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HUNGARIAN JOURNAL
OF INDUSTRIAL CHEMISTRY

VESZPRÉM
Vol. 39(1) pp. 107-112 (2011)

DETERMINING OPTIMAL STOCK LEVEL
IN MULTI-ECHELON SUPPLY CHAINS

A. KIRÁLY1, G. BELVÁRDI2, J. ABONYI1

1University of Pannonia, Department of Process Engineering, P.O. Box 158., Veszprém H-8200, HUNGARY
E-mail: abonyij@fmt.uni-pannon.hu

2IBS Consulting Ltd Szőlősi 16a Siófok H-8600, Hungary

Inventory control of Multi-echelon supply chains is a widely researched area. In most cases researchers choose analytic
methods to analyze such logistics systems. Simulation is a very useful alternative for analyzing supply chain systems and
a well-constructed model can provide a better approach and can give more realistic picture of the complex situation.
In this article the authors introduce an interactive, configurable simulator to analyze stock levels in a complex supply
chain. This new modeling approach (SIMWARE) is capable to simulate complex multi-echelon supply chains where the
frequency of stock transfer between the individual levels of the supply chain can be optimized. The authors evaluate different
optimization strategies and methods. The introduced SIMWARE method can be used to minimize the environmental
impact of the supply chain by minimizing the transportation between the nodes of the supply chain hierarchy. The model
provides an optimization methodology where the objective function is the total cost of the supply chain.

Keywords: supply chain, simulation, safety stock, optimization.

Introduction

Supply chains present a complex decision situation for
management. Supply chain performance impacts the
financial performance of the company therefore it is
important to find a way to optimize the performance of
the supply chain. Simulation provides a way to get closer
to real life situations and use less simplifications and
assumptions than you need with analytical solutions.

Our objective is to build a simulator that can use
simple building blocks to construct models of complex
supply chain networks. Supply chains processes can be
simulated using these models, where parameters of Key
Performance Indicators are analyzed by sensitivity
analysis. The result is a decision support tool flexible
enough to handle complex situations and straightforward
and simple to use enough for management purposes.

Literature review

The research of inventory management or the investigation
of supply chain is widespread in the scientific literature.
It is a seriously researched area since the fifties; Simpson
[6] was the first one who formulated the serial-line
inventory problem. Graves and Willems extend Simpsons
work to spanning trees in [2], while in [3] they give a
comprehensive review of the previous approaches for
safety stock placement, the same as they propose two
general approaches and introduce the supply chain
configuration problem. There exist good overviews in the

literature for supply chain and inventory management,
like in [11], where authors give an overview of different
approaches of supply chain modeling, and outline future
opportunities. In [9] Lau et al. give an overview of
various average inventory level (AIL) expressions and
presents two novel expressions which are simpler and
more accurate than previous ones. A comprehensive
overview can be found in [13], where a simulation model
for a real problem with lost sales is presented, where
authors consider several types of inventory policies.

The determination of safety stock in an inventory
model is one of the key actions in the management,
Miranda and Garrido include both cycle and safety stock
in the inventory model in [12], and the resulting model
in this article has a non-linear objective function. Authors
in [4] gives a model for positioning safety stock in a
supply chain subject to non-stationary demand, and they
show how to extend their former model to find the optimal
placement safety stocks under constant time service
(CST) policy. Prékopa in [16] gives an improved model
for the so called Hungarian inventory control model to
find the minimal safety stock level that ensures the
continuous production, without disruption.

The bullwhip effect is an important phenomenon in
supply chains, authors in [10] show how a supply chain
can be modeled and analyzed by colored petri nets (CPN)
and CPN tools and they evaluate the bullwhip effect, the
surplus of inventory goods, etc. using the beer game as
demonstration. More recent research can be found in [1],
which shows that an order policy applied to a serial
single-product supply chain with four echelons can reduce
or amplify the bullwhip effect and inventory oscillation.

108

Miranda et al. investigate the modeling of a two
echelon supply chain system and optimization in two
steps [15], while a massive multi-echelon inventory
model is presented by Seo [19], where an order risk
policy for general multi-echelon system is given, which
minimizes the system operation cost. A really complex
system is examined in [20], where it is necessary to
apply some clustering for similar items, because detailed
analysis could become impossible considering each item
individually.

The simulation-based approach was published only
in the last decade. Jung et al. [7] make a Monte Carlo
based sampling from real data, and apply a simulation–
optimization framework while looking for managing
uncertainty. They use a gradient-based search algorithm,
while authors in [8] discuss how to use simulation to
describe a five-level inventory system, and optimize this
model by genetic algorithm. Schwartz et al. [18]
demonstrate the internal model control (IMC) and
model predictive control (MPC) algorithms to manage
inventory in uncertain production inventory and multi-
echelon supply/demand networks. The stability of the
supply chain is also a seriously researched area recently,
[14] shows that a linear supply chain can be stabilized
by the anticipation of the own future inventory and by
taking into account the inventories of other suppliers,
and Vaughan in [21] presents a linear order point/lot
size model that with its robustness can contribute to
business process modeling.

A complex instance of inventory model can be
found in [5], where orders cross in time considering
various distributions for the lead time. Sakaguchi in [17]
investigates the dynamic inventory model in which
demands are discrete and varying period by period. The
author gives an algorithm to solve several examples.

Based on the previous review it is clear that most of
the multi-echelon supply chain optimization and analysis
are based on analytical approach. Simulation however
provides a very good alternative, because it can model
real life situations with accuracy, more flexible in terms
of input parameters and therefore it is more easy to use
in decision support. The simulation results can be
analyzed with various statistical methods and numerical
optimization algorithms. To analyze complex, especially
multi-echelon systems, multi-level simulation models
can be used, where the results of optimized high level
model feeds into the lower level more detailed models.

The structure of the paper is the following:
Section 2 is a general introduction to the problem

describing the multi-echelon supply chain and the
relevant cost structure. Section 3 introduces the flexible
modeling tool to build complex multi-echelon supply
chain models using simple, easy to understand modules.
Section 4 represents the main results through a case
study, while section 5 concludes our work. The reader
can follow the solution of a problem and we demonstrate
the use of sensitivity analysis as a decision support tool.
Finally we talk about the integration of this methodology
to a complex decision support system which is part of
the company Management Information System.

The proposed method

Our intent was to create a Monte-Carlo simulator which
uses probability distributions based on material usage
data posted in the logistic module of an ERP system.
This new methodology has two main objectives.

SIMWARE can be used as a verification tool to
analyze and evaluate inventory control strategies pinpoint
problems due to MRP parameter determinations. The
simulation of “actual” inventory controlling strategies
provides the most important KPI-s of these strategies.
On the other hand we can use the simulator as part of
optimization and determine the optimal values of the
key inventory control parameters.

SIMWARE provides a framework to analyze the
cost structure and optimize inventory control parameters
based on cost objectives. We are in the process to
finalize the costing model therefore we used a simple
cost function at this point. We have minimized the
inventory holding cost by changing the parameters of
our operational space while keeping the service level at
the required value.

Simple warehouse model

Figure 1: The classic model of inventory control.

In Fig. 1, Q is the theoretical demand over cycle

time T and this is the Order Quantity; R is the Reorder
point, which is the maximum demand can be satisfied
during the replenishment lead time (L). S is the Safety
stock; this is needed if the demand is higher than the
expected (line d). We have a special type of Safety stock
in our example. Utility companies use this stock to cover
demand due to system failures.

Now we give a summary of the most important
parameters of inventory control and their relevance and
connection to our model.

Lead time L is the time between the Purchase order
and the goods receipt. d̄ L denotes the average demand
during the replenishment lead time. d̄ L = d̄ ·L, where d̄ is
the daily average demand. Using the same logic dL is a
special case; it yields consumption if the service level is
100%. We will use dL to denote the consumption during
the manuscript. A Cycle time (T) is the time between
two purchase orders. The Order Quantity is Q where
Q = d¯·T. This is the ordered quantity in a purchase order.

ZS Lσ=

tout
outt

L
LLL tL

d
ZdESdR −+=−+= σ

d
out

L t
L

d
E =

109

Q is equal to the Expected demand and the Maximum
stock level. Maximum stock level is the stock level
necessary to cover the Expected demand in period T;
therefore it has to be the quantity we order.

Reorder point is the stock level when the next purchase
order has to be issued. It is used for materials where the
inventory control is based on actual stock levels.

In an ideal case R equals to total of safety stock and
average demand over lead time: ( SdR L += ).

S is the Safety stock which is to cover the stochastic
demand changes and for a given Service Level this is the
maximum demand can be satisfied over the Lead time.

Assuming constant demand pattern over the cycle
time, Average Stock (K) can be calculated as follows.

S
Q

K +=
2

(1)

It is calculated as a weighted average of stock levels
over the cycle time.

Service Level (SL) is the ratio of the satisfied and
the total demand (in general this is the mean of a
probability distribution), or in other words it is the
difference between the 100% and the ration of unsatisfied
demand:

Rd
SL L

)(
100100

−
−= (2)

We assume that all demand is satisfied from stock
until stock exists. When we reach stock level R the
demand over the lead time (dL) will be satisfied up to R.
Consequently if dL > R, we are getting a stock out
situation and there will be unsatisfied demand therefore
the service level will be lower than 100%. dL is not
known and it is a random variable. The probability of a
certain demand level is P(dL). Based on this, the service
level is formed as shown in the next equation.

dRddP
SL

d
LLL

∫ −
−=

max

))((
100100 , (3)

where dL is continuous random variable, and dmax is the
maximum demand over Lead time.

Stochastic model

Based on our experience in analyzing actual supply chain
systems we discovered that the probability functions of
material flow and demand are different from the theoretical
functions.

As Fig. 3 shows the distribution function of an actual
material consumption is significantly different from the
theoretical one (Fig. 2). This makes us believe that there
is a difference between the theoretical (calculated) and
the actual inventory movements, therefore it makes sense
using an approach based on “actual” distribution function.
We will demonstrate the results of the simulation using
this approach.

Figure 2: Theoretical Cumulative Distribution function

Figure 3: Actual Cumulative Distribution function for a

raw material based on its consumption data

Inventory movements can be modeled much better

using stochastic differential equations than modeling
based on the theoretical assumption that movements are
following normal distribution.

We propose the following model:

),,(
1 uiLL

tropxuWxx
ii

+−=
+

, (4)

where xi is stock level on the i
th week, Wi is a stochastic

process to represent consumption. This stochastic process
is based on the empirical cumulative distribution function
we described in the previous section. u is the quantity of
material received on week i, based on purchase orders.
Purchase orders are calculated based on the actual
inventory level (x), and reorder point (rop), and the
replenishment lead-time (tu).

Optimization of the system

Optimization process is changing the reorder point
while keeping the service level at the required value. As
a result we find the optimal reorder point for the given
service level. This determines an optimal inventory
level to minimize our current cost objective function.

In the first phase we created the simulator using the
Sequential Quadratic Programming (SQP) functionality
of MATLAB’s Optimization Toolbox.

During the optimization we calculate the minimum
value of a restricted non-linear multi-variable function.
In our case we use the reorder point as optimization
variable (parameter), we are seeking for the minimum
of the average days of inventory. Our restriction is the
required value of the service level.

110

In the basic case we use one set random consumption
data for the optimization therefore the optimum is related
to this dataset.

We applied the Monte-Carlo process to overcome
this problem. This is a robust methodology which
generates empirical distribution functions of consumption.
We use these data sets as input for the simulation model
to simulate the stock movements. We use many random
paths like this in every optimization step. Based on the
proven convergence of the Monte-Carlo process the
calculated stock movements, stock turnover and service
level are good estimations of the actual process at a
given reorder point.

In the current example the optimization process
executes 100 simulation runs for each parameter set and
uses the average of the results as objective function or
constraint. We use the reorder point as optimization
variable (parameter), we are seeking for the minimum
of the average days of inventory. Our restriction is the
required value of the service level. We demonstrate the
results in the next section.

Results

In this chapter we show a complete process for the
optimization of a 2-echelon supply chain, as well as we
demonstrate how to solve a more complex problem by
the help of our proposed SIMWARE program.

Analysis of a 2-level system

In this demonstrated example we used the simulator to
analyze a system with two connected warehouses. The
simulator is capable to optimize the two warehouses in
the same time and calculate the optimum for the supply
chain as a whole.

The following diagram shows the supply chain, i.e.
the structure of the analyzed 2-level system.

Figure 4: The analyzed 2-level system

Where the objective function is:

f(z) = mean(h1) + 1.3*mean(h2), (5)

i.e. the holding cost in the second Warehouse is 30%
higher than in the first Warehouse.

In Fig. 5, the values of the objective function (i.e.
cost) is presented as a function of the reorder point of
the two Warehouses. Fig. 6 shows the service level of
Warehouse 1 in the 2-level system. The constraint for
the service levels is 95% in this case.

Cost

Reorder point – Storage 2 Reorder point – Storage 1

C
os
t

Reorder point – Warehouse 1Reorder point – Warehouse 2
Figure 5: The values of the objective function for the

2-level system

Se
rv
ic
e
L
ev
el

Service Level – Warehouse 1

Reorder point – Warehouse 1 Reorder point – Warehouse 2
Figure 6: The values of service level for Warehouse 1.

The figure above shows only the service level for the

first warehouse, but in the optimization problem both
service level has to take into consideration as well as the
constraints. In the next picture, we investigate all of these.

Figure 7: The result of the optimization.

Fig. 7 shows the result of the optimization using the

SQP method. The optimal solution is highlighted with
the green square. It satisfies the 95% constraints and
ensures the minimal holding cost in the warehouses.
The cost function can be seen in Fig. 5.

The result of the simulation run is presented in the
next figure. The diagram shows weekly inventory levels
in the Warehouses at the optimal case.

111

Weeks

Figure 8: Inventory levels in the 2-level system.

In the rest of this section, we present a more complex,

3-level inventory model for illustration. This model can
be seen in the next figure, which contains 9 warehouses
in a 3-echelon structure.

In the SIMWARE program, users have to define the
structure of the problem, i.e. the connection between the
warehouses. The parameters of each warehouse has to be
defined also, e. g. the lead time or the average demand for
a product as well as the cost values, like the holding cost.

Fig. 10 represents the results of the simulation for a
3-level inventory model, where each inventory level is
depicted by lines with different style.

WH1

WH2

WH3

WH4

WH43WH42

WH41

WH31

WH21

Figure 9: The analyzed 3-level system.

Weeks

In
ve
n
to
ry
L
e
ve
l

Figure 10: The inventory levels in the 3 level multi-

echelon system.

Note that we have used separate distribution
functions for the central warehouse and the other regional
warehouses to simulate the consumption. We constructed
these distribution functions based on actual data. The
central warehouse consumption figures are calculated as
a total of the downstream warehouse consumptions.

Using different distribution functions in a multi-
echelon supply chain, or using different consumption
values for each warehouse is not a trivial, but necessary
task if we want to construct a more realistic model or
simulator. The SIMWARE program offers an easy-to-use
interface to build even complex supply chains, and propose
a novel component based structure. Using this program,
users can easily optimize the supply chain, for example
by an SQP method, but the proposed methodology gives
a chance to use more effective optimization algorithms.

Conclusions

A framework for modeling a multi-echelon supply chain
has been defined. The MM and WM modules of SAP
system provide data for the model for the whole supply
chain. In our project we simulated a central purchasing
scenario. In this supply chain the central warehouse
supplies the other companies in the group. Our
methodology can be used to calculate stock turnover
ratio and safety stock based on the agreed service level
of the warehouses. Our methodology can handle the
stochastic behavior of the replenishment lead time. We
used Monte-Carlo based simulation to evaluate the results
and also were able to evaluate the MRP parameters used
in the ERP system. Extended the model to analyze and
optimize the inventory levels at the other companies in
the group as well. The MATLAB simulation model can
be used to determine the optimal parameters for a
required service level. The SIMWARE model is available
at the following website: www.folyamatmernok.hu

ACKNOWLEDGEMENT

This work was supported by the TAMOP-4.2.1/B-
09/1/KONV-2010-0003 and TAMOP-4.2.2/B-10/1-2010-
0025 projects and the E.ON Business Services Kft.

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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice