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Scientific article – JMR Vrouenraets 1
Selecting the right inventory target strategy for minimizing
inventory levels for products with high variation in the forecast
error using Monte Carlo simulation
Jeroen Marc Roger Vrouenraetsa
03-04-2013
a Faculty of Technology, Policy and Management, Delft university of Technology, Jaffalaan 5, Delft, The
Netherlands
Article info
Keywords:
Inventory management,
Monte Carlo simulation,
supply chain management,
safety time, safety stock
Abstract
In supply chain management, analytical inventory models are used to
prescribe a target safety stock to achieve optimal inventory levels.
These models prescribe either a fixed safety stock or dynamic safety
time. In the existing literature little attention is given to the impacts of
either settings on the inventory performance based on the analytical
inventory models. In this paper is determined: What is the optimal
target safety setting strategy for products with a high variation in the
forecast errors using distribution resource planning, considering
inventory size and service levels? First, a literature review is discussed
on the little known effects on the use of either setting. Based on a case
study, a simulation model is developed to determine the optimal
target setting. The results of the simulation study quantify the
negative effects when using safety time for products which have a high
variation in forecast errors. Furthermore a valid inventory simulation
model for experimenting purposes is proposed.
1 Introduction
The fast moving consumer goods (FMCG) sector
is one in which having an optimal supply chain
(SC) is a great competitive benefit. The goal in
each SC as described by Goldratt & Cox (2004 ),
is to (1) maximize the rate at which the system
generates money through sales, while (2)
minimizing the money that the system has
invested in purchasing things that it intends to
sell and (3) minimize the money that is spend in
order to turn inventory into throughput.
In line with this goal, inventory management is
pursuing to achieve target service levels (1) by
balancing inventory (2), supply chain
capabilities and demand (3). Selected measures
of performance are the service level and the
average inventory size.
To achieve target service levels, literature
describes the use of analytical models. These
models describe reality with mathematical
equations. They attempt to capture the
stochastic nature of inventory drivers in order to
calculate a minimum inventory buffer which
covers for uncertainties in the SC (Silver &
Peterson, 1985).
There are two type of strategies to set the target
minimum inventory buffer: fixed safety stock or
dynamic safety time. A fixed safety stock reflects
a minimum number of products to keep in stock.
The safety time targets inventory replenishment
a certain period before the actual need is
expected(van Goor, Kruijtzer, & Esmeijer,
Goederenstroombesturing, voorraadbeheer en
materials handling, 1990).
Scientific article – JMR Vrouenraets 2
Although both strategies can be calculated using
the same analytical equations, it is unclear what
benefits and risks are related to which strategy.
In the literature little has been written on the
different effects of the different strategies on the
SC performance.
The objective of this paper therefore is to fill the
gaps in literature by adding knowledge on the
effects on performance measures in the SC
because of the use of different stock setting
strategies. The following question is central in
this paper: What is the optimal target safety
setting strategy for products with a high variation
in the forecast errors using distribution resource
planning, considering inventory size and service
levels?
This paper will give examples on the use of the
different stock setting strategies through a case
study conducted within an international FMCG
SC.
By means of combining a literature review on
the use of the setting strategies, the use of an
inventory Monte Carlo simulation model for
testing several scenarios and the experiences
during the case study, the effects are analyzed.
The contribution of this paper is the
quantification of the negative effects when using
safety time for products which have a high
variation in forecast errors. Furthermore a valid
inventory simulation model for experimenting
purposes is proposed.
The structure of this paper is as follows. Section
2 provides a literature review to define the
inventory management concepts necessary to
understand the importance of inventory safety
settings. Also, considerations on the use of the
two safety setting strategies will be given.
Furthermore the concept and limitations of
analytical inventory models are discussed.
Section 3 discusses the use of a Monte Carlo
simulation model for inventory management
after which section 4 describes the development
of such a simulation model based on a case study
and explores the use of different safety setting
strategies. Section 5 states the conclusions and
answers the research question. Section 6
provides a discussion on the performed
research.
2. Literature review
2.1 Inventory replenishments cycle
Schneider (1981) summarizes several
definitions for the service level of a supply chain.
A convenient measure to express service is the
so-called case fill rate (CFR), which is the
fraction of demand which can be directly
satisfied from the shelf.
A way to improve the CFR is to keep an optimal
amount of safety stock. This is a part of the
inventory that is held in excess of expected
demand due to variable demand rate and/or
lead time (Stevenson, 2005). Once a target safety
stock level is determined for the SC using
accurate data, the setting would not have to
change until high impact changes in the SC take
place (e.g. extra trade lane which significantly
shorten the lead time).
Once a target is set, one should focus on
maintaining the stock levels near the target
safety stock level by means of replenishing
optimal quantities of stock, called cycle stock.
Multiple methods are available for calculating
the right amount of cycle stock, which depends
on economical and technical constraints(van
Goor, Kruijtzer, & Esmeijer, 1990). E.g. the lot for
lot strategy replenishes the minimum quantity
which is necessary to align with the target safety
stock and technical constraints, but this strategy
does not consider economical constraints like
ordering cost(Silver, Pyke, & Peterson, 1998).
A system to control the time-phased
requirements of replenishment for distribution
centers is called distribution requirements
planning (DRP) (van Goor, van Amstel, & van
Amstel, 1989). By monitoring the inventory
levels, (forecasted) demand and replenishment
lead times, it suggest moments to replenish so to
keep inventory around the target safety stock.
Wrong safety settings fail to cover for demand
due to unexpected variation during the lead
time, resulting in lower than targeted CFR.
Hence, for the ongoing replenishment cycle, the
accurate planning of the initial safety setting is
crucial. Widely used are analytical inventory
models to calculate an optimal safety setting
with.
Scientific article – JMR Vrouenraets 3
2.2 Analytical inventory models
Analytical models describe a system using a set
of multiple equations. Analytical equations or
numerical algorithms are used to find one, point-
estimate solution for a problem (Haugh, 2004).
The analytical inventory model is used to
calculate the target safety settings with,
developed by Silver & Peterson (1985). Others,
such as de Kok et al (2012), have improved this
model by eliminating certain constraints of the
model.
Equation 1 shows the key elements in the model
to calculate the safety stock with.
Equation 1
SS = safety stock (product) K = risk factor (unit less) = Standard deviation of the estimated error during the replenishment lead time (product)
(1) The target safety stock. This should be used
as a target when the inventory is replenished so
that a certain target CFR is achieved.
(2) The standard deviation of the estimated
errors resembles the variation in demand and
supply chain elements during the replenishment
lead time. Little variations in these elements
make a supply chain relative predictable (in a
utopia, deterministic), thus a reason to keep less
safety stock.
The errors are driven by multiple elements in
the SC such as forecasting or production issues.
Silver et al (1998) propose an accurate method
to combine these elements.
(3) The k-factor, or the risk factor, determines
how many times the standard deviation should
be kept in stock in order to achieve a target CFR .
The value for k is driven by three parameters
(Equation 2) and a special function of the unit
normal distribution which eliminates the
cumulative density for k smaller than 0 (Rosen,
2013).
Equation 2
= Target service …………….level Q = Average order …………….quantity per …………….cycle = special
…………….function of …………….the standard …………….normal …………….distribution
A short reasoning; if a higher CFR target is
selected or the standard deviation of the
estimated errors increases, the function
returns a higher value for k and thus the safety
factor is relative high.
On the other hand, if the order quantity
increases, the safety factor decreases. This might
seem counterintuitive but with a higher order
quantity, the cycle time increases and thus the
number of replenishments in a period decreases
(ceteris paribus). Given the fact that the risk of a
stock out is higher near the end of a cycle (when
the stock is low), this risk now is reduced due to
the fewer replenishments and thus a smaller
safety factor can be used.
With a unit less risk factor and a standard
deviation expressed in number of products, the
safety stock setting resembles the number of
products that should be kept as safety inventory
to cover for the expected variation in demand
and lead time in order to achieve a target case fill
rate. This is called the safety stock and is
constant unless reviewed.
By dividing the safety stock by the average daily
demand, a safety time (expressed in days) is
calculated. Instead of using a fixed safety stock to
trigger replenishment, safety time sets the target
stock equal to the expected (forecasted) demand
over the safety time. This results in replenishing
inventory an amount of days before it is actual
needed, creating a safety time buffer(van Goor,
van Amstel, & van Amstel, 1989).
2.3 Differences between safety strategies
Literature on the use of safety time is limited, let
alone the differences in performance between
both safety strategies. While the conventional
analytical model simply suggest safety stock to
Scientific article – JMR Vrouenraets 4
be used as optimal safety setting, the first
question is why even bother to use safety time?
General reasoning on safety time returns that it
fluctuates with the expected demand. Thus, in
periods of low demand, inventory benefits from
safety time because it reduces total inventory,
whereas, in times of high demand, it provides
extra security.
Rosen (2013) favors this reasoning, suggesting
that safety time is useful near the end of a
product life cycle is preferable. While evidently
the demand will decrease, the safety time will
make the total inventory decrease with the same
pace, resulting with little non-performing
inventory at the end of the life cycle.
Whybark and Williams (1976) pose that
uncertainty in timing of demands should be
dealt with using safety time, whereas
uncertainty in quantity should be dealt with
using safety stock.
Chang (1985) argues that safety stock and safety
time are interchangeable. However, his modeling
of production and demand in essence is
deterministic which makes is less valuable to
apply in practice. Yano (1987) focused on
finding the optimal planning lead time but also
only considered deterministic demand.
Buzacott and Shanthikumar (1994) with the use
of stochastic modeling that safety time is
preferred over safety stock given the condition
that forecast are accurate. Moreover, with
changing customer orders during the lead time
or bad forecasts, inventory performance would
result from fixed safety stock.
Although is general safety stock seems a robust
choice to optimize supply chain performance
with, no studies have been found to quantify the
impact.
3. Monte Carlo simulation
Simulation models mimics the operating
behavior of a system (Verbraeck & Valentin,
2006). To understand the most important
behavior and to create a simulation model often
is more time consuming than the use of an
analytical model.
However, the effects of choosing between safety
stock and safety time are not clear. As stated by
Aguilar et al (1999), for such cases simulation is
an effective tool to communicate results and
performance dynamics.
Monte Carlo (MC) simulation is based on two
mathematical theorems which make it a very
useful simulation type to analyze inventory
behavior with. (1)the law of large numbers and
(2) the central limit theorem. Not only can MC
simulation, considering the first theorem, show
an estimate of the expected result, it also returns
an estimate of the uncertainty in this estimate
(Dunn & Shultis, 2011). These characteristics
make MC useful to account for risk in
quantitative analysis (Palisade, 2013).
Recent studies have been using Monte Carlo
simulation for inventory management problems
(Cáceres-Cruz, Grasman, Bektas, & Faulin,
2012)(Jaio & Du, 2010). However little effort has
been put specifically in using MC simulation to
explore the uncertainty involved with the
analytical model and the use of different safety
setting strategies.
Key differences between the use of an analytical
model and the MC simulation are shown in Table
1.
Analytical model
MC simulation
Input para-meters
Static parameters to describe stochastic behavior with
Used to draw samples from
System characteristics
Analytical equations
Defines operating behavior and functional relationships
Results Point estimates Range estimates
Table 1: Difference between analytical models and MC simulation models
Scientific article – JMR Vrouenraets 5
4. Case study
In order to develop a Monte Carlo simulation
model to simulate inventory management
processes with, key system characteristics have
to be selected. In order to do so a system
analysis is performed at an FMCG supply chain.
For the design approach of the simulation model
three viewpoints have been combined. First, the
simulation method of Banks (1999) for the
general modeling methods. Secondly, the META
model (Herder & Stikkelman, 2004) is used. The
model starts with defining the model
requirements and the possible solution space to
answer to these requirements with. Last, the
spiral model (Boehm, 1988) is used for the
approach. The spiral model consists of several
iterative stages in which the model is designed,
tested and adapted resulting in a robust model
design.
To obtain requirements for the model, the TIP-
framework as proposed by Koppenjan and
Groenenwegen (2005) is used to analyze
inventory management from a technology,
institutional and process point of view.
The selected performance measures for the
strategies are (1) the case fill rate (%), (2) and
the average inventory (Statistical units and
days).
The conceptual model consists of four parts: (1)
the sampling demand and a adhering forecast of
demand, (2) the daily inventory position, (3) the
DRP planning of order replenishments and (4)
the production and actual replenishments of
products. These four parts together are able to
simulate the behavior of inventory over time. Of
key interest is the use of safety settings as a
trigger for the DRP planning to replenish.
For the specification of the model, all data could
be retrieved from data sources located within
the company.
However, for the sampling of stochastic values
for the demand, forecast and lead times, accurate
probability distributions need to be selected.
Statistical test on sample data showed that a
gamma distribution fits the distribution of
demand (Vrouenraets, 2013). Silver et al (1998)
show that for both the variation in forecast
errors and lead time errors a normal
distribution can be used. The simulation model
has the property to set the average variation for
the distribution. This allows for experimenting
with accurate and useless forecast and likewise,
high lead time variations.
Multiple experts with different expertise within
the SC, were interviewed for the validation of the
simulation model (Sarikaya, 2013)(van der Oost,
2013)(Alves, 2013). Each expert is asked to
check the dynamics by looking at the specific
parts of the model and the output graphs. All
acknowledged the validity of the model. Alves
recognized the fact that no inventory is
simulated at the plant but agreed to leave it out
of the simulation while it concerns a single stage
simulation model. Moreover, using a binary
search, the average optimal safety stock setting
(after a 1000 iterations) for a 99% case fill rate
according to the simulation model was
compared with the suggested safety stock
setting of the analytical model. The range did not
show any significant deviation from the
analytical model which echoes the validity of the
simulation model.
Two experimental designs are conducted. For
both experiments, 4 product classes (table 3)
were defined using the criteria of daily volume
(threshold; 100 SU/day) and COVFE (threshold;
100%). Products from the company were
selected. All supply chain parameters are
selected in such a way that it would reflect
reality.
For one experiment, the analytical optimal safety
setting was used expressed as safety stock. For
the other experiment, safety time was used.
After using the method for calculating the
number of iterations as described by Verbraeck
and Valentin(2006), both experiments were
executed a 1000 times.
The results were analyzed both on the level of 1
iteration and on the MC simulation as a whole.
Now, the results for the product class with high
daily volume and high COVFE are discussed.
The analysis of 1 iteration using safety time
(Graph 1) show four dynamics:
Scientific article – JMR Vrouenraets 6
(1) The first peak in the target safety stock
(4) is exactly matched mainly because of
the good forecasting. The inventory goes
up to match the demand and after the
peak in demand, it lowers again.
(2) The creation of yellow NPI (1) increases
rapidly when only safety time is used.
E.g. due to the last extreme over
forecast, a lot of yellow NPI is created.
The duration of the yellow NPI wave is
larger because it is replenished already
a few days earlier compared with the
previous scenario. The peak of yellow
NPI is higher because the target safety
setting goes up as it ‘looks’ further in the
future.
(3) One of the assumed benefits of safety
time is that it should lower inventory in
case no demand is expected. On the one
hand, this is true. With low forecast of
demand, the target safety setting gets
near 0 which saves inventory costs (3a).
However, at this point the difference
between forecast and actual demand is
rather low. Point (3b) gives an example
where no demand is forecasted which
takes the inventory to 0, but
unfortunately gets surprised by a peak
in demand (1, in graph 3). Now, no stock
can satisfy the unexpected demand
during the lead time: and this is exactly
the type of demand safety stock should be
used for. Demand hurts the most, when
it is least expected. And that is why
there should be safety stock, which now
is not the case.
(4) An example of where the safety time
does prove its benefits is the moment
when the forecast of demand is accurate
(2 in graph 3). The first weeks (4) the
inventory goes up and down with the
accurate safety setting which provides
safety.
The analysis of 1 iteration using safety stock
(Graph 2), using the same initial conditions
(Graph 3) as with the previous iteration, show
other dynamics. Most important findings from
this graph on the dynamics are:
(1) Compared with both of the previous
graph of the inventory results, it
immediately becomes clear the
inventory with the use of safety stock is
more stable: Peaks are less extreme and
only need a shorter time to restore
around the inventory target.
(2) With the use of fixed safety stock, also
there is creation of some yellow NPI.
The largest peak happens near (2),
driven by the over forecast of demand.
Given the fact that in all three scenarios,
an over forecast of demand leads to
yellow NPI, means that yellow NPI can`t
be prevented with high COVFE.
Although the inventory position also reaches
zero in the scenario with the fixed safety stock,
the final score on average total inventory and
case fill rate, show better numbers compared
with safety time.
Both iteration already explain some of the
dynamic behavior, however results of the MC
simulation tell even more on the risk profile of
both safety settings.
Graph 4 and Graph 5 show the ranges of the
performance measures after a 1000 iterations. It
clearly shows how both the average inventort
and CFR score are better when safety stock is
used. Moreover, the ranges wherein these value
lie are more accurate with the use of safety
stock, in other words, the extremes are less
likely to be expected.
5. Discussion of results
Table 1 compares the ranking of the average CFR
score for each product class for the safety
settings scenarios. It clearly shows that in case of
s.k.u. with a high COVFE (Class I and III), a fixed
safety stock setting results in the best CFR score.
This also counts for class II s.k.u. Only, for the
class IV s.k.u. the current situation slightly
outperforms the fixed stock strategy.
Not only does the use of fixed safety stock out-
perform other strategies on the average CFR
score, it also makes the scoring range more
narrow (not shown in table below). This means
Scientific article – JMR Vrouenraets 7
that less (extreme) outliers are expected when
fixed safety stock is used.
Table 2 compares the ranking of the average
inventory levels for each class for the safety
setting scenarios. Also for the inventory, all s.k.u.
classes benefit most from a fixed stock setting.
Not only does the use of fixed safety stock out-
perform other strategies on the average
inventory level, it also makes the inventory
range more narrow (not shown in table below).
This means that less (extreme) outliers are
expected when fixed safety stock is used.
6. Conclusion and further research
The use of safety time has positive effects when
the forecast is accurate. Inventory benefits from
accurate low forecasts, which result a decrease
of the total inventory whereas high CFR score
are obtained in case of accurate forecasted peaks
in demand.
However, the negative effects of safety time
perhaps are bigger than currently known at P&G.
On the one hand, when over forecast are made,
the dynamics of safety time enlarges the creation
of excessive inventory. On the other hand, when
the forecast of demand is too low, there is not
enough safety stock to cover for the forecast
error. This leads to the reasoning why inventory
is now at its most vulnerable: Demand hurts the
most, when it is least expected. And that is
exactly when (and why) there should be safety
stock. This is not the case with the use of safety
time.
Safety stock is a robust approach for covering for
errors during the replenishment lead time.
Extreme low values for CFR scores or high
values for total inventory, are not common like
with the use of safety time. Moreover, the
average scores for both CFR and inventory are
better than when safety time is used according
to the simulation study.
Before this paper, little quantitative research
had been performed aiming at quantifying the
differences between the use of two commonly
used safety settings: safety stock and safety time.
Moreover, little effort had been put specifically
in using MC simulation to explore the
uncertainty involved with the analytical model
and the use of different safety setting strategies.
By developing and validating a simulation model
for inventory management, the possibility was
created to test the effects of both settings and
quantify the differences on performance
measures such as inventory and case fill rate.
The conclusion is clear. For DRP system the use
of safety stock for products with a high variance
of the forecast error, safety stock is preferred
over safety time.
The development of a valid simulation model for
inventory management can be used for many
purposes. The focus of this research was to point
out the difference between two types of safety
setting for products with a high variation in
forecast errors. However, by changing initial
parameters in the model, different values for
demand, the forecast errors or lead time errors
can be simulated. Moreover, single iteration
results can be used for educating purpose to
make clear the effects of both using wrong safety
settings values and the use of wrong safety
strategies.
7. Reflection
The design of the simulation model has been
performed with care. Moreover, the final
simulation model has been validated both by the
use of experts validation and a quantitative
validation using the results of the analytical
model. However, still some distinct choices have
been made that can influence the results.
First, the choice for the use of a lot for lot
replenishment strategy was driven by the case
study and the objective to minimize inventory.
However, many supply chains are driven by cost
of the inventory as well, therefore likely to use
other replenishment strategies. Adaption of the
model then is needed to experiment for
analyzing those effects.
Second, the use selection of sued probability
distribution can affect the dynamics in the
results. Especially the gamma distribution has
the feature of simulating a long tail, which might
not be accurate for certain products. This would
make the current simulation model results more
Scientific article – JMR Vrouenraets 8
extreme than would be the case for the product
types.
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Graph 1 Results of iteration with use of safety time
Graph 2 Results of iteration with use of safety stock
Graph 3 Input for demand and forecast of demand for both iterations shown above
(1)
(1)
(3b)
(2)
(3a) (4)
(1) (2)
(1) (2)
Scientific article – JMR Vrouenraets 10
Graph 4: Results average inventory score of Monte Carlo simulation using safety time and safety stock
Graph 5: Results of CFR score of Monte Carlo simulation using safety time and safety stock
Average CFR ranking Class I Class II Class III Class IV
Scenario 1 (safety time) 2 (80,6%) 2 (87,1%) 2 (92,5%) 2 (98,3%) Scenario 2 (fixed stock) 1 (95,8%) 1 (97,8%) 1 (99,0%) 1 (99,9%) Table 1: Ranking of scenarios on the average CFR score per s.k.u. class
Average inventory ranking Class I Class II Class III Class IV
Scenario 1 (safety time) 2 (6928 SU) 2 (1304 SU) 2 (1768 SU) 2 (44 SU) Scenario 2 (fixed stock) 1 (3640 SU) 1 (985 SU) 1 (705 SU) 1 (41 SU) Table 2: Ranking of scenarios on the average inventory level per s.k.u. class
High COVFE (group A) Low COVFE (group B)
High average demand (Group 1) Class I Class II Low average demand (Group 2) Class III Class IV Table 3: Recap of s.k.u. classification
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