European Journal of Operational Research 222 (2012) 216–228

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European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

A continuous approximation approach for the integrated facility-inventory

allocation problem
Yu-Chung Tsao a, Divya Mangotra b, Jye-Chyi Lu ⇑,b, Ming Dong a
a
Sino-US Global Logistics Institute, Shanghai Jiao Tong University, Shanghai, China
b
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, USA

a r t i c l e i n f o a b s t r a c t

Article history: In today’s retail business many companies have a complex distribution network with several national and
Received 15 December 2009 regional distribution centers. This article studies an integrated facility location and inventory allocation
Accepted 25 April 2012 problem for designing a distribution network with multiple distribution centers and retailers. The key
Available online 3 May 2012
decisions are where to locate the regional distribution centers (RDCs), how to assign retail stores to RDCs
and what should be the inventory policy at the different locations such that the total network cost is min-
Keywords: imized. Due to the complexity of the problem, a continuous approximation (CA) model is used to repre-
Allocation
sent the network. Nonlinear programming techniques are developed to solve the optimization problems.
Facility location
Inventory management
The main contribution of this work lies in developing a new CA modeling technique when the discrete
Supply chain design data cannot be modeled by a continuous function and applying this technique to solve an integrated facil-
ity location-allocation and inventory-management problem. Our methodology is illustrated with the net-
work from a leading US retailer. Numerical analysis suggests that the total cost is significantly lower in
the case of the integrated model as compared with the non-integrated model, where the location-
allocation and inventory-management problems are considered separately. This paper also studies the
effects of changing parameter values on the optimal solutions and to point out some management
implications.
Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Companies in the US spend approximately $14 billion per year

on inventory interest, insurance, taxes, depreciation, obsolescence
Manufacturing outsourcing in the global environment has never and warehousing. Their logistics activities account for 15–20% of
been stronger than it is today. Increased outsourcing has led to sig- the total cost of finished goods (Menlo, 2007). With such a huge
nificant changes in the design of the retail distribution network. inventory investment and the growing demand for warehouses,
While the traditional distribution network had manufacturing it is important to make optimal decisions for facility locations
plants supply goods to retail stores directly, off-shore manufactur- and inventory allocation in a supply chain.
ing has increased the network’s demand for transportation and Inventory is a key driver of supply chain performance, which is
warehousing to deliver the goods. When the goods arrive at sea- measured in terms of fill-rate and in-stock probability. The optimal
ports in a country, they must be consolidated by region at national inventory levels at the different distribution locations is known as
(import) distribution centers. From these national distribution cen- the multi-echelon inventory problem (see Roundy, 1985; Deuer-
ters (NDCs), the goods are shipped to regional distribution centers meyer and Schwarz, 1981; Ganeshan, 1999, for a detailed review).
(RDCs), from which they are delivered to retail stores. These RDCs In the past, the literature assumes that distribution centers have
help pool risk by consolidating shipments from the NDCs. Most been located optimally between the manufacturer and the stores
companies have complex distribution networks with several im- prior to the inventory decisions, and hence ignores the potential
port and regional DCs. For example, Target, Inc. has three import impact of facility location issues. On the other hand, without con-
warehouses, 22 regional distribution centers, and 1300 retail sidering the inventory management issue, the facility location-
stores. Frito-Lay, Inc. has operated its distribution network with allocation problem has been extensively studied in Daskin
42 plants, one national DC, and 325 regional DCs before 2000 (1995), Drezner (1995) and Brandeau and Chiu (1989).
(see Erlebacher and Meller, 2000). The two key decisions of facility location and inventory allocation
are dependent on each other. One of the key cost components for the
facility location problem is the transportation cost, which depends
⇑ Corresponding author. on the frequency of inventory replenishment at different facilities.
E-mail address: jclu@isye.gatech.edu (J.-C. Lu). This replenishment frequency is a function of the inventory policy.

0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ejor.2012.04.033
 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228 217

Similarly, the inventory allocation problem models the inventory fine-tuning the continuous approximation technique when the in-
cost at the distribution center (DC) for a known value of demand put variables cannot be approximated by a smooth function. Third,
served by each DC. This requires information regarding which retail- the model takes a complex nonlinear form and this paper provides
ers are assigned to which DC. a two-stage solution approach to solve the nonlinear programming
The interrelations between the facility location and inventory problem efficiently.
policy problems suggest that an integrated model with the facility,
transportation costs, and inventory costs is needed to solve
network design problems. In spite of this well understood depen- 2. Literature review
dence, most studies deal with these two problems individually
because of the sheer complexity of integrating them. Traditionally 2.1. Integrated network design and inventory policy decisions
inventory decisions were not considered in facility location models
mainly because the former is considered as a medium-term deci- There are several papers in the area of integrated facility loca-
sion that can be altered whereas the latter is a long-term decision. tion and single location inventory control. Shen (2007) has made
The integrated problem takes a non-linear form and the number of a complete review of the supply chain design literature and of
decision variables is enormous, making it computationally current practices. The research in this area considers distribution
challenging to solve the problem for a real network. The studies networks with a single plant serving multiple retailers. The distri-
by Teo and Shu (2004) and Romeijn et al. (2007) are the only bution center (DC) location and inventory policy at the DC are both
two papers we have found in the area of integrated logistic decision variables. It is assumed that each retailer has a variable
network design. Though these models are very detailed, the prob- demand process. Since the addition of inventory terms makes the
lem is only solved on a small scale and it does not account for objective function nonlinear, researchers have looked at approxi-
supply uncertainty in the network. mations to linearize it.
Unlike previous studies, which use discrete models, this study Nozick and Turnquist (1998) approximated the safety stock cost
adopts a continuous approximation (CA) approach to model the at each DC by a linear regression function of the number of DCs,
supply network design problem. The key idea under the CA ap- and uses this to estimate the inventory cost function. In their mod-
proach is to define decision variables using continuous functions el, inventory is stocked at the DC and replenished using a one-for-
and hence reduce the complexity of the problem. While the CA ap- one policy. The fixed-charge facility location model defined in Da-
proach does not determine the exact location of the distribution skin (1995) uses the linear inventory cost function to determine
centers, it defines a service area for each distribution center in the least cost set of DC locations. Nozick and Turnquist (2001) ex-
terms of circular influence areas. Earlier studies by Newell (1973) tended their previous model by adding service responsiveness and
and Dasci and Verter (2001) show that influence areas with central uncertainty in delivery time to the DC. Service responsiveness is
distribution nodes is a near optimal solution. The goal of this study defined in terms of stock-outs and time-based delivery. Stock-outs
is to provide logistics network planners with a high level solution are incorporated in the safety stock function, while the time-based
for the integrated facility location and inventory allocation prob- delivery constraint is modeled explicitly as coverage distance.
lem. To the best of our knowledge, this problem area is new, and Shen et al. (2003) studied a distribution network in which some
our work presents the most detailed non-linear cost model. We of the retailers are allowed to act as distribution centers to achieve
propose a solution technique that preserves the interrelation be- risk-pooling benefits in terms of inventory cost savings. Their prob-
tween facility and inventory decisions. lem determines which retailers should serve as DCs and how much
Most papers in the area of network design use a discrete model inventory these stocking points should hold. The inventory model
to formulate the problem. Discrete models provide managers with in their work is the continuous review (Q, r) model with a Type-I
optimal solutions, but their data and computational requirements service constraint. They reformulate the nonlinear problem as a
increase tremendously as they try to capture real operational net- set-covering model, and propose a column generation algorithm
works. Also, data reliability and model accuracy decrease as the that can solve the problem exactly for two special cases in O
amount of data increases. Continuous approximation could be a (jnj2logjnj).
remedy to these shortcomings, as it requires less data to generate Miranda and Garrido (2004) presented an integrated model for
closed or near-closed form solutions (Dasci and Verter, 2001). Even capacitated facility location problem (CFLP) and inventory control
though our modeling approach requires us to sacrifice the level of decisions. Their model determines the location of each distribution
detail by replacing discrete variables with smooth functions, the center based on the (Q, r) inventory policy at each DC location.
resulting model more closely depicts real logistic networks. Their solution methodology involves a lagrangian relaxation and
This paper presents a continuous approximation model for the sub-gradient method. In another study, Erlebacher and Meller
solving the location-inventory problem and answers the following (2000) examined a distribution system design problem in which
questions: (1) which RDC locations should be open, (2) which retail customer demand is distributed uniformly along a grid network.
store should be served from which RDC location, and (3) how much They proposed a two-stage heuristic procedure that fixes the num-
inventory should be held at the RDCs and the NDCs? The motivation ber of DCs in the first stage to estimate DC demand, which the
for this approximation is that if we can cut down the size of data, second stage then uses to estimate the number of DCs.
then we can solve larger-scale problems to get some meaningful in- More recently, Teo and Shu (2004) studied an integrated logistic
sights. Our solution defines the input data in terms of continuous network problem that considers inventory cost for multiple eche-
functions and is capable of formulating these functions for a data lons of inventory stocking locations. Their approach models the
set of any size. inventory cost at each DC and retailer. They use the convex inven-
The main purpose of this paper is to meet a threefold goal: First, tory minimization function proposed by Roundy (1985) along with
this paper highlights the importance of integrating facility location transportation and facility costs, to formulate a MIP problem. They
decisions with inventory decisions. We show that a non-integrated proposed a column generation technique to solve their model.
approach generates results that have a significantly higher total Their solution is solvable in O (nlog(n)) time, and is within 2% of
network cost compared to an integrated approach. Second, this the optimal solution when the problem instance is small (20 ware-
paper presents a continuous approximation method for solving houses and 100 retailers). However, their model does not include
the integrated facility location and inventory allocation problem demand or supply uncertainty. In another study, Romeijn et al.
with non-homogenous data. We propose a methodology for (2007) extended their previous model by adding safety stock terms
218 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228

to account for demand variability. In our work, we solve the inte- (1985) studied logistics planning models that used continuous
grated logistic design (i.e., facility location and inventory alloca- space models under general conditions, i.e., by relaxing the
tion) problem using two different approximation models for a assumption of uniform density for stores. They developed an ana-
scenario with 284 retail stores representing the southeastern re- lytical framework for estimating the transportation costs for dis-
gion of a major US retail chains distribution network. tributing goods from a single origin to multiple destinations,
using clusters to account for dense customer destinations. These
2.2. Data approximation for logistic network design clusters can be analyzed as sub-regions of the main region. Wang
and Lu (2006) studied spatial modeling and proposed smoothing
This line of research began to appear in the early 1970s in a techniques for non-homogeneous processes by considering details
seminal paper by Newell (1973) that uses data approximation at different levels of the distribution network. Their work proposes
techniques for warehouse location problem. Geoffrion (1976) fine refinements to the approximation models based on the level of
studied a continuous model for warehouse location in which ware- details captured by the data. We present a two-phase approxima-
house serves demand that is distributed uniformly over a plane. tion technique to solve the integrated facility location and inven-
Drezner and Wesolowsky (1980) determined the optimal location tory allocation problem. The proposed approach captures the
of a facility relative to area demands. Iri et al. (1984) formulated non-homogeneity of input parameters discussed by Blumenfeld
a class of location problems and showed that using the Voronoi- and Beckmann (1985) and Wang and Lu (2006).
diagram algorithm can numerically solve considerably large
problems within a practicable time. Cavalier and Sherali (1986)
considered the Euclidean distance location-allocation problems
3. Total network cost function
with uniform demand over convex polygons. Erlenkotter (1989)
used a General Optimal Market Area (GOMA) model to determine
The network under study in this thesis is a three-level distribu-
the optimal area served by a single production point when the de-
tion system with retail stores at level zero meeting demand of the
mand is assumed to be distributed uniformly. This was an exten-
end customers. The regional distribution centers (RDCs) are located
sion of a previous study by Geoffrion (1976) and Newell (1973),
at level one to help consolidate shipments and pool risk. The na-
with more detailed expressions for the production cost. Rutten
tional distribution centers (NDCs) are located at level two and they
et al. (2001) further refined the GOMA model by considering a
help consolidate shipments arriving from overseas manufacturers
distribution network and adding inventory cost terms.
and deliver them to the RDCs. The goods flow from the facilities
Burn et al. (1985) studied a distribution network with a single
at the higher level to the facilities at the lower level until they
supplier and multiple customers. They proposed an analytic meth-
reach level zero (see Fig. 1). We will refer to this three-level
od that uses the spatial density of customers to minimize the
network as a logistic network in the rest of the analysis.
inventory and transportation cost of freight. Their work considers
Before we model and solve the integrated facility location and
two different distribution strategies, direct shipping and peddling.
inventory allocation problem, we want to make some assumptions
Campbell (1992) considered a location-allocation problem for dis-
around the network structure, demand pattern and inventory
tribution to a uniform demand with transshipments. Campbell
replenishment policies. Our assumptions help us frame the different
(1993) discussed the hub location problems with continuous and
logistics cost functions and hence create a practical model to ana-
discrete demand. Daganzo (1996) presented continuum approxi-
lyze. These assumptions are needed to simplify the complex logistics
mation techniques for the network design problem, focusing in
model without sacrificing our understanding of the problem. The
particular on vehicle dispatch scheduling. Francis et al. (1996) pre-
simplified model structure also allows us to explore the different
sented a median-row-column aggregation algorithm with provable
decision issues in detail. Most of our assumptions are consistent
properties including an error bound for rectilinear distance p-med-
with those used extensively in the literature (see Ganeshan, 1999;
ian problems. Langevin et al. (1996) presented an extensive review
of continuous approximation models developed for freight distri-
bution problems. Carrizosa et al. (1998) was devoted to the study Retail stores
of the Regional Weber Problem, an extension of the Weber prob-
lem which allows the demand not be concentrated onto a finite
set of points.
Dasci and Verter (2001) studied a production and distribution
RDCs
design problem using the continuum approximation technique.
Their work explicitly models facility costs by looking at the
operational and acquisition cost components. The model presented
NDCs
in their work is an extension of a continuous approximation model
for the facility design problem. However, their approach does not
consider inventory costs. Pujari et al. (2008) utilized a CA proce-
dure to determine optimal number and size of shipments while
Manufacturer
considering issues of location, production, inventory, and transpor-
tation. Murat et al. (2010) provided a CA framework for solving
location-allocation problems with dense demand. Their methodol-
ogy is suitable for problems where allocation decisions can be rea-
sonably approximated by polygons.
Level 2
2.3. Spatial approximation
Level 1
To the best of our knowledge, there are only two papers that
study fine refinements of the continuous space models. These Level 0
refinements are necessary when the underlying assumptions for
the continuous models fail to hold. Blumenfeld and Beckmann Fig. 1. A multi-level distribution network.
 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228 219

Teo and Shu, 2004; Dasci and Verter, 2001). The mathematical  Total facility cost: A fixed rent, Fr, is paid for opening and oper-
model in this paper is developed around the following assumptions. ating each RDC. The total facility cost TF(x) is given by multiply-
ing the facility cost of opening each RDC with the number of
(1) The distribution network is an arborescence network (see RDCs; namely,
Fig. 1) in which each facility can serve multiple facilities in
the lower level but can be served by only one facility from TFðxÞ ¼ F r Nri ðxÞ: ð1Þ
the upper level. Therefore, each retailer is assigned to a par-
 Inbound transportation and outbound delivery cost: We consider
ticular RDC and served only by that RDC; each RDC is
two components for the transportation cost-outbound and
assigned to a particular NDC and served only by that NDC.
inbound costs. For the RDC, the outbound cost is the cost of
(2) The location of the NDCs are known and fixed.
shipping goods to the retailers located within its influence area.
(3) Demand per unit time for retail store in cluster Ci is an inde-
Inbound cost is the cost of sending shipments from the NDC to
pendent and identically distributed Poisson process with
the RDC. For the NDC, the outbound cost is the same as inbound
rate ki.
cost for the RDC. The inbound cost from the outside supplier is
(4) Each product can be analyzed independent of other prod-
not modeled explicitly at the NDC. Instead this cost is factored
ucts. The demand for a single product is considered in our
in the reorder cost at the NDC. Each transportation cost compo-
study.
nent consists of a fixed cost and a variable cost. The fixed com-
(5) The demand process at each RDC is a Poisson process as it is
ponent of cost can be associated with managing the fleet,
generated by the demand coming from the stores in its influ-
drivers, etc. The variable cost is the cost per item. Let Cf be
ence area. There is no reorder cost at the stores so that the
the fixed cost per inbound shipment and Cv be the variable cost
demand at the store gets passed over to the RDC on a per
per item for each inbound shipment. Then the total inbound
item basis.
transportation cost, TIT(x), is given by:
(6) There is no lateral shipment of goods, i.e., movement of
!
goods between facilities in the same echelon. Moreover, nE½Dri ðxÞ
each facility serves its immediate lower echelon facilities TITðxÞ ¼ ðC f þ C v Q ri ðxÞÞ Nri ðxÞ; ð2Þ
Q ri ðxÞ
via direct shipment.
(7) We do not consider the pipeline inventory cost for where Q ri ðxÞ is the ordering quantity for RDC r in cluster Ci and
units in transit from NDC to RDC or from suppliers to the C f þ C v Q ri ðxÞ is the transportation cost incurred in a single inbound
NDC. shipment to a single RDC. The expected demand per unit time faced
(8) Each RDC’s influence area is close to circular. Service regions by RDC r in cluster Ci is given by E½Dri ðxÞ, n is the length of the plan-
have somewhat irregular shapes as opposed to circles, hexa- nE½Dr i ðxÞ
ning horizon and is the expected number of inbound ship-
gons, or squares in the economics literature. This irregular Q r ðxÞ
i

service area is shown to have little effect on the optimal ments to a single RDC in cluster Ci during the planning horizon.
solution (Dasci and Verter, 2001). Moreover, each RDC is Nri ðxÞ is the number of RDCs in cluster Ci. E½Dri ðxÞ ¼ ki ðxÞdi ðxÞAri ðxÞ,
located in the center of the influence area. where Ari ðxÞ is the influence area for RDC r in cluster Ci. Let Cl be the
(9) The distances between the NDCs and the RDCs, and between delivery cost per mile per item and fr be the constant that depends
the RDCs and the retail stores are calculated using the on the distance metric and shape of the RDC service region (see Dag-
Euclidean norm. anzo and Newell, 1986; Dasci and Verter, 2001). Then the total
(10) The constraints from capacity limitations at the NDCs and outbound local delivery cost, TOT(x), is given by
the RDCs are not considered. qffiffiffiffiffiffiffiffiffiffiffiffi
(11) The inventory policy at the RDCs and the NDCs is a continu- TOTðxÞ ¼ C l ðfr Ari ðxÞÞnE½Dri ðxÞNri ðxÞ
ous review policy. qffiffiffiffiffiffiffiffiffiffiffiffi
(12) Both RDC and NDC operate under a Type-I service level ¼ C l ðfr Ari ðxÞÞðnki ðxÞdi ðxÞRÞ: ð3Þ
policy.
The total customer demand during the planning horizon n in the
R
service area R is given by R nki ðxÞdi ðxÞdx. Since ki(x)di(x) is a slow
All the cost functions are modeled using a continuous approx- R
varying function of x 2 R, we get R nki ðxÞdi ðxÞdx ¼ nki ðxÞdi ðxÞR. The
imation technique. The key idea under this technique is to ex-
average
pffiffiffiffiffiffiffiffiffiffiffiffi outbound distance traveled by each item is given by
press the entire distribution network in terms of smooth
fr Ari ðxÞ (see Dasci and Verter, 2001).
continuous functions. Let the distribution network under study
 Average inventory cost for RDC: Each RDC r orders in batches of
be represented by a continuous service area R, and the discrete
Qr(x) and there is a reorder cost, Or (x), associated with each
store location in cluster Ci be expressed as a spatial density slow
batch. The total reorder cost, TOr (x), for all the RDCs over the
varying function di(x), x 2 R. If the demand at stores in cluster Ci is
planning horizon is given by
expressed as a spatial density slow varying function ki(x), x 2 R, !
then the customer demand at each point x 2 R can be expressed nE½Dri ðxÞ
as a product of the store density and the store demand, and is gi- TOr ðxÞ ¼ Nri ðxÞOr ðxÞ : ð4Þ
Q ri ðxÞ
ven by di(x)ki(x), x 2 R. It is argued in Daganzo (1996) that if the
customer demand is a slow varying function of x then the influ- Average inventory at the RDC is given as the sum of the cycle inven-
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ence area of each RDC can be approximated by a circular region tory Qr(x)/2 and safety stock Z ar Var½Dri ;LT , where Var½Dri ;LT  ¼
and it is a slow varying function of x. Influence area in this anal-
lr Var½Dri  þ r2r E½Dri 2 and Var½Dri  ¼ ki ðxÞdi ðxÞAri ðxÞ. lr and r2r are
ysis is a region such that all the stores located within this region
the mean and the variance of the lead time respectively and ar is
are served by the RDC located at the center. Let Ari ðxÞ be the influ-
the service level at RDCs. Let hr be the RDC inventory holding cost
ence area associated with RDC r in cluster Ci. If we cover the en-
per item over the planning horizon n. Then the total RDC inventory
tire area of the distribution network with influence areas of size
holding cost, TIr(x), is given by
Ari ðxÞ, then the total number of RDCs (N ri ðxÞÞ is given by
N ri ðxÞ ¼ R=Ari ðxÞ.  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q ri ðxÞ
In our model, the components of the total network cost are TIr ðxÞ ¼ hr Nri ðxÞ þ Z ar Var½Dri ;LT  þ TOr ðxÞ: ð5Þ
2
calculated as follows:
220 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228

 Average inventory cost for NDC: For a NDC n, it orders in batches where Qn(x), Q ri ðxÞ, Ari ðxÞ are the decision variables in this problem.
of Qn(x) and there is a reorder cost, On (x), associated with each Eq. (9) is the area coverage constraint. It ensures that the entire
batch. The reorder cost for each NDC, TOn(x), over the planning service region is covered by the sum of the RDC influence areas.
horizon is given by Eqs. (10)–(12) are the non-negativity constraints for the decision
  variables. Eq. (13) guarantees integer values for Qn(x) and Q ri ðxÞ.
E½Dn ðxÞ
TOn ðxÞ ¼ On ðxÞ ; ð6Þ Note that any feasible solution, ðAri ; Q ri ; Q n Þ for the optimization
Q n ðxÞ
problem defined above should be strictly greater than 0. However,
where E[Dn(x)] is the total expected demand at the NDC during the adding the equality condition in constraints Eqs. (10)–(12) does
planning horizon and is given by nk(x)d(x)R. Define the cycle inven- not change the solution. It follows from the observation that when
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tory and safety stock for the NDC as Qn(x)/2 and Z an Var½Dn;LT , where ðAri ; Q ri ; Q n Þ ¼ ð0; 0; 0Þ, the value of the objective function explodes
P
Var½Dn;LT  ¼ r ln ki ðxÞdi ðxÞAri ðxÞ=Q ri ðxÞ2 (see Deuermeyer and Sch- (tends to infinity). Any feasible solution to the optimization prob-
warz, 1981), ln is the expected lead time and an is the service level lem above will be away from (0, 0, 0), so adding this point to the
at the NDC. We can interpret this result as follows. The demand rate constraint set does not change the nature of the problem.
at RDC r is ki ðxÞdi ðxÞAri ðxÞ and each RDC r orders Q ri units to the NDC
every time. Thus, the demand rate at the NDC in units of orders from 4. Solution methodology-two-phase approximation model
RDC r can be approximated by ki ðxÞdi ðxÞAri ðxÞ=Q ri ðxÞ. The demand
rate during the lead time ln is ln ki ðxÞdi ðxÞAri ðxÞ=Q ri ðxÞ. The total de- We now describe a two-phase approximation technique that is
mand is obtained by taking the sum of the demand rate due to each used to solve the facility location and inventory allocation problem
individual RDC. When we take the variance over this new expression modeled in the previous section. Two-phase approximation is an
we get the result. Let hn be the inventory holding cost per item during extension to the continuous approximation (CA) approach (see
the planning horizon n. Then the total NDC inventory holding cost is Daganzo, 1996). This extension is applicable when discrete data
given by TIn(x) as cannot be approximated with a smooth function as seen in the
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distribution network under study in this work (see Fig. 2). The
Q n ðxÞ distribution network given in Fig. 2 shows the store locations for
TIn ðxÞ ¼ hn þ Z an Var½Dn;LT  þ TOn ðxÞ: ð7Þ
2 a leading automotive company in US. Clearly the store density in
The cost expression derived in this section are in terms of each this figure is a non-homogenous Poisson process. This violates
point x in the service region R. The total cost for the entire region is the slow varying property for the input function that is key for
R the analysis using the CA technique. A more detailed analysis of
given by R (TNC(x))dx, where TNC(x) is the total network cost and is
given by the sum of the facility, transportation and inventory cost the store density data suggests that there are smaller areas over
functions. Each expression for the various cost components cap- which these functions are smooth. Thus the main idea for a two-
tures fine details of the network geometry. We can now define phase approximation method is to divide the network into smaller
our integrated facility location and inventory allocation problem as regions over which the discrete variable can be modeled using the
slow varying functions. In phase-I the network is divided into
Z Z
smaller regions such that the distribution of store density over
minimize min ðTNCðxÞÞdx ¼ ðTFðxÞ þ TITðxÞ
R R these sub-regions satisfies the slow varying property. The problem
þ TOTðxÞ þ TIr ðxÞ þ TIn ðxÞÞ ð8Þ is modeled over the sub-regions using the cost functions described
in Section 3 and it is solved using the CA approach in phase-II.
s:t: Nri ðxÞ  Ari ðxÞ ¼ R ð9Þ
Q n ðxÞ P 0 ð10Þ
4.1. Phase-I approximation: NDC service area and grid Cover-Couple
Q ri ðxÞ P 0 ð11Þ Approach
Ari ðxÞ P 0 ð12Þ
Q n ðxÞ; Q ri ðxÞ 2 Z þ ð13Þ A Grid Cover-Couple approach is used to partition the service
region into sub-regions. Suppose there are n NDCs in the service

Fig. 2. Example of a supply chain.

 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228 221

Fig. 3. Three grid cover.

Fig. 4. NDC sub-region.

region. It is reasonable to assume that the total demand over the within each grid square the demand is slow varying. A trial and error
service region R is distributed equally amongst the NDCs. This method is used to choose a feasible size for the grid, e.g., we can focus
problem of assigning equal demands to each NDC is a special case on all the county level demands and choose a county with the most
of the classic Transportation problem (see Appendix A). The costs in variable demand. Then a square grid cover is designed for this county
this problem are modeled in terms of the distance from the NDC such that the store density within each grid is nearly constant. These
and a solution can be obtained by a greedy heuristic. square grids cover the entire NDC partition. This idea is illustrated in
Let (A1, A2, . . . , An) be the areas corresponding to the NDC parti- Fig. 3. Note that a density can be assigned to each square on the grid
tions obtained after solving the assignment problem. The next step because the store density for each county is known and county is
is to design a grid cover for each of these NDC sub-regions. It is this the lowest level of detail captured by this grid-cover model.
grid-cover that helps divide each NDC partition into regions with Within each NDC partition, there are grids and each grid has a
slow varying functions. A mesh of equal sized squares is designed density associated with it. The grids with similar densities can be
to cover each NDC partition. The geometry of the square-mesh is clustered together to form areas over which the store density
an important decision and needs to satisfy the following conditions: function is slow varying. In order to form the clusters, a tolerance
(1) the smallest level of detail is captured at the county1 level and (2) limit for similarity needs to be specified. The tolerance limit defines
the amount of variability in the store density data that is acceptable
1 while treating them as similar. Let e be the desired tolerance limit.
The term county is used in 46 of the 50 states of the United States for the tier of
state government authority immediately below the statewide tier and above the This means that the grids with density at most e apart are consid-
township tier, in those states that sub-divided counties into civil townships. ered similar. Choice of e depends on the store density pattern in
222 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228

Fig. 5. Coupling.

the existing distribution network. Using the tolerance limit the en- tition is obtained by summing over the number of RDCs in each
tire NDC sub-region is covered with clusters. Figs. 4 and 5 illustrate cluster.
this idea. Clusters (C1, C2, . . . , Cn) exist within each NDC region Ai
such that the store density is nearly constant over each cluster. 4.3. Continuous approximation model
For example, in Fig. 5 there are three clusters (yellow, purple and
green) after coupling. In each cluster, the amount of variability in Since each cluster within a given NDC partition has slow vary-
the store density data is smaller than e while treating them as ing demand, we can ignore the dependence of all continuous func-
similar. tion on parameter x. For the rest of this study, the variables are
represented as Ari ; Q ri ; Q n ; ki and di. Let us focus on a given NDC par-
4.2. Phase-II approximation: RDC influence area using CA approach tition, says An, and suppose C n1 ; C n2 ; . . . ; C nN be the clusters within
An that are obtained using the grid Cover-Couple Approach. Let
The phase-I approximation divides the service region R into NDC Ari be the size of the influence area for each RDC in cluster C ni .
partitions (sub-regions) (A1, A2, . . . , An) and each partition Aj has The integrated facility location and inventory allocation problem
clusters ðC j1 ; C j2 ; . . . ; C ji Þ with slow varying demand. The CA tech- is:
nique can be used to model and solve the facility location and Minimize
  X !
inventory allocation problem over each cluster within the NDC XN N
C ni nki di C ni
partition. The optimization model developed in Section 3 is used TNCðAri ;Q n ;Q ri Þ ¼ Fr þ ðC f þ C v Q ri Þ
Ar i Q ri
for modeling the total logistic costs in each cluster. The solution i¼1 i¼1
! !
to this optimization problem will give the size of the circular influ- X N q ffiffiffiffiffi
ffi X N
nki di C ni
ence area for each RDC (see Fig. 6 for an illustration) and the opti- þ C l fr Ari nki di C ni þ Or
i¼1 i¼1
Q ri
mal values of (Q ri , Qn) for the RDCs and the NDC. Further, using the
X N    q ffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
size of the optimal influence area along with the information on C ni Q ri
þ hr þ Z ari lr ki di Ari þ r2r ðki di Ari Þ2
the area for each cluster, the total number of RDCs in each cluster Ari 2
i¼1
can be calculated. The total number of RDCs in the entire NDC par-  
X N
nki di C ni
þ On
i¼1
Qn
0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
u N
Q uX ln ki di C n
þ hn @ þ Z an t
n
2
iA
;
2 i¼1 Q r i

ð14Þ

s.t.

Q ri ðxÞ P 0; ð15Þ
Ari ðxÞ P 0; ð16Þ
Q n P 0; ð17Þ
Q n ; Q ri 2 Z þ : ð18Þ

C ni is the area of cluster i in the given NDC partition An and C ni =Ari is

the number of RDCs in cluster i. The decision variables are Ari , Q ri ,
Fig. 6. Influence area for a RDC. for i = 1, 2, . . ., N and Qn.
 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228 223

4.4. Solution approach dTNC E ðQ r jAri Þ

dQ r
i
¼ 0:
i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
The problem analyzed here is to determine the optimal influ- 2Ari ðC f þ Or Þnki di
ence area for each RDC Ari , ordering quantity for each RDC Q Eri ðAri Þ ¼ ; ð22Þ
hr
Q ri and ordering quantity for NDC Q n to minimize total network
cost TNCðAri ; Q n ; Q ri Þ. To solve the problem, we first deal with the Substituting Eq. (22) into Eq. (21) yields
decision Qn. Given Ari and Q ri , we have   X
XN
C ni N
 
2 PN TNC E ðAri Þ ¼ Fr þ C v nki di C ni
d TNCðQ n jAri ; Q ri Þ 2On i¼1 ðnki di C ni Þ i¼1
A ri i¼1
2
¼ > 0: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
dQ n Q 3n XN
2hr ðC f þ Or Þnki di
þ C ni
Therefore, TNCðQ n jAri ; Q ri Þ is a convex function of Qn. This means i¼1
Ar i
that the optimal Q n can be obtained by solving
dTNCðQ r ;Q n jAri Þ
i
¼ 0: N 
X qffiffiffiffiffiffi  X N  
dQ n C ni
þ C l frAri nki di C ni þ hr  Z ar i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ar i
P i¼1 i¼1
2On Ni¼1 nki di C ni vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q n ¼ : ð19Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u X N
hn
 lr ki di Ari þ rr ðki di Ari Þ þ t2hn On nki di C ni
2 2

i¼1
Substituting Eq. (19) into Eq. (14) yields
! vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
  X u N
XN
C ni N
nki di C ni uX
TNCðAri ;Q ri Þ ¼ Fr þ ðC f þ C v Q ri Þ þ hn Z an t ln ki di C ni : ð23Þ
i¼1
A ri i¼1
Q ri i¼1
!
X N  q ffiffiffiffiffi
ffi  XN
nki di C ni Based on the discussion above, the following algorithm deter-
þ C l fr Ari nki di C ni þ Or mines the optimal values for Q Eri and AEri .
i¼1 i¼1
Q ri
X N    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C ni Q ri Algorithm 1
þ hr þ Z ari lr ki di Ari þ r2r ðki di Ari Þ2
i¼1
A r i
2
v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Step 1: Determine local minimum points by solving for
u u N
u X N uX ln ki di C n dTNC ðAri Þ
E 2
d TNC E ðAr Þ
t
þ 2hn On nki di C ni þ hn @Z an t iA
: ð20Þ dAri
¼ 0 and by verifying that 2
dAr
i
> 0.
i¼1 i¼1 Q 2ri   i

Step 2: Let TNC E AEri associate with the local minimum point,
The model becomes a non-linear function with 2N variables (Ari
and Q ri , for i = 1, 2, . . . , N). It is still a very complex problem. The which gives the smallest value of TNC E ðAri Þ.
  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Ari ðC f þOr Þnki di
expressions for safety stocks at the RDCs and the NDC make the Step 3: Determine Q Eri AEri ¼ by Eq. (22).
hr
objective function hard to evaluate the convexity directly. This pa-
per provides a two-stage solution approach to solve the problem. Step 4: Adjust Q Eri ; i ¼ 1  N, to get the nearest integer values.
The first stage uses Algorithm 1 to determine an initial solution
which is close to the optimal solution. Based on the initial solution, Stage 2: The objective of this paper is to determine the optimal
the second stage uses Algorithm 2 to find the optimal solution. Ari , Q ri and Qn to minimize total network cost TNCðAri ; Q n ; Q ri Þ.
Stage 1: In Stage 1, we determine an initial solution for the Prob- Based on the initial solution Q Eri which we found in Algorithm 1,
lem (Eq. (20)). We delete Q 2ri in the safety stock at the NDC in Eq. a search-based algorithm (Algorithm 2) is developed to solve the
(20). The mode becomes original problem. Since Eq. (21) is similar to Eq. (20) except the
  X ! safety stock at the NDC, the optimal solution of TNC E ðAri ; Q ri Þ is
E
XN
C ni N
nki di C ni very close to that of TNCðAri ; Q ri Þ. This means Algorithm 2 can find
TNC ðAri ; Q ri Þ ¼ F r þ ðC f þ C v Q ri Þ
i¼1
Ar i i¼1
Q ri the optimal solution efficiently.
!
X N  qffiffiffiffiffiffi  X N
nki di C ni
þ C l fr Ari nki di C ni þ Or
Q ri Algorithm 2
i¼1 i¼1
X N   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C ni Q ri Step 1: Let Q j¼1 ¼ Q Eri , find the value of Aj¼1 to minimize
þ hr þ Z ari lr ki di Ari þ r2r ðki di Ari Þ2  ri
  ri

i¼1
A r i
2 1 1 1
v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! TNC Ari Q ri and compute TNC Ari ; Q ri by Eq. (20).
u u N
u X N uX
t
þ 2h O nk d C þ h Z t l k d C :ð21Þ Step 2: Let Q jþ1
ri ¼ Q jri þ 1, find the value of Ajþ1 ri to minimize
n n i i ni n an n i i ni    
i¼1 i¼1 jþ1 jþ1 jþ1
TNC Ari Q ri and compute TNC Ari ; Q ri by Eq. (20).
   
jþ1 jþ1 j
Note that Eq. (21) is similar to Eq. (20) except the safety stock at the Step 3: If TNC Ari ; Q ri < TNC Ari ; Q ri , then let Q jri ¼ Q jþ1
j
ri
NDC. The model TNC E ðAri ; Q ri Þ is much easier than the original prob- and go to Step 2; otherwise, go to Step 4.
lem (Eq. (20)). We first determine the optimal solution of
Step 4: Let Q jþ1 ¼ Q jri  1, find the value of Ajþ1 to minimize
TNC E ðAri ; Q ri Þ. The optimal solution of TNC E ðAri ; Q ri Þ is the initial  ri
  ri

solution in Stage 2. Given Ari , the second-order derivatives of
jþ1
TNC Ari Q ri and compute TNC Ajþ1 ri ; Q jþ1
ri by Eq. (20).
TNC E ðQ ri jAri Þ with respective to Q ri are    
Step 5: If TNC Ajþ1 jþ1
ri ; Q ri < TNC Ajri ; Q jri , then let Q jri ¼ Q jþ1
ri
2
d TNC E ðQ ri jAri Þ 2ðC f þ Or Þnki di C ni and go to Step 4; otherwise, go to Step 6.
¼ > 0; i ¼ 1; 2; . . . ; N:
2
dQ ri Q 3ri Step 6: Compute Q n by Eq. (19) and adjust Q n to get the
nearest integer value.
   
This means TNC E ðQ ri jAri Þ is a convex function of Q ri . The optimal Q Eri Step 7: Let TNC Ari ; Q ri ; Q n ¼ TNC Ajri ; Q jri ; Q n . Ari ; Q ri and

of the model TNC E ðAri ; Q ri Þ can be obtained by solving Q n is the optimal solution.
224 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228

Table 1
Store density and average distance data.

GA FL TN AL KY VA NC SC
Store density 0.0059 0.0039 0.0038 0.002 0.0052 0.0471 0.0041 0.0019

Table 2   pffiffiffiffiffi
Cn   nkdC n
Comparison for the three cases. TNCðAr ; Q n ; Q r Þ ¼ F þ Cf þ Cv Q r þ C l fr Ar nkdC n
Ar i Qr
Case 1 Case 2 Case 3   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nkdC
 n Cn Qr
Integrated Non-integrated Average þ Or þ hr þ Z ar lr kdAr þ r2r ðkdAr Þ2
Qr Ar 2
Q n 2583 2583 2533  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nkdC
 n Q n Z an
Q r Q r1 ¼ 343 Q r1 ¼ 291 Q = 361 þ On þ hn þ ln kdC n ; ð27Þ
Q r2 ¼ 308 Q r2 ¼ 262
Qn 2 Qr
Q r3 ¼ 306 Q r3 ¼ 261 The decision variables in this case are Ar, Qr and Qn. The optimal Qr
Q r4 ¼ 262 Q r4 ¼ 223 n jAr Þ
and Qn can be obtained by solving @TNCðQ@Qr ;Q ¼ 0 and
Q r5 ¼ 336 Q r5 ¼ 285 @TNCðQ r ;Q n jAr Þ
r

Q r6 ¼ 489 Q r6 ¼ 412 @Q n
¼ 0:
Q r7 ¼ 317 Q r7 ¼ 269 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q r8 ¼ 273 Q r8 ¼ 232 2Ari ½ðC f þ Or ÞnkdC n þ hn Z an ðnkdC n Þ1=2 
Q r ðAr Þ ¼ ; ð28Þ
Ar Ar 1 ¼ 20764:3 Ar 1 ¼ 19983:2 Ar = 17028.2 hr C n
Ar 2 ¼ 31634:0 Ar 2 ¼ 30558:3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ar 3 ¼ 32179:7 Ar 3 ¼ 31092:1 2On nkdC n
Qn ¼ : ð29Þ
Ar 4 ¼ 59517:3 Ar 4 ¼ 57780:1 hn
Ar 5 ¼ 22571:4 Ar 5 ¼ 21738:5
Ar 6 ¼ 5278:6 Ar 6 ¼ 5002:86 Substituting Eqs. (28) and (29) into Eq. (27), the procedure for
Ar 7 ¼ 28313:5 Ar 7 ¼ 27324:3
obtaining optimal Ar ; Q r and Q n for the average model is similar
Ar 8 ¼ 50900:3 Ar 8 ¼ 49355:6
to that for the integrated model. Following the solution approach
TF 1,519,360 1,586,270 1,773,930
described in the first stage of the integrated model, we can easily
TIT 1,024,950 1,029,420 989,347
TOT 3,241,740 3,172,540 3,768,640 find the optimal Ar ; Q r and Q n .
TIn 38,743 38,745 38,006 The analysis for the integrated facility location and inventory
TIr 189,269 191,514 204,117 allocation problem sheds light on some important issues. Eqs.
TNC 6,014,062 6,018,489 6,774,040 (25) and (28) lead to Observation 1 and Eqs. (26) and (29) lead
to Observation 2.

5. Discussion Observation 1. The ordering quantity for RDC Q ri increases as the

inventory holding cost for RDC hr decreases, or the ordering cost
The integrated model is compared with the non-integrated for RDC Or and the RDC influence area Ari increase.
model and the average model. Observation 2. The ordering quantity for NDC Qn increases as the
inventory holding cost for NDC hn decreases or the ordering cost
5.1. The non-integrated model for NDC On increases.

The non-integrated model is the one where the facility location

6. Numerical illustration
and inventory decisions are made in isolation of each other. The
model is first solved for the optimal influence area Ari using
In the numerical study in this section, the distribution network
information on the facility location cost and the transportation
for a leading US retailer is considered. This is a large home
cost. Using this value of the influence area Ari , the optimal inven-
improvement retail-chain in US. The entire US mainland has five
tory decision at NDC Q n is made to minimize the inventory cost
sub-regions, namely, south-eastern, south-western, north-eastern,
at NDC and the optimal inventory decisions at RDC Q ri is made
north western and mid-west. The distribution network has a total
to minimize the inventory cost at RDC.
 2=3 of five NDCs each serving one of the sub-regions. The numerical
2F r
Ar i ¼ ; ð24Þ analysis of this section is carried out using data for the southeast-
C l fr nki di ern (SE) region with the NDC located at Savannah, GA. Table 1 gives
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Ari Or nki di the store density for the eight states served by the Savannah DC.
Q ri ðAri Þ ¼ ; ð25Þ Clearly there is a significant amount of variation in the store den-
hr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sity data across states.
P
2On Ni¼1 nki di C ni For a fixed value of the inventory parameters Q ri and On, the
Qn ¼ : ð26Þ
hn number of RDCs increase (decrease) with an increase (decrease)
in the store density. Similarly, for a fixed number of RDCs, increase
5.2. The average model or decrease of the store density changes the value of the inventory
parameters Q ri and On. Thus, the density function affects the facil-
The average model is where the entire distribution region is ity, transportation and inventory cost.
assumed to be a smooth continuous region. It is also assumed that The goals of the numerical study are as follows:
the store density and demand density functions are smooth over
this region. In this model, the store density function for the entire 1. To illustrate the two-stage solution approach.
region is defined by theaverage
 value of individual store densities, 2. To compare the results of integrated, non-integrated and aver-
P
d ¼ Ni¼1 dNi and the demand rate for retail store
i.e., each di ¼  age models.
P   P
ki ¼ k ¼ Ni¼1 kNi . Then, Ni¼1 C ni is replaced by Cn, the size of entire 3. To discuss the impacts of the related parameters on decisions
distribution network. The model becomes and cost.
 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228 225

Fig. 7. Graphic illustrations of TNC versus different decision variables.

TNC⁄ = 6014070. Fig. 7 shows the graphic illustrations of TNC

6.1. Solution approach illustration
versus different decision variables. It shows that the two-stage
solution approach finds the optimal solution.
Case 1 in Table 2 shows the computed results after applying our
two-stage solution approach (Algorithms 1 and 2). The optimal
solution Ar , Q r and Q n are shown in Case 1 in Table 2. The total 6.2. Comparisons for integrated, non-integrated and average case
facility cost is TF⁄ = 1,519,360, the inbound transportation cost is
TIT⁄ = 1,024,950, the outbound transportation cost is TOT⁄ = 3,241, The results for the integrated, non-integrated and average cases
740, the total inventory cost for NDC is TIn ¼ 38; 743, the inventory of the facility location and inventory allocation problem are
cost for RDCs is TIr ¼ 189; 269 and the total network cost is presented in Table 2 and Fig. 8. The integrated case is the one with
226 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228

Fig. 10. Safety stock for each RDC in each zone-integrated vs. non-integrated.

Fig. 8. Total network cost.

the minimum value of total network cost. This example observes

that the TNC is 0.07% higher in the case of non-integrated problem
and 12.64% higher for the average case (see Table 2). These results
justify the need for a two-phase approximation approach.
Considering the up-stream facility bears the transportation cost.
The related cost for RDCs includes the facility cost, the outbound
transportation cost and the total inventory cost for RDCs. A focus
on the related costs for RDCs in each zone shows an interesting
trend. The related costs for RDCs in each zone for the integrated
case are less than those for the non-integrated case (see Fig. 9). If
this problem was modeled with a decentralized decision maker,
then the RDCs in these zones have no incentive to participate in
an integrated activity. This opens a new direction for our research
where game theory can be used. This interesting research proposal
is left for future work. Fig. 11. Safety stock for each zone-integrated vs. non-integrated.
As shown in Fig. 10, the safety stock at each RDC for the inte-
grated case is greater than that for the non-integrated case. This
may look counter intuitive initially. However, a careful inspection account at the RDCs. Reduction in the number of RDCs means more
shows that each zone has fewer RDCs in the integrated case. As inbound shipments to each RDC and thus more supply variability.
the number of RDCs increases, the safety stock at each RDC de- Hence when both types of variabilities are taken into account it is
creases. This result is quite unlike the Square Root law which says possible to see this reverse relation between the number of RDCs
that the total inventory in a system is proportional to the square and safety stock. Thus, each zone in the integrated problem has a
root of the number of locations at which a product is stocked higher value for total safety stock and reorder point as compared
(see Chopra and Meindl, 2003). The reason for this is while apply- to each zone under the non-integrated case. A zone wise compar-
ing the square root law we observe that reducing the number of ison of the safety stock in each zone is presented in Fig. 11. The to-
RDCs reduces the risk by pooling demand variability. But in our tal safety stock in each zone for the integrated case is greater than
model, both the demand and supply variability are taken into that for the non-integrated case.

Fig. 9. Cost for each zone.

 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228 227

Table 3 Table 5
Effects of Cf and Cv. Effects of On and Or.

Cf = 8, Cv = 0.8 Cf = 10, Cv = 1 Cf = 12, Cv = 1.2 On = 40, Or = 20 On = 50, Or = 30 On = 60, Or = 40

Q n 2583 2583 2583 Q n 2310 2583 2829

Q r Q r1 ¼ 334 Q r1 ¼ 343 Q r1 ¼ 352 Q r Q r1 ¼ 296 Q r1 ¼ 343 Q r1 ¼ 384

Q r2 ¼ 300 Q r2 ¼ 308 Q r2 ¼ 316 Q r2 ¼ 266 Q r2 ¼ 308 Q r2 ¼ 345
Q r3 ¼ 299 Q r3 ¼ 306 Q r3 ¼ 314 Q r3 ¼ 265 Q r3 ¼ 306 Q r3 ¼ 343
Q r4 ¼ 255 Q r4 ¼ 262 Q r4 ¼ 268 Q r4 ¼ 226 Q r4 ¼ 262 Q r4 ¼ 293
Q r5 ¼ 327 Q r5 ¼ 336 Q r5 ¼ 344 Q r5 ¼ 290 Q r5 ¼ 336 Q r5 ¼ 376
Q r6 ¼ 476 Q r6 ¼ 489 Q r6 ¼ 501 Q r6 ¼ 421 Q r6 ¼ 489 Q r6 ¼ 548
Q r7 ¼ 309 Q r7 ¼ 317 Q r7 ¼ 325 Q r7 ¼ 274 Q r7 ¼ 317 Q r7 ¼ 355
Q r8 ¼ 266 Q r8 ¼ 273 Q r8 ¼ 279 Q r8 ¼ 236 Q r8 ¼ 273 Q r8 ¼ 305

Ar Ar 1 ¼ 20744:7 Ar 1 ¼ 20764:3 Ar 1 ¼ 20783:9 Ar Ar 1 ¼ 20661:7 Ar 1 ¼ 20764:3 Ar 1 ¼ 20853:5

Ar 2 ¼ 31607:3 Ar 2 ¼ 31634:0 Ar 2 ¼ 31660:7 Ar 2 ¼ 31493:7 Ar 2 ¼ 31634:0 Ar 2 ¼ 31757:4
Ar 3 ¼ 32156:0 Ar 3 ¼ 32179:7 Ar 3 ¼ 32206:9 Ar 3 ¼ 32040:3 Ar 3 ¼ 32179:7 Ar 3 ¼ 32305:3
Ar 4 ¼ 59473:0 Ar 4 ¼ 59517:3 Ar 4 ¼ 59555:3 Ar 4 ¼ 59289:3 Ar 4 ¼ 59517:3 Ar 4 ¼ 59713:3
Ar 5 ¼ 22550:0 Ar 5 ¼ 22571:4 Ar 5 ¼ 22590:3 Ar 5 ¼ 22462:2 Ar 5 ¼ 22571:4 Ar 5 ¼ 22666:1
Ar 6 ¼ 5271:6 Ar 6 ¼ 5278:6 Ar 6 ¼ 5285:1 Ar 6 ¼ 5241:8 Ar 6 ¼ 5278:6 Ar 6 ¼ 5310:5
Ar 7 ¼ 28289:6 Ar 7 ¼ 28313:5 Ar 7 ¼ 28337:3 Ar 7 ¼ 28185:0 Ar 7 ¼ 28313:5 Ar 7 ¼ 28426:7
Ar 8 ¼ 50862:5 Ar 8 ¼ 50900:3 Ar 8 ¼ 50932:7 Ar 8 ¼ 50700:3 Ar 8 ¼ 50900:3 Ar 8 ¼ 51073:0
TNC 5,809,010 6,014,062 6,219,002 TNC 5,983,616 6,014,062 6,040,928

Table 4 Table 6
Effect of Cl. Effects of hn and hr.

Cl = 2 Cl = 3 Cl = 4 hn = 12, hr = 8 hn = 15, hr = 10 hn = 18, hr = 12

Q n 2583 2583 2583 Q n 2887 2583 2358

Q r Q r1 ¼ 394 Q r1 ¼ 343 Q r1 ¼ 311 Q r Q r1 ¼ 383 Q r1 ¼ 343 Q r1 ¼ 313

Q r2 ¼ 353 Q r2 ¼ 308 Q r2 ¼ 279 Q r2 ¼ 344 Q r2 ¼ 308 Q r2 ¼ 281
Q r3 ¼ 352 Q r3 ¼ 306 Q r3 ¼ 278 Q r3 ¼ 342 Q r3 ¼ 306 Q r3 ¼ 280
Q r4 ¼ 300 Q r4 ¼ 262 Q r4 ¼ 238 Q r4 ¼ 292 Q r4 ¼ 262 Q r4 ¼ 239
Q r5 ¼ 385 Q r5 ¼ 336 Q r5 ¼ 304 Q r5 ¼ 375 Q r5 ¼ 336 Q r5 ¼ 307
Q r6 ¼ 4561 Q r6 ¼ 489 Q r6 ¼ 443 Q r6 ¼ 545 Q r6 ¼ 489 Q r6 ¼ 447
Q r7 ¼ 363 Q r7 ¼ 317 Q r7 ¼ 287 Q r7 ¼ 353 Q r7 ¼ 317 Q r7 ¼ 290
Q r8 ¼ 312 Q r8 ¼ 273 Q r8 ¼ 247 Q r8 ¼ 304 Q r8 ¼ 273 Q r8 ¼ 249

Ar Ar 1 ¼ 27356:3 Ar 1 ¼ 20764:3 Ar1 ¼ 17082:0 Ar Ar 1 ¼ 20679:1 Ar 1 ¼ 20764:3 Ar 1 ¼ 20840:6

Ar 2 ¼ 41652:2 Ar 2 ¼ 31634:0 Ar2 ¼ 26031:8 Ar 2 ¼ 31516:7 Ar 2 ¼ 31634:0 Ar 2 ¼ 31739:2
Ar 3 ¼ 42375:3 Ar 3 ¼ 32179:7 Ar3 ¼ 26483:6 Ar 3 ¼ 32061:7 Ar 3 ¼ 32179:7 Ar 3 ¼ 32289:4
Ar 4 ¼ 78311:3 Ar 4 ¼ 59517:3 Ar4 ¼ 49001:9 Ar 4 ¼ 59324:0 Ar 4 ¼ 59517:3 Ar 4 ¼ 59687:6
Ar 5 ¼ 29731:1 Ar 5 ¼ 22571:4 Ar5 ¼ 18568:5 Ar 5 ¼ 22480:2 Ar 5 ¼ 22571:4 Ar 5 ¼ 22653:9
Ar 6 ¼ 6968:3 Ar 6 ¼ 5278:6 Ar6 ¼ 4336:7 Ar 6 ¼ 5248:4 Ar 6 ¼ 5278:6 Ar 6 ¼ 5305:8
Ar 7 ¼ 37283:7 Ar 7 ¼ 28313:5 Ar7 ¼ 23296:9 Ar 7 ¼ 28203:1 Ar 7 ¼ 28313:5 Ar 7 ¼ 28412:9
Ar 8 ¼ 66986:8 Ar 8 ¼ 50900:3 Ar8 ¼ 41898:6 Ar 8 ¼ 50727:5 Ar 8 ¼ 50900:3 Ar 8 ¼ 51051:3
TNC 4,860,716 6,014,062 7,041,050 TNC 5,985,741 6,014,062 6,039,964

6.3. Numerical analysis

(2) Table 4 shows that Q ri and Ari will decrease, TNC will
increase as Cl increases. When the outbound delivery cost
Several numerical analyses were conducted to gain quantitative
per mile per item (the cost of shipping goods from RDCs to
insights into the structures of the proposed policies. The following
the retailers) increase, the RDCs will decrease their ordering
numerical analyses are used to show the effects of Cf, Cv, Cl, On, Or,
quantities and influence areas to reduce the number and
hn and hr on the optimal influence area for each RDC Ari , the order-
delivery distance of the goods. Then the total outbound cost
ing quantity for each RDC Q ri , the ordering quantity for NDC Q n and
can be reduced.
the total network cost TNC. We summarize the results shown in
(3) Table 5 shows that Ari ; Q n ; Q ri and TNC will increase as On
Tables 3–6 as follows:
and Or increase. When the reorder costs increase, the NDC
and RDCs will increase their ordering quantities to reduce
(1) Table 3 shows that Ari ; Q ri and TNC will increase as Cf and Cv
the ordering frequency. Since the reorder costs increase, it
increase. When the inbound transportation cost (the cost of
is reasonable to increase the influence area for each RDC to
sending shipments from the NDC to the RDC) increase, the
reduce the number of RDC opening and then reduce the
RDCs will increase their ordering quantities to reduce the
reorder times.
inbound transportation times. It is also reasonable to
(4) Table 6 shows that Q n and Q ri will decrease, Ari and TNC will
increase the influence area for each RDC to reduce the num-
increase as hn and hr increase. When the inventory holding
ber of RDC opening and operating. This means the total
costs increase, the NDC and RDCs will decrease their order-
inbound transportation cost can be reduced by distributing
ing quantities to reduce the cost loss.
goods to fewer RDCs.
228 Y.-C. Tsao et al. / European Journal of Operational Research 222 (2012) 216–228

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