (2001) extend the restriction-decomposition heuristic in Gallego and Zipkin to
the echelon-stock setting under linear holding cost and give newsvendor type upper
and lower bounds to the optimal echelon policies.
Most of the research work on multi-echelon inventory systems has been based
on stationary demand processes that are independent over time. Although empiri-
cally, time-correlated demands are commonly observed (e.g. see Erkip et al., 1990
and Lee et al., 1999), there has been only limited work with time-correlated demand
processes. Chen and Zheng (1994) construct a simple proof of Clark and Scarf’s
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results in an infinite-horizon case by finding a lower bound on the system-wide cost
and constructing a feasible policy to achieve such a lower bound. Using this lower-
bounding approach, Chen and Song (2001) relax the independent demand assumption
in the classic model and consider a demand process whose demand distribution is de-
termined by the state of an underlying finite-state Markov chain. They show that
echelon base-stock policies are still optimal when order-up-to levels are adjusted based
on the state of the underlying Markov chain, and provide an algorithm to compute
the optimal base-stock levels.
For time-series based demand processes, the martingale model of forecast evo-
lution (MMFE) developed by Graves et al. (1986, 1998) and Heath and Jackson
(1994) offers a powerful descriptive framework that incorporates both past demands
and other influential factors to characterize the forecast processes. Readers are re-
ferred to Heath and Jackson for an excellent discussion of motivation of MMFE,
to Toktay and Wein (2001) for a review on researches (Güllü, 1996) built upon it.
A common demand process used in the literature is the order-one auto-regressive, or
AR(1), process with positive auto-correlation (see, for example, Johnson and Thomp-
son, 1975; Lovejoy, 1990&1992; Fotopoulos and Wang, 1988; Reyman, 1989; Scarf,
1959&1960; Miller, 1986; Kahn, 1987; Erkip et al., 1990; Lee et al., 1997; and Lee
et al., 1999). In fact AR(1) process with minimum mean-square error forecast is a
special case of MMFE. More complex time series demand processes have also been
modeled more recently, such as the random walk model of Graves (1999) and Lee and
Whang (1998).
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This paper develops three key results for the Clark and Scarf model. First,
we develop a simple approximation of the optimal echelon inventory stocking levels
that is easy to compute for serial multi-echelon systems with more than two echelons.
This approximation provides a good lower bound to the optimal stocking levels for the
system. This approximation applies to any convex inventory holding and penalty cost
functions, while in the linear function case, the lower bound is same as that of Shang
and Song (2001). Second, we show that Clark and Scarf’s results can be carried over
to the time-correlated demand process using MMFE, but with greater computation
complexity compared to its counterpart in the independent demand environment.
Finally, we extend the approximation in the independent demand case to the time-
correlated demand process and show that the approximation again provides a lower
bound to the optimal stocking levels. The approximation is used in particular for the
AR(1) case to explore the role of the cross-time demand correlation in affecting the
performance of the system.
The remainder of this paper is organized as follows: in section 2 we introduce
the serial inventory system; we then present the approximation of optimal inventory
levels under an i.i.d. demand process in section 3; in section 4 we establish Clark and
Scarf’s results under time-correlated demand processes, present the approximation,
and illustrate the impacts of the echelon leadtimes and demand auto-correlation on
the system performance; in section 5 we show through numerical examples that the
approximations in sections 3 and 4 are very accurate and are useful analytical tools of
unveiling the fundamental relationship between system parameters and the inventory
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cost; we conclude our findings in section 6.
2 The Serial Multi-echelon System
Consider an M-echelon serial periodic-review inventory system where customer de-
mand arises from echelon 1 and orders are placed by echelon 1 to echelon 2, by echelon
2 to echelon 3, ..., and echelon M is the highest echelon that orders from an external
supplier with infinite supply. The event sequence is as follows: (a) at the beginning
of a period, shipments due in this period arrive, followed by demand occurrence; (b)
demand is satisfied from on-hand inventory with complete backlogging; (c) at the end
of the period, inventory holding and shortage costs are charged against stocks and/or
shortages followed by ordering and shipment decisions. The shipment leadtime from
echelon m + 1 to echelon m is Tm, i.e., an order placed by echelon m at the end of
period t will arrive at echelon m at the beginning of period t+Tm+1. Such an event
sequence, different from that of Clark and Scarf but the same as that of Federgruen
and Zipkin (1984) and Lee and Whang (1999), makes the exposition easier. The
results of this paper hold for other sequences of events with appropriate adjustments.
Assume that there are no ordering setup costs at all sites. The following
notation will be used:
ξ : demand in a period, assumed to be i.i.d. in section 3;
um : on-hand inventory level at site m at the end of a period, just before ordering
decisions are made;
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wm : vector of amounts due to arrive from site m+1 to site m in future periods (the
dimension of the vector equals Tm, and the ith element, wim, is the amount due
to arrive in i periods);
w0m : vector wm without the first element;
em : column vector of ones with dimension of Tm;
xm : echelon inventory position at levelm at the end of a period, just before ordering
decisions are made; xm = vm +wmem;
vm : echelon inventory at levelm at the end of a period, just before ordering decisions
are made, i.e., vm = xm−1 + um;
ym : echelon inventory position at level m at the end of a period, after ordering
decisions are made;
cm : unit shipment or processing cost from site m+ 1 to site m;
Lm (vm) : holding and shortage cost in a period at level m, given that the echelon
inventory at the end of a period is vm;
α : discount factor per period;
v : column vector (vm)Mm=1;
x : column vector (xm)Mm=1;
y : column vector (ym)Mm=1;
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W : a collection of column vectors given by (wm)Mm=1;
W0 : a collection of column vectors given by (w0m)
Mm=1;
(W0, z) : a collection of column vectors given by³¡
w0mzm
¢´Mm=1
, where z =(zm)Mm=1, is
any vector of dimension M .
Let C (v,W) be the system minimum expected discounted cost, given initial
echelon inventory levels v and shipmentsW. The overall inventory control problem
can be stated as:
C (v,W) (1)
= miny:xm≤ym≤vm+1
nXM
m=1[cm (ym − xm) + Lm (vm)] + αE
£C¡v +w1 − ξ, (W0,y− x)¢¤o ,
where vM+1 =∞.
Clark and Scarf show that the optimal inventory policy for the overall system
follows an echelon base-stock policy at each echelon. Let S∗m be the optimal order-
up-to level at echelon m. Hence, at the end of each period, echelon m places an
order to its upstream echelon to bring its echelon inventory position up to S∗m. The
actual amount that gets shipped depends on the on-hand inventory at the upstream
site. Clark and Scarf’s results state that we can decompose (1) into a series of single-
echelon inventory control problems with the proper penalty cost function Γm added
to the original holding and shortage cost expression, with S∗m being the respective
solution for echelon m. We state this result explicitly as follows:
C (v,W) =XM
m=1fm (vm,wm) , (2)
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where
fm (vm,wm)
= miny:xm≤y
©cm (y − xm) + Lm (vm) + Γm (vm) + αE
£fm¡vm +w
1m − ξ, (w0
m, y − xm)¢¤ª
.
Define γ (j), a random variable, as the sum of demands in j periods, where γ (0) = 0.
Hence γ (1) = ξ. Since the decision made at the end of period t will not affect the
system cost until period t + Tm + 1, we consolidate the state space and rewrite the
function fm (vm,wm) :
fm (vm,wm) (3)
= Lm (vm) + Γm (vm) + αE£Lm¡vm +w
1m − ξ
¢+ Γm
¡vm +w
1m − ξ
¢¤+α2E
£Lm¡vm +w
1m +w
2m − γ (2)
¢+ Γm
¡vm +w
1m +w
2m − γ (2)
¢¤+...+ αTmE [Lm (vm +wmem − γ (Tm)) + Γm (vm +wmem − γ (Tm))]
+gm (xm) ,
where
gm (xm) (4)
= miny:xm≤y
{cm (y − xm) + αTm+1E [Lm (y − γ (Tm + 1)) + Γm (y − γ (Tm + 1))]
+αE [gm (y − ξ)]}.
We interpret the function fm (vm,wm) as the adjusted (through the penalty function
Γm (vm)) discounted expected cost for echelon m, given the current state of vm and
wm. For m = 1, the penalty function is defined as Γ1 (v1) = 0 for all values of v1. For
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m ≥ 1, the penalty function is defined as:
Γm+1 (vm+1) (5)
=
cm (vm+1 − S∗m) + αTm+1E [Lm (vm+1 − γ (Tm + 1)) + Γm (vm+1 − γ (Tm + 1))]
−αTm+1E [Lm (S∗m − γ (Tm + 1)) + Γm (S∗m − γ (Tm + 1))]
+αE [gm (vm+1 − ξ)− gm (S∗m − ξ)] if vm+1 ≤ S∗m;
0 otherwise.
For computation purpose, the cost function Lm (·) has to be explicitly specified.
We follow the linear cost specification used by Federgruen and Zipkin (1984) and Lee
and Whang (1999). Let
Hm = per-period unit holding cost for inventory at sitem, or in transit to sitem−1.
π = per-period unit backorder cost at site 1.
hm = Hm −Hm+1, m = 1, ...,M , where HM+1 = 0. hm is referred to as the echelon
inventory holding cost at level m.
We will specify the holding and shortage costs associated with echelons as:
Lm (vm) = hmvm, m > 1; (6)
L1 (v1) = h1v1 + (π +H1) v−1 ; (7)
where
x− = (−x)+ and x+ = max (0, x) .
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Lee and Whang (1999) show that such an echelon cost definition fully allocates the
total system cost to all sites, i.e.,
H1v+1 + πv−1 +
XM
m=2Hm (um +wm−1em−1) =
XM
m=1Lm (vm) .
3 An Approximation of the IID Demand Model
Federgruen and Zipkin develop a closed-form solution for a two-echelon system when
inventory holding and shortage cost is linear and the end demand follows a Normal
distribution. However, generalizing their results to serial multi-echelon systems with
more than two echelons involves calculating multi-variate Normal distribution func-
tions and is computationally challenging. Since the difficulty of calculating optimal
stocking level S∗m is caused by the complexity of the penalty function Γm (·), we pro-
ceed to develop an approximation for Γm (·), denoted by bΓm (·). Define: bΓ1 (·) = 0;and, for m ≥ 1,
bΓm+1 (vm+1) (8)
= cm³vm+1 − bSm´+ αTm+1E
hLm (vm+1 − γ (Tm + 1)) + bΓm (vm+1 − γ (Tm + 1))
i−αTm+1E
hLm³bSm − γ (Tm + 1)
´+ bΓm ³bSm − γ (Tm + 1)
´i+αE
hbgm (vm+1 − ξ)− bgm ³bSm − ξ´i,
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where bSm is the solution ofbgm (xm) (9)
= miny:xm≤y
{cm (y − xm) + αTm+1EhLm (y − γ (Tm + 1)) + bΓm (y − γ (Tm + 1))
i+αE [bgm (y − ξ)]}.
Hence, we have the following approximation for fm (vm,wm):
bfm (vm,wm) (10)
= Lm (vm) + bΓm (vm) + αEhLm¡vm +w
1m − ξ
¢+ bΓm ¡vm +w1
m − ξ¢i
+α2EhLm¡vm +w
1m +w
2m − γ (2)
¢+ bΓm ¡vm +w1
m +w2m − γ (2)
¢i+...+ αTmE
hLm (vm +wmem − γ (Tm)) + bΓm (vm +wmem − γ (Tm))
i+bgm (xm) .
Functions bΓm, bgm, and bfm together constitute an approximate formulation of the
inventory control of the multi-echelon system. Although the approximation is still
inductive, i.e., for echelonm+1, the definition of bΓm+1 requires the approximate lowerechelon target stocking level, bSm, whose calculation involves bΓm, the computation isgreatly simplified. Using this approximate penalty function, echelonm+1 is penalized
even when it can fulfill the order from echelon m, i.e., echelon m+ 1 is penalized for
overstocking! Theorem 1 states an intuitive result: the stocking level bSm derived fromthe approximate inventory control system is a lower bound to the optimal stocking
level S∗m for each echelon m, m = 1, ...,M . The proofs of Theorem 1 and others are
given in the Appendix.
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Theorem 1 Let bSm be the solution of (9), then bSm ≤ S∗m, m = 1, ...,M .It turns out that bSm can be computed easily for the specific cost functions
Lm (·) of (6) and (7), and an approximate cost function can be derived in closed-
form.
Define:
τ (i, j) = (Ti + 1) + (Ti+1 + 1) + ...+ (Tj + 1) for i ≤ j, and 0 otherwise.
Φ (·; t), Φ (·; t) : the cumulative distribution function (CDF) and the complementary
CDF of demand in t periods, respectively.
Θ (x) = xΦ (x)+φ (x), where Φ (·) and φ (·) are the CDF and the probability density
function (PDF) of the standard Normal distribution, respectively.
µ (t) ,σ (t) : mean and standard deviation of demand in t periods.
Theorem 2 For m = 1, 2, ...,M , bSm satisfies:Φ̄³bSm; τ (1,m)´ = Km,
where
Km =³Xm
i=1α−τ(1,i)
£(1− α) ci + αTi+1hi
¤´/ (π +H1) .
Theorem 2 thus gives us a very simple, non-inductive way to calculate bSm.Let bAα
m be the cost per period at echelon m by following the order-up-to bSm policy,calculated with the approximate induced penalty cost function bΓm and zero initialinventory. Let bAα be the corresponding system cost per period, i.e., bAα =
PMm=1
bAαm.
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Lemma 1
bAα1 = (1− α) c1 bS1 + αc1µ+ αT1+1h1
hbS1 − µ (T1 + 1)i+αT1+1 (π +H1)
Z ∞
t=bS1³t− bS1´ dΦ (t;T1 + 1) ;
for m > 1,
bAαm = (1− α) cm bSm + αcmµ+ αTm+1hm
hbSm − µ (Tm + 1)i+αTm+1E
hbΓm ³bSm − γ (Tm + 1)´i.
Lemma 2
bΓm (y) =³y − bSm−1´Xm−1
i=1ατ(i+1,m−1) £(1− α) ci + αTi+1hi
¤−ατ(1,m−1) (π +H1)
Z y
t=bSm−1 Φ (t; τ (1,m− 1)) dt.
Furthermore, if the end demand per period is Normal with mean µ and variance σ2,
then
bΓm (y) =³y − bSm−1´Xm−1
i=1ατ(i+1,m−1) £(1− α) ci + αTi+1hi
¤+ατ(1,m−1)σ (τ (1,m− 1)) (π +H1)
hΘ (ι (y))−Θ
³ι³bSm−1´´i ,
where
ι (y) = − [y − µ (τ (1,m− 1))] /σ (τ (1,m− 1)) .
Suppose the end demand per period is Normal with mean µ and variance σ2.
Let bAm be the approximate average cost per period, i.e., α→ 1, for echelon m when
the cost function are given by (8), (9), and (10) and when the stocking levels bS0ms are13
followed. Let bA be the corresponding average cost per period for the whole system,i.e., bA =PM
m=1bAm.
Lemma 3 With Normal demands,
bA1 = c1µ+ h1Φ−1 (1−K1)σ (T1 + 1) + (π +H1)σ (T1 + 1)Θ¡−Φ−1 (1−K1)
¢;
for m > 1,
bAm = cmµ+ hmµ (τ (1,m− 1))
+σ (τ (1,m))£(H1 −Hm+1)Φ−1 (1−Km) + (π +H1)Θ
¡−Φ−1 (1−Km)¢¤
−σ (τ (1,m− 1)) £(H1 −Hm)Φ−1 (1−Km−1) + (π +H1)Θ¡−Φ−1 (1−Km−1)
¢¤.
The telescoping structure of seriesn bAmoM
m=1leads to:
Theorem 3 With Normal demands,
bA = µXM
m=1cm +
XM
m=1hmµ (τ (1,m− 1)) + σ (τ (1,M)) (π +H1)φ
¡Φ−1 (1−KM)
¢= µ
XM
m=1cm +
XM−1m=1
Hm+1µ (Tm + 1) + σ (τ (1,M)) (π +H1)φ¡Φ−1 (1−KM)
¢.
Corollary 1 ∂ bA/∂Tm > 0 and ∂ bA/∂Tm ≥ ∂ bA/∂Tm+1.Theorem 3 provides a closed-form expression of the approximate system cost.
The system cost can be viewed as consisting of three parts: (1) the average ship-
ping/processing cost; (2) the average inventory holding cost; and (3) the safety
stock cost caused by the demand randomness within the total system-wide leadtime
τ (1,M) . Corollary 1 illustrates the role that leadtimes play in affecting the system
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cost. In general, longer leadtime leads to higher cost. Since holding inventory is more
expensive at the downstream sites, reducing lower-echelon leadtimes would have a
bigger impact to the system cost than reducing the higher-echelon leadtimes.
We are now ready to examine the true cost performance of following the
stocking levels of bSm in the multi-echelon system. Let Aαm denote the echelon-m
cost per period of following the order-up-to bSm policy, calculated with the true
penalty cost Γm. Note that Aαm = (1− α) cm bSm+αcmµ+αTm+1hm
hbSm − µ (Tm + 1)i+ αTm+1E
hΓm³bSm − γ (Tm + 1)
´i. Denote Aα as the corresponding true system-
wide per-period cost for policy bSm. For a two-echelon system, we were able to derivethe cost dominance relationship between bAα and Aα:
Corollary 2 For a two-echelon system:(1) bAα2 ≥ Aα
2 ; (2) bAα ≥ Aα.
For the two-echelon case with α→ 1, i.e., the average cost case, we see that bAprovides an upper bound for the true cost of implementing the approximate policy bSm.This is helpful, since the average cost bA is easily computable, as shown in Theorem3. In section 5, we will show through numerical examples that both bSm and bAα are
very accurate approximations of the optimal stocking levels and the optimal system
cost.
4 The Time-correlated Demand Model
When demands are correlated across time periods, the optimal control of the inventory
system will be based on the demand forecasts which are revised from period to period
15
as information accumulated over time. Let Dt be the end-customer demand in period
t. Let Dt,t+i be the forecast made in period t for demand in period t + i. Hence
we can define a vector of demand forecasts made in period t for future periods,
Dt = (Dt,t,Dt,t+1, ...), with Dt,t = Dt as the demand realized in period t. Then
²t,t+i = Dt−1,t+i −Dt,t+i,
represents the forecast update made for period t + i from period t − 1 to t, i =
0, 1, ..., and ²t = (²t,t+i)∞i=0 represents the forecast update vector in period t. We
adopt the MMFE developed by Heath and Jackson (1994), which treats forecasts
as the conditional expectation of the future demands on the current information set
and assumes that the demand random variables form a martingale relative to the
corresponding information set. We make the following assumptions explicitly:
(1) ²t vectors are i.i.d. multi-variate Normal random vectors with mean 0;
(2) there are only finite number of random variables in ²t that are linearly inde-
pendent, i.e., there exists a set of finite number of random variables in ²t such
that each ²t,t+i can be written as a linear combination of the random variables
in the set.
Assumption (1), essential to MMFE, requires the demand process to be sta-
tionary and forecasts to be unbiased. Assumption (2) implies that only finite number
of uncertainty factors of the information set affect the forecasts.
MMFE is indeed a powerful model of forecast process in that it captures both
historical demands and potential information available through other sources in one
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framework (see Aviv, 2001a, for an exploration of MMFE). In fact, an ARMA (auto-
regressive and moving average) process with minimum mean-square error forecast is
a special case of MMFE. To illustrate, we take a look at an AR(1) demand process
Dt = d+ ρDt−1 + εt,
with d > 0, −1 < ρ < 1, and εt being i.i.d. Normal with mean zero and variance
σ2 (σ ¿ d). Here ρ is the auto-correlation coefficient of demand in two consecutive
periods. In this case, Dt,t+i = dPi−1
j=0 ρj + ρiDt, ²t,t+i = ρiεt, and clearly ²t,t+i is
linearly dependent on εt.
Theorem 4 For a time-correlated demand process that satisfies assumptions (1) and
(2), a state-dependent base-stock policy {S∗ (D)} is optimal for a single-echelon in-
ventory system, where the state D is the demand forecast made before making the
ordering decision.
Let C (v,W,D0) be the minimum expected discounted cost for the serial in-
ventory system, given the forecastD0 made in the current period, the initial inventory
levels v and shipmentsW. The overall inventory control problem can be stated as
C (v,W,D0) (11)
= miny:xm≤ym≤vm+1
{XM
m=1[cm (ym − xm) + Lm (vm)]
+αED0
£C¡v+w1 −D1, (W0,y− x) ,D1
¢¤},where vM+1 = ∞. The expectation in (11) is taken over the next period’s forecast
D1, but conditional on the demand forecast D0 in the current period. In the rest
17
of this section, we will drop D0 in the conditional expectation for ease of exposition
whenever it is unambiguous.
Define D (i, j) as a random variable representing demand in j periods that
starts from period i, i.e., the cumulative demand in time interval [i, i+ j − 1]. Simi-
lar to the i.i.d. demand case, we define the adjusted discounted expected cost func-
tions, fm (vm,wm,D0) and gm (xm,D0), and the corresponding penalty cost function
Γm (vm,D0) for echelon m, given that the current period forecast is D0.
fm (vm,wm,D0) (12)
= miny:xm≤y
{cm (y − xm) + Lm (vm) + Γm (vm,D0)
+αE£fm¡vm +w
1m −D1, (w0
m, y − xm) ,D1
¢¤}= Lm (vm) + Γm (vm, D0)
+αE£Lm¡vm +w
1m −D1
¢+ Γm
¡vm +w
1m −D1, D1
¢¤+ ...
+αTmE [Lm (xm −D (1, Tm)) + Γm (xm −D (1, Tm) , DTm)] + gm (xm,D0) ,
gm (xm,D0) (13)
= miny:xm≤y
{cm (y − xm)
+αTm+1E [Lm (y −D (1, Tm + 1)) + Γm (y −D (1, Tm + 1) ,DTm+1)]
+αE [gm (y −D1,D1)]}.
18
For m = 1, Γ1 (·, ·) = 0; for m ≥ 1,
Γm+1 (vm+1, D0) (14)
=
cm [vm+1 − S∗m (D0)]
+αTm+1E [Lm (vm+1 −D (1, Tm + 1)) + Γm (vm+1 −D (1, Tm + 1) ,DTm+1)]
−αTm+1E [Lm (S∗m (D0)−D (1, Tm + 1)) + Γm (S∗m (D0)−D (1, Tm + 1) ,DTm+1)]
+αE [gm (vm+1 −D1,D1)− gm (S∗m (D0)−D1,D1)] if vm+1 ≤ S∗m (D0) ;
0 otherwise;
where S∗m (D0) is the solution of (13).
Under the assumptions (1) and (2), the dynamic programming formulation of
the inventory problem differs from the i.i.d. demand case in that more dimensions
are added to the state space, namely the forecasts for future periods, and that the
conditional expectation is used. The following theorem shows that the key result of
Clark and Scarf for i.i.d. demand system, i.e., the property of echelon decomposition
for the optimal policy using the induced echelon penalty cost, remains true in the
time-correlated demand setting.
Theorem 5 C (v,W,D0) =PM
m=1 fm (vm,wm,D0) .
19
We again define the approximate penalty cost function bΓm, similar to theprevious section. Define: bΓ1 (·, D0) = 0; and, for m ≥ 1,
bΓm+1 (vm+1, D0) (15)
= cmhvm+1 − bSm (D0)
i+αTm+1E
hLm (vm+1 −D (1, Tm + 1)) + bΓm (vm+1 −D (1, Tm + 1) ,DTm+1)i
−αTm+1EhLm³bSm (D0)−D (1, Tm + 1)
´+ bΓm ³bSm (D0)−D (1, Tm + 1) , DTm+1
´i+αE
hbgm (vm+1 −D1,D1)− bgm ³bSm (D0)−D1,D1
´i,
where bSm (D0) is the solution to:
bgm (xm,D0) (16)
= minxm≤y
{cm (y − xm) + αTm+1EhLm (y −D (1, Tm + 1)) + bΓm (y −D (1, Tm + 1) ,DTm+1)i
+αE [bgm (y −D1,D1)]}.
Also,
bfm (vm,wm,D0)
= Lm (vm) + bΓm (vm,D0) + αEhLm¡vm +w
1m −D1
¢+ bΓm ¡vm +w1
m −D1, D1¢i
+α2EhLm¡vm +w
1m +w
2m −D (1, 2)
¢+ bΓm ¡vm +w1
m +w2m −D (1, 2) , D2
¢i+...+ αTmE
hLm (vm +wmem −D (1, Tm)) + bΓm (vm +wmem −D (1, Tm) ,DTm)i
+bgm (xm,D0) .
When demand is time-correlated, it is possible that the physical inventory level
at a site is higher than the target stocking level, such that the target inventory level
20
may not always be realizable. Indeed, it arises in inventory problems in other settings
as well. For example, when demands are Normal, stock levels can exceed the target
even when demands are i.i.d., since it is theoretically possible that a negative demand
can occur. Another example is the classic paper by Eppen and Schrage (1981) where
the optimal allocation of stocks among multiple sites may not be feasible due to stock
imbalance. The traditional approach to this problem is to assume that the probability
of such occurrence is negligible. In fact, when demand follows an AR(1) process with
ρ > 0, the probability of excess inventory (overshoot) is smaller than the standard
i.i.d. demand case (see Lemma 1 in Lee et al.,1999, Aviv, 2001b) and the myopic
policy is quite an accurate approximation to the optimal policy. Another approach
is to make the “costless return” assumption (see Lee et al., 1997). Specifically, at
echelon m < M , if the physical echelon inventory level is higher than the target
echelon inventory level, then the excessive inventory will not be charged as echelon m
inventory, and echelon m gets a refund at the original purchasing price which is paid
by an outside source (for example, a bank); however, the excessive inventory does not
leave the system physically and is still available as inventory for the upper echelon,
m + 1, for future replenishment to echelon m; at echelon M , excessive inventory is
returned to the outside supplier with a refund at the original purchasing cost. Under
this assumption the myopic policy is indeed optimal. We will use such an assumption
in the remainder of the analysis.
Again the approximate inventory levels, which can be computed easily, are
lower bounds of the optimal stocking levels.
21
Theorem 6 bSm (D) ≤ S∗m (D), m = 1, ...,M .We now study the AR(1) demand process. Since in any given period t, the
forecasts of future periods are functions only of demand realization Dt, we need only
Dt to represent forecasts made in period t. Given the current period’s realized demand
D0, let Φ0 (·; t) and Φ0 (·; t) be the CDF and the complementary CDF of demand in
the subsequent t periods whose mean and standard deviation are given by µ0 (t) and
σ (t) respectively.
Theorem 7 For m = 1, 2, ...,M , bSm (D0) satisfies:Φ̄0³bSm (D0) ; τ (1,m)´ = Km,
where
Km =³Xm
i=1α−τ(1,i)
£(1− α) ci + αTi+1hi
¤´/ (π +H1) .
Hence bSm (D0) = µ0 (τ (1,m)) + Φ−1 (1−Km)σ (τ (1,m)) .
Let bAαm (D0) be the cost per period at echelon m by following the order-up-
to bSm (D0) policy and using bΓm as the penalty cost function. Let bAα (D0) be the
corresponding system-wide cost per period, i.e., bAα (D0) =PM
m=1bAαm (D0).
Lemma 4
bAα1 (D0) = (1− α) c1 bS1 (D0) + αc1µ0 (1) + αT1+1h1
hbS1 (D0)− µ0 (T1 + 1)i+αT1+1 (π +H1)
Z ∞
t=bS1(D0)ht− bS1 (D0)i dΦ0 (t;T1 + 1) ;
22
for m > 1,
bAαm (D0) = (1− α) cm bSm (D0) + αcmµ0 (1) + αTm+1hm
hbSm (D0)− µ0 (Tm + 1)i+αTm+1E
hbΓm ³bSm (D0)−D (1, Tm + 1) ,DTm+1´i .Lemma 5
bΓm (y,D0) =³y − bSm−1 (D0)´Xm−1
i=1ατ(i+1,m−1) £(1− α) ci + αTi+1hi
¤+ατ(1,m−1) (π +H1)σ (τ (1,m− 1))
hΘ (ι (y,D0))−Θ
³ι³bSm−1 (D0) , D0´´i ,
where
ι (x,D0) = − [x− µ0 (τ (1,m− 1))] /σ (τ (1,m− 1)) .
As the discount factor α→ 1, the average system-wide cost per period bA (D0)has the following closed-form expression:
Theorem 8
bA (D0) = µ0 (1)XM
m=1cm +
XM
m=1hm [µ0 (τ (1,m))− µ0 (Tm + 1)]
+σ (τ (1,M))φ¡Φ−1 (1−KM)
¢= µ0 (1)
XM
m=1cm +
XM
m=1hmE
£µTm+1 (τ (1,m− 1))
¤+σ (τ (1,M)) (π +H1)φ
¡Φ−1 (1−KM)
¢.
where µTm+1 (τ (1,m− 1)) represents the mean demand in the subsequent τ (1,m− 1)
periods starting from period Tm + 1.
Corollary 3 bA (D0) is nondecreasing in T1, T2, ..., TM and ∂ bA (D0) /∂Tm ≥ ∂ bA (D0) /∂Tm+1.23
The above lemmas and theorems share similar structures as those in the i.i.d.
and Normal demand case. Indeed, when ρ is set to zero, Theorem 8 is the same as
Theorem 3.
We now further assume that demands across periods are positively correlated,
i.e., ρ > 0. Erkip et al. (1990) and Lee et al. (1999) both report that positively
correlated demands were commonly observed in industry.
Theorem 9 If ρ > 0, then:
(1) bSm (D0) is nondecreasing in ρ and T1, T2, ..., Tm;
(2) bA (D0) is nondecreasing in ρ and T1, T2, ..., TM ;
(3) ∂ bSm (D0) /∂ρ is nondecreasing in T1, T2, ..., Tm and ∂ bA (D0) /∂ρ is nondecreasingin T1, T2, ..., TM .
Hence, as expected, both the approximate stocking levels and the approximate
average cost are higher with higher auto-correlated demands, or longer leadtimes,
ceteris paribus. Moreover, the sensitivity of stocking levels and cost to the auto-
correlation of demand is greater when leadtimes are long. This seems to imply that
the interaction effect of demand auto-correlation and leadtime is significant.
Similarly, let Aαm (D0) be the actual echelon-m cost using bSm (D0) as the target
stocking level, i.e., Γm is used to calculate the penalty cost. Then:
Aαm (D0) = (1− α) cm bSm (D0) + αcmµ0 (1) + αTm+1hm
hbSm (D0)− µ0 (Tm + 1)i+αTm+1E
hΓm³bSm (D0)−D (1, Tm + 1) ,DTm+1´i .
24
The corresponding system cost is Aα (D0) =PM
m=1Aαm (D0). For a two-echelon sys-
tem, we have the following cost dominance relationship.
Corollary 4 For a two-echelon system: (1) bAα2 (D0) ≥ Aα
2 (D0); (2) bAα (D0) ≥
Aα (D0).
5 Numerical Examples
In this section, we first present examples to show that the approximations developed
in sections 3 and 4 are accurate for both the i.i.d. and AR(1) demand processes. We
then illustrate how demand auto-correlation of the AR(1) process affects the decisions
and performance of the system.
In the i.i.d. demand scenario, the base case has the following parameters: the
demand per period is Normal with mean µ = 100 and standard deviation σ = 20;
replenishment leadtimes, T1 = 2, T2 = 1; transportation or processing costs, c1 = 10,
c2 = 5; inventory holding costs, H1 = 4, H2 = 2; site 1 shortage cost, π = 10; per-
period discount factor, α = 0.9. The true optimal stocking levels and system cost are
calculated by the method given by Federgruen and Zipkin (1984) for a two-echelon
system.
Figure 1 shows the approximate and optimal inventory stocking levels under
varying σ/µ. We observe, as stated in Theorem 1, that the approximate stocking
level, bS2, is a lower bound of the optimal stocking level, S∗2 . Moreover, the differencebetween the two levels is very small (within 3% of the optimal stocking level S∗2).
25
Figure 2 shows the comparison of three system costs: (1) bAα is the system cost using
the approximate optimal stocking levels bS, calculated with the approximate penaltycost function bΓ; (2) Aα is the system cost using bS, but calculated with the penaltycost function Γ; (3) A
αis the system cost using optimal stocking levels S∗, calculated
with the true penalty cost Γ. As shown in Corollary 2, we observe that bAα ≥ Aα.
Moreover, the three cost curves are very close (within 1.01% of the optimal system
cost Aα). Hence, Figure 2 shows that both bAα and Aα are very good approximations
of Aα. Comparisons based on other parameter ranges, such as c2/c1, H1/c1, H2/c2,
π/c1, and T2/T1 (a total of 61 examples, see Table 1), all show the similar accuracy
of the approximation.
Figures 1 and 2 about here.
To verify the accuracy of this approximation for systems with more than two
echelons, we also run comparison for a three-echelon system. The three-echelon sys-
tem parameters are set as: Normal distribution with mean µ = 100 and standard
deviation σ = 20; replenishment leadtimes, T1 = 2, T2 = 1, T3 = 1; transportation or
processing costs, c1 = 10, c2 = 5, c3 = 2; inventory holding costs, H1 = 4, H2 = 2,
H3 = 1; site 1 shortage cost π = 10, per-period discount factor, α = 1. Figures 3 and 4
show that the gap between our approximation and the optimal result is very small. In
fact, comparisons based on other parameter ranges (a total of 59 examples, see Table
2) also shows similar accuracy (gaps are within 2.55% of the optimal stocking level
S∗3 and 0.36% of the optimal system cost A, respectively). Hence, this approximation
26
is reasonably accurate for a three-echelon inventory system as well.
Figures 3 and 4 about here.
For the AR(1) demand process, the base case is the same as that of the i.i.d.
case, with α = 1 and the AR(1) process specified as d = 100, ρ = 0.1, D0 = 10,
and σ = 20. We extended the method of Federgruen and Zipkin (1984) for the i.i.d.
case to the AR(1) case to calculate the optimal stocking levels and system costs
(see Dong, 1999, for details). Figure 5 shows the approximate and optimal stocking
levels under varying σ/d. Again, the approximate stocking level bS2 (D0) is a lowerbound of optimal S∗2 (D0), and their difference is very small (within 2.20% of A (D0)).
Similarly, Figure 6 confirms that bA (D0) ≥ A (D0). The cost curves are very close toone another, with gaps that are within 1.12%. A total of 71 examples (see Table 3)
based on different parameter changes also show similar degree of accuracy. Hence,
for both the i.i.d. and AR(1) demand processes, the approximation not only provides
an effective and simple computation for the target stocking levels, also generates
an accurate representation of the actual system cost by following the approximate
stocking levels that is close to optimal.
Figures 5 and 6 about here.
We further explore the effectiveness of the simple closed-form expression of
the approximate model. Figures 7 and 8 show the optimal inventory levels and
system costs under varying auto-correlation coefficient ρ and replenishment leadtime
T1. As expected, both the optimal inventory levels and system cost increase as ρ or T1
27
increases. In addition, the cost increments due to leadtime increase, is greater when
ρ is higher. Leadtime and demand auto-correlation are thus shown to have significant
interaction effects. Leadtime reduction has greater impact on system cost when the
demand auto-correlation is high. Such observations match the results of Theorem
9. Hence, the analytical results and the associated qualitative insights that we draw
from the approximate model are shown to be valid under the true cost model, based
on this set of examples.
Figures 7 and 8 about here.
6 Conclusion
In this paper, we revisit the Clark and Scarf’s model and develop a simple approxima-
tion of the induced penalty cost function. This approximation leads to a lower bound
on the optimal stocking levels and an upper bound on the average system cost. The
approximation is then extended to the time-correlated demand process with MMFE,
where Clark and Scarf’s decomposition result is shown to hold. In particular, under
the AR(1) process the approximation provides a simple, easy to compute closed-form
expression for the stocking levels and the average system cost. The closed-form ex-
pression from the approximation allows us to investigate how the underlying demand
process affects the performance of the inventory system. As noted earlier, an AR(1)
model may be a better representation of the underlying demand process in many
high-tech and consumer goods industries, and it is indeed used in recent supply chain
28
management research studies. Our study of the AR(1) demand process shows that
both the system cost and target inventory stocking levels increase as the demand
auto-correlation coefficient increases. Hence, in a highly time-correlated demand en-
vironment, simplifying the demand assumption to i.i.d. would understate the actual
system cost and result in target inventory levels that are too low. We have also shown
that the impact of leadtime reduction is greater when the auto-correlation coefficient
ρ, is higher. Hence it is more worthwhile to invest in leadtime reduction in a highly
time-correlated demand environment.
The approximation developed in this paper offers a benchmark for further
study of the forecast/information sharing in supply chains. An important avenue
of future research is on fine tuned MMFE, which offers great flexibility in modeling
information and collaboration in multi-echelon systems.
Acknowledgments: The authors gratefully acknowledge the helpful comments
of the editor and two anonymous referees. The authors also thank Paul Zipkin, Jing-
Sheng Song, Yossi Aviv for insightful discussions. Useful feedback was provided by
participants at the 2001 Multi-echelon Inventory Systems Conference at Berkeley.
Appendix
Proof of Theorem 1. First observe that, in (4), dE [gm (y − ξ)] /dy = −cm.
Hence, the first derivative of the minimand on the R.H.S. of (4) w.r.t. y is given by
(1− α)cm + αTm+1 (d/dy)E [Lm (y − γ (Tm + 1)) + Γm (y − γ (Tm + 1))] . (A.1)
29
Similarly, the first derivative of the minimand on the R.H.S. of (9) w.r.t. y is given by
(1− α)cm + αTm+1 (d/dy)EhLm (y − γ (Tm + 1)) + bΓm (y − γ (Tm + 1))
i. (A.2)
To show bSm ≤ S∗m, it suffices to show that:(d/dy)E
hbΓm (y − Z)i ≥ (d/dy)E [Γm (y − Z)]for any random variable Z. We show this inequality by induction. When m = 1, bΓ1 = Γ1,
and so equality holds. Assume that the inequality holds up to m and bSm ≤ S∗m. Let F (·)be the CDF of Z. First, note that, when bSm ≤ S∗m < y−z, bΓm+1 (y − z)−Γm+1 (y − z) =bΓm+1 (y − z) is nondecreasing in y. Second, for S∗m ≥ y − z, from (5), we have:
(d/dy)Γm+1 (y) = (1− α) cm+αTm+1 (d/dy)E [Lm (y − γ (Tm + 1)) + Γm (y − γ (Tm + 1))]
and from (8), we have
(d/dy) bΓm+1 (y) = (1− α) cm+αTm+1 (d/dy)E
hLm (y − γ (Tm + 1)) + bΓm (y − γ (Tm + 1))
i.
Hence, we have:
(d/dy)EhbΓm+1 (y − Z)− Γm+1 (y − Z)
i=
Zz≤y−S∗m
(d/dy)hbΓm+1 (y − z)− Γm+1 (y − z)
idF (z)
+
Zz>y−S∗m
(d/dy)hbΓm+1 (y − z)− Γm+1 (y − z)
idF (z)
=
Zz≤y−S∗m
hdbΓm+1 (y − z) /dyi dF (z)
+
Zz>y−S∗m
αTm+1 (d/dy)EhbΓm (y − z − γ (Tm + 1))− Γm (y − z − γ (Tm + 1))
idF (z)
≥ 0.
30
Assume that Eh(d/dy)
³bΓm+1 (y − Z)− Γm+1 (y − Z)´i< ∞. The interchange of dif-
ferentiation and integration is justified by the Lebeague Dominance Convergence Theorem.
Proof of Lemma 1. Let bAαm (xm) be the cost per period at echelon m by
following the order-up-to bSm policy with initial inventory of xm. Clearly bAαm (xm) =
(1− α) bgm (xm). Define egm (·) as:egm (xm) = bgm (xm) + cmxm
= (1− α) cm bSm + αcmµ+ αTm+1EhLm³bSm − γ (Tm + 1)
´+ bΓm ³bSm − γ (Tm + 1)
´i+αE
hegm ³bSm − ξ´i.
And
bAαm (xm) = (1− α) [egm (xm)− cmxm] .
Note that egm (xm) is actually independent of xm, hencebAαm = bAα
m (0) = (1− α)egm (0)= (1− α) cmbSm + αcmµ+ αTm+1E
hLm³bSm − γ (Tm + 1)
´+ bΓm ³bSm − γ (Tm + 1)
´i.
By definition of Lm (·), the desired result follows.
Lemma A.1 For all m > 1 and for a random variable Z, we have:
(d/dy)EhbΓm (y − Z)i
= (1− α)Xm−1
i=1ατ(i+1,m−1)ci +
Xm−1i=1
ατ(i,m−1) ddyE [Li (y − Z − γ (τ (i,m− 1)))] .
31
Proof of Lemma A.1. The proof is by induction. We refer the reader to details
in Dong (1999).
Proof of Theorem 2. For m = 1, the desired result is obtained by observing
that, setting (A.2) to zero yields
(1− α) c1 + αT1+1 (d/dy)E [L1 (y − γ (T1 + 1))]
= (1− α) c1 + αT1+1£h1 − (π +H1) Φ̄ (y;T1 + 1)
¤= 0.
For m ≥ 2, using Lemma A.1, (A.2) becomes
(1− α) cm + αTm+1hm + αTm+1 (d/dy)EhbΓm (y − γ (Tm + 1))
i= (1− α) cm + αTm+1hm + αTm+1{(1− α)
m−1Xi=1
ατ(i+1,m−1)ci
+m−1Xi=1
ατ(i,m−1) ddyE [Li (y − γ (Tm + 1)− γ (τ (i,m− 1)))]}
= (1− α) cm + αTm+1hm + (1− α)Xm−1
i=1ατ(i+1,m)ci
+Xm−1
i=1ατ(i,m) d
dyE [Li (y − γ (τ (i,m)))]
= (1− α)Xm
i=1ατ(i+1,m)ci + αTm+1hm +
Xm−1i=2
ατ(i,m)hi
+ατ(1,m)£h1 − (π +H1) Φ̄ (y; τ (1,m))
¤= (1− α)
Xm
i=1ατ(i+1,m)ci +
Xm
i=1ατ(i,m)hi − ατ(1,m) (π +H1) Φ̄ (y; τ (1,m))
Setting (A.2) to zero immediately yields the desired result.
Proof of Lemma 2. Using Lemma A.1 by setting Z to be zero and integrating
32
both sides from bSm−1 to y gives:bΓm (y)− bΓm ³bSm−1´
=
Z y
t=bSm−1{(1− α)m−1Xi=1
ατ(i+1,m−1)ci +m−1Xi=1
ατ(i,m−1) ddtE [Li (t− γ (τ (i,m− 1)))]}dt
=³y − bSm−1´ (1− α)
Xm−1i=1
ατ(i+1,m−1)ci
+Xm−1
i=1ατ(i,m−1)
Z y
t=bSm−1 hidt− ατ(1,m−1) (π +H1)Z y
t=bSm−1 Φ̄ (t; τ (1,m− 1)) dt=
³y − bSm−1´Xm−1
i=1ατ(i+1,m−1) £(1− α) ci + αTi+1hi
¤−ατ(1,m−1) (π +H1)
Z y
t=bSm−1 Φ̄ (t; τ (1,m− 1)) dt.
The result follows from noting that bΓm ³bSm−1´ = 0.For Normal demand, note that Θ0 (x) = Φ (x), so that
Z y
t=bSm−1 Φ̄ (t; τ (1,m− 1)) dt=
Z y
t=bSm−1 Φµ− (t− µ (τ (1,m− 1)))
σ (τ (1,m− 1))¶dt
= −σ (τ (1,m− 1))hΘ (ι (y))−Θ
³ι³bSm−1´´i .
Here we state two lemmas that are useful for calculations involving Normal demands.
The proofs of these two lemmas are quite tedious and are given in Dong (1999).
Lemma A.2
(1/Σ)
Z ∞
t=−∞(− (a− t) /b)Φ (− (a− t) /b)φ ((t− µ) /Σ) dt
=Σ2
b√Σ2 + b2
φ
µµ− a√Σ2 + b2
¶+µ− ab
Φ
µµ− a√Σ2 + b2
¶.
33
Lemma A.3
(1/Σ)
Z ∞
t=−∞φ (− (a− t) /b)φ ((t− µ) /Σ) dt
=³b/√b2 + Σ2
´³1/√2π´exp
Ã− (a− µ)22 (b2 + Σ2)
!.
Lemma A.4
EhbΓm ³bSm − γ (Tm + 1)
´i=
³bSm − µ (Tm + 1)− bSm−1´Xm−1i=1
ατ(i+1,m−1) £(1− α) ci + αTi+1hi¤
−ατ(1,m−1) (π +H1)E
"Z bSm−γ(Tm+1)t=bSm−1 Φ (t; τ (1,m− 1)) dt
#.
If the end demand per period is Normal with mean µ and variance σ2, then
EhbΓm ³bSm − γ (Tm + 1)
´i=
(m−1Xi=1
ατ(i+1,m−1) £(1− α) ci + αTi+1hi¤)ש
Φ−1 (1−Km)σ (τ (1,m))− Φ−1 (1−Km−1)σ (τ (1,m− 1))ª
+ατ(1,m−1) (π +H1)×©Θ¡−Φ−1 (1−Km)
¢σ (τ (1,m))−Θ
¡−Φ−1 (1−Km−1)¢σ (τ (1,m− 1))ª .
Proof of Lemma A.4. The first part is a consequence of Lemma 2. For Normal
demand, we can write
bSm = µ (τ (1,m)) + Φ−1 (1−Km)σ (τ (1,m)) ,
34
for m = 1, ...,M . Hence
E
"Z bSm−γ(Tm+1)t=bSm−1 Φ (t; τ (1,m− 1)) dt
#= −σ (τ (1,m− 1))×
{Z ∞
t=−∞Θ³−hbSm − t− µ (τ (1,m− 1))i /σ (τ (1,m− 1))´ dΦ (t;Tm + 1)
−Θ ¡−Φ−1 (1−Km−1)¢}.
Let a = bSm − µ (τ (1,m− 1)), b = σ (τ (1,m− 1)), eµ = µ (Tm + 1), Σ = σ (Tm + 1),
then Z ∞
t=−∞Θ³−³bSm − t− µ (τ (1,m− 1))´ /σ (τ (1,m− 1))´ dΦ (t;Tm + 1)
= (1/Σ)
Z ∞
t=−∞Θ (− (a− t) /b)φ ((t− eµ) /Σ) dt
= (1/Σ)
Z ∞
t=−∞(− (a− t) /b)Φ (− (a− t) /b)φ ((t− eµ) /Σ) dt
+(1/Σ)
Z ∞
t=−∞φ (− (a− t) /b)φ ((t− eµ) /Σ) dt
=Σ2
b√Σ2 + b2
φ
µ eµ− a√Σ2 + b2
¶+eµ− ab
Φ
µ eµ− a√Σ2 + b2
¶+³b/√b2 + Σ2
´³1/√2π´exp
(−1/2(a− eµ)2
b2 + Σ2
)(by Lemmas A.2 & A.3)
= σ2 (Tm + 1) / (σ (τ (1,m− 1))σ (τ (1,m)))φ¡−Φ−1 (1−Km)
¢− (σ (τ (1,m)) /σ (τ (1,m− 1)))Φ−1 (1−Km)Φ
¡−Φ−1 (1−Km)¢
+σ (τ (1,m− 1)) /σ (τ (1,m))φ ¡Φ−1 (1−Km)¢
= (σ (τ (1,m)) /σ (τ (1,m− 1)))Θ ¡−Φ−1 (1−Km)¢.
The desired result follows.
Proof of Lemma 3. The result follows from taking the limit of bAαm as α→ 1.
See Dong (1999) for the algebraic details.
35
Proof of Theorem 3.n bAmoM
m=2is a telescope series. Using Lemma 3, we get
bA = µXM
m=1cm +
XM
m=1µ (τ (1,m− 1))
+σ (τ (1,M))£H1Φ
−1 (1−KM) + (π +H1)Θ¡−Φ−1 (1−KM)
¢¤.
Note that
H1Φ−1 (1−KM) + (π +H1)Θ
¡−Φ−1 (1−KM)¢
= H1Φ−1 (1−KM)
+ (π +H1) {φ¡−Φ−1 (1−KM)
¢− Φ−1 (1−KM)Φ¡−Φ−1 (1−KM)
¢}= (π +H1)φ
¡−Φ−1 (1−KM)¢,
since Φ (−Φ−1 (1−KM)) = KM = H1/ (π +H1) as α → 1, and the desired result
follows. The alternative expression can be obtained by using hm = Hm−Hm+1,HM+1 = 0.
Proof of Corollary 1. ∂ bA/∂Tm = Hm+1+ 12σ(τ(1,M))
(π +H1)φ (Φ−1 (1−KM)) ≥
0 form = 1, ...,M−1; ∂ bA/∂TM = 12σ(τ(1,M))
(π +H1)φ (Φ−1 (1−KM)) ≥ 0. ∂ bA/∂Tm ≥
∂ bA/∂Tm+1, since H1 ≥ H2 ≥ ... ≥ HM .Proof of Corollary 2. (1)
bAα2 −Aα
2 = αT2+1EhbΓ2 ³bS2 − γ (T2 + 1)
´− Γ2
³bS2 − γ (T2 + 1)´i≥ 0,
since bΓ2 (·) ≥ Γ2 (·) on <. (2) follows for the fact that bAα1 = A
α1 .
For the AR(1) case, we will make use of the property that
Dn = dXn−1
i=0ρi + ρnD0 +
Xn
i=1ρn−iεi,
36
µn = dXn−1
i=0ρi + ρnD0, (A.3)
σ2n = σ2¡1− ρ2n
¢/¡1− ρ2
¢,
D (1, n) = d/ (1− ρ)Xn
i=1
¡1− ρi
¢+D0
Xn
i=1ρi +
Xn
i=1
Xi
j=1ρi−jεj
= dXn
i=1
Xi−1j=0
ρj +D0Xn
i=1ρi + 1/ (1− ρ)
Xn
i=1
¡1− ρi
¢εi,
µ0 (n) = dXn
i=1
Xi−1j=0
ρj +D0Xn
i=1ρi,
σ20 (n) = σ2Xn
i=1
³Xi−1j=0
ρj´2.
Notice that the variances of Dn and D (1, n) are independent of historical demand D0,
hence we can drop subscript 0 from the variance notation.
Proof of Theorem 4. We first study the finite horizon case. With zero leadtime,
the dynamic programming formulation of the problem is
fn (xn,Dn) = minxn≤y
{c (y − xn) + Ln (xn) + αE [fn+1 (y −Dn+1,Dn+1)]} ,
where Ln (x) is convex in x for all n and fN+1 (·, ·) = 0. Note that the expectation is taken
over a finite number of multi-variate random variables, no anomaly should be encountered.
Let V ∗ be the set of functions v (·, ·) that are convex w.r.t. the first variable;
π∗ = ∆∗×∆∗× ...×∆∗ (∆∗ appears N times), where ∆∗ is the set of base-stock policies,
and for δ ∈ ∆∗, δ (x,D) = S∗ (D) if x < S∗ (D), δ (x,D) = x otherwise.
The following three steps are sufficient to show that a base-stock policy is optimal
for period n and for every n, and the optimal base-stock level is a function of demand in
period n.
37
• It can be easily shown that, if fn+1 ∈ V ∗, then E [fn+1 (y −Dn+1,Dn+1)] is convex in
y.
• If fn+1 ∈ V ∗, then there exists δ ∈ ∆∗ that is optimal for period n. To see this, suppose
fn+1 ∈ V ∗. Now, we have cyn + Ln (xn) + αE [fn+1 (yn −Dn+1,Dn+1)] being convex
in yn. Let S∗n be a minimizer of cyn + Ln (xn) + αE [fn+1 (yn −Dn+1,Dn+1)], which is
clearly a function of Dn. Let δ (xn,Dn) = S∗n (Dn) if xn < S∗n (Dn), δ (xn,Dn) = xn
otherwise. Hence, following δ (xn,Dn) is optimal for period n.
• If fn+1 ∈ V ∗, then fn ∈ V ∗. This follows from:
fn (xn,Dn)
=
c (S∗n (Dn)− xn) + Ln (xn) + αE [fn+1 (S
∗n (Dn)−Dn+1,Dn+1)] if xn < S∗n (Dn) ;
Ln (xn) + αE [fn+1 (xn −Dn+1,Dn+1)] otherwise,
and that fn (xn,Dn) is convex and has continuous derivative with respect to xn.
The nonzero leadtime case can be extended as in standard inventory literature.
Finally, we can extend the result to the infinite horizon case using Proposition 1.6 and
Proposition 1.7 in Section 3.1 of Bertsekas (1995), such that the limit of S∗n(D) converges
to an optimal stationary policy S∗(D) as n→∞.
Proof of Theorem 6. Similar to that of Theorem 1. See Dong (1999) for
details.
Lemma A.5 For all m > 1 and for a random variable D (1, T ), we have:
(d/dy)EhbΓm (y −D (1, T ) ,DT )i
= (1− α)Xm−1
i=1ατ(i+1,m−1)ci +
Xm−1i=1
ατ(i,m−1) ddyE [Li (y −D (1, T + τ (i,m− 1)))] .
38
Proof of Lemma A.5. The proof is similar to that of Lemma A.1. See Dong
(1999) for details.
Proof of Lemmas 4 and 5 .
Similar to that of Lemmas 1 and 2. See Dong (1999) for details.
Lemma A.6
EhbΓm ³bSm (D0)−D (1, Tm + 1) , DTm+1´i
=nXm−1
i=1ατ(i+1,m−1) £(1− α) ci + αTi+1hi
¤oשΦ−1 (1−Km)σ (τ (1,m))− Φ−1 (1−Km−1)σ (τ (1,m− 1))
ª+ατ(1,m−1) (π +H1)שσ (τ (1,m))Θ
¡−Φ−1 (1−Km)¢− σ (τ (1,m− 1))Θ ¡−Φ−1 (1−Km−1)
¢ª.
Proof of Lemma A.6. The proof is similar to that of Lemma A.4. See Dong
(1999) for details.
Proof of Theorem 8. The first equality is similar to the proof of Theorem 2.
For the second equality, we note that
µ0 (τ (1,m))− µ0 (Tm + 1) (A.4)
= dXτ(1,m)
i=1
Xi−1j=0
ρj +D0Xτ(1,m)
i=1ρi − d
XTm+1
i=1
Xi−1j=0
ρj −D0XTm+1
i=1ρi
= dXτ(1,m)
i=Tm+1+1
Xi−1j=0
ρj +D0Xτ(1,m)
i=Tm+1+1ρi,
39
and
E£µTm+1 (τ (1,m− 1))
¤(A.5)
= E
·dXτ(1,m−1)
i=1
Xi−1j=0
ρj +DTm+1Xτ(1,m−1)
i=1ρi¸
= dXτ(1,m−1)
i=1
Xi−1j=0
ρj +³dXTm
j=0ρj + ρTm+1D0
´Xτ(1,m−1)i=1
ρi
= dXτ(1,m−1)
i=1
³Xi−1j=0
ρj +XTm
j=0ρj+i
´+D0
Xτ(1,m−1)i=1
ρi+Tm+1
= dXτ(1,m−1)
i=1
Xi+Tm
j=0ρj +D0
Xτ(1,m)
i=Tm+1+1ρi
= dXτ(1,m)
i=Tm+1+1
Xi−1j=0
ρj +D0Xτ(1,m)
i=Tm+1+1ρi.
Proof of Corollary 3. Since σ (τ (1,M)) is nondecreasing in T1, ..., TM and
∂σ (τ (1,M)) /∂Tm = ∂σ (τ (1,M)) /∂Tm+1, we only need to check the termPM
m=1 hm×
E[µTm+1 (τ (1,m− 1))]. Note that
XM
m=1hmE
£µTm+1 (τ (1,m− 1))
¤=
XM
m=1hmE
·dXτ(1,m−1)
i=1
Xi−1j=0
ρj +DTm+1Xτ(1,m−1)
i=1ρi¸
which is nondecreasing in T1, ..., TM . To show ∂ bA (D0) /∂Tm ≥ ∂ bA (D0) /∂Tm+1, weexamine:
∂³XM
m=1hmE
£µTm+1 (τ (1,m− 1))
¤´/∂Ti
= ∂hiE£µTi+1 (τ (1, i− 1))
¤/∂Ti + ∂
³XM
m=i+1hmE
£µTm+1 (τ (1,m− 1))
¤´/∂Ti
40
It follows that
∂³XM
m=1hmE
£µTm+1 (τ (1,m− 1))
¤´/∂Ti
−∂³XM
m=1hmE
£µTm+1 (τ (1,m− 1))
¤´/∂Ti+1
= ∂¡hiE
£µTi+1 (τ (1, i− 1))
¤¢/∂Ti + ∂
³XM
m=i+1hmE
£µTm+1 (τ (1,m− 1))
¤´/∂Ti
−∂ ¡hi+1E £µTi+1+1 (τ (1, i))¤¢ /∂Ti+1−∂
³XM
m=i+2hmE
£µTm+1 (τ (1,m− 1))
¤´/∂Ti+1
= ∂¡hiE
£µTi+1 (τ (1, i− 1))
¤¢/∂Ti + ∂
¡hi+1E
£µTi+1+1 (τ (1, i))
¤¢/∂Ti
−∂ ¡hi+1E £µTi+1+1 (τ (1, i))¤¢ /∂Ti+1.Since
∂E£µTi+1+1 (τ (1, i))
¤/∂Ti − ∂E
£µTi+1+1 (τ (1, i))
¤/∂Ti+1
= ∂ [µ0 (τ (1, i+ 1))− µ0 (Ti+1 + 1)] /∂Ti − ∂ [µ0 (τ (1, i+ 1))− µ0 (Ti+1 + 1)] /∂Ti+1
= ∂µ0 (τ (1, i+ 1)) /∂Ti − ∂µ0 (τ (1, i+ 1)) /∂Ti+1 − µ0 (Ti+1 + 1) /∂Ti
+∂µ0 (Ti+1 + 1) /∂Ti+1
and
∂µ0 (τ (1, i+ 1)) /∂Ti = ∂µ0 (τ (1, i+ 1)) /∂Ti+1,
µ0 (Ti+1 + 1) /∂Ti = 0,
∂µ0 (Ti+1 + 1) /∂Ti+1 ≥ 0,
41
it suffices to show that ∂E£µTi+1 (τ (1, i− 1))
¤/∂Ti ≥ 0, which is true because
E£µTi+1 (τ (1, i− 1))
¤= d
Xτ(1,i−1)k=1
Xk−1j=0
ρj +E [DTi+1]Xτ(1,i−1)
j=1ρj.
Lemma A.7 (1) E£µTm+1 (τ (1,m− 1))
¤is nondecreasing in ρ;
(2) ∂E£µTm+1 (τ (1,m− 1))
¤/∂ρ is nondecreasing in T1, T2, ..., Tm.
Proof of Lemma A.7. (1) Taking first derivative w.r.t. ρ:
∂E£µTm+1 (τ (1,m− 1))
¤/∂ρ
= dXτ(1,m)
i=1
Xi−1j=0jρj−1 + ∂E [DTm+1] /∂ρ
Xτ(1,m−1)i=1
ρi +E [DTm+1]Xτ(1,m−1)
i=1iρi−1
≥ 0.
where ∂E [DTm+1] /∂ρ ≥ 0 follows from (A.3). (2) It is now obvious that
∂E£µTm+1 (τ (1,m− 1))
¤/∂ρ is nondecreasing in T1, T2, ..., Tm.
Lemma A.8 (1) σ (τ (1,M)) is nondecreasing in ρ;
(2) ∂σ (τ (1,M)) /∂ρ is nondecreasing in T1, T2, ..., TM .
Proof of Lemma A.8. (1) In general, for any n,
∂σ (n) /∂ρ
= ∂
rσ2Xn
i=1
³Xi−1j=0
ρj´2/∂ρ
= σnXn
i=1
³Xi−1j=0
ρj´³Xi−1
j=0jρj−1
´o/
rXn
i=1
³Xi−1j=0
ρj´2≥ 0.
42
(2) We need to show
nXn+1
i=1
³Xi−1j=0
ρj´³Xi−1
j=0jρj−1
´o/
rXn+1
i=1
³Xi−1j=0
ρj´2
≥nXn
i=1
³Xi−1j=0
ρj´³Xi−1
j=0jρj−1
´o/
rXn
i=1
³Xi−1j=0
ρj´2.
Let ai =³Pi−1
j=0 ρj´2, bi =
³Pi−1j=0 ρ
j´³Pi−1
j=0 jρj−1´, we will show that
nXn+1
i=1bio/
rXn+1
i=1ai ≥
nXn
i=1bio/
rXn
i=1ai.
It is equivalent to show that
³Xn
i=1bi + bn+1
´2Xn
i=1ai ≥
³Xn
i=1bi´2 ³Xn
i=1ai + an+1
´or
2bn+1Xn
i=1biXn
i=1ai + b
2n+1
Xn
i=1ai ≥
³Xn
i=1bi´2an+1
or
2bn+1Xn
i=1ai ≥
³Xn
i=1bi´an+1
or Xn
i=1(2bn+1ai − bian+1) ≥ 0.
To show that 2bn+1ai − bian+1 ≥ 0 for i = 1, ..n, note that
2bn+1ai − bian+1
=³Xi−1
j=0ρj´³Xn
j=0ρj´×n
2³Xn
j=0jρj−1
´³Xi−1k=0
ρk´−³Xi−1
k=0kρk−1
´³Xn
j=0ρj´o
≥ 0,
43
since
2³Xn
j=0jρj−1
´³Xi−1k=0
ρk´−³Xi−1
k=0kρk−1
´³Xn
j=0ρj´
=Xn
j=0
Xi−1k=0(2j − k) ρj+k−1
=Xi−1
j=0
Xi−1k=0(2j − k) ρj+k−1 +
Xn
j=i
Xi−1k=0(2j − k) ρj+k−1
≥Xi−1
j=0
Xi−1k=0(j − k) ρj+k−1
=Xi−1
j=0
Xj−1k=0(j − k) ρj+k−1 +
Xi−1j=0
Xk=j(j − k) ρj+k−1
+Xi−1
j=0
Xi−1k=j+1
(j − k) ρj+k−1
=Xi−1
j=0
Xj−1k=0(j − k) ρj+k−1 +
Xi−1j=0
Xi−1k=j+1
(j − k) ρj+k−1
=Xi−1
j=0
Xj−1k=0(j − k) ρj+k−1 +
Xi−1k=0
Xk−1j=0(j − k) ρj+k−1
= 0.
Proof of Theorem 9. Straightforward by Lemmas A.7 and A.8.
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49
498
499
500
501
502
503
504
505
σ /µ
Inve
ntor
y le
vel
S2^ 500.15 500.18 500.21 500.24 500.27 500.3 500.33 500.37 500.4 500.43 500.46
S2* 501.47 501.76 502.06 502.35 502.65 502.94 503.23 503.53 503.82 504.12 504.41
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Figure 1: Inventory Level Comparison: Optimal vs. Approximation (Normal Distri-
bution)
2350
2400
2450
2500
2550
2600
2650
2700
σ /µ
Syst
em C
ost
A^ 2470.2 2487.05 2503.89 2520.73 2537.57 2554.41 2571.25 2588.09 2604.93 2621.77 2638.61
A 2469.04 2485.65 2502.26 2518.87 2535.48 2552.09 2568.7 2585.31 2601.92 2618.53 2635.13
A - 2468.94 2485.52 2502.11 2518.7 2535.28 2551.87 2568.46 2585.05 2601.63 2618.22 2634.81
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Figure 2: System Cost Comparison: Optimal vs. Approximation (Normal Distribu-
tion)
50
710
720
730
740
750
760
770
σ/ µ
Inve
ntor
y le
vel
S3^ 714.97 717.97 720.96 723.96 726.95 729.95 732.94 735.94 738.93 741.93 744.92
S3* 721.47 725.77 730.06 734.36 738.65 742.95 747.24 751.54 755.83 760.13 764.42
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Figure 3: Inventory Level Comparison for A Three-echelon System: Optimal vs.
Approximation (Normal Distribution)
2600
2650
2700
2750
2800
2850
2900
σ/ µ
Syst
em C
ost
A^ 2625.9 2651.1 2676.3 2701.4 2726.6 2751.8 2777 2802.2 2827.3 2852.5 2877.7
A 2618.4 2642.1 2665.8 2689.5 2713.1 2736.8 2760.5 2784.2 2807.9 2831.6 2855.2
A_ 2615 2638 2661 2683.9 2706.9 2729.9 2752.9 2775.9 2798.9 2821.9 2844.9
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Figure 4: System cost Comparison for A Three-echelon System: Optimal vs. Approx-
imation (Normal Distribution)
51
541
542
543
544
545
546
547
548
549
550
σ/d
Inve
ntor
y le
vel
S2^ 544.487 544.52 544.553 544.586 544.619 544.652 544.685 544.718 544.751 544.785 544.818
S2* 546.042 546.386 546.731 547.075 547.419 547.763 548.108 548.452 548.796 549.14 549.484
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Figure 5: Inventory Level Comparison: Optimal vs. Approximation (AR(1) Process)
2450
2500
2550
2600
2650
2700
2750
2800
σ/ d
Syst
em C
ost
A^ 2589 2607 2626 2662 2680 2699 2717 2753 2772A 2588 2606 2624 2660 2678 2696 2714 2750 2768A- 2587 2605 2623 2659 2677 2695 2713 2749 2767
0.1 0.12 0.14 0.18 0.2 0.22 0.24 0.28 0.3
Figure 6: System Cost Comparison: Optimal vs. Approximation (AR(1) Process)
52
0
500
1000
1500
2000
2500
3000
ρ
S2*
T1=2 529.4971 576.6876 632.1269 697.6646 775.4779 868.0883 978.3771 1109.6 1265.402 1449.824
T1=3 631.2226 689.7189 759.2915 842.9569 944.6014 1069.154 1222.77 1413.024 1649.114 1942.058
T1=4 732.9189 802.7237 886.4328 988.2616 1113.977 1271.417 1471.125 1727.104 2057.692 2486.562
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 7: Inventory Level vs. Auto-correlation for Various Leadtimes (AR(1) Process)
2000
3000
4000
5000
6000
7000
ρ
T1=2 2306.189 2404.918 2521.571 2660.098 2825.148 3022.093 3257.068 3537.001 3869.644 4263.598
T1=3 2528.266 2652.329 2800.806 2980.314 3199.336 3468.593 3801.426 4214.202 4726.742 5362.754
T1=4 2748.097 2897.046 3076.753 3296.564 3569.218 3911.968 4347.936 4907.721 5631.267 6569.989
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
A
Figure 8: System Cost vs. Auto-correlation for Various Leadtimes (AR(1) Process)
53
Param eter R a t io bS2 S∗2O p t im a lG ap
bAα Aα AαO p tim a lG ap
σ/µ
0.10.120.140.160.180.200.220.240.260.280.30
500.15500.18500.21500.24500.27500.30500.33500.37500.40500.43500.46
501.47501.76502.06502.35502.65502.94503.23503.53503.82504.12504.41
0.26%0.32%0.37%0.42%0.47%0.52%0.58%0.63%0.68%0.73%0.78%
2470.202487.052503.892520.732537.572554.412571.252588.092604.932621.772638.61
2469.042485.652502.262518.872535.482552.092568.702585.312601.922618.532635.13
2468.942485.522502.112518.702535.282551.872568.462585.052601.632618.222634.81
0.05%0.06%0.07%0.08%0.09%0.10%0.11%0.12%0.13%0.14%0.14%
c2/c1
0.10.20.30.40.50.60.70.80.91
510.28507.38504.51501.66498.82495.97493.10490.21487.27484.28
527.52516.55509.68504.55500.37496.73493.43490.32487.30484.28
3.27%1.77%1.01%0.57%0.31%0.15%0.07%0.02%0.01%0.00%
1629.971892.272153.722414.312674.032932.853190.733447.613703.373957.78
1618.691885.002149.272411.762672.682932.213190.473447.533703.353957.78
1613.741883.062148.532411.502672.602932.193190.473447.533703.353957.78
1.01%0.49%0.24%0.12%0.05%0.02%0.01%0.00%0.00%0.00%
H1/c1
0.10.20.30.40.50.60.70.80.91
522.24513.37506.26500.30495.15490.59486.48482.73479.27476.05
527.31517.26509.42502.94497.39492.52488.16484.21480.57477.20
0.96%0.75%0.62%0.52%0.45%0.39%0.34%0.30%0.27%0.24%
2137.272279.902418.632554.412687.862819.402949.363077.973205.403331.81
2133.942276.962416.022552.092685.802817.582947.743076.543204.143330.69
2133.442276.592415.742551.872685.632817.442947.643076.453204.073330.64
0.18%0.15%0.12%0.10%0.08%0.07%0.06%0.05%0.04%0.04%
H2/c2
0.10.20.30.40.50.60.70.80.91
504.78503.28501.79500.30498.82497.33495.83494.34492.83491.32
515.60510.09506.08502.94500.37498.17496.25494.50492.88491.32
2.10%1.33%0.85%0.52%0.31%0.17%0.08%0.03%0.01%0.00%
2192.982313.872434.352554.412674.032793.192911.843029.933147.353263.87
2184.772308.232430.632552.092672.682792.472911.513029.803147.313263.87
2182.212307.032430.102551.872672.602792.442911.503029.803147.313263.87
0.49%0.30%0.17%0.10%0.05%0.03%0.01%0.00%0.00%0.00%
π/c1
0.51.01.52.02.53.03.54.04.55.0
466.43500.30515.27524.74531.58536.88541.19544.80547.90550.60
467.46502.94518.67528.60535.75541.28545.76549.50552.71555.51
0.22%0.52%0.66%0.73%0.78%0.81%0.84%0.86%0.87%0.88%
2473.912554.412598.552628.672651.332669.392684.332697.032708.062717.78
2473.532552.092594.462623.122644.562661.572675.612687.512697.822706.89
2473.522551.872594.012622.472643.742660.612674.522686.322696.532705.51
0.02%0.10%0.18%0.24%0.29%0.33%0.37%0.40%0.43%0.45%
T2/T1
0.51.01.52.02.53.03.54.04.55.0
500.30599.51698.48797.20895.64993.771091.561188.971285.941382.41
502.94603.70704.02803.90903.321002.271100.721198.621295.951392.65
0.52%0.69%0.79%0.83%0.85%0.85%0.83%0.81%0.77%0.74%
2554.412562.652572.832585.232599.952617.002636.322657.822681.382706.86
2552.092558.842567.902579.552593.852610.752630.142651.872675.782701.71
2551.872558.402567.272578.782593.002609.862629.252651.022674.992700.99
0.10%0.17%0.22%0.25%0.27%0.27%0.27%0.26%0.24%0.22%
Table 1: Stocking Levels and System Costs Comparisons: IID Normal Demand
54
Param eter R a t io bS3 S∗3O p t im a lG ap
bA A AO p t im a lG ap
σ/µ
0.10.120.140.160.180.200.220.240.260.280.30
714.97717.97720.96723.96726.95729.95732.94735.94738.93741.94744.92
721.47725.77730.06734.36738.65742.95747.24751.54755.83760.13764.42
0.90%1.07%1.25%1.42%1.58%1.75%1.91%2.08%2.24%2.39%2.55%
2625.902651.082676.262701.442726.622751.802776.992802.172827.352852.532877.71
2618.412642.102665.782689.462713.142736.832760.512784.192807.872831.562855.24
2614.972637.962660.962683.952706.942729.942752.932775.922798.922821.912844.90
0.13%0.16%0.18%0.21%0.23%0.25%0.28%0.30%0.32%0.34%0.36%
c2/c1
0.30.40.50.60.7
714.97714.97714.97714.97714.97
722.67721.47720.77720.27720.07
1.07%0.90%0.80%0.74%0.71%
2275.902525.902775.903025.903275.90
2267.812518.412769.363020.203270.74
2261.592514.972767.193018.583269.34
0.28%0.14%0.08%0.05%0.04%
H1/c1
0.30.40.50.60.70.80.91.0
719.48714.97711.40708.43705.90703.70701.75700
724.88721.47718.80716.63714.70713.00711.55710.3
0.74%0.90%1.03%1.14%1.23%1.30%1.38%1.45%
2454.632625.902794.302960.523125.033288.153450.113611.10
2448.992618.412785.352950.353113.813276.023437.183597.46
2445.992614.972781.252945.563108.323269.853430.353590.00
0.12%0.13%0.15%0.16%0.18%0.19%0.20%0.21%
H2/c2
0.30.40.50.60.7
714.97714.97714.97714.97714.97
722.67721.47720.77720.27720.07
1.07%0.90%0.80%0.74%0.71%
2475.902625.902775.902925.903075.90
2467.812618.412769.362920.203070.74
2461.592614.972767.192918.583069.34
0.25%0.13%0.08%0.06%0.05%
π/c1
0.51.01.52.02.53.03.54.04.55.0
703.70714.97721.29725.60728.83731.40733.52735.33736.89738.26
709.20721.47728.19732.80736.13738.90741.02742.93744.49745.96
0.78%0.90%0.95%0.98%0.99%1.02%1.01%1.02%1.02%1.03%
2594.072625.902645.092658.652669.052677.452684.462690.462695.702700.33
2589.782618.412635.332647.152656.142663.352669.342674.452678.892682.81
2587.712614.972631.042642.262650.812657.672663.372668.242672.482676.22
0.08%0.13%0.16%0.18%0.20%0.21%0.22%0.23%0.24%0.25%
T2/T1
0.20.40.60.81.01.21.41.61.82.0
654.32694.76735.19775.60816.01856.40896.79937.16977.541017.91
660.52701.16741.89782.40823.01863.70904.29944.86985.441026.00
0.94%0.91%0.90%0.87%0.85%0.85%0.83%0.81%0.80%0.79%
2560.392604.092647.692691.192734.602777.922821.172864.342907.442950.48
2553.332596.762640.032683.162726.182769.092811.902854.632897.292939.87
2550.492593.522636.372679.082721.672764.152806.542848.842891.082933.25
0.11%0.12%0.14%0.15%0.17%0.18%0.19%0.20%0.21%0.23%
T3/T1
0.20.40.60.81.01.21.41.61.82.0
654.32694.76735.19775.60816.01856.40896.79937.17977.541017.90
659.52700.86742.09783.30824.51865.60906.69947.67988.741029.70
0.79%0.87%0.93%0.98%1.03%1.06%1.09%1.11%1.13%1.15%
2620.392624.092627.692631.192634.602637.922641.172644.342647.442650.48
2614.802617.242619.562621.772623.902625.942627.922629.842631.712633.52
2612.032614.022615.882617.642619.312620.902622.432623.902625.322626.69
0.11%0.12%0.14%0.16%0.18%0.19%0.21%0.23%0.24%0.26%
Table 2: Three-echelon Stocking Levels and System Costs Comparisons: IID Normal Demand
55
Param eter R a t io bS2 S∗2O p tim a lG ap
bA A AO p tim a lG ap
σ/µ
0.10.120.140.160.180.200.220.240.260.280.30
544.49544.52544.55544.59544.62544.65544.69544.72544.75544.78544.82
546.04546.39546.73547.07547.42547.76548.11548.45548.80549.14549.48
0.28%0.34%0.40%0.45%0.51%0.57%0.62%0.68%0.74%0.79%0.85%
2588.972607.252625.532643.812662.082680.362698.642716.922735.192753.472771.75
2587.572605.562623.562641.562659.552677.552695.542713.542731.542749.532767.53
2587.432605.402623.372641.32659.302677.272695.242713.212731.172749.142767.11
0.06%0.07%0.08%0.09%0.10%0.12%0.13%0.14%0.15%0.16%0.17%
ρ
00.10.20.30.40.50.60.70.80.9
500.30544.65596.62657.89730.40816.44918.561039.671183.001352.13
502.94547.76600.37662.51736.23823.89928.211052.231199.361373.40
0.52%0.57%0.62%0.70%0.79%0.90%1.04%1.19%1.36%1.55%
1244.291331.301435.791561.851714.321898.842121.842390.672713.553099.65
1241.971328.491432.291557.371708.451890.962111.162376.092693.663072.63
1241.751328.211431.921556.871707.741889.962109.692373.942690.523068.11
0.20%0.23%0.27%0.32%0.39%0.47%0.58%0.70%0.86%1.03%
c2/c1
0.10.20.30.40.50.60.70.80.91
555.50552.35549.23546.13543.03539.94536.82533.67530.48527.23
575.02562.84555.22549.53544.89540.88537.24533.82530.52527.23
3.40%1.86%1.08%0.62%0.34%0.17%0.08%0.03%0.01%0.00%
1694.171975.342255.602534.922813.303090.713367.103642.413916.514189.15
1681.131966.832250.312531.832811.633089.903366.763642.293916.494189.15
1675.341964.512249.402531.512811.533089.873366.763642.293916.494189.15
1.12%0.55%0.28%0.13%0.06%0.03%0.01%0.00%0.00%0.00%
H1/c1
0.10.20.30.40.50.60.70.80.91
568.50558.85551.13544.65539.05534.09529.62525.55521.78518.28
574.36563.38554.83547.76541.71536.40531.64527.33523.36519.68
1.02%0.80%0.67%0.57%0.49%0.43%0.38%0.34%0.30%0.27%
2218.392376.282529.922680.362828.272974.113118.223260.873402.243542.49
2214.442372.752526.772677.552825.762971.873116.233259.093400.663541.09
2213.832372.302526.422677.272825.532971.693116.083258.983400.573541.02
0.21%0.17%0.14%0.12%0.10%0.08%0.07%0.06%0.05%0.04%
H2/c2
0.10.20.30.40.50.60.70.80.91
549.52547.89546.27544.65543.03541.41539.79538.16536.53534.88
561.88555.74551.26547.76544.89542.45540.31538.38536.59534.89
2.20%1.41%0.91%0.57%0.34%0.19%0.10%0.04%0.01%0.00%
2278.772413.082546.952680.362813.302945.743077.643208.933339.503469.11
2269.192406.422542.512677.552811.632944.843077.213208.763339.463469.11
2266.152404.972541.852677.272811.532944.803077.203208.763339.463469.11
0.56%0.34%0.20%0.12%0.06%0.03%0.01%0.01%0.00%0.00%
π/c1
0.51.01.52.02.53.03.54.04.55.0
507.82544.65560.92571.22578.65584.42589.11593.03596.40599.34
509.09547.76564.89575.70583.47589.49594.36598.43601.92604.96
0.25%0.57%0.70%0.78%0.83%0.86%0.88%0.90%0.92%0.93%
2592.912680.362728.312761.022785.642805.252821.482835.282847.262857.81
2592.432677.552723.402754.402777.582795.982811.142824.012835.142844.94
2592.412677.272722.842753.602776.582794.812809.832822.572833.592843.29
0.02%0.12%0.20%0.27%0.33%0.37%0.41%0.45%0.48%0.51%
T2/T1
0.51.01.52.02.53.03.54.04.55.0
544.65654.90764.88874.58983.971093.031201.701309.951417.711524.93
547.76659.78771.31882.32992.821102.791212.201321.021429.181536.64
0.57%0.74%0.83%0.88%0.89%0.89%0.87%0.84%0.80%0.76%
2680.362689.952701.442715.322731.722750.692772.162796.032822.182850.47
2677.552685.412695.622708.642724.592743.402764.962789.122815.692844.50
2677.272684.872694.852707.712723.562742.342763.912788.112814.762843.66
0.12%0.19%0.24%0.28%0.30%0.30%0.30%0.28%0.26%0.24%
Table 3: Stocking Levels and System Costs Comparisons: AR(1) Demand
56
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