Analysis of a two-echelon inventory system with two supply modes

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DOI: 10.1016/j.ejor.2012.09.043
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Abstract
In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We extend results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and illustrate that dual-sourcing can lead to significant cost savings in cases with high demand uncertainty, high backlogging cost or long lead times.
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Production, Manufacturing and Logistics
Analysis of a two-echelon inventory system with two supply modes
Joachim Arts
a,
, Gudrun P. Kiesmüller
b
a
Eindhoven University of Technology, School of Industrial Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
b
Christian-Albrechts-University at Kiel, Olshausenstraße 40, 24098 Kiel, Germany
article info
Article history:
Received 19 December 2010
Accepted 24 September 2012
Available online 17 October 2012
Keywords:
Inventory
Dual-sourcing
Dual-index policy
Markov chain
Lead times
Multi-echelon
abstract
In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes
at the most upstream stock point. As control policy for this system, we propose a natural extension of the
dual-index policy, which has three base-stock levels. We consider the minimization of long run average
inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested
newsboy characterizations for two of the three base-stock levels involved and show a separability result
for the difference with the remaining base-stock level. We extend results for the single-echelon system to
efficiently approximate the distributions of random variables involved in the newsboy equations and find
an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based
on these results, we provide an algorithm for setting base-stock levels in a computationally efficient man-
ner. In a numerical study, we investigate the value of dual-sourcing in supply chains and illustrate that
dual-sourcing can lead to significant cost savings in cases with high demand uncertainty, high backlog-
ging cost or long lead times.
Ó2012 Elsevier B.V. All rights reserved.
1. Introduction
Modern supply networks are complex and often consist of many
manufacturing facilities and inventory locations spread over the
continents. In recent decades, many European companies have
switched production to Asia and have built new production plants
there, adding to the globalization of supply chains. However, in or-
der to stay competitive and be flexible, they maintain the possibil-
ity of manufacturing in Europe as well, albeit at a higher price. As a
consequence, inventory managers have two different options for
replenishments, differing mainly in costs and lead times. With
the growing complexity of supply chains, situations with one buyer
and several supply options have become increasingly common.
Nevertheless, quantitative modeling approaches to analyze these
supply networks are limited. Although there is a large body of
literature on inventory management in supply chains, most
authors consider single vendor/single buyer relationships or single
vendor/multiple buyer relationships. Furthermore, most research
on multiple supplier inventory systems is restricted to a single
inventory location.
In this article, we extend the existing literature on inventory
systems with two supply modes, mainly focussing on single stage
systems, and present a two-echelon inventory system with two
supply options where the leadtimes can differ arbitrarily. More
precisely, we consider a supply chain composed of a retailer and
a warehouse, both belonging to the same company. We restrict
ourselves to a supply chain offering only one single product. While
the retailer is always replenished by the warehouse, there are two
alternatives for refilling the stock of the warehouse. Ordering at the
first option results in a low per unit price but a long leadtime has to
be taken into account. We call this supply option the regular sup-
plier. In contrast to this, the second supply option is able to deliver
products within a short time period, but charges a higher per unit
price. In general, this second supply option is not used very often
and therefore, an additional effort is necessary to release an order,
resulting in fixed cost for each order placed. This supply option is
called the emergency supplier.
Situations as described above can for example be found in real-
ity if the warehouse can place orders at two different manufactur-
ing facilities (van Geest, 2007). As mentioned before, many
companies have built production plants in China, where labor costs
are low. However, the goods are often transported to Europe by
ship resulting in a long leadtime. Therefore, in order to remain flex-
ible, not all production facilities in Europe are closed and the op-
tion to order products at a production facility close by is still
available, but for a higher price. Additionally, the developed model
can also be applied if the warehouse is only supplied by one man-
ufacturing facility, but two different transportation modes are
available, a fast and expensive one (plane) or a slow and cheap
one (ship).
The flexibility of the fast supply mode may be necessary,
because the retailer has to satisfy uncertain demand, modeled as
a discrete random variable. We assume that customer demand
0377-2217/$ - see front matter Ó2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ejor.2012.09.043
Corresponding author. Tel.: +31 402415822.
E-mail address: j.j.arts@tue.nl (J. Arts).
European Journal of Operational Research 225 (2013) 263–272
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Author's personal copy
only occurs at the retailer and due to the stochastic nature of the
demand, it can happen that there is no stock available at the retai-
ler when a customer wants to buy the product. We assume that in
such a situation the customer is willing to wait and demand is
backordered.
In order to avoid such situations, the retailer as well as the
warehouse have to keep safety stocks. Due to the low product
price, the inventory manager of the warehouse probably prefers
to place orders at the cheaper supplier with the long leadtime. As
a consequence, a high safety stock is needed resulting in high hold-
ing costs. A reduction of the safety stock and the corresponding
costs can be obtained if more orders are placed at the supplier with
the shorter leadtime, resulting in larger procurement cost. We are
interested in the optimal balance between holding costs and pro-
curement cost and the orders to be placed at the emergency sup-
plier and the regular supplier.
Many planning systems for supply chain management support
periodic decision making and therefore we assume that inventories
are reviewed periodically and as a consequence, ordering decisions
are also made periodically at the beginning of each review period
where for both stockpoints the same review period is applied. In or-
der to replenish the inventories at the retailer and the warehouse, a
central control system is used. This means that decisions about the
order quantities at the warehouse are based on information of the
whole supply chain. Further, the order quantities at the retailers
are never larger than the available inventory at the warehouse.
The inventory manager has to make three decisions about the
quantities to be ordered at the warehouse, at the regular supplier
and at the emergency supplier. Since no restrictive assumptions
for the lead times are made, such as a one period lead time differ-
ence for the most upstream stockpoint, the optimal control policy
is not known and expected to have a complex structure. (Not even
for the single-echelon case can the optimal policy structure be ob-
tained or computed for general lead time differences (Fukuda,
1964; Whittmore and Saunders, 1977; Feng et al., 2006a,b)). There-
fore, we suggest a natural extension of the dual-index policy for the
control of the system, because this policy is optimal in case of a sin-
gle stockpoint with two supply modes with leadtime difference
one, and it has been shown to work well in case of dual sourcing
with general leadtime difference (Veeraraghavan and Scheller-
Wolf, 2008). This policy keeps track of three inventory positions
with different aggregated information. Each one is compared with
its corresponding order-up-to level to determine the amount to
be ordered to raise the inventory position to its order-up-to level.
In this paper we show how to compute near optimal order-up-
to levels in an efficient manner, minimizing the operational costs
of the system, composed of ordering costs, holding and backorder
costs. We use a similar approach as in Arts et al. (2011) and illus-
trate that the approximations do not conflict with the undershoot
and that the optimal order-up-to levels of a serial system with two
supply options can be computed sequentially, similar as the
decomposition approach as suggested in Clark and Scarf (1960).
This enables us to determine the optimal allocation of safety stocks
in a two echelon system with two supply modes and to compute
the added value of a second supply mode.
Our research is related to two streams of literature. In the first
stream of literature, dual-sourcing for single-echelon models is
studied. Since the excellent review of Minner (2003), much re-
search has been done to generate new knowledge and results in
this area. Due to the complexity of the optimal replenishment pol-
icy in case of two or more suppliers, most of the papers present
heuristic policies and the computation of good or optimal policy
parameters. The so-called constant order policy, where each period
a constant amount is ordered at the regular supplier, is studied in
Rosenshine and Obee (1976), Chiang (2007), Janssen and de Kok
(1999), and Allon and Van Mieghem (2010). Although the constant
order policy performs well when the regular lead time is long
(Klosterhalfen et al., 2011), the dual-index policy (DIP) performs
well in general (Veeraraghavan and Scheller-Wolf, 2008). A fast
algorithm to compute near optimal policy parameters for the
dual-index policy can be found in Arts et al. (2011). Extensions of
the dual-index policy are investigated in Sheopuri et al. (2010).
Zhang et al. (2012) study a model with two capacitated suppliers,
one of which has a higher per unit purchase price, and the other a
fixed set-up cost. Lead times in this model are assumed identical
and so the suppliers are distinguished based on their costing plans.
In addition to periodic review policies, models in continuous time
with two or more suppliers are studied in Song and Zipkin (2009)
and Plambeck and Ward (2007).
The second stream of literature related to our work is devoted
to serial multi-echelon systems where there is only one way of
replenishing each stockpoint. Since the seminal work of Clark
and Scarf (1960), many contributions have been added to this
stream of literature. While some researchers have derived bounds
(e.g. Chen and Zheng, 1994; Shang and Song, 2003; Chao and Zhou,
2007), others concentrate on computational efficiency (e.g. Gallego
and Özer (2006)). For an extensive discussion of the existing liter-
ature and important results in this field, we also refer to Axsäter
(2003) and van Houtum (2006).
Both research streams are merged in the new field of serial mul-
ti-echelon inventory systems with multiple supply modes. To the
best of our knowledge, there are only a few contributions in this
field. The first extension of the classical Clark and Scarf model is
presented in Lawson and Porteus (2000). They allow for two differ-
ent transportation modes between the stockpoints where the
emergency delivery mode has lead time zero and the regular mode
a lead time of one period. For such a system they are able to char-
acterize the optimal policy under linear holding and backorder
costs. Muharremoglu and Tsitsiklis (2003) also allow for super-
modular shipping costs and derive the optimal policy. The optimal
policy under physical storage constraints is derived in Xu (2009).In
a more recent paper by Zhou and Chao (2010), the case of arbitrary
regular lead times and a one period shorter emergency lead time is
studied. They also provide bounds and heuristics based on news-
vendor equations. Although their model is a clear extension of
the model of Lawson and Porteus, an environment where a product
can either be shipped in 3 weeks over sea or in 1 day by plane is
not included in the modeling approach. Therefore, there is a clear
need for models with more general lead time assumptions.
The main contribution of this paper to the literature is as fol-
lows. We provide a two-echelon inventory system with two supply
options for the most upstream stockpoint and, in contrast to the
papers discussed above, we allow for general lead time difference
between the two supply modes. We show how to compute near
optimal policy parameters in an efficient manner and how the
optimal safety stocks are allocated in the studied system. We fur-
ther quantify the added value of the emergency supply mode. By
means of a numerical study we also derive conditions where an
emergency supply source is beneficial.
The remainder of the paper is organized as follows. In Section 2
we present the model, which is analyzed in Section 3. In Section 4
numerical results are presented, and the benefit of the second sup-
ply source is analyzed. The paper concludes with a summary and
directions for future research in Section 5.
2. Model
We consider a two-echelon serial supply chain that faces sto-
chastic demand for a single stock keeping unit (SKU) at the lowest
echelon. (The model can easily be extended to nechelons, but the
notation becomes burdensome and cluttered). Time is divided in
264 J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272
Author's personal copy
periods of equal length and demand per period is a sequence of
non-negative i.i.d. discrete random variables fD
t
g
1
t¼0
, where tis a
period index. It is assumed that unsatisfied demand is backlogged
and we further need the regularity condition PðD>0Þ>0, where
Dis the generic single period demand random variable. Also for
notational convenience, let D
t
1
;t
2
¼P
t
2
t¼t
1
D
t
.
The most upstream stockpoint (stockpoint 2) has two different
supply options differing in leadtimes and costs. The price and the
deterministic lead time for an SKU, ordered via the regular (emer-
gency) supply mode, are c
r
(c
e
) [$/SKU] and l
2,r
(l
2,e
) [periods], with
c
e
>c
r
,l
2,r
>l
2,e
. For convenience we also define :¼l
2,r
l
2,e
P1.
Additionally, there is a fixed set-up cost k[$/emergency order]
for each order placed through the emergency supply mode. The
lead time from stockpoint 2 to stockpoint 1 is denoted as l
1
and as-
sumed to be deterministic as well.
Besides the procurement costs also holding and backorder costs
are charged. Each period a holding cost of h
2
[$/unit] is charged to
each unit in stockpoint 2 and downstream therefrom. For units in
stockpoint 1, an additional charge of h
1
[$/unit] is applied. If back-
orders exist at stockpoint 1 at the beginning of a period, a penalty
cost p[$/unit] is charged for each unit backlogged.
Since time is divided in periods, ordering decisions can only be
placed at the beginning of a period. The quantity ordered at the
regular (emergency) supplier by stockpoint 2 in period tis denoted
as Q
2;r
t
Q
2;e
t

, and the quantity ordered by stockpoint 1 is Q
1
t
. The
sequence of events in a period tcan be summarized as follows:
1. Inventory holding and backlogging costs are charged.
2. Stockpoint 2 places orders of sizes Q
2;e
t
and Q
2;r
t
and incurs
ordering costs.
3. Stockpoint 2 receives quantities Q
2;e
tl
2;e
and Q
2;r
tl
2;r
.
4. Stockpoint 1 places an order of size Q
1
t
.
5. Stockpoint 1 receives an order of size Q
1
tl
1
.
6. Demand at stockpoint 1 occurs and is satisfied except for possi-
ble backorders.
The system is centrally controlled and all information about
stock levels and pipeline inventory is available at the moment of
decision making. Thus, in order to determine the order quantities
a natural combination of the dual-index policy and an echelon or-
der-up-to policy is applied. This policy uses three inventory posi-
tions in its operation, which are defined at the beginning of a
period right before the corresponding orders are placed. The first
inventory position is needed for the ordering decision of stockpoint
1 and is simply defined as the echelon 1 inventory position IP
1
:
IP
1
t
¼I
1
t
þX
t1
n¼tl
1
Q
1
n
;ð1Þ
where I
1
t
denotes the physical stock minus backorders at stockpoint
1 at the beginning of period t. For the ordering decision at stock-
point 1, we assume that the quantity to be ordered is limited by
the available inventory at the second stockpoint, given as
I
2
t
þQ
2;e
tl
2;e
þQ
2;r
tl
2;r
, where I
2
t
denotes the physical stock at stockpoint
2 at the beginning of period t.
For the ordering decision for echelon 2, we distinguish between
the emergency (IP
2,e
) and the regular (IP
2,r
) echelon inventory posi-
tions defined as follows:
IP
2;e
t
¼IP
1
t
þI
2
t
þX
t1
n¼tl
2;e
Q
2;e
n
þX
t
n¼tl
2;r
Q
2;r
n
;ð2Þ
IP
2;r
¼IP
1
t
þI
2
t
þX
t
n¼tl
2;e
Q
2;e
n
þX
t1
n¼tl
2;r
Q
2;r
n
¼IP
2;e
t
þX
t1
n¼tþ1
Q
2;r
n
þQ
2;e
t
:
ð3Þ
Notice that the last term in IP
2;e
t
includes only regular orders
that will arrive at stockpoint 2 within the emergency lead time,
whereas IP
2;r
t
includes all outstanding regular orders. Moreover,
IP
2;r
t
includes the emergency order Q
2;e
t
placed in period t.
A graphical representation of these inventory positions in peri-
od tand the rest of the model and its notations is given in Fig. 1.
Orders that are placed in period thave a dashed line indicating
the place where they will be in the pipeline upon placement.
The ordering decisions in period tare determined by the
differences of the inventory positions with their corresponding
order-up-to-levels S
2,r
,S
2,e
and S
1
. For convenience we assume
the regularity condition that IP
1
t
6S
1
and IP
2;r
t
6S
2;r
for all tP0.
This can be assumed without loss of generality because if this
assumption is violated, the number of periods where this assump-
tion is violated is finite with probability 1 if we do not order until
these assumptions are met. Consequently we can always renumber
periods such that this assumption holds. Since the order quantity
for the first echelon is restricted by the available inventory we get:
Q
1
t
¼min S
1
IP
1
t
;I
2
t
þQ
2;e
tl
2;e
þQ
2;r
tl
2;r

:ð4Þ
The order decisions for the second echelon are obtained as:
Q
2;e
t
¼S
2;e
IP
2;e
t

þ
;ð5Þ
Q
2;r
t
¼S
2;r
IP
2;r
¼S
2;r
IP
2;e
t
þQ
2;e
t
þX
t
1
n¼tþ1
Q
2;r
n
!
:ð6Þ
Notice that Q
2;e
t
is not simply the difference between the inven-
tory position and its order-up-to-level S
2,e
. The reason is that IP
2;e
t
is
usually not below its order-up-to-level S
2,e
. In fact, it is usually lar-
ger and the excess is called the overshoot O
t
defined as:
Fig. 1. Graphical representation of the model at the beginning of period t.
J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272 265
Author's personal copy
O
t
¼IP
2;e
t
S
2;e

þ
¼IP
2;e
t
þQ
2;e
t
S
2;e
;ð7Þ
where x
+
= max(x, 0). As with the single stage DIP, the evaluation of
the stationary distribution of O
t
plays a crucial role in the evaluation
and optimization of the echelon DIP.
Ordering decisions should be made such that the procurement
costs and the average holding and backorder costs are minimized.
In order to derive the average holding costs, we define echelon 1
and 2 inventory levels at the beginning of a period as:
IL
1
t
¼I
1
t
;ð8Þ
IL
2
t
¼IL
1
t
þX
t1
n¼tl
1
Q
1
n
þI
2
t
:ð9Þ
We use the standard notations x
+
= max(0, x) and x
= max(0,
x). By using the fact that x=x
+
x
, we can write the total in-
curred holding and penalty costs in a period tas:
h
2
IL
2
t
þIL
1
t


þh
1
IL
1
t

þ
þpIL
1
t

¼h
2
IL
2
t
þh
1
IL
1
t
þðpþh
1
þh
2
ÞIL
1
t

:ð10Þ
For the emergency ordering costs, we observe that on average
EðDÞhas to be ordered per period by stockpoint 2. Thus, we have that
the average purchasing cost per period equals c
r
EðDÞþ
ðc
e
c
r
ÞEðQ
2;e
Þ, where we dropped the time index to indicate steady
state. Since c
r
EðDÞis a fixed cost term regardless of policy operation,
we omit it from the cost function. If we define c=c
e
c
r
, then the
relevant variable ordering costs in a period tare cQ
2;e
t
. Also, for nota-
tional convenience we allow Q
2;e
t
¼0 but only account for the fixed
emergency ordering cost kwhenever Q
2;e
t
>0. Thus, in a period tthe
total relevant ordering costs are given by:
kIQ
2;e
t
>0

þcQ
2;e
t
;ð11Þ
where IðnÞis the indicator function of the event n.
Summarizing, we can state the average cost function C(S
1
,S
2,e
,
S
2,r
) as:
CðS
1
;S
2;e
;S
2;r
Þ¼cEðQ
2;e
ÞþkPðQ
2;e
>0Þ
þh
2
EðIL
2
Þþh
1
EðIL
1
Þþðpþh
1
þh
2
ÞE½ðIL
1
Þ
:ð12Þ
In order to derive the optimal policy parameters (S
1
,S
2,e
,S
2,r
)
minimizing the average cost, we need expressions for the expecta-
tions and probability that appear in Eq. (12).
3. Analysis
The analysis will proceed along the following lines. In Sec-
tion 3.1 we show that when
D
:¼S
2,r
S
2,e
is given, the stationary
distributions of O,Q
2,r
and Q
2,e
are fixed. This result allows us to
decompose the problem and provide newsvendor characteriza-
tions for the optimal order-up-to levels S
1
and S
2,e
for a given value
of
D
. In Section 3.2 we show how the distribution of Oand
PðQ
2;e
>0Þcan accurately be approximated.
3.1. Optimization
With the definition of
D
, a dual-index policy is fully defined by
D
,S
2,e
and S
1
as (S
1
,S
2,e
,S
2,r
)=(S
1
,S
2,e
,S
2,e
+
D
). Furthermore, let us
consider the outstanding regular orders that will not arrive to
stockpoint 2 within the emergency lead time in period tjust after
all orders have been placed, A
t
:
A
t
¼X
t
n¼tþ1
Q
2;r
n
:ð13Þ
With these preliminaries we have the following result.
Lemma 3.1 (Separability result). The following statements hold:
(i) Suppose t P0. Then the following relations hold:
D
¼O
t
þA
t
;ð14Þ
O
tþ1
¼O
t
D
t
þQ
2;r
tþ1

þ
;ð15Þ
Q
2;e
tþ1
¼D
t
O
t
Q
2;r
tþ1

þ
;ð16Þ
Q
2;r
tþ1
¼D
t
Q
2;e
tþ1
:ð17Þ
(ii) The stationary distributions of O,Q
2,e
and Q
2,r
depend on S
2,e
and S
2,r
only through their difference
D
=S
2,r
S
2,e
.
Proof. For part (i), recall the regular echelon 2 inventory position
as given in Eq. (3). Adding Q
2;r
t
to both sides of this equation and
substituting Eq. (7) yields:
IP
2;r
t
þQ
2;r
t
¼S
2;e
þO
t
þX
t
n¼tþ1
Q
2;r
n
:
Now by supposition tP0 and so IP
2,r
6S
2,r
. Consequently
Q
2;r
t
¼S
2;r
IP
2;r
t
and with the definition of A
t
in (13) we obtain:
S
2;r
¼S
2;e
þO
t
þA
t
:
Rearrangement and substitution of
D
=S
2,r
S
2,e
proves (14).
For the proof of Eqs. (15)–(17), we rewrite the emergency
echelon 2 inventory position:
IP
2;e
tþ1
¼IP
2;e
t
þQ
2;e
t
D
t
þQ
2;r
tþ1
¼S
2;e
þO
t
D
t
þQ
2;r
tþ1
:ð18Þ
The second equality follows from the definition of the overshoot
(7). Now by rewriting we have:
O
tþ1
¼IP
2;e
tþ1
S
2;e

þ
¼S
2;e
þO
t
D
t
þQ
2;r
tþ1
S
2;e

þ
¼O
t
D
t
þQ
2;r
tþ1

þ
:ð19Þ
For the echelon 2 emergency order quantity, we can write
similarly:
Q
2;e
tþ1
¼S
2;e
IP
2;e
tþ1

þ
¼D
t
O
t
Q
2;r
tþ1

þ
:ð20Þ
Lastly, Q
2;r
tþ1
¼D
t
Q
2;e
tþ1
follows immediately from the fact that
the dual-index policy ensures that every period the total amount
ordered to stock point 2 equals demand from the previous period.
To prove part (ii), we substitute Eq. (14) into Eqs. (15)–(17) to
find
O
tþ1
¼
D
D
t
X
t
n¼tþ2
Q
2;r
n
!
þ
;ð21Þ
Q
2;e
tþ1
¼D
t
þX
t
n¼tþ2
Q
2;r
n
D
!
þ
;ð22Þ
Q
2;r
tþ1
¼D
t
Q
2;e
tþ1
:ð23Þ
From these equations, we see that the stochastic processes
fO
t
g;Q
2;e
t
no
, and Q
2;r
t
no
can be described completely using S
2,r
and S
2,e
only through their difference
D
. Consequently, their station-
ary distributions can depend only on
D
.h
From Lemma 3.1, we immediately have that the cost terms
cEðQ
2;e
ÞþkPðQ
2;e
>0Þcan be determined and are fixed, for a given
value of
D
. Thus, we only need to optimize the holding and penalty
266 J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272
Author's personal copy
costs h
2
EðIL
2
Þþh
1
EðIL
1
Þþðpþh
1
þh
2
ÞE½ðIL
1
Þ
. We now investi-
gate how to evaluate these costs for a given value of
D
. We do this
by analyzing replenishment cycles.
Let C
t;2
¼h
2
IL
2
t
denote the holding costs associated with echelon
2 that are incurred in period t. Suppose we start in some period
t
0
P0 and order according to the echelon dual-index policy, we
obtain:
EC
t
0
þl
2;e
þ1;2
jIP
2;e
t
0
þQ
2;e
t
0
¼S
2;e
þO
hi
¼E½h
2
ðS
2;e
þOD
t
0
;t
0
þl
2;e
Þ ¼ h
2
ðS
2;e
þEðOÞE½D
t
0
;t
0
þl
2;e
Þ:ð24Þ
Next, we let C
t;1
¼h
1
IL
1
t
þðpþh
1
þh
2
ÞIL
1
t

denote the hold-
ing and penalty costs associated with echelon 1 in period t. Then
we have:
EC
t
0
þl
2;e
þl
1
þ1;1
jIP
1
t
0
þl
2;e
þQ
1
t
0
þl
2;e
¼min S
1
;IP
2;e
t
0
þQ
2;e
t
0
D
t
0
;t
0
þl
2;e
1
hi
¼EC
t
0
þl
2;e
þl
1
;1
jIP
1
t
0
þl
2;e
þQ
1
t
0
þl
2;e
¼S
1
ðD
t
0
;t
0
þl
2;e
1
OðS
2;e
S
1
ÞÞ
þ
hi
¼h
1
ðS
1
EðD
t
0
;t
0
þl
2;e
1
OðS
2;e
S
1
ÞÞ
þ
hi
E½D
t
0
þl
2;e
;t
0
þl
2;e
þl
1
Þ þ ðpþh
1
þh
2
Þ
E½D
t
0
;t
0
þl
2;e
1
OðS
2;e
S
1
Þ
þ
þD
t
0
þl
2;e
;t
0
þl
2;e
þl
1
S
1

þ

:
ð25Þ
With the conventions
B
1
¼ðD
t
0
;t
0
þl
2;e
1
OðS
2;e
S
1
ÞÞ
þ
;ð26Þ
B
0
¼ðB
1
þD
t
0
þl
2;e
;t
0
þl
2;e
þl
1
S
1
Þ
þ
;ð27Þ
the average holding and penalty cost function for fixed
D
,C
hp
-
(S
1
,S
2,e
j
D
), can be written succinctly as:
C
hp
ðS
1
;S
2;e
j
D
Þ¼h
2
ðS
2;e
þEðOÞEðD
t
0
;t
0
þl
2;e
1
ÞÞþh
1
ðS
1
EðB
1
Þ
EðD
t
0
þl
2;e
;t
0
þl
2;e
þl
1
ÞÞþðpþh
1
þh
2
ÞEðB
0
Þ:ð28Þ
The following result follows immediately from Lemma 3.1, Eq.
(28), and results for serial supply chains with discrete demand
(Dog
ˆru et al., 2004, in press; van Houtum, 2006).
Theorem 3.2 (Newsvendor inequalities for S
2,e
and S
1
). Suppose
D
=S
2,r
S
2,e
has been fixed. Then the optimal choice for S
1
is the
smallest integer that satisfies the following newsvendor inequality:
PB
ð1Þ
0
¼0

Ppþh
2
pþh
2
þh
1
;ð29Þ
where
B
ð1Þ
0
¼ðD
t
0
þl
2;e
;t
0
þl
2;e
þl
1
S
1
Þ
þ
:
Now let S
1
denote the optimal S
1
and
let
e
ðS
1
Þ¼PD
t
0
þl
2;e
;t
0
þl
2;e
þl
1
6S
1

pþh
2
pþh
2
þh
1
. Then the optimal S
2,e
is
the smallest integer that satisfies the following newsvendor inequality:
PB
0
¼0

Pp
pþh
1
þh
2
þPB
1
¼0

e
ðS
1;
Þ;ð30Þ
where
B
1
¼ðD
t
0
;t
0
þl
2;e
1
OðS
2;e
S
1
ÞÞ
þ
and
B
0
¼B
1
þD
t
0
þl
2;e
;t
0
þl
2;e
þl
1
S
1

þ
:
Remark. Note that while Theorem 3.2 pertains to two-echelon
systems, analogous results can easily be obtained for n-echelon
systems as long as dual-sourcing only occurs at the most upstream
stock point.
3.2. Overshoot and emergency ordering probability
In Lemma 3.1, we have established that the stationary distribu-
tions of O,Q
2,e
, and Q
2,r
are uniquely defined for a fixed value of
D
.
However, to evaluate the performance of a given DIP, we need to
evaluate their distributions to find the cost terms
kPðQ
2;e
>0Þ;cEðQ
2;e
Þand apply Theorem 3.2. In this section, we
provide an accurate approximation for the needed results.
The overshoot distribution behaves in a manner entirely identical
to the overshoot distribution in a single stage system as an inspec-
tion of Lemma 3.1 will reveal. Consequently, we can adopt the
approximation proposed by Arts et al. (2011) to determine the over-
shoot distribution. To be self-contained, we briefly outline how to do
this.
Observe that Eq. (14) in Lemma 3.1 implies that that the sta-
tionary distribution of Ocan be obtained from the stationary distri-
bution of Aas PðO¼xÞ¼PðA¼
D
xÞ. For A
t
the following
recursion holds (Arts et al., 2011, provide a detailed derivation):
A
tþ1
¼min
D
;A
t
Q
2;r
tþ1l
þD
t

:ð31Þ
From this equation, it is readily verified that a one-dimensional
Markov chain for A
t
has transition probabilities
p
ij
¼PðA
tþ1
¼jjA
t
¼iÞ:
p
ij
¼X
j
k¼0
PQ
2;r
tþ1
¼iþkjjA
t
¼i

PðD¼kÞ;if j<
D
;
X
i
k¼0
PQ
2;r
tþ1
¼kjA
t
¼i

PðDP
D
þkiÞ;if j¼
D
:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
ð32Þ
Note that this is an aggregated Markov chain for A
t
. A full Mar-
kov chain for A
t
would require storing regular orders Q
2;r
tþ1
to Q
2;r
t
in the state space. Thus, the state-space grows exponentially in .In
making this aggregation, we require the probability
PQ
r
tþ1
¼kjA
t
¼i

, which is in fact unknown. However, this prob-
ability can be approximated using the following limiting result
which is proven in Arts et al. (2011).
Proposition 3.3. The following statements hold:
(i) As
D
!1;PQ
2;r
t
¼x

!PðD
t1
¼xÞ.
(ii) As
D
!1;PQ
2;r
tþ1
¼xjA
t
¼y

!PD
tþ1
¼xP
t
n¼tþ1
D
n
¼y

.
(iii) For
D
¼1;PQ
2;r
tþ1
¼xjA
t
¼y

¼PD
tþ1
¼xP
t
n¼tþ1
D
n
¼y:Þ.
Using the limiting results in Proposition 3.3 by approximating
PQ
2;r
tþ1
¼xjA
t
¼y

with PD
tþ1
¼xP
t
n¼tþ1
D
n
¼y

, we can
compute an approximation to
p
x
¼PðA¼xÞby solving the linear
balance equations together with the normalization equation:
p
x
¼X
D
j¼0
p
j
p
xj
x¼0;1;...;
D
1X
D
j¼0
p
j
¼1:ð33Þ
An approximation for the stationary distribution of Ois now
PðO¼xÞ¼
p
D
x
. This result can be used in the newsvendor charac-
terizations in Theorem 3.2.
Next, to evaluate the term cEðQ
2;e
Þwe note that EðAÞ¼EðQ
2;r
Þ
and EðDÞ¼EðQ
2;r
ÞþEðQ
2;e
Þ. Combining these relations, we have
cEðQ
2;e
Þ¼cðEðDÞEðAÞÞ/.
J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272 267
Author's personal copy
Evaluating kPðQ
2;e
>0Þcan be done by conditioning as follows:
kPðQ
2;e
>0Þ¼kPDOQ
2;r
tþ1

þ
>0jA
t
¼
D
O

¼kPDOQ
2;r
tþ1
>0jA
t
¼
D
O

¼kX
D
y¼0
PDQ
2;r
tþ1
>yjA
t
¼
D
y

PðO¼yÞ
¼kX
D
y¼0
X
D
y
z¼0
PðD>yþzÞPQ
2;r
tþ1
¼zjA
t
¼
D
y

PðO¼yÞ:
ð34Þ
From Eq. (34), one can readily compute an approximation for
kPðQ
2;e
>0Þusing Proposition 3.3 again in the same manner. The
procurement costs associated with emergency ordering C
e
(
D
) are:
C
e
ð
D
Þ¼cðEðDÞEðAÞ=‘ÞþkX
D
y¼0
X
D
y
z¼0
PðD>yþzÞ
PQ
2;r
tþ1
¼zjA
t
¼
D
y

PðO¼yÞ:ð35Þ
In Arts et al. (2011), it is shown that the approximations sug-
gested here are extremely accurate for a single stage system in that
the solutions obtained are statistically not distinguishable from
simulation estimates. An efficient algorithm to optimize the
parameters of the dual-index policy for the system under study
is now straightforward, and an outline for an algorithm to do this
is given in Fig. 2.
4. Numerical results
In this section, we investigate the effect of different problem
parameters on optimal costs, order-up-to-levels, and savings com-
pared to the equivalent single sourcing system that uses only the
best supply source, which can be solved to optimality. We show
that an emergency supplier provides safety time to hedge against
demand uncertainty. This safety time can be traded-off with the
use of safety stock. We discuss in what settings the availability of
safety time can be used to decrease safety stock and costs.
For the computation of the optimal policy parameters in a serial
system with single delivery modes, we use the approach described
in van Houtum (2006). The computations were implemented in
MATLAB and run on a 2.4 GHz dual core processor. The computa-
tion times for finding optimal dual-index policy parameters for
one instance were 0.5 s on average and always within 1 s. These
short computation times make the dual-index computationally
feasible for real life application.
For our numerical study, we define a base case and then unilat-
erally vary different parameters. The numerical values for the base
case are shown in Table 1.
Here cv
D
is the coefficient of variation of demand
r
ðDÞ=EðDÞ.
The demand distributions we use are either mixtures of negative
binomial or geometric distributions as fitted on the first two mo-
ments by the procedure of Adan et al. (1996).
Since we are interested in the added value of a second supply
mode, we compare our system with a system where an optimal
single sourcing policy is applied. The average costs for the optimal
two echelon order-up-to policy with single supply modes is de-
noted as C
SS
and the percentage saving is given as:
d¼C
SS
CðS
1
;S
2;e
;S
2;r
Þ
C
SS
100%:ð36Þ
We start with a discussion of the impact of the demand variabil-
ity. We let cv
D
range from 0.3 to 2 and plot the average cost
(Fig. 3a), the base-stock levels (Fig. 3b), and the percentage savings
compared to optimal single sourcing (Fig. 3c). (In fact, we for all
scenarios we investigate in this numerical section, we plot these
three quantities as a function of the parameter of interest.)
The plot of cost is further divided to see the share of different
cost terms. For all other scenarios we will generate the same type
of plots for overall comparison purposes. Fig. 3a shows that costs
increase rapidly with demand variability. Yet from Fig. 3c we ob-
serve that savings compared to single sourcing do, too. This is an
important finding that has also been shown to hold for the single
stage system: dual-sourcing is most beneficial when demand var-
iability is high. The explanation for this is that the additional sup-
ply source provides a mean, other than inventory, to hedge against
uncertainty. This effect is shown in Fig. 3a, where it is illustrated
that holding costs grow slower with demand variability than the
other costs. Thus we may think of a second supply source as a type
of safety time to buffer demand variability.
Furthermore, it is noteworthy that the optimal emergency base-
stock level S
2,e
is mostly below the optimal echelon 1 base-stock
level S
1
as shown in Fig. 3b. Further results show that this is usual
behavior. This implies that, even when echelon 1 cannot order up
to its base-stock level, the emergency supply source is not used un-
less the deficit is sufficiently great.
Second, we investigate the effect of the lead time difference
=l
2,r
l
2,e
on the system. Results are depicted in Fig. 4.Fig. 4a
shows that costs do not increase rapidly with the leadtime
Fig. 2. Outline of an algorithm to optimize the two-echelon dual-index policy.
Table 1
Numerical values of the base case parameters.
Demand Leadtimes Holding
and
backorder
costs
Procurement
costs
l
1
=2 h
1
= 0.6
E[D]=10 l
2,e
=1 h
2
= 0.4 c=10
cv
D
=1 =l
2,r
l
2,4
=10 p=19 k=0
268 J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272
Author's personal copy
0.5 1 1.5 2
0
20
40
60
80
100
120
140
160
180
Coefficient of variance
Average Costs
(a)
Holding echelon 1
Holding echelon 2
Penalty
Variable ordering
0.5 1 1.5 2
0
50
100
150
200
250
300
350
Coefficient of variance
Base-stock level
(b)
Echelon 1
Regular echelon 2
Emergency echelon 2
Single source echelon 2
0.5 1 1.5 2
0
2
4
6
8
10
12
Coefficient of variance
Percentage Cost Saving
(c)
Compared to optimal single sourcing
Fig. 3. Effect of demand variability.
5 10 15
0
20
40
60
80
100
120
Lead time difference
Average Costs
(a)
Holding echelon 1
Holding echelon 2
Penalty
Variable ordering
510 15
0
50
100
150
200
250
300
Lead time difference
Base-stock level
(b)
Echelon 1
Regular echelon 2
Emergency echelon 2
Single source echelon 2
510 15
0
1
2
3
4
5
6
7
8
Lead time difference
Percentage Cost Saving
(c)
Compared to optimal single sourcing
Fig. 4. Effect of lead time difference.
0.8 0.9 1
0
20
40
60
80
100
120
Service level
Average Costs
(a)
Holding echelon 1
Holding echelon 2
Penalty
Variable ordering
0.8 0.9 1
-50
0
50
100
150
200
250
300
Service level
Base-stock level
(b)
Echelon 1
Regular echelon 2
Emergency echelon 2
Single source echelon 2
0.8 0.9 1
0
1
2
3
4
5
6
7
8
Service level
Percentage Cost Saving
(c)
Compared to optimal single sourcing
Fig. 5. Effect of required service level.
J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272 269
Author's personal copy
difference . But from Fig. 4c we observe that savings compared to
single sourcing do. This saving is possible because more units are
ordered via the emergency channel as evidenced from the increase
in variable ordering costs shown in Fig. 4a. Thus, dual-sourcing is
especially efficient when regular lead times are long compared to
emergency lead times. In the previous paragraph, we already ob-
served that the second supply source provides a type of safety time
that allows the safety stock to decrease. Apparently the lead time
difference is a measure for the amount of safety time available.
Third, the service level defined as SL ¼
p
pþh
1
þh
2
is varied by vary-
ing p.InFig. 5c, we see that dual-sourcing is more beneficial when
customers require high service, while the added value when ser-
vice is relatively unimportant is negligible. In Fig. 5b, we observe
that the emergency echelon 2 base-stock level can become nega-
tive in case of low service levels. This means that the emergency
supply mode is not used unless a positive backlog position must
persist for more than l
1
+l
2,e
periods into the future.
Fourth, the unit emergency costs care varied. The results,
shown in Fig. 6a, indicate that variable ordering and echelon 2
holding costs act as substitutes. This highlights previous findings
of Veeraraghavan and Scheller-Wolf (2008), that the dual-index
policy saves money by reducing holding, not penalty costs. This
is in line with our finding that the emergency source provides
safety time: the benefit of an emergency supply source is the abil-
ity to trade-off safety stock with safety time. Also, as expected,
Fig. 6b shows that the dual-index policy approaches a single-sourc-
ing base-stock policy as cincreases. Evidently in this setting, the
trade-off between safety stock and safety time leans more to safety
stock.
In the next experiment, we keep h
1
+h
2
= 1 while varying h
1
.
This represents the incremental value added from stage 2 to stage
1 of the supply chain. In Fig. 7c, we see that, as the incremental va-
lue of stage 1 inventory increases, the value of dual sourcing de-
creases (although total costs decrease). The explanation for this
is that, as observed in Fig. 6a, the benefit from the dual-index pol-
icy lies in the reduction of echelon 2 stock. As the relative impor-
tance of echelon 2 holding cost decreases, so does the benefit of
the dual-index policy over single-sourcing. We can conclude that
10 20 30
0
10
20
30
40
50
60
70
80
90
100
110
Unit emergency costs
Average Costs
(a)
Holding echelon 1
Holding echelon 2
Penalty
Variable ordering
10 20 30
0
50
100
150
200
250
300
Unit emergency costs
Base-stock level
(b)
Echelon 1
Regular echelon 2
Emergency echelon 2
Single source echelon 2
10 20 30
2
4
6
8
10
12
14
16
18
Unit emergency costs
Percentage Cost Saving
(c)
Compared to optimal single sourcing
Fig. 6. Effect of unit emergency costs.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
20
40
60
80
100
120
140
Unit echelon 1 holding costs
Average Costs
(a)
Holding echelon 1
Holding echelon 2
Penalty
Variable ordering
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
50
100
150
200
250
300
Unit echelon 1 holding costs
Base-stock level
(b)
Echelon 1
Regular echelon 2
Emergency echelon 2
Single source echelon 2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
1
2
3
4
5
6
7
8
9
Unit echelon 1 holding costs
Percentage Cost Saving
(c)
Compared to optimal single sourcing
Fig. 7. Effect of holding cost division.
270 J. Arts, G.P. Kiesmüller / European Journal of Operational Research 225 (2013) 263–272
Author's personal copy
a combination of a PUSH strategy and a dual sourcing strategy at
the most upstream supplier in a supply chain is not an efficient
combination. Dual sourcing is only beneficial under a PULL
strategy.
Finally, we investigate the effect of having a fixed emergency
ordering cost k. In many cases, shipping tariffs are based mostly
on the modality and less so on the order quantity as long as the or-
der fits in a standard container. To model this, we reduce the var-
iable ordering costs cin the base instance to 1 and let the fixed
ordering costs krange from 5 to 95. The results are shown in
Fig. 8.Fig. 8c shows a large saving potential compared to single
sourcing. This is striking because the dual-index policy is not espe-
cially fitted for fixed ordering costs. For example, it is quite possi-
ble to place an emergency order of size one under this policy. This
effect is shown in Fig. 8a by the division of fixed and variable
ordering costs for large k. A more advanced policy may avoid fixed
costs associated with placing very small emergency orders. Despite
this, the dual-index policy makes large cost savings possible as
long as the fixed ordering costs are not getting too large. Note also
from Fig. 8b that when both cost parameters cand kare small, it is
possible for S
2,e
to exceed S
1
. In this situation, the emergency
source is used to ensure that echelon 1 can order up to its base-
stock level most of the time.
5. Summary and directions for future research
In this paper, we have studied a two-echelon serial inventory
system where the most upstream stockpoint has two suppliers.
Replenishment orders are placed following a combination of a
dual-index and an echelon base-stock policy. For the case of a dis-
crete demand distribution, we have derived newsvendor inequali-
ties for two base-stock levels. For the overshoot distribution, we
rely on an approximation based on a Markov chain. These results
enable us to compute near-optimal policy parameters with very
little computational effort.
We have further illustrated that the second supply option can
lead to considerable cost savings in case of high demand variabil-
ity, large lead time difference, and small cost difference. However,
the second supply option is less worthwhile for systems where the
larger part of the value is added at stockpoint one, since the second
supply option mostly results in inventory reduction at the most
upstream stockpoint. As a consequence, multiple sourcing strate-
gies should only be combined with PULL strategies. Furthermore
we have shown that the emergency supplier provides safety time
that can be traded-off against safety stock in an effort to hedge
against demand uncertainty.
Our model is the first serial multi-echelon model where two
supply options with arbitrary lead times are included in the model.
The short computation times, intuitive structure, and cost saving
potential of the dual-index policy make it especially fit for use in
practice. However, lead time flexibility for a higher price is only
considered at the most upstream stockpoint. In a next step we plan
to investigate two delivery options at intermediate stockpoints.
Acknowledgements
The authors thank an anonymous referee for his comments that
significantly improved the presentation of the paper. The authors
also acknowledge Frank Karsten for his detailed comments on
earlier drafts of this manuscript.
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