Int. J. Production Economics 135 (2012) 116–124
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Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
Inventory control in a two-level supply chain with risk pooling effect
Jae-Hun Kang, Yeong-Dae Kim n
Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Yusong-gu, Daejon 305-701, Korea
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 24 November 2009
Accepted 15 November 2010
Available online 23 November 2010
We consider an inventory control problem in a supply chain consisting of a single supplier, with a central
distribution center (CDC) and multiple regional warehouses, and multiple retailers. We focus on the
problem of selecting warehouses to be used among a set of candidate warehouses, assigning each retailer
to one of the selected warehouses and determining replenishment plans for the warehouses and the
retailers. For the problem with the objective of minimizing the sum of warehouse operation costs,
inventory holding costs at the warehouses and the retailers, and transportation costs from the CDC to
warehouses as well as from warehouses to retailers, we present a non-linear mixed integer programming
model and develop a heuristic algorithm based on Lagrangian relaxation and subgradient optimization
methods. A series of computational experiments on randomly generated test problems shows that the
heuristic algorithm gives relatively good solutions in a reasonable computation time.
& 2010 Elsevier B.V. All rights reserved.
Keywords:
Supply chain
Inventory control
Risk pooling
Lagrangian relaxation
Heuristic
1. Introduction
We consider a two-level supply chain consisting of a single
supplier and multiple retailers. In the supply chain, the supplier is
composed of a central distribution center (CDC) and multiple
candidate regional warehouses, from which up to a given number
of warehouses are selected and actually used. It is assumed that the
supplier is authorized to manage inventory levels of the retailers by
a vendor-managed inventory (VMI) contract. In a VMI system, the
supplier monitors inventory levels of the retailers as well as
demands from final customers, and determines when and how
much to deliver to the retailers as well as when and how much to
replenish its own inventory at the warehouses. That is, the retailers
do not place orders to the supplier, but the supplier controls the
inventory levels of the retailers by determining replenishment
timing and quantities for the retailers. It is known that by employing the VMI system, one can reduce the operating cost of the supply
chain and maintain or improve the service level for the customers
(C
- etinkaya and Lee, 2000).
The problem considered here is to select warehouses to be
actually used among a set of candidate warehouses, to assign each
retailer to one of the selected warehouses, and to determine
replenishment plans for the warehouses and the retailers, in the
two-level supply chain that employs the VMI system. The warehouses and the retailers are assumed to use the (r, q) policy. That is,
when the inventory level falls down to the reorder point, denoted
by r, an order for q units is issued. Also, it is assumed that demands
(per unit time) at the retailers (from final customers) are independent of each other and they follow normal distributions with
n
Corresponding author. Fax: + 82 42 350 3110.
E-mail address: ydkim@kaist.ac.kr (Y.-D. Kim).
0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijpe.2010.11.014
mean mj and variance vj for retailer j, and that distances and lead
times from CDC to warehouses as well as those from warehouses to
retailers are known and fixed.
As the safety stock is generally set to be proportional to the
standard deviation of the demand during the lead time, the safety
stock can be reduced if the demand variation is reduced. Also, since
demands from different retailers are independent, the variance of
the sum of the demands from a set of retailers is smaller than the
sum of the variances of the demands from those retailers. As a
result, the safety stock needed for the pooled demands is generally
less than the sum of the safety stocks for the individual demands.
Therefore, to reduce operating costs of the supply chain, especially
inventory holding costs, one may use the risk pooling strategy, the
strategy of reducing the demand variability by aggregating
demands from multiple retailers. Such aggregation can be done
by allocating more retailers to each warehouse or reducing the
number of warehouses to be selected and used.
In the example given in Fig. 1, which illustrates the supply chain
considered in this study, safety stocks of warehouses 1 and 2 are set
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
to kw ðv1 þ v2 ÞL1 and kw ðv3 þ v4 ÞL2 , respectively. Here, Li and kw
denote the lead time from CDC to warehouse i and the safety factor
at the warehouses, respectively. Assume L1 rL2. If we assign
retailers 3 and 4 to warehouse 1 by not using warehouse 2, the
safety stock of the supply chain can be reduced since the safety
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
stock of warehouse 1 is set to kw ðv1 þ v2 þ v3 þv4 ÞL1 , which is not
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
greater than kw ðv1 þ v2 ÞL1 + kw ðv3 þ v4 ÞL2 .
The risk pooling strategy should be carefully applied since it may
increase inventory levels of the retailers. As the number of warehouses that are selected and used is decreased, warehouse operation
costs may be decreased and so may the safety stocks at the
warehouses. However, lead times from the warehouses to the
retailers may increase due to the increase of transportation distances
J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
117
(r1, q1)
Retailer 1
(R1, Q1)
~ N (m1, v1)
(r2, q2)
Warehouse 1
Retailer 2
~ N (m2, v2)
(r3, q3)
(R2, Q2)
Retailer 3
Warehouse 2
~ N (m3, v3)
(r4, q4)
……
Central
distribution
center
Retailer 4
~ N (m4, v4)
……
(Ri, Qi)
Warehouse i
(rj,qj)
Retailer j
~ N (mj, vj)
Fig. 1. Network diagram of the supply chain.
between the selected warehouses and the retailers, and hence the
safety stocks at the retailers may be increased. Therefore, it is
necessary to determine the optimal level of risk pooling, i.e., the
optimal number of warehouses to be used, with the consideration of
the trade-off between the decrease in the inventory holding costs at
the warehouses and the warehouse operation costs and the increase
in the inventory holding costs at the retailers.
Fry et al. (2001), De Toni and Zamolo (2005), Hong and Park
(2006), Gumus et al. (2008), and Southard and Swenseth (2008)
show benefits of a VMI system by comparing it with a traditional
retailer-managed inventory system, and Wong et al. (2009) show
that the performance of a supply chain can be improved through a
sales rebate contract, which is devised to help centralize and
coordinate decentralized decisions of a supply chain. Also,
Szmerekovsky and Zhang (2008) investigated the effect of attaching radio frequency identification (RFID) tags at items on VMI
system consisting of one manufacturer and one retailer, and Xu and
Leung (2009) presented a stocking policy in VMI system with a
limit on the shelf space. In addition, C
- etinkaya and Lee (2000) and
Axsäter (2001) presented analytical results for the coordination
problem between inventory and transportation decisions in VMI
systems, where a supplier has information on the demands from a
group of retailers located in a given geographic region. On the other
hand, Bertazzi et al. (2005) considered a production and distribution planning problem in VMI system and presented a decomposition approach to the problem, and Kang and Kim (2010) develop
heuristic algorithms for an integrated inventory and transportation
problem, in which a supplier determines the replenishment
quantities and timing for retailers as well as the amount of products
to be delivered by each vehicle with limited capacity.
As mentioned earlier, the benefit of risk pooling can be obtained
through the consolidation of inventories of multiple locations into a
single one. Eppen (1979), Chen and Lin (1989), and Chang and Lin
(1991) show that a pooled system incurs less cost than a distributed
system, and the difference of the costs of the two systems depends
on the variance of demands and the correlation among the
demands. Also, Alfaro and Corbett (2003), Gerchak and He
(2003), and Benjaafar et al. (2005) investigated the benefits and
costs of inventory pooling, and Kulkarni et al. (2005) evaluated
trade-offs between logistics costs and risk pooling benefits in a
manufacturing network with component commonality. In addition, Shen et al. (2003), Miranda and Garrido (2004), and Romeijn
et al. (2007) used the risk pooling strategy in network design
problems. However, their decisions are made only from the
supplier’s point of view since they do not consider the inventory
holding costs at retailers. Also, without considering the inventory
holding costs at the retailers, Miranda and Garrido (2004) show
that as the inventory holding cost at warehouses, the variability of
demands at retailers, and/or the service level increase the effect of
risk pooling, i.e., cost reduction, increases. On the other hand, Gaur
and Ravindran (2006) determined the best level of inventory
aggregation for two conflicting objectives, maximizing responsiveness to customers’ demands and minimizing the total cost of a
supply chain, without considering inventory holding costs at the
retailers.
As another alternative for reducing inventory of the supply
chain, one can employ the policy of transshipment, i.e., replenishing inventories from locations at the same echelon level instead of a
location at an upper level, since lead times can be reduced by
employing the policy. Schwarz (1989) and Glasserman and Wang
(1998) investigated the relationship between the lead time and the
inventory level required for achieving a given service level for
customers. Also, Grahovac and Chakravarty (2001) show that
inventory sharing and lateral transshipment in a supply chain
often reduce inventory holding costs and waiting costs of customers. In addition, Tagaras (1989), Archibald et al. (1997), Herer and
Rashit (1999), Herer and Tzur (2001), Rudi et al. (2001), and Olsson
(2009) dealt with transshipment problems in two-location inventory systems, while Tagaras (1999), Kukreja et al. (2001), Herer and
Tzur (2003), Hu et al. (2005), Kukreja and Schmidt (2005), and
Archibald (2007) developed inventory stocking policies in multiple-location inventory systems considering transshipments. Meanwhile, Lee (1987) and Axsäter (1990) presented lateral
transshipment models for repairable items, and Evers (2001) and
Minner et al. (2003) provided heuristic algorithms for determining
transshipment timing.
In this study, we consider the problem of selecting warehouses
from a given set of candidate warehouses, assigning retailers to the
selected warehouses and determining replenishment plans at the
warehouses and the retailers in a two-level supply chain, in which
each member uses the (r, q) policy. We present a non-linear mixed
integer programming model for the problem and develop a
Lagrangian heuristic algorithm. In the next section, the problem
considered in this study is described in more detail and a non-linear
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J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
mixed integer programming formulation is given, and Section 3
presents the heuristic algorithm for the problem. For evaluation of
the performance of the suggested algorithm, a series of computational experiments are performed and results are reported in
Section 4. Finally, Section 5 concludes the paper with a short
summary.
2. Problem description
There are a supplier, composed of a central distribution center
(CDC) and multiple (candidate) regional warehouses, and multiple
retailers in a two-level supply chain, in which the supplier and the
retailers are under VMI contract. It is assumed that the supply chain
adopts an inventory management strategy, in which the costs
incurred at the supplier’s side as well as the retailers’ side are
considered simultaneously, since the partnership of the members
needs to be maintained or improved. In the problem considered
here, we select warehouses that are to be actually used from a given
set of candidate warehouses and assign the retailers to the selected
warehouses. In addition, we determine the reorder points and
the order quantities of the warehouses and the retailers for the
objective of minimizing the sum of warehouse operation costs, and
inventory holding costs and transportation costs of the whole
supply chain.
In this study, the following assumptions are made. Note that real
situations of a typical VMI system are reflected in these
assumptions.
(a) The cost resulting from the selection and operation of each
regional warehouse is given, and the cost may be different for
different candidate warehouses. Once a warehouse is selected,
it is used indefinitely (throughout the planning horizon).
(b) The transportation cost is composed of a fixed cost and a
variable cost proportional to the quantity and distance.
(It includes the wage of the driver of the vehicle, the material
handling costs for loading and unloading, and the fuel cost of
the vehicles, or shipping charges paid to third-party shipping
companies.)
(c) Lead times from CDC to warehouses as well as those from
warehouses to retailers are given as integer multiples of a unit
time, and may be different from each other.
(d) The per-unit-time demands at each retailer are independent
and identically distributed. They follow normal distributions
with means of integer values. Therefore, the mean and the
variance of the demand that occurs for n time-units are n times
those of the demand that occurs for a unit time.
(e) The demands at the retailers are independent of each other.
(f ) The demand information is known to the supplier (as a result of
VMI contract). Therefore, the demands at the retailers assigned
to a warehouse can be pooled at the warehouse.
(g) Each member of the supply chain uses the (r, q) policy.
(h) Safety factors for the warehouses and the retailers are given.
These safety factors are computed from the required service
levels, which are determined by the management policy of the
supply chain. Reorder points can be computed from the safety
factors.
Assumptions (a)–(c) reflect real situations in the supply chain
considered in this study. The cost structure stated in (b) is
commonly found in real supply chains as well as in many academic
studies, and lead times are generally managed by the days in
practice, which are the units for time periods in this study. In an
assumption (d), we use a normal distribution for the distribution
form of demands since demand quantities may be approximated
with normal distributions if the parameter values are appropriately
estimated, although the normal distributions do not closely fit the
demands. The normal distribution has useful properties that can be
used for mathematical handling of inventory models. The (r, q)
policy is assumed to be used, since it is one of the most commonly
used policies for inventory replenishment.
For a clearer description of the problem, we present a non-linear
mixed integer programming formulation. For the formulation, we
use the following notation.
Ar
Aw
Ci
D
drij
dw
i
hrj
hw
i
i
j
kr
kw
Li
lij
Mi
mj
MS
Oi
Qi
qj
Ri
rj
Vi
vj
VS
Xij
Yi
fixed transportation cost for a delivery to a retailer (from a
warehouse)
fixed transportation cost for a delivery to a warehouse (from
CDC)
storage capacity of warehouse i
variable transportation cost per unit distance
distance between warehouse i and retailer j
distance between CDC and warehouse i
inventory holding cost at retailer j per unit time
inventory holding cost at warehouse i per unit time
index for warehouses (i¼1, 2, y, I)
index for retailers (j¼ 1, 2, y, J)
safety factor corresponding to the required service level at
the retailers
safety factor corresponding to the required service level at
the warehouses
lead time for an order issued by warehouse i to CDC
lead time for an order issued by retailer j to warehouse i
mean of the demand per unit time at warehouse i
P
(Mi ¼ j mj Xij )
mean of the demand per unit time for retailer j
P
sum of the mean demands over all retailers (MS ¼ j mj )
fixed operation cost for warehouse i
order quantity of warehouse i
order quantity of retailer j
reorder point of warehouse i
reorder point of retailer j
variance of the demand per unit time at warehouse i
P
(Vi ¼ j vj Xij )
variance of the demand per unit time for retailer j
sum of the variances of the demands over all retailers
P
(V S ¼ j vj )
a binary variable that is equal to 1 if retailer j is served by
warehouse i, and 0 otherwise
a binary variable that is equal to 1 if warehouse i is open, and
0 otherwise
In the formulation for the problem, the expected values for the
inventory holding costs and the transportation costs of the supply
chain are approximated as follows.
2.1. Average inventory holding cost of a regional warehouse
The reorder
pffiffiffiffiffiffiffiffi point of warehouse i, denoted by Ri, is set to
Mi Li þkw Vi Li since the mean and the standard deviation p
of ffiffiffiffiffiffiffiffi
the
demand during the lead time at the warehouse are MiLi and Vi Li ,
respectively. Also, since the minimum and the maximumpinventory
ffiffiffiffiffiffiffiffi
levels p
atffiffiffiffiffiffiffiffi
the warehouse are approximated as kw Vi Li and
w
Qi þk
Vi Li , respectively, the expected value of the
inventory
pffiffiffiffiffiffiffiffi
level of the warehouse can be estimated as Qi =2 þ kw Vi Li . Therefore, the expected inventory holding
pffiffiffiffiffiffiffiffi cost of the warehouse can be
w
approximated as hw
Vi Li Þ.
ðQ
=2þ
k
i
i
The above approximation of the expected inventory level may
be inaccurate, if there are lumpy demands from the retailers and if
not many retailers are assigned to the warehouse. However, in
J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
many practical applications including a supply chain for personal
computers, by which this research was motivated, the order sizes
from the retailers are not very large because of relatively high
inventory holding cost. Also, in many cases, demands at the
retailers are mutually independent, and so are the order cycles
of the retailers. In such cases, there is no specific time period (in an
order cycle of the warehouse) with very high frequencies of orders
from the retailers. Since the inventory level decreases somewhat
linearly in those cases, the above approximation will not result in a
large error.
2.2. Average transportation cost from CDC to a regional warehouse
When an order for Qi units is delivered from CDC to warehouse i,
the transportation cost for a single delivery is Aw þ DQi dw
i . Also,
since the expected number of deliveries per unit time is Mi/Qi, the
expected transportation cost to warehouse i is given as
ðAw þDQi dw
i ÞMi =Qi .
2.3. Average inventory holding cost of a retailer
If we assume that retailer j is assigned to regional warehouse i,
the reorder point of the retailer, denoted by rj, is set to
qffiffiffiffiffiffiffiffi
mj lij þ kr vj lij , since the mean and the standard deviation of the
qffiffiffiffiffiffiffiffi
demand during the lead time at the retailer are mjlij and vj lij ,
respectively. Also, since the minimum and the maximum inventory
qffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffi
levels at the retailer are approximated as kr vj lij and qj þ kr vj lij ,
respectively, the expected value of the inventory level of the
qffiffiffiffiffiffiffiffi
retailer can be estimated as qj =2 þkr vj lij . Therefore, the expected
qffiffiffiffiffiffiffiffi
inventory holding cost at the retailer is given as hrj ðqj =2 þ kr vj lij Þ.
2.4. Average transportation cost from a warehouse to a retailer
Assume retailer j is assigned to the regional warehouse i. The
transportation cost from the warehouse to the retailer for a single
delivery is given as Ar þDqj drij , when the delivery quantity is qj. Also,
the expected number of deliveries per unit time is mj/qj. Hence, the
expected transportation cost to the retailer can be given as
ðAr þDqj drij Þmj =qj .
Note that the reorder points of the warehouses and the retailers
are affected by the assignment of the retailers to the warehouses
since the mean and the variance of the demand for a warehouse as
well as the lead time to a retailer are determined by the assignment.
Using the above costs, we can formulate the problem considered
in this paper as the following non-linear mixed integer program.
X
pffiffiffiffiffiffiffiffi
w
w
½P Min
½Oi þðAw þ DQi dw
Vi Li ÞYi
i ÞMi =Qi þ hi ðQi =2 þk
i
qffiffiffiffiffiffiffiffi
XX
þ
½ðAr þ Dqj drij Þmj =qj þ hrj ðqj =2 þ kr vj lij ÞXj
i
ð1Þ
j
X
mj Xij ¼ Mi
subject to
8i
ð2Þ
j
X
vj Xij ¼ Vi
8i
ð3Þ
j
X
mj Xij rCi Yi
8i
ð4Þ
j
X
Xij ¼ 1
i
8j
119
Yi A f0, 1g
8i
ð6Þ
Xij A f0, 1g
8i,j
ð7Þ
Mi Z 0
8i
Vi Z 0
8i
ð8Þ
ð9Þ
w
þDQi dw
i ÞMi =Qi
In the objective function, Oi, ðA
and
pffiffiffiffiffiffiffiffi
w
hw
ðQ
=2
þk
L
V
Þ
denote
the
operation
cost
of
the
warehouse
i,
i
i i
i
the transportation cost from CDC to the warehouse and the
inventory holding cost at the warehouse, respectively, while
qffiffiffiffiffiffiffiffi
ðAr þDqj drij Þmj =qj and hrj ðqj =2 þ kr vj lij Þ represent the transportation cost from warehouse i to retailer j and the inventory holding
cost at the retailer, respectively. Constraint sets (2) and (3) ensure
that the mean and the variance of the demand at a warehouse are
the sums of the means and the variances of the demands at the
retailers assigned to the warehouse, respectively. Also, constraint
set (4) is the capacity constraint related to the warehouses, while
(5) ensures that each retailer can be assigned to one and only one
P
P
warehouse. Here, constraints i Mi ¼ M S and i Vi ¼ V S are intentionally omitted since they are redundant, i.e., they are always
satisfied in a solution that satisfies (2), (3), and (5).
3. Solution method
Since it is not easy to find optimal solutions for problem [P], in
which approximated costs are used, and it takes an excessive
amount of time to obtain optimal solutions using commercial
software even for small-sized problems without non-linear cost
terms, we present a heuristic algorithm in this study. The suggested
heuristic algorithm is based on Lagrangian relaxation and subgradient optimization methods, i.e., problem [P] is relaxed by using
Lagrangian multipliers and the solution of the problem is obtained
by finding the best Lagrangian multipliers with the subgradient
optimization method. In the algorithm, we modify the objective
function by replacing order quantities for warehouses and retailers
with the well-known economic order quantities (EOQs) of the
deterministic lot size model. After the replacements, the problem is
relaxed by dualizing a set of constraints with Lagrangian multipliers, and then the relaxed problem is decomposed into three
subproblems. Each subproblem is solved through updates of the
Lagrangian multipliers, until one of the following termination
conditions is satisfied: (1) the iteration count reaches a predetermined limit (to be denoted by U or U1); (2) the gap between an
upper and a lower bounds becomes less than a predetermined limit
(to be denoted by e or e1); or (3) the lower bound has not been
improved for a predetermined number of iterations (to be denoted
by B or B1).
We can obtain the EOQ for each warehouse by differentiation of
related terms in the objective function, i.e., AwMi/Qi +hw
i Qi/2. For
warehouse i, the EOQ is given as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð10Þ
Qi ¼ 2Aw Mi =hw
i
Note that the second derivative of the cost function is always
greater than 0. Also, note that the order quantity of a warehouse is
determined by the assignment of retailers to the warehouse since
the mean of the demand at the warehouse is determined by the
assignment.
Similarly, the EOQ for retailer j can be given as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qj ¼ 2Ar mj =hrj
ð11Þ
ð5Þ
and the second derivative is 40 as well.
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J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
By the replacements of these EOQs for Qi and qj, the objective
function to be minimized is modified to
Xnqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
XX
X
pffiffiffiffiffiffiffiffio
w w
2Aw Mi hw
þ DMi dw
Vi Li Yi þ
Oi Y i þ
eij Xij
i þ hi k
i
i
i
i
j
qffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where eij ¼ 2Ar mj hrj þ Dmj drij þ hrj kr vj lij . Also, since the values of
Mi and Vi are equal to 0 when Yi ¼0 in feasible solutions of [P], the
above objective function can be replaced with
Xnqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
pffiffiffiffiffiffiffiffio XX
w
w w
2Aw Mi hw
Vi Li þ
Oi Y i þ
eij Xij
i þ DMi di þ hi k
i
i
i
j
ð1uÞ
In addition, we replace constraints (2) and (3), which are
equality constraints, with the following inequality constraints,
(20 ) and (30 ), respectively.
X
mj Xij rMi 8i
ð2uÞ
Note that the first subproblem, [SP1], is the problem for
determining values of Yi and Xij, while [SP2] and [SP3] are the
problems for determining values of Mi and Vi for the given k and l,
respectively. In the following, we describe the methods used to
obtain upper and lower bounds for [P].
The upper bound on the solution value of [P] at iteration k,
k
denoted by Z , is given as
k
k
k1
, Z~ g
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffi
P
P
k
~ k hw þDM
~ k dw þ hw kw V~ k Li þ
where Z~ ¼ i Oi Yik þ i
2Aw M
i i
i i
i
i
PP
k
i
j eij Xij ,
X
~k¼
mj Xijk
M
i
Z ¼ Min fZ
j
and
k
V~ i ¼
X
vj Xijk
j
j
X
vj Xij r Vi
8i
ð3uÞ
j
Note that the optimal solution of the problem with Eqs. (20 ) and
(30 ) instead of Eqs. (2) and (3) is also optimal for the original
problem, as discussed in Miranda and Garrido (2004). Among the
solutions that satisfy Eqs. (20 ), (30 ), and (4)–(9), only the solution in
P
P
which
j mj Xij ¼ Mi and
j vj Xij ¼ Vi are satisfied minimizes the
0
objective function, (1 ). That is, since Eqs. (2) and (3) are always
satisfied by an optimal solution of the problem with inequality
constraints, they are replaced with Eqs. (20 ) and (30 ) in this study. In
addition, inequality constraints can be more effectively used in
Lagrangian relaxation methods.
The solution approach suggested in this research for problem
[P], after replacements of Eqs. (1), (2) and (3), is based on
Lagrangian relaxation and subgradient optimization methods. In
the solution approach, upper and lower bounds for the problem are
obtained iteratively until one of the aforementioned three termination conditions is satisfied. First, we derive the following relaxed
problem, [LR], by relaxing constraints (20 ) and (30 ) with Lagrangian
multipliers k and l, where k and l are vectors with nonnegative
elements, i.e., li Z0 and mi Z0.
Xnqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
pffiffiffiffiffiffiffiffi
w
w w
2Aw Mi hw
Oi Yi þ
Vi Li li Mi
½LR Min
i þ DMi di þhi k
i
i
XX
mi Vi þ
ðeij þ li mj þ mi vj ÞXij
i
j
subject to Eqs. (4)–(9)
Then, [LR] is decomposed into three subproblems as follows:
X
XX
½SP1 Min
Oi Y i þ
ðeij þ li mj þ mi vj ÞXij
i
i
j
subject to (4)–(7)
o
Xnqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w
2Aw Mi hw
½SP2 Min
i þ DMi di li Mi
Here, Yki and Xkij denote the solution of [SP1] at iteration k.
On the other hand, the lower bound on the solution value for [P]
at iteration k, denoted by Z k , is obtained as the sum of the lower
bound (or the optimal solution value) of [SP1] and the optimal
solution values of [SP2] and [SP3], since the subproblems are
mutually independent and can be solved independently without
consideration of the others. In the following, we describe the
methods used to solve the three subproblems in the suggested
solution approach.
Solving [SP1] for the given k and l
For the given k and l, [SP1] is solved by Lagrangian relaxation
and subgradient optimization methods. That is, [SP1] is relaxed by
using Lagrangian multipliers, and the best values for the Lagrangian
multipliers are found by using the subgradient optimization
method, as described below. By dualizing constraint set (5) with
Lagrangian multipliers ht, the vector of which the jth element, ytj ,
corresponds to retailer j at iteration t, and adding another constraint, we obtain the following problem.
X
XX
X t
t
½LRðqt Þ Min
Oi Yi þ
ðeij þ li mj þ mi vj þ yj ÞXij yj
i
Min
Xn
i
ð14Þ
j
Note that the additional constraint, (14), can provide tighter
lower bounds for the problem although it is redundant, as shown in
Nauss (1978).
Bitran et al. (1981) show that [LR(ht)] can be decomposed into I
independent (0,1) knapsack problems for the I warehouses, and the
problem corresponding to warehouse i, denoted by [LR0 i(ht)], can be
solved with a dynamic programming approach. The ith knapsack
problem can be given as follows:
X
t
ðeij þ li mj þ mi vj þ yj ÞXij
½LRui ðqt Þ Min Oi þ
subject to
X
mj Xij rCi
ð4uÞ
j
ð12Þ
w
hw
i K
o
pffiffiffiffiffiffiffiffi
Vi Li mi Vi
Xij A f0, 1g
8j
ð7uÞ
By using solutions for the above problems, we can restate
[LR(ht)] as follows. Here, Z() denotes the optimal solution value of
the problem [].
X
X t
ZðLRui ðht ÞÞYi yj
½LR3 ðqt Þ Min
i
subject to Eq. (9)
X
Vi ¼ V S
i
j
j
i
½SP3
j
subject to Eqs. (4), (6), (7) and
X
X
Ci Yi Z
mj
i
subject to (8)
X
Mi ¼ M S
i
ð13Þ
i
subject to Eqs. (6) and (14)
j
J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
Note that this new expression for [LR(ht)] is also in the form of
the knapsack problem, and this can be solved with a dynamic
programming method.
From the optimal solutions (X0 ij) for the independent I knapsack
problems, [LR0 i(ht)], i¼1, y, I, and the optimal solution (Y0 i) for the
t
t
knapsack problem, [LR1(ht)], the optimal solution (X^ and Y^ ) of
ij
i
[LR(ht)] at iteration t can be obtained as follows:
t
t
if Y0 i ¼1, Y^ ¼Y0 i and X^ ¼ X0 ij; and
i
ij
t
if Y0 i ¼0, X^ ij ¼ 0
As a result, a lower bound on the optimal solution value of [SP1]
at iteration t can be given as
X
XX
t
t X t
t
Oi Y^ i þ
ðeij þ li mj þ mi vj þ yj ÞX^ ij yj
ð15Þ
i
i
j
j
which is the optimal solution value of [LR(ht)].
A feasible solution (Yti and Xtij) for [SP1] can be found from the
solution of [LR(ht)] as follows. Before describing the procedure for
obtaining a feasible solution for [SP1] at iteration t, we define the
t
t
following sets, using the solution (X^ and Y^ ) of [LR(ht)]:
ij
IO ¼
t
fi A I9Y^ i
i
¼ 1g;
t
IC ¼ fi A I9Y^ i ¼ 0g;
X t
J L ¼ fj A J9 X^ ij ¼ 0g;
i
X t
J E ¼ fj A J9 X^ ij ¼ 1g; and
i
X t
J M ¼ fj A J9 X^ ij 4 1g
i
Also, let Di denote the leftover capacity of warehouse i, i.e.,
P
t
Di ¼ Ci j mj X^ ij . Then, a feasible solution for [SP1] can be obtained
with the following procedure.
Procedure 1. (Obtaining feasible solutions for [SP1])
t
Step 1. Assign retailer jAJE to warehouse i for i–j pairs with X^ ij ¼ 1.
Step 2. If JM ¼ +, go to step 4. Otherwise, assign retailers j’s in JM
to a warehouse with the minimum value of eij + limj + mivj among
t
warehouses i’s with X^ ij ¼ 1.
L
Step 3. If J ¼+, terminate. Otherwise, for a retailer with the maximum mean demand among retailers in JL, say retailer jn, check if there
are warehouses i’s with Di mnj Z0 among the warehouses in IO.
3.1) If there exist open warehouses with Di mnj Z0, assign
retailer jn to a warehouse with the minimum value of
enij + limnj + mivnj among warehouses iAIO with Di mnj Z0, say warehouse in. Set Dni ’Dni mnj and JL’JL\{jn}, and go to step 4.
3.2) If there is no open warehouse with Di mnj Z0, assign retailer
jn to a warehouse with the minimum value of enij + limnj + mivnj among
nn
n L
L
n
warehouses iAIC, say warehouse inn. Set Dnn
i ’Di mj , J ’J \{j }
t
and Y^ i ’1, and go to step 4.
Step 4. If JL ¼+, terminate. Otherwise, go to step 3.
The subgradient method is used to find the best Lagrangian multipliers. For a given multiplier corresponding to retailer j at iteration t, ytj,
the Lagrangian multiplier at the next iteration is set as
P t
ytj + 1 ¼ ytj + bt( i X^ ij 1), in which bt is a positive step size set as
. P t
2
bt ¼ rt ðzt ZðLRðht ÞÞ 99 i X^ ij 199 , where zt is the best upper bound,
i.e., the solution value of the best feasible solution obtained so far. The
value of rt 40 is set to 2 initially and is reduced by a half when the
121
lower bound of [SP1] is not improved for a given number of iterations
(20 iterations, in the algorithm suggested in this research). The
procedure for solving [SP1] can be summarized as follows. In the
procedure, U, e, and B are parameters needed for the stopping
conditions mentioned earlier. Also, U0 is another parameter that
specifies the number of times the relaxed problem of [SP1] is to be
solved for the given h.
Procedure 2. (Solving [SP1])
Step 0. Set u ¼0, b ¼0, and h ¼0.
Step 1. If u 4U or b4B, stop; otherwise, go to step 2.
Step 2. Obtain an optimal solution of [LR(h)], and let u’u+ 1. If
the solution is feasible to [SP1], terminate. The current solution is
optimal. Otherwise, go to step 3.
Step 3. Obtain a lower bound from the solution obtained in step 2.
If the lower bound is better than the best lower bound obtained so
far, let b’0; otherwise, let b’b+ 1 and update Lagrangian multipliers using the subgradient optimization method. If u is a multiple
of U0 , go to step 4; otherwise, go to step 1.
Step 4. Find a feasible solution for [SP1]. Let UBn and LBn be the
solution value of the best feasible solution and the best lower bound
obtained so far. If (UBn LBn)/LBn o e, stop; otherwise, go to step 1.
Solving [SP2] for the given k
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S w
S
To solve [SP2], we compute 2Aw M S hw
i þDM di li M for each
warehouse i, and find a warehouse with the minimum value, say
warehouse in. If the minimum value is less than or equal to 0, the value
of Mi is set to MS for i¼in, and 0 for iain. If the minimum value is
positive, the value of Mi is set to 0 for all warehouses. Therefore,
the optimal solution value of the problem is equal to min
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S w
S
[ 2Aw M S hw
i þ DM di li M , 0]. Note that the optimality of the
solution can be proven by induction as in Miranda and Garrido (2004).
Solving [SP3] for the given l
First, we select a warehouse with the minimum value of
pffiffiffiffiffiffiffiffiffi
w
hw
V S Li mi V S among all warehouses, say warehouse in. If the
i k
minimum value is less than or equal to 0, the value of Vi is set to VS
for i¼in, and 0 for iain. If the value is positive, the value of Vi is set to
0 for all i. The optimality of the resulting solution can be proven as
in Miranda and Garrido (2004). The optimal solution value is equal
pffiffiffiffiffiffiffiffiffiffiffi
w
V S Li mi V S , 0].
to min[hw
i k
Overall procedure
The overall procedure for solving the original problem is given
below. Here, U1, e1, and B1 are parameters needed for stopping
conditions, and U00 is another parameter that specifies number of
times the three subproblems are to be solved for the given k and l.
Procedure 3. (Solving [P])
Step 0. Set u’0, b’0, k’0, and l’0.
Step 1. If u 4U1 or b4B1, stop; otherwise, go to step 2.
Step 2. Solve [LR], and let u’u + 1. If the solution is feasible to [P],
stop. The current solution is optimal. Otherwise, go to step 3.
Step 3. Obtain a lower bound by adding the optimal solution
values of the subproblems. If the lower bound is better than the best
lower bound obtained so far, let b’0; otherwise, let b’b+ 1 and
update Lagrangian multipliers with the subgradient optimization
method. If u is a multiple of U00 , go to step 4; otherwise, go to step 1.
Step 4. Obtain a feasible solution for [P]. Let UBn and LBn be the
solution value of the best feasible solution and the best lower
bound obtained so far, respectively. If (UBn LBn)/LBn o e1, stop;
otherwise, go to step 1.
122
J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
In the above procedure, the Lagrangian multipliers are updated
as follows. For the given multipliers of warehouse i at iteration k, lki ,
and mki , the Lagrangian multipliers at the next iteration are set as
X
lki þ 1 ¼ maxf0, lki þ ak ð mj Xijk Mik Þg
j
and
mki þ 1 ¼ maxf0, mki þ ak ð
X
vj Xijk Vik Þg
j
P
P
k
2
2
where ak ¼ rk ðZ Z k Þ=ð99 j mj Xijk Mik 99 þ 99 j vj Xijk Vik 99 Þ, and Mki
and Vki denote the solutions of [SP2] and [SP3], respectively, at
iteration k. The value of rk is set to 2 initially and halved when the
lower bound of the problem is not improved for a given number of
iterations (20 iterations, in the suggested algorithm).
4. Computational experiments
To evaluate the performance of the heuristic algorithm suggested in
this study, we compare results of the algorithm with those obtained
from the method, currently used in a real logistics system for personal
computers in Korea, as well as lower bounds on the optimal solutions.
For the comparison, we tested the algorithm on two sets of problems: a
set of smaller problems with 5 warehouses and 20 retailers and a set of
larger problems with 10 warehouses and 40 retailers. Note that the
larger problems represent the real system more closely. For each
problem set, we generated 128 test problems, 3 problems for each of all
combinations of two levels (high and low) for the fixed transportation
cost from the central distribution center (CDC) to a warehouse, the
fixed transportation cost from a warehouse to a retailer, the unit
inventory holding cost at a warehouse, the unit inventory holding cost
at a retailer, the mean of the demand at a retailer, the variance of the
demand at a retailer, and the lead time. Other related data were
generated as follows. Here, U(x, y) and DU(x, y) denote the uniform
distribution with range (x, y) and the discrete uniform distribution with
range [x, y], respectively.
(1) Locations of the (candidate) warehouses and the retailers
were generated randomly as follows: x-coordinates of the
locations were generated from U(–400, 400) and y-coordinates of the locations were generated from U(–500, 500). The
location of CDC was set to (0, 0).
(2) The fixed operation costs of the warehouses were generated
from U(300000, 500000).
(3) The mean of the demand at a retailer was generated from
DU(20, 40) and DU(40, 60) for low and high levels, respectively.
(4) The variance of the demand at a retailer was generated from
U(20, 40) and U(60, 80) for low and high levels, respectively.
(5) The unit inventory holding cost at a warehouse was generated
from U(60, 80) and U(120, 160) for low and high levels,
respectively.
(6) The unit inventory holding cost at a retailer was generated
from U(200, 250) and U(400, 500) for low and high levels,
respectively.
(7) The distances between CDC and regional warehouses as well
as those between regional warehouses and retailers were
given as the Euclidean distance between them.
(8) The fixed transportation cost from CDC to a warehouse was
generated from U(1000, 2000) and U(2000, 4000) for low and
high levels, respectively.
(9) The fixed transportation cost from a warehouse to a retailer
was generated from U(300, 500) and U(600, 1000) for low and
high levels, respectively.
(10) The variable transportation cost per unit distance was generated from U(1, 5).
(11) The lead time was generated from DU(0.01d0 , 0.03d0 ) and
DU(0.04d0 , 0.06d0 ) for low and high levels, respectively, where
d0 is the distance between locations, i.e., between CDC and a
warehouse or between a warehouse and a retailer.
(12) The storage capacity of a warehouse was generated from
J
J
P
P
DU(3
mj =I, 5
mj =I).
j¼1
j¼1
(13) The safety factors for the warehouses and the retailers were
set to 1.96, representing 97.5% service level.
The heuristic algorithm was coded in C programming language,
and computational experiments were performed on a personal
computer with Pentium 4 processor operating at 3.2 GHz clock speed.
We set the values of parameters needed for stopping conditions after a
series of tests on several candidate values for them. Although detailed
results of these tests are not given here, selected values were: U¼300,
e ¼0.01, B¼100, and U0 ¼25 for procedure 2; and U1¼600, e1¼0.01,
B1¼200, and U00 ¼25 for procedure 3.
The performance of the suggested algorithm was shown with
the percentage of cost reduction from the cost resulting from the
method currently used in a real system. In that system, the retailers
are assigned to the nearest warehouses of which the storage
capacity constraint is not violated by the assignment. Also, to
see the absolute performance of, or the quality of the solutions
obtained from, the algorithm, we show the results in terms of
percentage gap of heuristic solutions from the lower bounds that
are obtained from the Lagrangian relaxation method.
Results of the test on smaller-sized problems (5 warehouses and
20 retailers) are given in Table 1. The average percentage gap of the
solution values of the suggested algorithm from lower bounds was
o4%. Also, the suggested algorithm could save over 24% of the costs
compared with the currently used method. This may be because in
the suggested algorithm, we take account of the warehouse
operation costs as well as the inventory holding costs and the
transportation costs of the whole supply chain simultaneously to
select warehouses to be used and to assign the retailers to the
selected warehouses. Note that in the currently used method, the
retailers are assigned to the warehouses by considering only
distances between retailers and warehouses and demand quantities (from the customers) at the retailers. The suggested heuristic
algorithm required 1 h of CPU time on an average for a problem.
Table 2 shows the results of the test on larger-sized problems
(10 warehouses and 40 retailers). The average percentage gap from
lower bounds was o8%, and the average percentage reduction of
solutions from those of the currently used method was about 45%.
The outperformance of the heuristic algorithm over the currently
used method was more significant in these larger-sized problems,
possibly because the benefit of employing the risk pooling strategy
becomes more significant when the numbers of candidate warehouses and retailers are larger. In other words, the risk pooling
strategy can be more effectively used in supply chains with more
candidate warehouses and retailers. It took about 4 h of CPU time
on an average to solve a problem with the suggested heuristic
algorithm. Although the computation time does not seem to be
very short, it may be reasonable since the problem does not have to
be solved very quickly or on a real-time basis in practice. From the
above results, one can argue Lagrangian heuristic algorithm
suggested in this study is a viable tool for inventory management
in the supply chain considered in this study.
To see which factors affect the relative performance (percentage
reduction) of the heuristic algorithm, analyses of variance were
performed on results of tests on two problem sets, and the results
are given in Table 3. The results show that the relative performance
was affected by the unit inventory holding costs of the retailers, the
lead time, and the mean and the variance of the demands at the
J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
123
Table 1
Performance of the algorithm on smaller-sized problems.
Factors
Levels
Percentage gapa (%)
Percentage reductionb (%)
Average CPU time (s)
Fixed transportation cost to a warehouse (Aw)
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
3.35
3.34
3.45
3.25
3.33
3.36
3.47
3.23
4.44
2.26
1.58
5.11
3.33
3.36
24.84
24.92
24.95
24.80
24.76
24.99
25.85
23.90
27.16
22.60
25.60
24.16
25.46
24.30
(6.17)
(6.12)
(6.16)
(6.12)
(6.18)
(6.10)
(6.07)
(6.06)
(6.29)
(5.05)
(6.03)
(6.18)
(6.19)
(6.05)
3065
3101
3226
2940
3015
3151
3160
3005
2721
3445
2500
3666
3058
3107
24.88 (6.14)
3083
Fixed transportation cost to a retailer (Ar)
Unit inventory holding cost at a warehouse (hw
i )
Unit inventory holding cost at a retailer (hrj )
Mean of the demand at a retailer (mj)
Variance of the demand at a retailer (vj)
Lead time
Average
a
b
(2.74)
(2.67)
(2.66)
(2.75)
(2.74)
(2.67)
(2.79)
(2.62)
(3.04)
(1.72)
(1.20)
(2.63)
(2.73)
(2.68)
3.35 (2.70)
Average and standard deviation (in parentheses) of the percentage gap of the heuristic solution value from the best lower bound.
Average and standard deviation (in parentheses) of the percentage reduction of the cost obtained with the heuristic from the cost obtained with the current method.
Table 2
Performance of the algorithm on larger-sized problems.
Factors
Levels
Percentage gapa (%)
Percentage reductionb (%)
Average CPU time (s)
Aw
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
7.31
8.01
7.58
7.74
7.47
7.85
7.83
7.49
9.65
5.67
3.34
11.98
7.87
7.45
45.40
45.24
45.49
45.16
45.30
45.34
47.17
43.48
48.02
42.62
46.87
43.77
46.40
44.24
(5.72)
(5.81)
(5.80)
(5.73)
(5.77)
(5.76)
(5.50)
(5.43)
(4.80)
(5.38)
(5.68)
(5.43)
(5.63)
(5.70)
14,000
15,320
16,228
13,092
13,655
15,665
16,024
13,296
12,518
16,802
13,146
16,174
13,937
15,383
45.32 (5.76)
14,660
Ar
hw
i
hrj
mj
vj
Lead time
Average
a
b
(6.23)
(6.36)
(6.16)
(6.45)
(6.26)
(6.35)
(6.34)
(6.27)
(6.85)
(4.97)
(2.93)
(5.78)
(6.45)
(6.15)
7.66 (6.30)
Average and standard deviation (in parentheses) of the percentage gap of the heuristic solution value from the best lower bound.
Average and standard deviation (in parentheses) of the percentage reduction of the cost obtained with the heuristic from the cost obtained with the current method.
Table 3
Results of the analyses of variance.
Source of variation
Sum of squared error
Mean squared error
(a) Results for smaller-sized problems
Aw
1
Ar
1
hw
1
i
hrj
1
mj
1
vj
1
Lead time
1
Error
376
Total
383
0.54
2.06
5.16
363.96
1997.71
198.28
129.51
11,723.03
14,420.25
0.54
2.06
5.16
363.96
1997.71
198.28
129.51
31.18
0.02
0.07
0.17
11.67b
64.07b
6.36a
4.15a
(b) Results for larger-sized problems
Aw
1
Ar
1
w
hi
1
hrj
1
mj
1
vj
1
Lead time
1
Error
376
Total
383
2.29
10.51
0.21
1306.32
2792.92
924.26
450.23
7228.38
12,715.11
2.29
10.51
0.21
1306.32
2792.92
924.26
450.23
19.22
0.12
0.55
0.01
67.95b
145.28b
48.08b
23.42b
a
b
Degree of freedom
There is difference in the effects at the significance level of 0.05.
There is difference in the effects at the significance level of 0.01.
F-value
124
J.-H. Kang, Y.-D. Kim / Int. J. Production Economics 135 (2012) 116–124
retailers for both problem sets. That is, the percentage reduction is
larger when the unit inventory holding costs of the retailers, the
lead time, and/or the mean and the variance of the demands at the
retailers are smaller. This may be due to the fact that the increase in
the inventory holding costs of the retailers (that are supposed to
increase due to the risk pooling strategy) is smaller when values of
the above factors are smaller.
5. Concluding remarks
In this paper, we considered an inventory management problem
in a two-level supply chain, in which there are a single supplier
(composed of a central distribution center and multiple regional
warehouses) and multiple retailers. Assuming the supply chain is
operated under a vendor-managed inventory contract, we presented
a heuristic algorithm based on Lagrangian relaxation and subgradient optimization methods for the problem of selecting warehouses
to be used among a set of candidate warehouses, assigning each
retailer to one of the selected warehouses, and determining replenishment plans of the warehouses and the retailers. Results of
computational experiments showed that the heuristic algorithm
gave relatively good solutions in a reasonable computation time.
To the best of our knowledge, this research is the first attempt to
solve a supply chain planning problem considering the risk pooling
strategy and the trade-offs between the inventory holding costs at
the retailers and the inventory holding costs and operation costs at
the warehouses. Although the risk pooling strategy is considered
for supply chain management in the previous research such as
Gerchak and He (2003), Shen et al. (2003), and Miranda and Garrido
(2004), they do not consider the inventory holding costs at retailers,
and argue that the effect of risk pooling becomes more significant
as the inventory holding costs at the warehouses and the variances
of the demands increase. Note that the risk pooling strategy may
cause an increase in the inventory holding costs at the retailers and
the transportation costs from the warehouses to the retailers, since
the lead times of the retailers and the distances between warehouses and retailers may be increased. Since the bargaining power
of the retailers is not negligible but rather significant nowadays, the
costs incurred at both the retailers’ side and the supplier’s side
should be considered simultaneously (as was done in this study) to
reduce the overall cost of the whole supply chain and to improve
the partnership between the supplier and retailers.
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