Main characteristics of inventory
systems
In general, inventory models can be broadly classify according to the
demand and the type deterioration. Depending on the type of demand,
there are deterministic inventory models or stochastic one. If the
demand is deterministic, the variation of inventories over time on each
inventory cycle may be affected by a prediction of a constant
demand\(\ \left(\frac{\text{dI}\left(t\right)}{\text{dt}}=-D_{1}\right)\),
or by the combined effects of a constant demand \(D_{1}\) and a fixed
fraction \(D_{2}\) of the instantaneous stock level\(\left(\frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-D_{2}I(t)\right)\).
In more elaborate inventory models, the depletion of the inventory can
also occur due to a known function of demand depending on time or
further depending on the selling price and/or one of various marketing
parameters (e.g., selling price, frequency of advertisement, credit, and
freshness of products). In turn, when the demand is uncertain, it may
follow a known probability distribution, or it may be represented
through an additive or multiplicative functional-form with random
components. When the accuracy of the stochastic demand
distribution/function is unknown, modeling a fuzzy/hybrid demand
[32-35] can be useful to address this
type of uncertainty.
According to Raafat [23], any stocked
items restrained by any process from being used for its original
intended use is known as inventory subject to deterioration or decay.
This definition encompasses many different types of products. However,
they have been traditionally classified into three main categories:
items with fixed lifetime, items with random lifetime, and items subject
to obsolescence.
Fixed lifetime refers to the best-before date (BBD) of most packaged
products. Although these types of products are not usually spoiled at
the end of its BBD, sellers discard them in order to follow regulations.
Random lifetime refers to the uncertainty in the spoiled time of items
like fresh produce. Here, the time to spoilage may be uncertain for each
individual stocked item, but, in practice, or from a modeling
perspective, the total amount of spoiled items within any specific
interval of time may follows either a deterministic or probabilistic
function. Finally, obsolescence refers to the rapid loss in value of
unsold items due to the introduction of a new product or the end of a
shopping season. Unlike fixed lifetime items, this type of products does
not suffer physical degradation due to its own nature, and thus, they do
not necessarily need to be removed from the inventory throughout their
selling season. Typical examples are found in the fashion and technology
industry, and almost all the industry applications of studies further
investigating or extending the classic newsvendor problem.
Table 4 shows the classification of deteriorating inventory items
adopted for most authors in the literature. Note that some of the most
common terms used in the literature of inventory models such as
“perishable items” and “random lifetime products” may be used in
different context. For example, the term “perishable products” may be
used for either items with fixed life time or items subject to
obsolescence, and the “random lifetime” term may be utilized for
models that do not necessarily consider deterioration as a stochastic or
random process. As a result, for the sake of transparency, in the
present review we use three categories that represent the way in which
spoiled items are model or represented into the mathematical models.
These categories are fixed lifetime items, constant deterioration rate,
and time-varying deterioration rate.
The first category, inventory models with a known fixed lifetime,is used for products discarded in a particular point of time, i.e., when
their expiration date has lapsed (e.g., 2 days, 1 week, etc.). The
second category, inventory models with a constant deterioration
rate, is used to those models where the variation of spoiled items at
each period or instant of time \(t\) is represented by a constant
fraction \(\theta\) of the instantaneous stock level\(\left(e.g.,\ \frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-\text{θI}\left(t\right)\right)\).
The third category, inventory models with a varying deterioration
rate, is used to discuss those models where the variation of spoiled
items at each period or instant of time \(t\) is represented by a
non-uniform fraction over time of the instantaneous stock level\(\left(e.g.,\ \frac{\text{dI}\left(t\right)}{\text{dt}}=\ {-D}_{1}-\theta\left(t\right)I\left(t\right)\right)\).
Note that papers such as [36,
37] that claim the inclusion of a
deterioration rate following a probability distribution are classified
in the second category when, in the mathematical model, the mean of a
probability density function is used as a constant rate, and thus, the
amount of spoiled items is the same over time. Contributions such as
[38-50], in which the rate of
deterioration can be reduced by investing \(\xi\) monetary units in a
preservation technology, are classified in the second or third category
depending on the variability over time (or not) of spoiled items in the
inventory model\(\left(e.g.,\ \frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-\left[\theta-m\left(\xi\right)\right]I\left(t\right)\rightarrow category\ 2;\ \frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-\left[1-m\left(\xi\right)\right]\theta\left(t\right)I\left(t\right)\rightarrow category\ 3\right)\).
Inventory models in which an item can randomly expire before or up until
their maximum lifetime (uncertain lifetime) are classified in the third
category due to the amount of the same item perishing at each period or
instant of time \(t\) explicitly varies with respect to time. Meanwhile,
studies such as [51-74], in which the
inventory loses value but it is not physically destroyed are classified
in the first category.
Among other interactions, the assumptions or restrictions to be
considered for the correct application of inventory models for
deteriorating items include the lead time (negligible, constant, or with
a known or unknown distribution), the inventory review policy (periodic
or continuous), the existence of shortages (lost sales or backorders),
the inclusion of multiple products (with complementary and/or substitute
items), the production rate (finite, infinite or uncertain), the selling
price (fixed, variable or uncertain), the time value of money, the
number of echelons within the supply chain (closed loop supply chain,
VMI, non-cooperative, etc.), a permissible delay in payments (fixed or
conditioned), and the set of uncertain parameters or variables.