Revista EIA, ISSN 1794-1237 Número 16, p. 43-60. Diciembre 2011
Escuela de Ingeniería de Antioquia, Medellín (Colombia)
Artículo recibido 2-V-2011. Aprobado 21-IX-2011
Discusión abierta hasta junio de 2012
FINANCIAL RISK ASSESSMENT OF DIFFERENT
INVENTORY POLICIES
Héctor Hernán toro*
Leonardo rivera**
diego Fernando Manotas**
ABSTRACT
This work addresses the valuation of economic effects of different inventory holding policies. We study the
VaR over the value of a company and the variations induced on this indicator by the changes made to the working
capital, related to inventory policies. Three typical different inventory systems are studied and comparisons are
drawn between different policies. Policies derived from net present value (NPV) maximization are contrasted against
cost minimization, as well as against arbitrary inventory policies derived from market conditions. Every inventory
system under study is evaluated using two performance indicators: NPV and VaR over NPV. The inventory policies
are derived in a deterministic scenario, but are tested under the risk conditions that the inventory systems have to
face. This is done by using Monte Carlo simulation. In all three of the inventory systems under study, the difference
between price and variable cost is what causes the greatest variation on the NPV indicator. An important result of
this work is that for the cases studied, which are rather common in the real world, the optimal inventory policies
obtained by using the cost minimization approach are equally good from a risk minimization perspective than
those obtained by using the profit maximization approach.
KEY WORDS: investment analysis; inventory management; Monte Carlo simulation; Value at Risk.
* Ingeniero Industrial y Magíster en Ingeniería Industrial, Universidad del Valle. Profesor Auxiliar, Escuela de Ingeniería
Industrial y Estadística, Universidad del Valle, Cali, Colombia. htorodi@clemson.edu
** Ingeniero Industrial, Universidad del Valle; Ph.D. in Industrial and Systems Engineering, Virginia Polytechnic Institute
and State University. Jefe de Departamento de Ingeniería Industrial, Universidad Icesi. Cali, Colombia. leonardo@
icesi.edu.co
*** Ingeniero Industrial, Universidad del Valle; Magíster en Gestión Financiera, Universidad de Chile. Profesor Asociado,
Escuela de Ingeniería Industrial y Estadística, Universidad del Valle. Cali, Colombia. diego.manotas@correounivalle.
edu.co
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Financial risk assessment oF diFFerent inventory policies
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EVALUACIÓN DE RIESGO FINANCIERO PARA DIFERENTES POLÍTICAS
DE INVENTARIO
RESUMEN
Este trabajo se concentra en la evaluación de los efectos económicos asociados a diferentes políticas de
inventario. Se estudia el valor en riesgo del valor de una compañía, además de las variaciones inducidas por
los cambios en el capital de trabajo asociados a diferentes políticas de inventario. Se estudian tres sistemas de
inventario y se comparan diferentes políticas. Se contrastan las políticas derivadas de la maximización del valor
presente neto (VPN) contra la minimización de costos y contra políticas de inventario arbitrarias derivadas de las
condiciones de mercado. Para cada sistema de inventario se estiman dos indicadores de desempeño: VPN y VaR
del VPN. Las políticas de inventario se construyeron en un escenario de certeza, pero se evalúan incorporando
diferentes condiciones de riesgo, para lo cual se utiliza simulación de Monte Carlo. Las conclusiones se obtienen
a partir del desempeño observado en diferentes casos de estudio, usando ambos indicadores. En los tres sistemas
de inventario bajo estudio, la diferencia entre precio y costo variable es lo que causa la mayor variación en el
indicador del VPN. Un resultado importante de este trabajo es que para los casos estudiados, más bien comunes
en el mundo real, las políticas de inventario óptimo obtenidas por minimización de costos son, desde la perspectiva
de la minimización de riesgo, tan buenas como las obtenidas mediante maximización del beneficio.
PALABRAS CLAVE: análisis de inversiones; administración de inventarios; simulación de Monte Carlo;
valor en riesgo.
AVALIAÇÃO DE RISCO FINANCEIRO PARA DIFERENTES POLÍTICAS
DE ESTOQUE
RESUMO
Este trabalho concentra-se na avaliação dos efeitos econômicos associados a diferentes políticas de esto-
que. Estuda-se o valor em risco do valor de uma companhia, além das variações induzidas pelas mudanças no
capital de trabalho associadas a diferentes políticas de estoque. Estudam-se três sistemas de estoque e compa-
ram-se diferentes políticas. Contrastam-se as políticas derivadas da maximização do valor presente neto (VPN)
contra a minimização de custos e contra políticas de estoque arbitrárias derivadas das condições de mercado.
Para cada sistema de estoqueestimam-se dois indicadores de desempenho: VPN e VaR do VPN. As políticas de
estoque construíram-se em um cenário de certeza, mas avaliam-se incorporando diferentes condições de risco,
para o qual se utiliza simulação de Monte Carlo. As conclusões obtêm-se a partir do desempenho observado em
diferentes casos de estudo, usando ambos indicadores. Nos três sistemas de estoque baixo estudo, a diferença
entre preço e custo variável é o que causa a maior variação no indicador do VPN. Um resultado importante
deste trabalho é que para os casos estudados, mais bem comuns no mundo real, as políticas de estoque ótimo
obtidas por minimização de custos são, desde a perspectiva da minimização de risco, tão boas como as obtidas
através da maximização do benefício.
PALAVRAS-CHAVE: análise de investimentos; administração de inventários; simulação de Monte Carlo;
valor em risco.
45Escuela de Ingeniería de Antioquia
1. INTRODUCTION
The usual approach for the economic valu-
ation of projects begins with the creation of a cash
flow, in which the analyst makes some estimation
of future incomes and expenses. This makes it pos-
sible to account for net cash flows. Many of the cash
flow parameters are random variables for most real
projects, which turns the valuation into a complex
problem. Simulation is usually applied to handle
this situation, but the analysis of simulation outputs
re quires statistical tools and statistical indicators
designed for comparisons between different alterna-
tives and scenarios.
The Value at Risk (VaR) indicator is widely
used today in financial institutions, as well as in the
economic valuation of portfolios within the financial
sector. VaR is a single number that measures the maxi-
mum potential loss in the present value of a portfolio,
as long as the market conditions behave in the usual
way (i.e., the market exhibits conditions similar to its
historical behavior) (Linsmeier and Pearson, 1996).
The idea of having a “portfolio” comes from the
financial sector, but in a general way, it is possible
to see a real project as a portfolio, in the sense that
you need to make an investment to have the proj-
ect in operation, and after the investment has been
made, some economic returns can be expected in
the future. For example, buying some treasury bills
now and selling them after a while at a different price,
hopefully higher.
The effort required for valuing a real project is
usually higher than the required for valuing a finan-
cial portfolio, especially if the portfolio is composed of
standardized financial instruments. For a real project,
the analyst should understand the intrinsic nature
of the business, identifying the main elements that
affect the economic value of the project, model their
behavior, and forecast them. Even if just a couple of
alternatives that sell the same final product in the
same market are being considered, modeling each
of them is a different exercise. Each alternative differs
in their internal processes and decisions.
2. PROBLEM PRESENTATION
Real projects, as well as financial instruments
and portfolios, are affected by the uncertainty and
risk observed in various factors, including markets,
political environment, natural environment, just to
mention a few. By uncertainty we refer to events
that cannot be represented by probability distribu-
tion functions, and by risk we refer to those that can.
Traditional methods for valuing a project use a static
representation of critical future parameters, especially
those that are outside the control of the project owner.
This lack of control may considerably influence the
main performance indicators. A single static forecast
of critical parameters is not recommended for the
valuation of real projects, since projects will face risk.
The final values for parameters can be very different
from the ones forecasted in a single scenario.
One approach to deal with risk has been the
formulation of deterministic scenarios instead of the
use of a single forecast. Scenario generation has at
least two hurdles in its implementation: the identi-
fication of a set of scenarios that truly represent the
future possibilities of the real world system and the
assignment of a probability of occurrence to each
scenario, since not all are equally likely to happen.
Since the possibilities of behavior in a real system are
basically endless, it is not an easy task to determine
a subset of scenarios that represent future behavior.
The assignment of probabilities for each scenario
is also sometimes a subjective process, therefore
exposed to human misperceptions.
Simulation is a practical approach to the
analysis of the real system, assuming that it is possible
to set up a formal representation of the system
(mathematically and computationally). Simulation
can be used to calculate performance indicators
given a set of parameters constructed from the real
world. The basic idea is to generate a large quantity
of scenarios (each scenario is determined by the
set of values of the parameters), to calculate the
performance indicators for each resulting scenario.
At the end, the analysis is focused on the different
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observed values for those indicators. Analysis is
usually done by identifying probability distributions
functions associated to the performance indicators.
Analysis of real projects under risky conditions
is a topic that has been studied by academics and
practitioners for some time. This paper builds upon
ideas presented in previous documents related to
the development of a performance indicator that
deals with risk quantification, while at the same time
measures how good a project is from a financial
point of view (Ye and Tiong, 2000). Specifically,
the problem is to quantify the financial risk of a
manufacturing system that applies a given inventory
holding policy. The expected result is an indicator of
VaR type that will be used for selecting an inventory
policy that maximizes the present value of a project,
while exposing the company to an acceptable level
of risk. As in previous works (see Section 3), the
economic analysis is performed in a continuous
time framework. The cash flow is constructed in this
continuous framework, and a discount rate is used
to estimate the net present value of the project over
an infinite time horizon. A simulation tool is used
to generate random scenarios and calculate a new
estimation of the NPV for each of them. Finally, a VaR
indicator over the NPV accounts for the implicit risk
for the system, related with the inventory policy and
the parameters of the system.
3. PRELIMINARY RESEARCH
Hadley (1964) developed a comparison
between optimal order sizes for an inventory system
with fixed order quantity, when those orders are
derived using two approaches: the first one is the
usual minimization of average long run cost and the
second one is the minimization of a cost function
that accounts for the time value of money, using a
discount rate. Hadley proposed this methodology
some forty years ago, and his conclusion was that
for the inventory systems that he studied there was
no difference in the order size when using any of
those approaches. However, works presented after
this seminal paper have shown that differences do
occur, unless some assumptions can be considered.
These assumptions include parameters that do not
change their value for long periods and very low
discount rates (usually less than 1 %). None of these
two conditions are observed in the industrial world
today, so the conclusions suggested by Hadley are
no longer applicable today.
Kim, Philippatos and Chung (1986), Roumi
and Schnabel (1990), and Chung and Lin (1995)
proposed a valuation of inventory decisions under
a financial framework treating the inventory like any
other asset of a company. The shareholders usually
expect the investment in assets to be productive,
generating an adequate financial return. The usual
explanation for not valuing the inventory from a
financial framework is that inventory decisions are
usually beyond the reach of the financial manager,
and production managers do not use a financial
approach for making decisions. They usually use
a costing approach. These works conclude that
there are differences between inventory policies,
specifically the order sizes, when those order sizes
are derived using the NPV maximization instead of
cost minimization.
Hill and Pakkala (2005) proposed a discounted
cash flow approach for finding a good inventory
policy in a system that uses periodic revision (every
R units of time). The amount ordered each time is
variable, calculated as the difference between a
maximum inventory level previously determined
(denoted as S) and the inventory available by the
time the revision is made (this inventory policy
is often referred as R,S). The authors focus their
attention on the cash flow derived from applying the
inventory policy, instead of just calculating the cost
related to it. Demand is modeled by using a Poisson
distribution. They considered a deterministic lead
time and allowed backorders for unsatisfied demand.
The optimal inventory policy is found using NPV (net
present value) as the performance indicator.
47Escuela de Ingeniería de Antioquia
Disney and Grubbström (2004) studied a
manufacturing system also with an R,S inventory
policy, in which the demand is modeled using a
stochastic autoregressive process. The production/
distribution lead time is set to one period, and the
demand is forecasted using exponential smoothing.
They approached the problem of determining the
cost performance of an inventory holding policy
including new cost elements, in particular, the cost
of requiring additional production capacity, as well as
the cost of having extra production capacity. Those
two cost elements are not considered as necessarily
symmetric. Every cost element is modeled using the
lineal form of the functions. Their main conclusion
is that optimal policies coming from the usual cost
approach are not necessarily optimal in the presence
of new cost elements. Also, they have found that if
one considers the cost of the bullwhip effect as a
part of the cost of applying some inventory policy,
then the optimal values derived from the usual cost
minimization approach are also invalid.
Bulinskaya (2003) introduces the analysis
of inventory management in the framework of
corporate risk management. Her work starts
criticizing the usual cost approach, specifically the
static nature of the framework employed to calculate
cost. She emphasizes that real systems are dynamic
by nature. Also, previous models do not include
capacity constraints, therefore assuming that it will be
possible to acquire any amount of product suggested
by an inventory policy, while in the practice there are
budget and availability constraints. The point of view
used by Bulinskaya is that the inventory is just another
investment option to the decision maker, and thus,
the optimum inventory policy can be determined
by solving an investment selection problem that
can include other possibilities (assets) besides the
inventory. The author uses the approximation of
Cox-Ross-Rubinstein (1979) for modeling the financial
market, and proposes some optimal strategies
derived from that formulation.
Naim, Wikner and Grubbström (2007) studied
the problem of selecting an order policy in a planning
and control production system. The study is carried
over using a simulation model widely used, referred
as IOBPCS (inventory and order-based production
control system). NPV is again the performance in-
dicator to be optimized, but some additional cost
elements are included. The concept of variance
cost is introduced, defined as the cost in which a
system incurs when it is not in perfect control with
respect to some previously defined goals. Two kinds
of variance costs are considered: The cost of not
being able to reach a production goal and the cost
of holding more or less inventory than the expected
amount. Authors used a formal representation of
the IOBPCS model using differential equations and
then solved it using a Laplace transformation. The
performance indicator including variance cost is used
in the valuation of make-to-order and make-to-stock
systems, concluding that the specific cost structure
of every manufacturing system is a critical factor in
the selection of an optimal inventory holding policy.
Luciano, Peccati and Cifarelli (2003) explored
the possibility of using VaR in the context of inventory
management. They used both cost minimization and
profit maximization. Ordering and holding costs are
included in valuation of inventory policies. Since
probabilistic parameters are allowed, the authors
study the resulting probability distributions of
performance indicators. Upper and lower bounds are
calculated for VaR as a way to estimate its confidence
level. Numerical procedures are recommended,
due to the complexity related to modeling real
manufacturing systems. The main conclusion is that
there is a good chance for obtaining large errors
if normality is assumed a priori for optimization
criteria. Numerical exploration of solutions, including
simulation, is highly advised.
Ahmed, Çakmak and Shapiro (2007) analyzed
an extension of the classical multi-period, single-item,
linear cost inventory problem where the objective
function is to minimize a coherent risk measure.
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Tapiero (2005) introduced the use of VaR
(Value at Risk) in the optimal selection of inventory
policies. The main idea is to introduce an ex-post
analysis, instead of the usual ex-ante approach. The
ex-post analysis is possible by measuring the deviations
observed from a performance indicator related to its
expected values. With respect to inventory, Tapiero
argues that a manager would value differently the
situation of having more inventory than the expected,
than the undesirable outcome of losing demand for
not having enough stock. Tapiero introduces the
concept of a right decision cost, which is supposed to
be the cost of taking a decision from which no future
deviations occur. The VaR is then suggested for this
ex-post valuation, but some difficulties appear due
to the complexity of mathematical expressions, some
of them without analytical solutions. The VaR is also
identified as a good indicator because the parameters
required for its calculations (probability and a
confidence level) can be related with the technical
characteristics of inventory systems. Borgonovo and
Peccati (2009) also proposed a research about the
quantitative implications of the risk measure choice
on the optimal inventory policies. Those authors
developed a method to select the most adequate
risk indicator to use in the evaluation of different
inventory problems.
Singh et al. (2010) developed a model based
on the theory of Capital Asset Pricing Model (CAPM)
with the purpose to determine the lot size and
reorder points from minimizing present value of total
cost subject to risk adjusted discount rate. Some other
research that use financial indicators as performance
measurement or optimization objectives has been
done for the evaluation of projects within the
manufacturing sector. For further reading please
refer to Followill and Dave (1998), Grubbström
(1999), Luciano and Peccati (1999), Van der Laan
and Teunter (2001), Van der Laan (2003) and Giri
and Dohi (2004).
4. PROBLEM MODELING
Following the work of Kim, Philippatos and
Chung (1986), three inventory systems are studied:
batch sales with finite uniform production rate; uni-
form demand with finite uniform production rate,
and uniform demand with infinite replenishment rate
and stockouts. The following notation will be used:
NPV = Net present value of the cash flows for the
first operative cycle ($)
NPV(∞) = NPV over an infinite planning horizon ($)
C = Variable production/procurement cost ($/unit)
D = Demand rate (units/day)
E = Sales expenses per batch ($/batch)
I = Maximum inventory level (units)
P = Selling price ($/unit)
Q = Order quantity or batch sales volume (units)
S = Fixed ordering/setup cost ($/order)
T = Cycle time (days)
U = Production rate (units/day)
f = Out of pocket shortage cost ($/unit/day)
h = Out of pocket inventory carrying cost ($/unit/
day)
k = Discount rate (cost of money) per period
(% daily)
Case I: Batch sales, finite uniform production
rate inventory system
A firm produces at a uniform rate and stores
the products until they are shipped (sold) in batch
at the end of each inventory cycle. It is assumed that
there will be continuous cash outflows of production
cost and an out-of-pocket inventory carrying cost that
is proportional to the inventory level during each cy-
cle. It is also assumed that there will be cash outflows
of sales expenses such as shipping and packaging
when the products are sold at the end of the cycle.
Income from sales is received in full and in cash at
49Escuela de Ingeniería de Antioquia
the moment of the sale. The NPV for cash flows in
this situation for one operation cycle is:
(1)
For the infinite planning horizon, the NPV
will be:
(2)
The previous expression can be simplified as:
(3)
Making (to search for the
maximum NPV value) results in:
, and Q* = DT*
(4)
The optimal batch sales volume will be
(derived from NPV maximization):
(5)
While by minimizing the cost, the optimal
batch sales will be:
(6)
Case II: Uniform demand, finite uniform pro-
duction rate inventory system
In this case, production and demand rate are
both finite. The factory produces in lots at a uniform
rate and delivers the products to a warehouse
where demand happens at another uniform rate.
There will be a cash outflow of setup cost at the
beginning of each cycle. From time zero to τ, there
will be continuous cash outflows of production cost.
Throughout the cycle, there will be continuous cash
outflows of inventory carrying cost and inflows of
product sales. The NPV for cash flows in this situation
for one operation cycle is:
(7)
(9)
(8)
For the infinite planning horizon, the NPV will be:
The optimality condition results in:
and Q* = DT*
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(10)
(11)
While by minimizing the cost, the optimal
batch sales will be:
Case III: Uniform demand, infinite replenish-
ment rate, stockouts inventory system
In this case, a reseller purchases products in
lots for resale at the known demand rate. All demands
must be satisfied, but it is allowed to be out of stock
when a particular demand occurs. Demands that
occur when there is a stockout are backlogged until
the next procurement arrives. Backorders are met
before the procurement can be used to meet any
other demands. It is assumed that procurement cost
for backorders will occur at the end of the cycle and
outflows of ordering cost and procurement will occur
at the beginning. From time zero to τ there will be
continuous cash outflows of inventory carrying cost
and inflows of product sales. All sells are assumed to
be in cash. There will be cash outflows of backorder
cost during time τ to T and a cash inflow of product
sales for backordered items at the end of the cycle.
The NPV for cash flows in this situation for one oper-
ation cycle is:
For the infinite planning horizon, the NPV will be:
The optimality conditions and results in:
(14)
(13)
(15)
While by minimizing the cost, the optimal batch sales will be:
(16) (17)
(12)
51Escuela de Ingeniería de Antioquia
Where f ’ is defined as f plus other costs asso-
ciated with the delayed revenue, excluding any loss
of goodwill.
5. BUILDING THE VaR
PERFORMANCE MEASUREMENT
The main idea is to evaluate, from a financial
framework, the impact of different inventory policies
over a company’s value. To do so a general equation
for a company’s value is presented:
(18)
CV stands for company’s value, estimated from
the free cash flow FCF and assuming an infinite num-
ber of flows that are discounted at a rate i. The factor
g is used as a decreasing or increasing rate over the
FCF along the time horizon. The FCF expression is
constructed adding the income I less costs of sales C
and the expenses of sales management CMS. The fac-
tor 1-T is related to government taxes. Depreciation
and amortization are considered in the FCF taking
into account that they both are non-monetary flows,
and finally there are the investments in capital assets
CAPEX as well as investment in working capital WK.
This last term is where the inventory investments
are located.
As it is clear from the CV equation, the invest-
ment related with the inventory policy has an effect
over the performance measurement. At first sight,
reducing the investment in WK will increase the value
of CV. However, this is not always the case, especially
when considering investments in inventory. The
main objective of holding inventory is to anticipate
demand variations, as well as changes in production
conditions and lead time variability, among others.
Reducing the investment in inventory can cause
some demands to be not satisfied on time and hence
the value of CV will decrease. Increasing inventory
investment is not always a solution, because the
holding cost of the inventory can be higher than the
benefit of satisfying variable and uncertain demands.
In this regard, finding an optimal set of operation
parameters related with the inventory policy is a
relevant problem for company’s valuation.
In this work, the set of equations mentioned
in section 4 for several inventory systems is used to
build a VaR type performance indicator. For every
inventory system there is an equation that measures
its resulting NPV. Using a set of parameters taken
from Kim, Philippatos and Chung (1986), the op-
timal order quantities are obtained under the two
optimization perspectives: NPV maximization and
cost minimization. After those optimal quantities are
calculated from a deterministic point of view, their
performance is evaluated under risk assumptions. In
particular, for a subset of parameters (price, cost of
sales, variable production cost, demand and setup/
ordering cost) some probability distributions are
assumed. If there were data available, the logical
step would be to perform a statistical fit to a specific
distribution, however, since this is not the case, a
priori distributions will be used for these calculations.
The benefits of using these types of distributions are
documented in Johnson (1997) and Williams (1992).
By using Monte Carlo simulation multiple sce-
narios are generated and the NPV value is calculated
for each one while the inventory policy is kept fixed
using the values obtained from the deterministic
scenario. Finally, the multiple results obtained for
the NPV from every run of the simulation process are
used to create an empirical probability distribution
of the NPV. The computational software Crystal Ball
automatically performs this process.
5.1 Understanding the basic idea of
the VaR indicator
According to Jorion (1996), the VaR indicator
makes an effort to consolidate the total risk exposures
faced by a financial portfolio into a single number.
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The VaR calculates the maximum expected loss, or
equivalently, the worst scenario for the portfolio’s
value, given a certain probability and a period of
time. Generally speaking, if the planning horizon is
N periods and α % is the confidence level, then the
VaR indicator can be obtained as the loss for the
(100-α) percentile in the left tail, given the probability
distribution for the changes in portfolio valuation for
the next N periods. VaR can be seen as the answer
for a single question: How bad can things go? It is
measured in currency units so it is easy to understand.
Additional details on several methods to calculate
the VaR can be reviewed in Pritsker (1996). Figure
1 illustrates the VaR concept in a graphical way. The
shadowed area below the curve denotes the proba-
bility α %, while the NPVα value corresponds to the
VaR, worte scenario expected over the NPV indicator
given the confidence level.
Figure 1. VaR over NPC
6. CALCULATING AND
INTERPRETING THE VaR
INDICATOR
For every one of the three inventory systems
described and modeled, the first step is to calculate
the optimal order size under the assumption of some
values for the parameters required. In the tables 1,
2 and 3 the following set of values are used (Kim,
Philippatos and Chung, 1986): U = 1, k = 0.0005
(20 % a year), h = 0.0005 and D = 1. The price is
expressed as a function of variable cost.
In table 1, for every combination of the param-
eters E, C and P, the first line shows the order size
derived from cost minimization perspective (hence,
it is a constant number, because price variation does
not affect it) and the second line shows the order size
derived from NPV maximization. For those cases
where there is an order size equal to zero it is because
the NPV obtained is a negative value. It is possible
to see that the difference between the order sizes
derived from NPV and cost minimization is higher
when the difference between price and variable
cost increases.
In table 2, for every combination of the
parameters S, C and h, again the first line shows
the order size derived from cost minimization per-
spective and the second line shows the order size
derived from NPV maximization. It is possible to see
that the difference between the orders size derived
from NPV and cost minimization is lower than in
the previous case. This can be explained because
the process of demand occurrence is independent
from the inventory policy.
In table 3, the value for parameter f is 0.0003.
For every combination of the parameters S, C and
P, once again the first line shows the order size de-
rived from cost minimization perspective and the
second line shows the order size derived from NPV
maximization. It is possible to see, as in the case I,
that the difference between the orders size derived
from NPV and cost minimization is higher when
higher is the difference between price and vari-
able cost.
53Escuela de Ingeniería de Antioquia
Table 1. Optimal order size, Case I
Sal. Exp Cost Price
E C 1.2C 1.6C 2C 2.6C 3C
25
0.1 707 707 707 707 707
0 0 607 552 522
1 224 224 224 224 224
0 200 186 170 161
10 71 71 71 71 71
68 63 58 53 51
100 22 22 22 22 22
22 20 19 17 16
125
0.1 1581 1581 1581 1581 1581
0 0 0 0 1233
1 500 500 500 500 500
0 456 423 385 365
10 158 158 158 158 158
153 141 131 119 113
100 50 50 50 50 50
48 45 41 38 36
Table 2. Optimal order size, Case II
Sal. Exp. Cost H
S C 0.0003 0.0016 0.008 0.04 0.06
25
0.1 884 546 272 125 102
846 531 268 124 101
1 280 173 86 40 33
276 172 86 40 33
10 89 55 28 13 11
88 55 28 13 11
100 28 18 9 4 4
28 18 9 4 4
125
0.1 1977 1220 607 278 228
1790 1148 589 274 225
1 625 386 192 88 72
606 379 190 88 72
10 198 122 61 28 23
196 122 61 28 23
100 63 39 20 9 8
63 39 20 9 8
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In all the above three cases the order size
has been obtained for a deterministic scenario. In
contrast, every inventory system will have to face
the variability of critical parameters. To represent
this situation, the next step is to include uncertainty
by allowing a subset of parameters to behave in a
probabilistic way. In particular, the following subset
of parameters will be modeled as random variables:
price, sales expenses, variable production cost, de-
mand, and setup/order cost. Demand is modeled as
a normally distributed random variable, with mean 1
and variation coefficient 0.1. Sales expenses are mod-
eled with a triangular distribution with parameters
lower limit 23, most likely 25 and upper limit 29. Price
is modeled by a uniform distribution with parameters
low limit 20 and upper limit 30. Finally, the variable
production cost is modeled by a normal distribution
Table 3. Optimal order size, Case III
Sal. Exp. Cost P
S C 1.2C 1.6C 2C 2.6C 3C
25
0.1 1514 1514 1514 1514 1514
0 1235 1126 1031 990
1 479 479 479 479 479
439 385 354 328 317
10 151 151 151 151 151
138 121 112 104 101
100 48 48 48 48 48
44 39 36 33 32
125
0.1 3385 3385 3385 3385 3385
0 0 0 2317 2205
1 1070 1070 1070 1070 1070
0 866 794 730 703
10 339 339 339 339 339
309 271 251 232 224
100 107 107 107 107 107
98 86 80 74 71
with mean value 10, and variation coefficient 0.1. It
is assumed that there is no correlation between the
random variables.
For the first case and the set of parameters
mentioned (the mean value for the random vari-
ables), the optimal order sizes will be 71 and 53 units
(see table 1), depending whether NPV maximization
or cost minimization is used as optimization criteria.
The order size derived from NPV maximization is
higher than the other, derived from cost minimiza-
tion, by approximately 34 %. Given this difference in
the order size, it was expected that the NPV would
present a different behavior under the application
of the two inventory policies. The results obtained
for the NPV indicator in this case are shown in the
graphics of figures 2, 3 and 4.
55Escuela de Ingeniería de Antioquia
Figure 2. NPV probability density functions for Case I, VaR Indicator
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In the graphics, the NPV has been calculated
under three inventory policies (order sizes). The
first one is NPV maximization; the second one, cost
minimization and the last one is an arbitrary policy in
which the order size is four times the order calculated
under NPV maximization. This last policy represents
a case in which the company purchases more than
the regular order size to take advantage of market
discounts. For the particular case the discount has
been assumed as 8 % over the variable cost and it is
applied to all units. The confidence level has been
arbitrarily selected to be 97.5 % (the usual confidence
level used in hypothesis testing, for example, is 95 %).
Five thousand trials have been used in every case
after checking that the coefficient of variability does
not change by increasing the number of runs (stability
of the response). Furthermore, running 5.000 trials
only takes a couple of minutes of computation time.
From the graphics and the statistical informa-
tion, it is possible to realize that the mean value of
the NPV is similar in the three cases studied. In fact,
the difference between the mean NPV for the two
first scenarios (NPV maximization or cost minimiza-
tion) is as much as 0.2 %. The third scenario shows
a variation of approximately 5 % when compared
against the first one. This is interesting because the
variation in mean NPV is just 5 %, while the variation
in order size is four times the original value. Hence,
the three different policies are similar when they are
evaluated from NPV perspective. The other concern
is the risk, which in this case is measured through the
VaR. For the three scenarios under consideration the
resulting VaR values are 17.535, 17.518 and 16.676,
respectively. In all cases the confidence level used
was 97.5 %. Given the interpretation of the VaR in-
dicator as a worse case value for the NPV, then it is
necessary to select the policy in which the worst sce-
nario has the highest value, so the best policy in this
context will be to use the order size derived from NPV
maximization. However, it is possible to see that the
VaR for the second scenario is very close to the VaR
obtained in the first one. Basically, it can be said that
the first and second policy present a similar behavior
also when compared using the VaR indicator. The
third scenario shows a VaR of 16.676, which means
a variation when compared against the VaR of the
first scenario of about 900 monetary units. However,
instead of making the comparison directly between
VaR it is better to compare the VaR indicator against
the mean value of NPV, because the VaR is a value
for the NPV, the value for the worst expected case
given a certain probability. For the three scenarios,
the ratio between VaR and its respective mean NPV
is about 62.5 %. This consistency among the three
scenarios means that, from a risk perspective, the
three policies are also similar.
For the second inventory system, the set
of parameters that are now assumed to have
probabilistic behavior are: demand, normally
distributed with mean 1 and variation coefficient
0.1; setup/ordering cost is assumed to have a
uniform distribution, with parameters (112, 138);
the price is modeled with a triangular distribution
with parameters (250, 300, 330) and the variable
cost is modeled by a normally distributed variable
with mean value 100 and variation coefficient 0.04.
k = 0.0005, h = 0.0008 and U = 5. In this case, the
order size resulting from NPV maximization and
cost minimization is the same quantity, 20 units. The
comparison will be made against an arbitrary policy,
similar to the one described before and related with
market discounts. The order size is again 4 times the
regular order, and the discount is again 8 % applied
to all units. The results are shown in the figure 3.
57Escuela de Ingeniería de Antioquia
Figure 3 shows the results for the second case.
Once again, the resulting NPV values from both inven-
tory policies are similar, since the variation observed
in NPV indicator for the two scenarios is just about
3 %, as well as the VaR indicator where the variation
is about 4 %. The ratio between VaR and mean NPV
is again consistently equal to 73 %. Once again, the
mean behavior of the system measured with the NPV
indicator and also the risk position, measured by the
VaR, are very close for the two inventory policies used,
although those policies are quite different.
Figure 3. NPV probability density functions for Case II, VaR indicator
Finally, for the third inventory system the
following set of parameters is used. Demand is
modeled by a normal distribution with mean value
1 and variation coefficient 0.1. Setup/ordering cost is
modeled with a uniform distribution with parameters
(112, 138). Price is modeled with a triangular
distribution with parameters (17, 20, 25). Variable
cost is modeled with a normal distribution with mean
value 10 and variation coefficient 0.1. Simulation
outputs are shown in the graphics of figure 4.
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Figure 4. NPV probability density functions for Case III, VaR Indicator
59Escuela de Ingeniería de Antioquia
From figure 4 and the statistical information
it is possible to see that the behavior of this inven-
tory system is quite similar to the exhibited for the
two previous. Once again, three different inventory
policies lead to a similar behavior in terms of mean
NPV but also VaR. The ratio between mean NPV and
VaR is in this case about 65 %.
In all three inventory systems studied it has
been identified the critical parameters that had the
biggest influence on the NPV indicator. The general
conclusion is that the difference between price and
variable cost is what causes the greatest variation
over NPV indicator. As the margin increases (price
less variable cost), so does the variability over NPV,
measured by its standard deviation.
7. CONCLUSIONS AND FUTURE
RESEARCH
After reviewing the results that have been
shown regarding the different inventory systems
studied, it can be concluded that although different
optimization criteria are used to make decisions about
inventory policies, the system performance when
applying different inventory policies is similar when it
has to face the risk conditions that are inherent to its
nature. The usual approach to determine inventory
policies is the average long term cost minimization,
approach that have been criticized because its lack of
representation of the genuine conditions of the real
manufacturing systems. Instead, some authors have
claimed that NPV maximization should be used, since
this approach is consistent with the maximization
of company’s value. The present work has used
both approaches to derive optimal operation
conditions, and in fact the inventory policies coming
from every approach differ. However, when those
different policies are applied to the system, and it is
simulated by using Monte Carlo simulation, assuming
a probabilistic behavior for a subset of critical
parameters, the mean behavior observed for the
system have been quite similar. In fact, from a risk
perspective, it has been observed that the different
policies are consistently equivalent, since for every
system studied the ratio between VaR and mean
NPV is a constant. From a return perspective, the
analysis of the inventory systems under deterministic
assumptions reveals a difference between the NPV
indicators when using different inventory policies.
However, the mean NPV of the system when it is
calculated from the simulation process results in a
very close value, especially when comparing the
policies derived from NPV maximization and cost
minimization. These results suggest that the usual
approach used in the real world of manufacturing
systems, consisting in determining inventory policies
from cost minimization perspective is a robust
procedure, as good as using NPV maximization,
which is claimed to be more precise.
Further studies can be conducted to evaluate
other inventory systems, or the same inventory
systems studied in this work, but including different
operation conditions, such as credit policy, for
example. It is also possible to conduct studies
intended to find optimal inventory policies derived
from risk minimization, using the VaR indicator, or
some other risk measurement.
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