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Two-echelon supply chain inventory model with controllablelead time and service level constraint
J.K. Jha
*
, Kripa Shanker
Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208 016, India
a r t i c l e i n f o
Article history:
Received 26 November 2008Received in revised form 29 April 2009Accepted 29 April 2009Available online 6 May 2009
Keywords:
Vendor–buyer integrated systemControllable lead timeContinuous review policyService level
a b s t r a c t
This paper considers a two-echelon supply chain inventory problem consisting of a single-vendor and asingle-buyer. In the system under study, a vendor produces a product in a batch production environmentand supplies it to a buyer facing a stochastic demand, which is assumed to be normally distributed. Also,buyer’s lead time is controllable which can be shortened at an added cost and all shortages are backor-dered. A model has been formulated for an integrated vendor–buyer problem to jointly determine theoptimal order quantity, lead time and the number of shipments from the vendor to the buyer during aproduction cycle while minimizing the total expected cost of the vendor–buyer integrated system. It isoften difﬁcult to estimate the shortage cost in inventory systems. Therefore, instead of having a shortagecost term in the objective function, a service level constraint (SLC) is included in the model that requires acertain proportion of demands to be met in each cycle. An efﬁcient procedure has been suggested to ﬁndthe bounds on number of shipments and then, an algorithm is developed to obtain the optimal solution of the proposed model. A numerical example is included to illustrate the algorithmic procedure and theeffects of key parameters are studied to analyze the behavior of the model. Finally, the savings of buyerand vendor are investigated from implementation of joint optimization model over the model in whichthey minimize their own cost independently.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Nowadays, companies can no longer compete solely as individ-ual entities in the constantly changing business world. Globaliza-tion of market and increased competition force organizations torely on effective supply chains to improve their overall perfor-mance. Successful supply chain management requires a changefrom managing distinct function to integrating activities into keysupply chain processes. Integration betweentwo different businessentities is an important way to gain competitive advantages as itlowers the joint total cost of the system. Therefore, in modernenterprises, the integration of vendor–buyer inventory system isan important issue.The joint optimization concept for buyer and vendor was initi-ated by Goyal (1976). Subsequently, numerous scholars developedintegratedinventorymodelsundervariousassumptions.For exam-ple, Banerjee (1986) assumed that the vendor manufactures at a ﬁ-nite rate and considered a joint economic-lot-size model in whicha vendor produces to order for a buyer on a lot-for-lot basis. Goyal(1988) relaxed the lot-for-lot policy and suggested that vendor’seconomic production quantity should be an integer multiple of buyerpurchasequantity.Asaresultofusingtheapproachsuggestedin the Goyal’s (1988) model, signiﬁcant reduction in inventory costcan be achieved. Later, Pan and Yang (2002) improved Goyal’s
(1988) model by considering lead time as a controllable factor inthemodelandobtainedalowerjointtotalexpectedcostandshorterleadtime.Goyal(2003)suggestedasimpleprocedurefordetermin-ingtheoptimaloperatingpolicyforthePanandYang’s(2002)model.Yang and Pan’s (2004) extended Pan and Yang’s model (2002) by
incorporating quality related issue. Hoque and Goyal (2006) pro-posed a generalized model for the same system as in Pan and Yang(2002) by transferring the lot from the vendor to the buyer withequalorunequalsizedbatches.Ouyang,WuandHo(2007)extendedYangandPan’s(2004)modelbyallowingforshortagesandusingthereorderpointasadecisionvariable.Severalresearchershaveshownthatinintegratedmodels,onepartner’sgainexceedstheotherpart-ner’sloss.Thus,thenetbeneﬁtcanbesharedbybothpartiesinsomeequitable fashion (Goyal & Gupta, 1989).When the demand is stochastic, lead time becomes an impor-tant issue and its control leads to many beneﬁts. Shorter lead timereduces the safety stock and the loss caused by stock-out,improvescustomer service level and increases the competitive advantage of business (Ouyang & Wu, 1997). As stated by Tersine (1994), lead
timeusually comprises several components, such as order prepara-tion, order transit, supplier lead time, delivery time and setup time.In many practical situations, lead time can be reduced using an
0360-8352/$ - see front matter
2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2009.04.018
*
Corresponding author. Tel.: +91 512 2597101; fax: +91 512 2597553.
E-mail address:
jkjha@iitk.ac.in (J.K. Jha).Computers & Industrial Engineering 57 (2009) 1096–1104
Contents lists available at ScienceDirect
Computers & Industrial Engineering
journal homepage: www.elsevier.com/locate/caie
added crashing cost. Inventory models incorporating lead time as adecision variable were developed by several researchers. Liao andShyu (1991) ﬁrst devised a probabilistic inventory model in whichlead time was the unique decision variable. Later, several research-ers (e.g., Ben-Daya & Raouf, 1994; Hariga & Ben-Daya, 1999; Lee,Wu, & Hou, 2004; Moon & Choi, 1998; Ouyang & Chuang, 2001)developed various analytical inventory models to explore the leadtime reduction problem. Ouyang, Chen and Chang (2002) extendedMoon and Choi’s (1998) model to include the possible relationshipbetween quality and lot size and then investigate the joint effectsof quality improvement and setup cost reduction in the model.Lee (2005) extended the work of Ouyang and Chuang (2001) by
considering the mixtures of distribution of the lead time demandand controllable backorder rate. Lee, Wu and Lei (2007) consideredthe continuous review inventory systems with variable lead timeand backorder discount, where the ordering cost can be reducedby capital investment. The underlying assumption in the abovestudies was that lead time could be decomposed into severalmutually independent components, each with a different but ﬁxedcrash cost and so lead time crash cost is a function of the reducedlead time. Also, other authors (e.g., Chang, Ouyang, Wu, & Ho,2006; Ouyang & Chang, 2002; Pan & Hsiao, 2005; Pan, Hsiao, &Lee, 2002; Pan, Lo, & Hsiao, 2004) explored the lead time reductionproblem assuming crash cost of lead time component as a functionof the order quantity and hence lead time crash cost is a functionof both the order quantity and the reduced lead time. The lead timecrash cost deﬁnition considered in the present study belongs tothe earlier category and lead time crash cost is the function of the reduced lead time only.The study of integratedinventorymodelsin a single-vendor anda single-buyer can be divided into two broad groups: the full costmodel and the service level approach model. In the full cost model,the objective is to ﬁnd the optimal inventory policy which mini-mizes the joint total relevant cost of the vendor–buyer integratedsystem including the cost of shortages (Chang et al., 2006; Ouyang,Wu, & Ho, 2004; Ouyang et al., 2007). The service level approachintroduces a service level constraint in place of the shortage costwhich implies that the stock-out level per cycle is bounded andthe availability of stock in a probabilistic or expected sense. There-fore,manyauthors(Chu,Yang,&Chen,2005;Lee,Wu,&Hsu,2006;Moon & Choi, 1994; Ouyang & Wu, 1997) replace the shortage costby a condition on the service level. All these authors focused ondetermining the optimal policy of the buyer only under differentsettings by optimizing the relevant cost of the buyer alone. How-ever, in present study of the vendor–buyer integrated system, it isworth investigating the optimal decisions of both the vendor andthe buyer by optimizing the cost of the vendor and the buyer to-gether, when there is a service level constraint on the buyer.Ouyangetal.(2004)extendedtheworkof PanandYang(2002)by
adding the shortage cost in the objective function and consideringreorder point as an additional decision variable. The present papermodiﬁes the model of Ouyang et al. (2004) by introducing a servicelevel constraint, instead of considering the shortage cost in theobjective function. Specifying a service level avoids the difﬁcultpractical issue of explicitly estimating the shortage cost. Moreover,a service level criterion is generally easy to interpret and establish.Therefore, SLC approach model is much more popular than full costmodelinspiteofitbeingperceivedaslessmathematicallytractable(Chen & Krass, 2001).In the previous works, the vendor–buyer integrated systemswith lead time crashing concept have been modeled as full costmodel including the cost of shortages. In this paper, we presentthe SLC approach to the vendor–buyer integrated inventory modelinvolving variable lead time, i.e. the lead time is controllable andreducible by adding additional crash costs. Consequently, thisstudy considers an integrated inventory model involving variablelead time in which the objective is to minimize the joint totalexpected cost of the vendor–buyer integrated system and subjectto a service level constraint on the buyer by simultaneously opti-mizing the order quantity, lead time and the number of shipmentsbetween the vendor and buyer in a production cycle. It is assumedthat the lead time demand follows a normal distribution. The fur-ther assumptions which are made to describe the system and toformulate the model are listed in the next section.The remainder of this paper is organized as follows: Section 2describes the notations and assumptions used throughout thisstudy. We formulate a vendor–buyer integrated inventory modelwith service level issue and lead time reduction in Section 3. In
Section 4, a solution technique is developed to obtain the optimalsolution. A numerical example is provided in Section 5 to illustratethe results. Finally, we draw some conclusions in Section 6.
2. Notations and assumptions
To establish the mathematical model, the following notationsand assumptions are used. Some additional notations will be listedlater when they are needed.
NotationsD
average demand per unit time on the buyer
P
production rate of the vendor,
P
>
D A
buyer’s ordering cost per order
S
vendor’s setup cost per setup
C
b
unit purchase cost paid by the buyer
C
v
unit production cost incurred by the vendor,
C
v
<
C
b
r
b
buyer’s holdingcost rate(per monetaryunitinvestedin inventory) per unit time
r
v
vendor’s holding cost rate (per monetary unitinvested in inventory) per unit time
Q
order quantity of the buyer (decision variable)
r
reorder point of the buyer
L
length of lead time for the buyer (decision variable)
a
proportion of demands that are not met from stockso (1
a
) is the service level
B
(
r
) expected demand shortage at the end of cycle
m
the number of lots in which the product is deliveredfrom the vendor to the buyer in one productioncycle, a positive integer (decision variable)
X
the lead time demand, which follows a normaldistribution with ﬁnite mean
DL
and standarddeviation
r
ﬃﬃﬃ
L
p
; where
r
denotes the standarddeviation of demand per unit time,
X
N
ð
DL
;
r
ﬃﬃﬃ
L
p Þ
E
(
) mathematical expectation
x
+
maximum value of
x
and 0, i.e.
x
+
= max {
x
, 0}
Assumptions
1. There is a single-vendor and a single-buyer and they deal with asingle item.2. The buyer orders a lot of size
Q
and the vendor produces
mQ
units with a ﬁnite production rate
P
(
P
>
D
) in one setup butships in quantity
Q
to the buyer over
m
times.3. Buyer uses a continuous review inventory policy and the orderis placed whenever inventory level falls to the reorder point
r
.4. Shortages are completely backordered.5. The reorder point
r
= the expected demand during lead time(
DL
) + safety stock (
s
), and
s
=
k
(standard deviation of leadtime demand), i.e.
r
¼
DL
þ
k
r
ﬃﬃﬃ
L
p
where
k
is the safety factorand satisﬁes
Pr
(
X
>
r
) =
q
,
q
represents the allowable stock-outprobability during lead time.
J.K. Jha, K. Shanker/Computers & Industrial Engineering 57 (2009) 1096–1104
1097
6. The lead time
L
has
n
mutually independent components. The
i
th component has a minimum duration
a
i
, and normal duration
b
i
, and a crash cost per unit time
c
i
. Further, we assume that
c
1
6
c
2
6
. . .
6
c
n
.7. The lead time components are crashed one at a time startingwith the least
c
i
component and so on.8. Let
L
0
¼
P
n j
¼
1
b
j
and
L
i
be the length of lead time with compo-nents 1, 2,
. . .
,
i
crashed to their minimum duration, then
L
i
¼
P
n j
¼
i
þ
1
b
j
þ
P
i j
¼
1
a
j
;
i
¼
1
;
2
;
. . .
;
n
. The lead time crash cost
C
(
L
) per cycle for a given
L
e
[
L
i
,
L
i
1
], is given by
C
ð
L
Þ ¼
c
i
ð
L
i
1
L
Þþ
P
i
1
j
¼
1
c
j
ð
b
j
a
j
Þ
.9. If a shortened lead time is requested then the extra costsincurred by the vendor will be fully transferred to the buyer.Therefore, lead time crash cost is the buyer’s cost component.
3. Model formulation
In this section a model is developed for the vendor–buyer inte-grated system to minimize the joint total expected cost per unittime of the vendor–buyer integrated system, which is the sum of the ordering, holding and lead time crashing costs per unit timefor the buyer and the setup and holding costs per unit time forthe vendor, subject to a SLC on the buyer.The buyer places an order after every
Q
demands, therefore foraverage cycle time of
Q
/
D
, the expected ordering and lead timecrashing costs per unit time can be given by
AD
/
Q
and
DC
(
L
)/
Q
,respectively. The expected net inventory level just before arrivalof a procurement is the safety stock,
s
=
r
DL
. The expected netinventory level immediately after arrival of a procurement is
Q
+
s
. Hence, the average inventory over the cycle can be approxi-mated by (
Q
/2) +
s
(Hadley & Whitin, 1963), i.e.
ð
Q
=
2
Þþ
k
r
ﬃﬃﬃ
L
p
(Assumption 5). So the buyer’s expected holding cost per unit timeis
r
b
C
b
ð
Q
=
2
þ
k
r
ﬃﬃﬃ
L
p Þ
. Consequently, the total expected cost per unittime for the buyer comprising of ordering cost, holding cost andlead time crashing cost can be expressed as
TEC
b
ð
Q
;
L
Þ ¼
ADQ
þ
r
b
C
b
Q
2
þ
k
r
ﬃﬃﬃ
L
p
þ
DQ C
ð
L
Þ
:
ð
1
Þ
The vendor–buyer integrated system is designed for a vendor’sproduction situation in which once the buyer orders a lot of size
Q
,the vendor begins production with a constant production rate
P
,and a ﬁnite number of units are added to inventory until the pro-ductionrunhas been completed.The vendorproducesthe iteminalot of size
mQ
in each production cycle of length
mQ
/
D
, and thebuyer will receive the supply in
m
lots each of size
Q
. The ﬁrstlot of size
Q
is ready for shipment after time
Q
/
P
just after the startof the production, and then the vendor continues making the deliv-ery on average every
Q
/
D
units of time until the inventory levelfalls to zero (Fig. 1). The expected setup cost per unit time is givenby
SD
/(
mQ
).Vendor’s average inventory is evaluated as the difference of thevendor’s accumulated inventory and the buyer’s accumulatedinventory (see Fig. 1). That is,
mQ Q P
þð
m
1
Þ
Q D
m
2
Q
2
2
P
#
Q
2
D
ð
1
þ
2
þþð
m
1
ÞÞ
#( )
DmQ
¼
Q
2
m
1
DP
1
þ
2
DP
:
Therefore, the total expected cost per unit time for the vendorcomprising of setup cost and holding cost can be expressed as
TEC
v
ð
Q
;
m
Þ ¼
SDmQ
þ
r
v
C
v
Q
2
m
1
DP
1
þ
2
DP
:
ð
2
Þ
Accordingly, the joint total expected cost per unit time for thevendor–buyer integrated system is given by
JTEC
ð
Q
;
L
;
m
Þ ¼
TEC
b
ð
Q
;
L
Þ þ
TEC
v
ð
Q
;
m
Þ
JTEC
ð
Q
;
L
;
m
Þ ¼
DQ A
þ
S m
þ
C
ð
L
Þ
þ
Q
2
r
b
C
b
þ
r
v
C
v
m
1
DP
1
þ
2
DP
þ
r
b
C
b
k
r
ﬃﬃﬃ
L
p
;
L
2 ½
L
i
;
L
i
1
:
ð
3
Þ
3.1. Establishing the service level constraint
Usually it is not easy to quantify the penalty costs associatedwith a shortage, as a stock-out event may include intangible inﬂu-ences.Therefore,several authorsassume that buyerhas seta targetservice level corresponding to proportion of demands to be satis-ﬁed directly from available stock. Therefore, SLC puts a limit onthe proportion of demands not met from stock. For a given safetyfactor which satisﬁes probability that lead time demand at thebuyer exceeds reorder point (Assumption 5), the actual proportionof demands not met from stock should not exceed the desired va-lue of
a
. Therefore, the SLC can be established as
Expecteddemandshortagesattheendof cycleforagivensafetyfactorQuantityavailableforsatisfyingthedemandpercycle
6
a
;
i
:
e
:
B
ð
r
Þ
Q
6
a
:
Since the shortages occur when
X
>
r
, and then the expected de-mand shortages at the end of cycle is given by, according to Ravin-dran, Phillips and Solberg (1987),
B
ð
r
Þ ¼
E
ð
X
r
Þ
þ
¼
r
ﬃﬃﬃ
L
p
w
ð
k
Þ
where
w
(
k
) =
/
(
k
)
k
[1
U
(
k
)] > 0, and
/
,
U
are the standard nor-mal probability density function and cumulative distribution func-tion, respectively. Therefore, SLC is given by
r
ﬃﬃﬃ
L
p
w
ð
k
Þ
Q
6
a
;
L
2 ½
L
i
;
L
i
1
:
ð
4
Þ
Thus, the problem is to ﬁnd the optimal order quantity
Q
, leadtime
L
and the number of shipments in a production cycle
m
thatminimize the joint total expected cost (3), subject to SLC (4), that is
Min
JTEC
ð
Q
;
L
;
m
Þ ¼
DQ A
þ
S m
þ
C
ð
L
Þ
þ
Q
2
r
b
C
b
þ
r
v
C
v
m
1
DP
1
þ
2
DP
þ
r
b
C
b
k
r
ﬃﬃﬃ
L
p
Subject to
r
ﬃﬃﬃ
L
p
w
ð
k
Þ
Q
6
a
where
L
e
[
L
i
,
L
i
1
] and
w
(
k
) =
/
(
k
)
k
[1
U
(
k
)] > 0.
4. Solution technique
The problem formulated in the previous section appears as con-strained non-linear programming problem. To solve this problem,we temporarily ignore the SLC and try to ﬁnd the optimal solutionof
JTEC
(
Q
,
L
,
m
). First, for ﬁxed
m
we take the ﬁrst partial deriva-tives of
JTEC
(
Q
,
L
,
m
) with respect to
Q
and
L
e
(
L
i
,
L
i
1
), respec-tively, and obtain
@
JTEC
ð
Q
;
L
;
m
Þ
@
Q
¼
DQ
2
A
þ
S m
þ
C
ð
L
Þ
þ
12
r
b
C
b
þ
r
v
C
v
m
1
DP
1
þ
2
DP
;
ð
5
Þ
and
1098
J.K. Jha, K. Shanker/Computers & Industrial Engineering 57 (2009) 1096–1104
@
JTEC
ð
Q
;
L
;
m
Þ
@
L
¼
DQ c
i
þ
r
b
2
C
b
k
r
L
1
=
2
:
ð
6
Þ
Hence, for ﬁxed
m
and
L
e
[
L
i
,
L
i
1
],
JTEC
(
Q
,
L
,
m
) is convex in
Q
,since
@
2
JTEC
ð
Q
;
L
;
m
Þ
@
Q
2
¼
2
DQ
3
A
þ
S m
þ
C
ð
L
Þ
>
0
:
However, for ﬁxed (
Q
,
m
),
JTEC
(
Q
,
L
,
m
) is concave in
L
e
[
L
i
,
L
i
1
], because
@
2
JTEC
ð
Q
;
L
;
m
Þ
@
L
2
¼
r
b
4
C
b
k
r
L
3
=
2
<
0
:
Therefore, for ﬁxed (
Q
,
m
), the minimum joint total expectedcost occurs at the end points of the interval [
L
i
,
L
i
1
].On the other hand, for ﬁxed
m
and
L
e
[
L
i
,
L
i
1
], we obtain opti-mal order quantity
Q
=
Q
JTEC
by setting (5) to zero as
Q
JTEC
¼
2
D A
þ
S m
þ
C
ð
L
Þ
r
b
C
b
þ
r
v
C
v
m
1
DP
1
þ
2
DP
#
1
=
2
;
L
2 ½
L
i
;
L
i
1
:
ð
7
Þ
Thus, for ﬁxed
m
and
L
e
[
L
i
,
L
i
1
], when the SLC is ignored, (7)gives optimal value of
Q
such that the joint total expected cost isminimum. Now, the SLC (4) is taken into consideration and if
ð
r
=
a
Þ
ﬃﬃﬃ
L
p
w
ð
k
Þ
6
Q
for
Q
=
Q
JTEC
, then
Q
JTEC
is the local minimum of
JTEC
(
Q
,
L
,
m
) and SLC is inactive. Otherwise, optimal value of
Q
should be at least equal to
Q
SLC
¼ ð
r
=
a
Þ
ﬃﬃﬃ
L
p
w
ð
k
Þ
which is greaterthan
Q
JTEC
so that speciﬁed level of service can be achieved at min-imum joint total expected cost and
Q
SLC
is the local minimum of
JTEC
(
Q
,
L
,
m
). Therefore, for ﬁxed
m
and
L
e
[
L
i
,
L
i
1
], the optimal
Q
is given by max {
Q
JTEC
,
Q
SLC
}.Next, in order to examine the effect of
m
on the joint total ex-pected cost, we temporarily relax the integer requirement on
m
and take the ﬁrst and second partial derivatives of
JTEC
(
Q
,
L
,
m
)with respect to
m
and obtain
@
JTEC
ð
Q
;
L
;
m
Þ
@
m
¼
DS Qm
2
þ
Q
2
r
v
C
v
1
DP
ð
8
Þ
and
@
2
JTEC
ð
Q
;
L
;
m
Þ
@
m
2
¼
2
DS Qm
3
>
0
:
Therefore,
JTEC
(
Q
,
L
,
m
) is convex in
m
, for ﬁxed
Q
and
L
e
[
L
i
,
L
i
1
]. As a result, the search for the optimal shipment num-ber,
m
*
, is reduced to ﬁnd a local minimum.Now equating (8) to zero, we have
m
¼
1
Q
2
DS r
v
C
v
1
DP
#
1
=
2
:
ð
9
Þ
From (9) it can be observed that
Q
and
m
have inverse relation-ship but due to integer restriction on
m
this expression can not beused for calculating the optimal value of
m
. Therefore, we suggest aprocedure in the following section to construct a range for search-ing the value of optimal
m
.
Reorder point
r QQQQQQQ
B u y e r ’ s s t o c k
Time Safety stock
s LQ
Buyer’s accumulated inventory level (
m
-1)
Q/D Q/D QQ
V e n d o r ’ s s t o c k
Q Q Q QQ Q/PQ/D Q/D
Time Production and shipmentShipment
Q
Production cycle
Q/D Q/D
Vendor’s accumulated inventory level
mQ/D mQ/P mQ
Fig. 1.
The inventory pattern for the vendor and the buyer.
J.K. Jha, K. Shanker/Computers & Industrial Engineering 57 (2009) 1096–1104
1099

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