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.