Floating Stock

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1
Mathematical Modeling of Floating Stock Policy in
FMCG Supply Chains

Morteza Pourakbar
1
, Andrei Sleptchenko, Rommert Dekker
Econometric Institute, Erasmus University Rotterdam
P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands

Econometric Institute Report EI2007-54

Abstract
The Iloating stock distribution concept exploits inter-modal transport to deploy inventories in a supply chain
in advance oI retailer demand. It is appropriate in case oI batch production and containerized transport oI
standard product mixes. In this way response times are reduced and storage costs can be reduced as well by
having products in the transport-pipeline. This concept was earlier analysed using a simulation approach and
showed to be eIIicient under simpliIying assumptions Ior the demand distribution.
In this paper we present two mathematical models to analyse this policy while backlogging is allowed. The
Iirst one tries to optimize the advanced shipping time oI containers to inter-modal terminal, and the second
one optimizes the total number oI containers in pipeline and terminal. In Iact, in both policies containers are
shipped to a terminal beIore the demand is realized in order to beneIit Irom less storage cost at the terminal by
utilizing the shipping time and also Iree storage cost period at inter-modal terminals.
A comparison is made with the simulation outcomes oI applying previously developed strategies which shows
that this concept has advantages in inventories over other strategies.

Keywords: Supply chain, Floating Stock, Inter-modal Transport, Virtual Warehousing, Inventories.

1. Introduction
Fast delivery is used in many retail supply chains. The advantages are enjoyed mainly by
the retailers as they can operate in a just-in-time mode: they need Iewer inventories on-site
which reduces operational costs (both holding and storage costs) and investment costs

1
Corresponding author Tel: ¹31 10 40 88903; Fax: ¹31 10 40 89162
E-mail address: pourakbar¸Iew.eur.nl (M. Pourakbar)

2
(through less amount oI warehouse space required). When they call orders, they can rely on
rapid IulIilment. This works well iI the order lead times and production time allow the
manuIacturer to operate on a make-to-order basis. II this option is not available and
substantial batches are made, the burden oI keeping inventory is shiIted Irom the retailer to
the supplier. In this case the supplier has to either store it close to the retailer or use Iast
transport in order to meet the required order lead time (Ochtman et. al 2004). This leads to
many transport movements with Iew opportunities Ior loaded return trips. The considered
distribution concept in this paper, inter-modal Iloating stocks, supports just-in-time delivery
with shorter order lead times Ior manuIacturers that Iollow a make-to-stock strategy.

The Iloating stock concept exploits the opportunities inter-modal transport oIIers to deploy
inventories in the supply chain. The idea is that by advanced deployment and careIully
tuning demand with transport modes we can reduce non-moving inventories, shorten lead
times and improve the order Iill rate. This strategy beneIits Irom Iloating oI stocks and the
existence oI inter-modal terminals to postpone the selection oI the destination so that a
pooling eIIect can be obtained in comparison to direct road transport. We use inter-modal
transport with deIerred Iinal transport instead oI transshipments to achieve postponement oI
positioning in a sense built on Herer et al. (2002). By this setting, we create a kind oI
virtual warehouse at the inter-modal terminals, yet one diIIerent than commonly reIerred to
in literature (see e.g. Landers et al. 2000, as they stress real-time global visibility oI logistic
assets).

Although the term Iloating stock is relatively new, the concept is applied already Ior a long
time. It is used when shippers send their containers in advance oI demand Irom Asia to
Europe or to US and the Iinal destination is determined only in the Iinal port. Another
example is North American lumber, where lumber producers would ship loads to north
central and eastern customers beIore demand had Iinalized (Sampson et al. 1985). The
Ilatcars or boxcars were held at transit yards in the mid-west until a customer order was
received. This practice enabled western US producers to compete in the eastern markets
with their southern competitors in terms oI lead times. Yet, almost no literature is available
3
on Iloating stocks, as the terminology is not yet standardized. The exceptions are Teulings
& van der Vlist (2001) which do not deal with inventories and a companion paper by
Ochtman et al. (2004) which applies a simulation approach Ior studying the Iloating stock
concept, but do not propose a mathematical optimization model to deal with this policy.

To position our contribution in literature, we Iurther relate it to three streams, viz.
intermodal studies, inventory management and outbound dispatch policies. Inter-modal
transport can be deIined (ECMT 1993) as the movement oI goods in one and the same
loading unit or vehicle by successive modes oI transport without handling oI the goods
themselves during transIers between modes, e.g. container transport via rail and road. The
transIer points oIIer short term storage to decouple the successive steps in the transport
chain and they oIten Ieature a limited amount oI Iree time Ior which no storage costs are
charged. Nowadays this transport method is strongly advocated by many European
governments in order to reduce road congestion and pollution. Inter-modal transport is
however, on short distances more costly than road transport since it requires more handling.
Furthermore, its transit time is oIten longer than that oI direct road transport and its
reliability is not always high (Konings 1996). Transport studies such as Bookbinder & Fox
(1998) and Rutten (1995) typically make such comparisons between road transport and
inter-modal transport, but in these studies inventories are leIt out oI consideration.

Inventory management is another important topic in supply chains (Chopra & Meindl
2004). The main emphasis here is on determining how much inventory should be kept at
which stocking locations, while typically only one lead time (and hence transportation
mode) is considered. A well-known result is that centralization or pooling can reduce
inventories iI demands are uncorrelated, at the expense oI higher transportation costs and a
longer response time. This has led to the creation oI European Distribution Centres, Irom
which goods are trucked to clients throughout Europe directly upon client's calls. DiIIerent
transport modes are considered primarily in the case oI emergency shipments to take care
oI stock-outs (Moinzadeh and Schmidt, Moinzadeh & Nahmias 1988). Some studies also
consider lateral transshipments in multi-echelon chains, but mostly just in case oI stock-
4
outs (Minner 2003 and Diks et al. 1996). Herer et al. (2002) is an exception as they
consider lateral transshipments to enhance postponement and hence leagility (i.e. a
combination oI lean and agility) in supply chains. There are a Iew studies that integrate
transportation and inventory control (see e.g. Tyworth & Zeng (1998)), but they Iocus on
the relation between either transport Irequency or transit time reliability and inventory
control. A negative eIIect oI the Iloating stock concept is that Iew possibilities exist Ior
pooling, as products are shipped already towards their destination. Evers (1996), (1997) and
(1999) study risk pooling oI demand and lead times in relation to transhipments. However,
these studies do not consider transport costs. No studies seem to exist on integration oI
inter-modal transport and inventory control, according to recent reviews on inter-modal
research, such as Bontekoning et al. (2004) and Macharis & Bontekoning (2004).

The ideas in this paper can also be related to outbound dispatch policies Ior integrated stock
replenishment and transportation decisions. The logistic literature reports that two diIIerent
types oI such policies are popular in current practice. These are time-based and quantitv-
based dispatch policies. Considering the case oI stochastic demand, it has been shown that
the quantity-based policy has substantial saving over time based policy. Under a time-based
policy each order is dispatched by a pre-speciIied shipment release date even though the
dispatch quantity does not necessarily realize transportation scale economies. On the other
hand, under a quantity-based policy, the dispatch quantity assures transportation scale
economies, but a speciIic dispatch time cannot be guaranteed. An alternative to these two
policies is a hybrid routine aimed at balancing the trade-oII between the timely delivery
advantages oI time-based policies and the transportation cost savings associated with
quantity-based policies. Under a hybrid policy, the objective is to consolidate an
economical dispatch quantity, denoted by q
H
. However, iI this quantity does not accumulate
within a reasonable time window, denoted by T
H
, then a shipment oI smaller size may be
released. A dispatch decision is made either when the size oI a consolidated load exceeds
q
H
, or when the time since the last dispatch exceeds T
H
(Cetinkaya et al. 2006).

5
The main diIIerence between the problem discussed in this paper and the previously
discussed outbound dispatch logistic is that in those models Iirst demand is realized and
then a shipping is done by either time-based or quantity-based policies. But in the problem
discussed in this paper, it is intended to ship beIore demand realization. It is worth nothing
that when intermodal transport is used, the shipment time increases considerably and the
order lead time will usually be exceeded iI the order is shipped aIter it is received.
Increasing the order lead time Iorms a great problem, especially in the retail industry in
which even the short delays are not acceptable. To avoid this problem the shipment should
be sent in advance, beIore the order is placed, and then in this case the system can be
beneIited Irom a Iast delivery time to customer.

In this study we consider a Fast Moving Consumer Goods supply chain and we will present
mathematical models Ior Iloating stock policy in that supply chain. These models address
the question oI how to schedule shipment oI containers through intermodal channel. Next,
we will compare the results oI our models with other distribution strategies.

This paper is organized as Iollows. In section 2 the problem environment is explained and
based on that a quick review on possible distribution strategies Ior this problem is
presented. This review is done, to be able to compare the result oI developed policies to the
previously developed distribution strategies. Then in section 3 the Iloating stock strategy is
Iormulated and two diIIerent policies to deal with this problem are developed. Section 4
Iocuses on the calculation oI saIety stock in DS/CSS strategy. In section 5 some numerical
results oI applying the developed policies Ior a real world case are shown. Finally, section 6
ends up with some conclusions.

2. Problem definition
In this paper we consider a Fast Moving Consumer Goods (FMCG) supply chain with two
echelons (the manuIacturer`s warehouse and the intermodal terminals) and one type oI
product (or aggregated mix). The products can be stored in a storage location near the
Iactory (which we call the Iactory storage) or can be transported to an intermodal terminal
6
where they can wait beIore being sent to the Iinal destination (see Figure 1). The advantage
oI Iactory storage is that it is cheaper than intermodal terminal, but the traveling time Irom
that terminal to customer is shorter than in case oI Iactory storage.

Figure 1. Conceptual representation oI an FMCG supply chain with intermediary stocking points.

Generally Fast Moving Consumer Goods (FMCG), also known as Consumer Packaged
Goods (CPG), is a business term with diIIerent interpretations. The main characteristic oI
these products is having a high turnover and relatively low cost. Though the absolute proIit
margin made on FMCG products is relatively small, the large numbers they sell in can
yield a substantial cumulative proIit. Examples oI FMCG include a wide range oI
Irequently purchased consumer products such as cosmetics, batteries, paper products and
plastic goods. FMCG may also include pharmaceuticals, consumer electronics, packaged
Iood products and drinks. What is essential in our analysis is that a batch production is
applied and Iull containers are used Ior transportation. II turnover is low and product cost is
high then typically smaller shipments are sent. This is the reason we Iocus on FMCG.

We assume that in each production run a batch oI containerized products is produced and
they are packaged in the Iactory. Determining the set oI products that are packed into one
container is leIt out oI consideration. Moreover, we assume that the Iinishing oI the
production batch can be adjusted to the shipment oI the last container Irom the Iactory.

In practice, intermodal terminals are not meant Ior long-time storage, since they have
higher storage costs, but they beneIit Irom a short delivery time to customers. That is,
Intermodal connection
Factory
Storage
Terminal
DC
DC
DC
Terminal
7
delivering the whole production batch directly to the terminal will cause higher storage
costs but will also guarantee very Iast delivery to the customers and lower backorder costs.
In addition, once they have arrived, it is common in practice that intermodal terminals oIIer
a period oI Iree oI holding costs Ior each delivery (e.g. due to discounts given by the
terminal authorities), this Ieature is included in the proposed model. Moreover, we assume
that when a customer arrives while there is no container available in the terminal, the
demand is backlogged and will be satisIied as soon as a container becomes available.
ThereIore system endures shortage cost in this case.

Total costs are made up oI transportation, shortage and storage costs. Transportation costs
diIIer per transportation route. They contain all costs that result Irom using the speciIic
transportation route. ThereIore, transportation costs can cause diIIerences in the total costs
oI each strategy, but these are independent oI the inventory levels during a production
cycle. The storage costs are the direct costs Ior storing a certain number oI products Ior a
certain period. These costs depend on the storage tariII at the speciIic point, the storage
time, and volume oI the products (or load units) stored. Shortage cost is stock-out cost, and
is calculated base on the waiting time oI customer Ior a container to arrive.

The main question in this paper is how to schedule the shipments oI products Irom the
Iactory such that the total cost is minimized. In the next section all the possible strategies
Ior this problem are brieIly reviewed.

2.1 Distribution strategies
For an FMCG supply chain with the above mentioned characteristics, we consider Iour
distribution strategies. These strategies are based on two decisions. First, iI every container
will be stored in a centralized or a decentralized location, and second, iI road or inter-modal
is used Ior transportation.

Among these strategies, the Iirst strategy is based on the just-in-time concept and applies
direct road transport only. This is Irequently used in FMCG-supply chains. The second
8
strategy is completely based on distributed storage: all transports are inter-modal. This
strategy is especially popular in supply chains where an inter-modal connection has lower
transport costs than a road connection. The third and Iourth strategies aim to take as much
advantage oI Iloating stock as possible. Below we will explain these strategies in detail.

Strategv CS: Centrali:ed storage and unimodal transport: Using this just-in-time based
strategy means that the whole production batch and a possible saIety stock are stored on-
site at the Iactory storage. When an order arrives, it is always IulIilled using road transport
Irom the on-site inventory. In this strategy the emphasis is on Iast transportations and easy
coordination.

Strategv DS: Decentrali:ed storage and inter-modal transport: The complete production
batch is shipped to regional terminals using inter-modal transport. The saIety stock is also
stored in these regional terminals. This batch cannot be used to IulIill orders until it has
arrived at the regional terminal. Any order which comes in during this transit time is
assumed to be backlogged. This increases storage time and costs. A demand prediction is
used to determine the split oI the production batch over the regional terminals. Orders are
delivered by truck Irom these terminals to the DCs. The emphasis is on using inter-modal
transportation and short order lead times (because the orders lead time Irom the terminal
will be shorter than the Iactory). II the saIety stocks are depleted at a terminal, lateral
transshipments Irom other terminals are made.

Strategv DS/CSS: Decentrali:ed storage, inter-modal transport, and centrali:ed safetv
stock: In this case the saIety stock is stored at the Iactory storage, whereas the production
batch is shipped to the terminals using inter-modal transport and stored there. The saIety
stock takes care oI demands during shipment oI the batch to the intermodal terminal. As
soon as batches reach to terminals regular deliveries to the retailers are IulIilled Irom the
terminals. The emergency deliveries Irom Iactory are done by road, because the inter-modal
transit time is much longer.

9
The saIety stock storage costs will probably be lower in the DS/CSS strategy when
compared to the DS strategy. This is because long storage on-site is in general cheaper than
long storage in an intermodal terminal. Furthermore, demand Iill rate increases iI the saIety
stock is stored in a central location.

Strategv FS. Floating Stock with staged arrivals. The Floating Stock strategy stores part oI
the production batch in the Iactory storage (centralized) and part oI the production batch is
stored in decentralized terminals. The shipment to the terminals is done by inter-modal
transport. Once the products have arrived at the terminal, they are shipped to retailer Irom
that point upon demand occurrence with a shorter order lead time. This strategy is designed
to beneIit Irom costs advantages oI Iloating stock storage without having to increase the
total inventory level in the supply chain. The FS strategy we consider here has staged
arrivals, viz. Ior each container a shipment time is determined. This diIIers Irom the FS
strategies in Ochtman et al. (2004) where the containers are sent together.

Note that iI we assume the batching decision to be Iixed beIorehand and that the timing oI
the Iinishing oI the batch coincides with the shipment oI the last container, then the CS
strategy needs no saIety stock and is completely determined. For the DS/CSS strategy we
still need to determine the amount oI saIety stock needed. This can be done with standard
approaches such as marginal cost analysis and the approach will be described in section
Iour. For the Floating Stock strategy however, we need to determine when to ship each
container, which is the problem this paper Iocuses at. We will do that in the next section.

3. Formulation of the container shipping scheduling problem.
The shipment process can be explained as Iollows: when the last container has been sent
out oI the Iactory storage, say at time t, a batch oI size m is produced (the production time
is neglected) and thereIore we have m containers ready Ior shipment at the production
Iacility. Customers call oII containers according to a stochastic demand processes.
Production planning is outside the scope oI this paper, thus we assume the value oI m is
given. To IulIill the demand, containers are shipped according to FS strategy Irom the
10
manuIacturer site to the terminal beIore demand actually occurs. It takes T
fi
days Ior
delivery Irom Iactory to the intermodal terminal and during these days and the initial period
T
nh
at the terminal no storage costs occur. T
nh
is the number oI days without storage cost at
intermodal terminal. II a customer requests his container aIter termination oI the Iree charge
period, the system incurs storage cost h
i
at the intermodal terminal per container per time
unit. On the other hand, iI a customer requests his container beIore end oI the Iree oI charge
period, the container had spent long time at the Iactory and Iactory storage costs could be
reduced by sending it earlier to the terminal. II a customer arrives while there is no
container at the terminal the demand will be backlogged until the container arrives. This
shortage costs c
b
per container per time unit.

We assume an Erlang(k, ·) renewal process Ior demand, since the Erlang(k, ·) distribution
can be interpreted as the batching oI k exponentially distributed demands, which is simply
treatable and it is straightIorward to calculate the convolution oI the same Erlang random
variables. Other processes may need more computational eIIort to calculate the
convolution.

In the rest oI this section, we present two approaches to deal with the FS strategy. The Iirst
one is a time based policy, that tries to Iind the optimal shipping time and the second one is
a quantity based policy that ships a new container whenever the total number oI containers
in pipeline and terminal drops down to a certain level. It is worth noting that, without
loosing generality, the supply network is decomposed into serial chains including an
intermodal terminal and a warehouse, and each chain is treated independent oI other chains.

3.1 Time based policy
As we have already mentioned, we aiming at the optimization oI shipping moments such
that the average expected costs are minimal. As we assume a Iixed batch oI size m which is
repeated all the time, the long-term average costs equal the expected costs per
manuIacturing batch. That is, Ior each batch we will optimize the shipment moments r
1
, .,
r
m
oI the containers in the batch such that the average cost per container is minimal:
11
( )
( )
1
1 2
1 2
,..., 0
batch cost , , ...,
min , , ,
batch size n
m
m
r r
E r r r
C r r r
>
(
¸ ¸
. =
Let us construct now this cost Iunction.

3.1.1 Cost function
The assumed cost structure incorporates Iactory, and inter-modal terminal storage and
shortage cost. It is worth noting that during shipment oI a container system does not endure
holding cost. To Iormulate the total cost, assume that k-1 demands have happened so Iar
and A
k
as the arrival time oI the k
th
container arriving to inter-modal aIter replenishment
time and it is planned to satisIy k
th
demand. Considering this situation three possible
circumstances are likely to happen in backlogging case:

D
k
_ A
k

In this case, demand occurs beIore the arrival oI the container, i.e. the demand is
backlogged and the customer has to wait Ior the container to arrive. Total cost in this case
consists oI backlogging costs Ior the length oI delay and holding costs at the Iactory until
the shipment moment.

A
k
· D
k
_ A
k
·T
nh

Here, demand happens aIter arrival oI the container and beIore end oI the inter-modal
terminal Iree holding cost period. ThereIore, no holding cost is incured at inter-modal
terminal, and total cost in this case includes only the holding costs at the Iactory until the
shipment moment.

A
k
· T
nh
· D
k

In this circumstance, container arrives to inter-modal, spends the entire Iree holding charge
period at the terminal and then the demand is realized. This case leads to holding cost at
inter-modal terminal plus the holding costs at the Iactory until the shipment moment. Due
to deterministic transportation time, the arrival moments A
k
can be immediately computed
Irom the shipment moment r
k
as A
k
÷r
k
· T
fi
.
12

Note now that, due to the backlogging assumption, each shipped container will be in Iact
'assigned¨ to a certain demand moment D
t
. ThereIore, optimizing the shipment moments
r
1
, ., r
n
oI the containers in a certain batch, we can ignore the demand moment that were
'assigned¨ to other containers. This allows us to separate the Iunction oI the total expected
cost oI a batch and rewrite it as:
( ) ( )
1 2
1
, ,...,
m
m k k
k
E C r r r E C r
=
= ( (
¸ ¸ ¸ ¸
¿
(1)
Taking into account the demand model, the cost Iunctions ( )
k k
C r Ior each container will
be computed as:

( )
( ) ( )
( )
( ) ( )
,
,
,
f k b k fi k k k fi
k k f k k fi k k fi nh
f k i k k fi nh k k fi nh
h r c r T D P D r T
C r h r P r T D r T T
h r h D r T T P D r T T
¦
+ + ÷ s +
¦
¦
= + < s + +
´
¦
+ ÷ ÷ ÷ > + +
¦
¹
(2)
Then, expected cost Ior each container is the Iollowing:

( ) ( )
( ) ( )
0
| ( )|
k fi
k
k
k fi nh
r T
k k f k b k fi k D
i k k nh nh D
r T T
E C r h r c r T D f d
h D r T T f d
t t
t t
+
·
+ +
= + + ÷
+ ÷ ÷ ÷
}
}
(3)
Since total cost Iunction is separable, to minimize the total expected cost oI the batch, we
need to minimize the expected cost oI each container | ( )|
k k
E C r .
In this case the Iollowing lemma is valid.

Lemma 1.
Given continuous distribution of the demand time D
t
, optimal shipment moments r
k
, that
minimi:e the expected cost for each container | ( )|
k k
E C r either solve equation.

( )
( )
( )
i f
i
k k fi k fi k k fi nh
b i b i
h h
h
P D r T P r T D r T T
c h c h
÷
s + + + s s + + =
+ +
(4)
or should be set equal to the production moment, when there is no optimal r
k
after the
production moment.
13

Proof.
Assuming continuous distribution oI the demand D
t
, the expected cost Iunction is
continuous in r
k
. ThereIore, we can Iind the optimal shipment moment r
k
by analyzing the
Iirst derivative oI the cost Iunction:

( ) ( )
( ) ( ) ( )
( ) ( )
( )
| ( )|
1
k k
f b k k fi i k k fi nh
k
f b k k fi i k k fi nh
f i b i k k fi
i k fi k k fi nh
dE C r
h c P D r T h P D r T T
dr
h c P D r T h P D r T T
h h c h P D r T
h P r T D r T T
= + s + ÷ > + +
= + s + ÷ ÷ s + +
= ÷ + + s +
+ + s s + +
(5)
From this expression it is easy to see that the Iirst derivative is always greater than or equal
to 0 in cases with h
i
_ h
f
:

( ) ( )
( )
0
f b k k fi i k k fi nh
f i k k fi nh f i
h c P D r T h P D r T T
h h P D r T T h h
+ s + ÷ > + +
> ÷ > + + > ÷ >
(6)
That is, the cost Iunction | ( )|
k k
E C r is continuously increasing in r
k
. Then, in order to
minimize the costs, we have to set r
k
as low as possible. In other words, we get a very
natural conclusion that in the case with the holding cost at the Iactory higher than the
holding cost at the terminal, it will be cheaper to send everything to the terminal as soon as
it is produced. To Iind optimal shipping time r
k
Ior the other case (h
i
~ h
f
), we apply the
standard method oI optimization oI convex Iunctions and set the Iirst derivative oI the cost
Iunction
| ( )|
k
k
dE C r
dr
to 0, i.e. we have to solve the Iollowing equation:

( ) ( )
0
f b k k fi i k k fi nh
h c P D r T h P D r T T + s + ÷ > + + = (7)
It is easy to see that the Iirst derivative is continuous non-decreasing Iunction, since the
second derivative

( ) ( )
2
2
| ( )|
0
k k
b k k fi i k k fi nh
k
d E C r
c P D r T h P D r T T
dr
= = + + = + + > (8)
is always greater than or equal to 0.
14

This means that the solution oI the equation (4) minimizes the expected cost Ior each
container | ( )|,
k k
E C r iI r
k
is unbounded. However, in real liIe situations it is not possible to
ship an order that is not produced yet. ThereIore, optimal shipment moments r
k
that
minimize the expected cost Ior each container | ( )|
k k
E C r either solve equations
| ( )|
0
k
k
dE C r
dr
= or should be set equal to the production moment (in cases when there is no
optimal r
k
aIter the production moment). u

Note here, that Ior certain probability distributions, there is a possibility oI multiple
solution Ior the equation
| ( )|
0
k
k
dE C r
dr
= . However, since the Iirst derivative is non-
decreasing, all solutions oI
| ( )|
0
k
k
dE C r
dr
= will belong to one compact set and all oI them
will in Iact produce the same value oI | ( )|
k
E C r . The optimal solution can be easily Iound
by a bisection search or an enumeration method.

3.2 Quantity based policy
To deal with this approach we make the extra assumption that the demand moments D
t
are
modeled as D
t
÷ D
t-1
¹ p
t
, where D
t-1
is the arrival moment oI previous demand and p
t
is
stochastic interdemand time. As the second proposed policy, we will look Ior a shipment
schedule deIined by the system state. Namely, the schedule in which the shipment moments
are deIined by the total number oI containers in the delivery 'pipeline¨ and intermodal
terminal, i.e. shipped but not picked-up yet by customers.

The cost Iunction in this case can be deIined similarly to the cost Iunction (1) in the
previous section. However, estimation and minimization oI the expectation oI the total cost
Ior each container C
k
requires diIIerent approach. First oI all, we assume here that the
container k will be shipped aIter the number oI containers in the 'pipeline¨ has changed
15
Irom S
k
to S
k
1. S
k
is the optimal total number oI containers in the pipeline and intermodal
terminal beIore realization oI demand k. Since backorders are allowed, customers k S
k
·
1, ., k 1 will pick up their containers Irom the terminal beIore the customer k will arrive
there. That is, we know exactly that S
k
1 customers will come to the terminal beIore
customer D
k
. Then, the time Irom the moment oI the system change Irom S
k
to S
k
1 until
the moment oI arrival oI customer k will be distributed as
1
0
k
S
k i
i
q
÷
÷
=
¿
.

Taking into account these assumptions about shipment policy, the total cost Ior container k
can be deIined as:
( )
( ) ( )
( )
( ) ( )
1 1
0 0
1
0
1 1
0 0
,
, ,
,
k k
k
k k
S S
f k b k fi k i k i k fi
i i
S
k k f k k fi k i k fi nh
i
S S
f k i k i k fi nh k i k fi nh
i i
h r c r T P r T
C r S h r P r T r T T
h r h r T T P r T T
q q
q
q q
÷ ÷
÷ ÷
= =
÷
÷
=
÷ ÷
÷ ÷
= =
¦
+ + ÷ s +
¦
¦
= + < s + +
´
¦
¦
+ ÷ ÷ ÷ > + +
¹
¿ ¿
¿
¿ ¿
(9)

where r
k
has a diIIerent deIinition Irom that in the time based policy. It is deIined as the
time Irom the moment that the number oI container in the system changes Irom S
k
to S
k
1
(moment
k
k S
D
÷
) until the container is shipped to customer k (see Figure 2).
time
D
k
D
k-1
D
k - r
k
S
k
·1
D
k - S
k
p
k
time
D
k
D
k-1
D
k - r
k
S
k
·1
D
k - S
k
p
k

Figure 2. Arrivals oI the customers (moments Dt) and oI the containers to the terminal.

The expectation oI the total cost Ior container k is shown in expression (10) and
minimization oI the system cost is done by the parameters S
k
and r
k
.
16

( ) ( ) ( )
( ) ( )
0
0
0
,
k fi
S
k
k i
i
S
k
k i k fi nh
i
r T
k k k f k b k fi
i k nh nh
r T T
E C r S h r c r T f d
h r T T f d
q
q
t t t
t t t
÷
=
÷
=
+
·
+ +
= + + ÷ (
¸ ¸
¿
+ ÷ ÷ ÷
¿
}
}
(10)
Lemma 2.
Given continuous distribution of the interdemand times pt, optimal shipment moments
r
k
(S
k
), that minimi:e the expected cost for each container | ( , )|
k k k
E C r S either solve
equation.
( )
( )
( )
0 0
k k
S S
i f
i
k i k fi k k fi k k i k fi nh k
i i
b i b i
h h
h
P r T d P r T d r T T d
c h c h
q q
÷ ÷
= =
÷
s + ÷ + + ÷ s s + + ÷ =
+ +
¿ ¿
or equal to 0.

Proof. ProoI oI this lemma is very similar to the prooI oI lemma 1. u

This lemma, however, requires knowing convolutions oI the interdemand times
¿
=
÷
k
S
i
i k
0
q ,
which are not trivial in many cases. Still, Ior some distributions (such as normal or
exponential), this convolutions are relatively easy. Let us Iind now optimal S
k
and r
k
Ior an
Erlang demand process with identically distributed interdemand times.
3.2.1 Erlang distributed interdemand times
Let us assume that the interdemand times are Erlang distributed i.i.d`s with parameters ·
and k, say p
k
~ Erlang(k, ·) Since the interdemand times are identically distributed, it is
easy to conclude that Ior each arrival D
t
the optimal parameters S
t
and r
t
will be identical.
The convolutions
1
0
S
i
i
q
÷
=
¿
used in expression (10) will be then distributed according to
Erlang(kS, ·) distribution. This means that we can write expression (10) as:

( ) ( )
( )
( )
( )
( )
( )
1
1
1 !
0
1 !
,
kS
fi
kS
fi nh
r T
f b fi kS
i nh nh kS
r T T
E C r S h r c r T e d
h r T T e d
ì ìt
ìt
ì ìt
ìt
t t
t t
÷
÷
+
÷
÷
·
÷
÷
+ +
= + + ÷ (
¸ ¸
+ ÷ ÷ ÷
}
}
(11)
Further integration gives us the Iollowing close-Iorm expression:
17
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( )
( )
( ) ( ) ( )
( ) ( )
1
! !
0 0
1
! !
0 0
, 1 1
n n
fi fi
fi fi
n n
fi nh fi fi
fi fi fi fi
kS kS
r T r T
r T r T
f b fi b n n
n n
kS kS
r T T r T T
r T r T T T
i nh nh i n n
n n
kS
E C r S h r c r T e c e
kS
h r T T e h e
ì ì
ì ì
ì ì
ì ì
ì
ì
÷
+ +
÷ + ÷ +
= =
÷
+ + + +
÷ + + ÷ + +
= =
| | | |
= + + ÷ ÷ ÷ (
| | ¸ ¸
\ . \ .
÷ + + +
¿ ¿
¿ ¿
(12)
It is easy to see, that Ior a Iixed number S the optimal time r(S) has to satisIy the Iollowing
equation:
( )
( )
( )
( )
( )
( ) ( ) ( )
( ) ( )
1 1
1
1 ! 1 !
0
1 1
! !
0 0
| ( , )|
1 1 0
kS kS
fi
fi nh
n n
fi fi nh fi fi fi
r T
f b i kS kS
r T T
kS kS
r T r T T r T r T T
f i b i n n
n n
E C r S
h c e d h e d
r
h h c e h e
ì ìt ì ìt
ìt ìt
ì ì
ì ì
t t
÷ ÷
÷
+ ·
÷ ÷
÷ ÷
+ +
÷ ÷
+ + + ÷ + ÷ + +
= =
c
= + ÷
c
| | | |
= ÷ + ÷ + ÷ =
| |
\ . \ .
} }
¿ ¿
(13)
SimpliIying the last equation we obtain an equation Ior the time r(S) that is quite similar to
the equation in the previous section:

( )
( ) ( )
( )
( )
( ) ( ) ( ) ( )
1 1 1
! ! !
0 0 0
1
n n n
fi fi fi nh fi fi
nh
kS kS kS
r T r r T T T r T r T
i f T i
n n n
n n n
b i b i
h h
h
e e e
c h c h
ì ì ì
ì ì
ì
÷ ÷ ÷
+ + + + ÷ + ÷ +
÷
= = =
÷
| | | |
÷ + ÷ =
| |
+ +
\ . \ .
¿ ¿ ¿
(14)

and the optimal time r(S) is either equal to 0 or solves this equation. Substituting the
optimal r(S) into equation 8 Ior the expected cost per container we can simpliIy it Iurther
as:

( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
1
1
1
!
0
( 1)!
1
!
0
( 1)
, 1
n
fi
fi
kS
fi
fi
n
nh fi
fi nh
kS
nh fi
fi nh
kS
r S T
r S T
b fi n
n
r S T
r S T
b fi kS
kS
r S T T
r S T T
i nh if n
n
r S T T
r S T T
i nh fi kS
kS
E C r S S c T e
c r S T e
kS
h T T e
h r S T T e
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
÷
÷
÷
+
÷ +
=
+
÷ +
÷
÷
+ +
÷ + +
=
+ +
÷ + +
÷
| | | |
( = ÷ ÷
| |
¸ ¸
\ .\ .
+ +
| |
÷ + ÷
|
\ .
+ + +
¿
¿
!
(15)
Although, the Iunction ( ) ( )
, E C r S S (
¸ ¸
and equation
| ( , )|
0
E C r S
r
c
=
c
are not too complex
Ior numerical computation, analytical derivation oI the optimality conditions Ior S is not
trivial. It is also not clear whether the Iunction ( ) ( )
, E C r S S (
¸ ¸
is convex. On other hand, S
18
is a one-dimensional discrete variable and in the most oI real liIe situations it can not be
extremely high (e.g. it can be limited by the number oI available containers). ThereIore, in
many situations, optimization oI S by enumeration will be still easy.

4. Calculation of safety stock for DS/CSS strategy
In DS/CSS strategy saIety stock is hold in the Iactory and takes care oI demand while the
batch is being shipped to the intermodal terminal. The amount oI containers which hold as
the saIety stock is calculated based on a marginal cost analysis. We deIine C(ss) as the cost
oI stocking ss containers in Iactory and dispatching the rest oI production batch to
intermodal terminals. II this value is increased by one unit then extra holding cost should be
paid at Iactory while it reduces the risk oI shortage during shipment and decreases the
holding cost at intermodal terminal since less containers are shipped to that terminal.
0 ) ( ) ( )
2
( ) ( ) 1 (
) 1 ( 1
) 1
> ÷ ÷ ÷ ÷ ÷
÷
= ÷ +
+
+ ÷
+
+
+
nh fi ss m i ss fi b
ss ss
f
T T D E h D T E c
D D
E h ss C ss C
0 ) ( ) ( )
2
( ) 1 ( ) (
1
s + ÷ ÷ ÷ ÷
÷
= ÷ ÷
+
÷
+ ÷
nh fi ss m i ss fi b
ss ss
f
T T D E h D T E c
D D
E h ss C ss C (16)

5. Numerical Results (Case Study)
Below we present a real case which has been made together with the logistic service
provider Vos Logistics in the Netherlands to illustrate our model. Vos Logistics is primarily
a European trucking company, but provides inter-modal transport as well. It Iaced more and
more urgent transports with Iew opportunities Ior loaded return trips. Motivated by
increasing road taxes, like the MAUD in Germany, she was considering expanding her
inter-modal capabilities. The case is also described in Ochtman et al. (2004), but has been
slightly adapted to allow the application oI our models. The case is as Iollows. An FMCG-
manuIacturer runs a Iactory in Poznan (Poland) and distributes its products to two retail
DCs in Germany, viz. one serving Dortmund and Köln and the other one covering demands
Irom Rüsselsheim (near FrankIurt), and Appenweier (near Strassbourg). At this moment all
orders are transported FTL by truck. The load unit is a 40 It. container. An alternative inter-
modal route is a rail connection Irom a station in Gadki (15 km Irom Poznan) to two train
19
terminals in Duisburg and Mannheim. The conceptual network representation Ior this case
is depicted in Figure 3.

The transit time Ior all two direct truck routes is two days including handling time Ior in-
and outbound in the on-site DC. The inter-modal connection makes use oI the rail
connection. Due to the long time needed Ior shunting, the transit time oI the train transport

Figure 3. Conceptual network representation oI the case

to both terminals is 4 days. The shipping time Irom an intermodal terminal to a customer is
one day using road transport. II a stock-out happens in one oI the terminal, then the
customer has to wait until the container arrives to the terminal. The cost components which
are used to estimate the costs are linear per FTL container delivery and are detailed in Table
1. It is worth noting that direct truck transport is slightly cheaper than inter-modal transport.
The model presented in section three is now applied Ior each inter-modal terminal.

5.1 Simulation results
In order to be able to compare the developed policies with the other distribution strategies
we implemented a simulation program using MATLAB 7.0. In this simulation the
Iollowing criteria are considered: the expected total costs and the order Iill rate. The order
Iill rate is the percentage oI the orders that can be IulIilled in less than two days. This two
day period is the direct shipping time Irom Iactory to the customer. In calculating total cost
a Iixed transportation cost component has been excluded since it has no eIIect on the
optimal value. ThereIore we have no cost Ior direct transport and 20 units per container Ior
Poznan
Gadki
Duisburg
Mannheim
Koln &
Dortmund

Russelsheim
&
Appenweier


15 km
815 km
880 km
820 km
810 km
60 km
135 km
Road connection
Intermodal connection
20
intermodal transport. Assuming that the size oI the production batch is 80 containers and
demand at each terminal is exponentially distributed with rate ì ÷ 1.5 per day, we reach to
the results shown in table 2. Floating stock policies outperIorm the other strategies in total
cost. Comparing the Iill rates shows that CS strategy and time based policies have the
highest Iill rate. The Iill rate oI CS strategy is always 100° since in case oI direct shipment
all containers are delivered to customer in less than two days and moreover we assume a
zero production time oI the batch. The DS strategy has the lowest Iill rate, since Ior the Iirst
Iour days all demands that happen during this period have to wait Ior the whole production.

Table 1: Cost Components
Components: Costs (Euro per container):
Storage and Holding
Centralized holding cost at Iactory storage h
f
8 / day
Decentralized holding cost in terminal (Iirst 3 days are Iree) h
i
18 / day
Backlogging cost at inter-modal c
b
20 / day
Transportation
For the direct road connection Irom Iactory to DC 880
For the intermodal connection Irom Iactory to DC 900

batch to arrive at intermodal terminal. Extra analysis showed that iI we exclude the constant
transportation cost Irom total cost more than 95° oI the total cost is due to storage cost at
Iactory and intermodal terminal, 3° transportation cost and just 2° oI the total cost results
Irom backlogging. In table 3, the optimal shipping times which are the resultant oI the time
based policy are shown (rounded oII to an integer number oI days). It shows Ior each day,
how many containers should be shipped Irom Iactory to the each intermodal terminal. Since
the arrival rates are assumed to be identical, number oI containers that should be shipped to
each terminal in each day is identical as well.

Table 2: PerIormance criterion Ior all policies
Floating Stock
CS DS CS/DSS
Time based Quantity based
Total Cost 79054.2 81958.9 82551.5 59337 73025
Fill Rate (°) 100 84 98 99.7 87
21

Table 3: Optimal shipping time oI containers to each terminal
Shipping Day 0 1 2 3 4 5 6 7 8 9 10 11 12
Number oI containers 4 4 3 4 3 3 4 3 2 2 2 3 3

5.1.1 Sensitivity analysis
In this section, some sensitivity analysis is done on cost parameters. It is done Ior a case in
which demand Iollows an Erlang distribution process. The cost parameters are changed and
the results oI diIIerent policies are shown in the tables 4 and 5. As it is obvious in table 5,
the Iloating stock and particularly the time based policy outperIorm the other strategies in
minimizing total cost. But as it is shown in table 4 the Iloating stock strategy is not the best
one when we are dealing with slow moving items. We did Iurther analysis on the basic
problem with cost parameters shown in table 1 to Iind out under which circumstances it is
Ieasible to use Iloating stock regarding total arrival rate to the system. Since the time based
approach outperIorms the other one Ior both criteria, we excluded the quantity based policy
in Iigure 4. The results are depicted in Iigure 4 and it is obvious that Ior a total arrival rate
oI more than 0.6 per day, it is proIitable to implement a Iloating stock policy.

Table 4: PerIormance criterion Ior all policies ì
D
÷0.11 and ì
M
÷0.13 and k÷3
Floating Stock
h
i
h
f
c
b
PerIormance CS DS DS/CSS
Time based
Quantity
based
S
D
S
M

Total Cost 178720 278580 279350 240210 183040
16 8 20
Fill Rate(°) 100 99.9 100 94.5 73.4
1 1
Total Cost 178540 384190 382450 294490 190160
24 8 20
Fill Rate(°) 100 99.9 100 96.6 70.8
1 1
Total Cost 178120 176020 175600 185510 172210
8 8 20
Fill Rate(°) 100 99.1 100 96.7 93.4
1 1
Total Cost 97483 280800 278830 196410 139110
16 2 20
Fill Rate(°) 100 99.1 100 90.1 85.0
1 1
Total Cost 259010 280100 278770 281950 256380
16 14 20
Fill Rate(°) 100 99.9 100 95.6 99.5
1 1
Total Cost 178170 278430 279500 241560 179680
16 8 50
Fill Rate(°) 100 99.1 100 95.7 95.7
1 1
22
Total Cost 178580 279140 278850 239310 185420
16 8 18
Fill Rate(°) 100 99.9 100 99.3 82.6
1 1
D stands Ior Duisburg and M Ior Mannheim

Table 5: PerIormance criterion Ior all policies ì
D
÷2.5, ì
M
÷2 and k÷3
Floating Stock
h
i
h
f
c
b
PerIormance CS DS DS/CSS
Time based
Quantity
based
S
D
S
M

Total Cost 22130 37597 29767 12453 19781
16 8 20
Fill Rate(°) 100 95.6 100 92.5 84.8
3 2
Total Cost 22090 55773 40057 14698 19484
24 8 20
Fill Rate(°) 100 96 100 92.3 86.5
3 2
Total Cost 22250 19684 20326 14563 18847
8 8 20
Fill Rate(°) 100 96 100 100 81.2
3 2
Total Cost 5543.7 37511 22637 5202 5484
16 2 20
Fill Rate(°) 100 96 100 90.8 79.2
3 2
Total Cost 38933 37550 36966 14133 25345
16 14 20
Fill Rate(°) 100 96 100 99.4 80.3
3 2
Total Cost 22210 37958 29614 10242 18334
16 8 50
Fill Rate(°) 100 96.1 100 98.9 81.7
3 2
Total Cost 22140 37647 29855 11481 19758
16 8 18
Fill Rate(°) 100 95.9 100 96.6 81.4
3 2

It is worth noting that, CS strategy depends only on holding cost at Iactory but we can see
in both tables 4 and 5 that by changing other cost parameters the resultant total costs
become slightly diIIerent. To analyze the accuracy oI the simulation, 0.95° conIidence
intervals are derived. In table 4, Ior a case in which holding cost at Iactory is 8 the
conIidence interval is computed to be |178300, 178850| Ior 100 simulation runs. ThereIore
the diIIerence among total costs in this case is not signiIicant. This interval Ior arrival rates
considered in table 5 is |22050, 22340|, and again we can conclude that the diIIerences are
not signiIicant.

23

Figure 4. Total cost oI diIIerent policies vs. ì
t

6. Discussion and conclusion
Floating stock is a concept where a new production batch is (partly) pushed into the supply
chain, without determining the exact destination Ior each product beIorehand. Use oI this
concept may lead to lower storage costs and a shorter order lead time, without a demand Iill
rate decrease. This is possible iI the production batch is split up into some parts and being
shipped to intermodal terminal in advance oI demand realization at the right time. In this
paper we developed mathematical approaches Ior Iloating stock strategy to determine the
optimal shipping time oI containers through intermodal route. In the developed policies, the
production batch is split up into some parts. In the time based policy the shipping moments
are optimized while in the quantity based policy the optimal total number oI containers in
the pipeline and intermodal terminal is determined. The simulation results show that the
Iloating stock strategy oIIers the best opportunities to beneIit Irom low storage costs
without aIIecting Iill rate level. The popular just-in-time strategy oIten uses centralized
storage and road transport. The computational results show that the Iloating stock strategy
can reduce costs and lead times, in spite oI possibly higher transportation costs oI an inter-
modal connection. The main conclusion that can be drawn is that when an intermodal
transport is used even though it can be slower and more expensive, iI we integrate it with
inventory control then the results show that it leads to cost saving without aIIecting the
24
service level. So when considering a move Irom Iactory to DC, storage and holding costs as
well as transportation costs should be taken into account.

Acknowledgment
We grateIully acknowledge the eIIort oI E. van Asperen Irom Erasmus University
Rotterdam and also G. Ochtman and W. Kusters Irom Vos Logistics Company in
Netherlands Ior doing initial works and suggesting this problem to us.

References
Bontekoning, Y., Macharis, C. & Trip, J., 2004. Is a new applied transportation research Iield emerging? a
review oI intermodal rail-truck Ireight transport literature., Transportation Research Part A . Policy and
Practice 38(1), 1-34.
Bookbinder, J. & Fox, N., 1998. Intermodal routing oI Canada. Mexico shipments under NAFTA.
Transportation Research Part E: Logistics and Transportation Review, 34(4), 289-303.
Cetinkaya S., Mutlu F., Lee C., 2006. A comparison oI outbound dispatch policies Ior integrated inventory
and transportation decisions. European journal oI Operational Research, 171, 10941112
Chopra, S. & Meindl, P., 2004. Supply Chain Management, 2nd edition, Prentice-Hall, New-Jersey.
Diks, E., de Kok, A. & Lagodimos, A., 1996. Multi-echelon systems: A service measure perspective.
European Journal oI Operational Research 95, 241-263.
ECMT, 1993. Terminology on combined transport, OECD, Publications Services, Paris, France. URL:
http://www1.oecd.org/cem/online/glossaries/
Evers, P. T., 1996. The impact oI transshipments on saIety stock requirements. Journal oI Business Logistics
17(1), 109-133.
Evers, P. T., 1997. Hidden beneIits oI emergency transshipments, Journal oI Business Logistics 18(2), 55-76.
Evers, P. T. (1999). Filling customer order Irom multiple locations: A comparison oI pooling methods,
Journal oI Business Logistics 20(1), 121-139.
Herer, Y., Tzur, M. & Y¯ucesan, E., 2002. Transshipments: An emerging inventory recourse to achieve
supply chain leagility. International Journal oI Production Economics, 80(3), 201-212.
Kelton, W., Sadowski, R. & Sturrock, D., 2004. Simulation with Arena, 3rd edition, McGraw-Hill.
Konings, J. 1996. Integrated centers Ior the transshipment, storage, collection and distribution oI goods: A
survey oI the possibilities Ior a high-quality intermodal transport concept. Transport Policy 3(1.-2), 3-11.
Landers, T., Cole, M., Walker, B. & Kirk, R. (2000), The virtual warehousing concept. Transportation
Research Part E 36(2), 115-125
25
Macharis, C. & Bontekoning, Y., 2004. Opportunities Ior OR in intermodal Ireight transport research: A
review. European Journal oI Operational Research 153(2), 400-416.
Minner, S., 2003. Multiple-supplier inventory models in supply chain management: a review. International
Journal oI Production Economics 81.82, 265-279.
Moinzadeh, K. & Nahmias, S., 1988. A continuous review model Ior an inventory system with two supply
modes. Management Science 34(8), 761-773.
Ochtman, G., Dekker, R., van Asperen, E. & Kusters, W., 2004. Floating stocks in a Iast-moving consumer
goods supply chain: Insights Irom a case study, Technical Report ERS-2004-010-LIS, ERIM Report Series
Research in Management, Erasmus University Rotterdam, The Netherlands.
Rutten, B., 1995. On Medium Distance Intermodal Rail Transport, PhD thesis. Faculty oI Mechanical
Engineering, TU DelIt, DelIt University Press, DelIt, The Netherlands.
Sampson, R. J., Farris, M. T. & Shrock, D. L., 1985. Domestic transportation: practice theory and policy. 5th
edition, TODO ?.
Teulings, M. & van der Vlist, P., 2001. Managing the supply chain with standard mixed loads. International
Journal oI Physical Distribution & Logistics Management 31(3), 169-186.
Tyworth, J. & Zeng, A., 1998. Estimating the eIIects oI carrier transit-time perIormance on logistics cost and
service. Transportation Research Part A 32(2), 89-97.

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