Classification of inventory models
In this section, 317 articles that developed an inventory model for
deteriorating items are classified and analyzed. In Section 4.1, the
studies published between 2001 and 2018 are classified according to the
type of demand and deterioration assumed. Similarly, in Section 4.2, the
different papers referenced are also classified according to the
inclusion of the following characteristics: a determination of an
optimal price policy, considerations of a stock-out period, the
inclusion of multiple products, consideration of two warehouses, a study
of two or more echelon within the supply chain, considerations of delay
in payments, the inclusion of the time value of money, and the inclusion
of uncertain parameters or variables.
Classification according to the demand and
deterioration
Table 5 shows the type of demand
and deterioration considered in the models of our sample according to
the characteristics discussed in Section 3. Here, 230/317 of the papers
reviewed have included a deterministic demand, and 92/317 included an
uncertain demand. In addition, 153/317 considered a constant
deterioration rate, 77/317 considered a variable deterioration rate, and
94/317 studied deterioration with a fixed lifetime. Note that of all of
the authors who considered an inventory model for deteriorating items
with a fixed lifetime, 63/94 have taken an uncertain demand into
account. Also note that from all authors who considered a deterministic
demand, 148/230 did not include any of the marketing strategies that
companies commonly use to influence demand, such as pricing,
advertising, markdowns, post sales services, among others.
When marketing strategies are used to stimulate customers’ consumption,
the integration between marketing and inventory management decisions
becomes important to maximize mutual benefits and avoid potential
conflicts. Of the 230 deterministic inventory models, 82 incorporated an
optimal decision for marketing strategies (pricing and others marketing
dependent demand factors) in combination with an optimal inventory
replenishment policy. Of these 82 models, 32 considered that the demand
for an item depends on characteristics further than selling price, thus
affecting marketing decisions: in
[75-79], the demand rate is a
function of the trade credit period offered by the retailer to their
customers; in [53,
80-86], the demand rate is influenced
by the frequency of advertisement or the promotional expenditure; in the
work of [50,
86-88], the demand rate is affected by
the customer service level such as the warranty period and the
aftersales service expenditure; in
[52-54,
56, 59,
61-64, 71,
89-94], the consumption rate depends on
the product’s freshness or quality of an item during the planning
period. Finally, in [47,
80], customer’s environmental concern
affecting the demand are considered.
If environmental factors such as economic and marketing conditions
change during the product life and have a significant effect on the
demand, then the assumption that the demand in each period is a random
variable and is independent of environmental factors apart from time
will be incorrect. In such real life situations, the Markov chain
approach provides a flexible alternative for modeling the demand process
[55,
276,
321,
335,
336,
339-341,
343-346]; not only does it
significantly generalize the Poisson process
[55,
272,
278,
280,
281,
287,
290,
297,
303,
304,
307,
308,
316,
333,
338,
341], but it is also a convenient tool
for modeling both the renewal and non-renewal demand arrivals. However,
despite this fact, the Markovian assumption holds for demand processes
with relatively low variation coefficients, i.e., in cases where high
demand variances are observed, the non-stationary assumption does not
hold in the Markovian environment because standard periods of constant
length may introduce memory and generate correlated demand distributions
within periods. For these cases, the models in
[8, 285,
286,
292,
310,
319] provide a reasonable alternative.
Of the inventory models with stochastic demand explicitly guiding
marketing strategies, the models in
[66, 67,
69, 283,
289,
293,
300,
314,
325,
326,
328,
330,
334,
342] are developed to define both a
price decision and inventory control policies. In turn, the inventory
models in [51,
66, 67,
69, 283,
289,
300,
302,
307] take into account that the market
demand is random and sensitive to the freshness of the product, which is
more assertive for deteriorating items.
From studies with varying deterioration rates and deterministic demand,
in the inventory models [38,
89, 108,
199-202,
206,
207,
211,
212,
215,
219,
244,
263-266], the deterioration rate is
assumed to be a general or arbitrary function of time \(\theta(t)\); in
the models described in[268], the
deterioration rate is studied as time proportional\(\theta\left(t\right)=\theta t\); in the papers
[94,
155,
203,
204,
209,
262] a three-parameter Weibull
distribution\(\theta\left(t\right)=\alpha\beta\left(t-\gamma\right)^{\beta-1}\)is included; in the models described in
[87, 93,
157,
161,
205,
210,
213,
214,
217,
221,
222,
245,
269,
271], the deterioration rate follows a
two Weibull distribution\(\theta\left(t\right)=\alpha\beta\left(t\right)^{\beta-1}\);
and in the models developed in [76-79,
154,
158-160,
162,
218,
220,
246,
267,
270], the products deteriorate at a
rate \(\theta\left(t\right)=\frac{1}{1+m-t}\), where \(m\) is
the maximum lifetime at which the total on-hand inventory deteriorates.
In other particular works: [245]
present the deterioration rate as a function of both quality level and
time, where the time-related function follows a two-parameter Weibull
distribution; in [208] the time to
deterioration is assumed normally distributed over time; and
[255] introduce a deterioration rate
depending on the time and storage temperature to which food products are
exposed.
With respect to inventory models with an uncertain demand, it is
interesting to note that most of these studies assumed a fixed life
time, but few of them [34,
51, 55,
66-70, 73,
74, 90,
289] consider the influence of product
freshness over demand. From inventory models with uncertain demand and
constant deterioration, the deterioration of items in
[32, 33,
35, 112,
323-333] is assumed to be a constant
fraction \(\theta\) between [0,1] from the on-hand inventory.
Meanwhile, from inventory models with uncertain demand and time-varying
deterioration, the product lifetime in
[282,
337,
339,
340,
343] is random and can be described by
a discrete distribution, while the lifetime of each item in
[297,
334-336,
338,
341,
344-346] has negative exponential
distribution with parameter \(\gamma\ (>0)\), which may be suitable
for inventory systems where item lifetimes are typically small but
occasionally have long lifetime.
In nearly all of the inventory models considering deteriorating items,
the demanded items are immediately delivered to the customers from the
goods in stock. However, when the items from the on-hand inventory are
not delivered at the time of demand, but after some positive service
time that usually is random, the inventory managers need to consider the
replenishment policy as well as the effect of the formation of queues in
the inventory system, in order to implement suitable control policies.
Amirthakodi, Radhamani [335],
Manuel, Sivakumar [344], and
Yadavalli, Sivakumar [346] analyzed
an inventory system with a service facility assuming that the customers
arrive according to a Markovian arrival process with a product lifetime
following an exponential distribution, but
[335,
344] consider that the service
facility has a single server, and
[346] consider a multi-server
perishable inventory system.
Classification according to shared characteristics with
other research
streams
As can be observed in Table 6, 167/317 of the models allowed a stock-out
period in their models (category 1), 65/317 took into account an
inventory system with a permissible delay in payment (category 2),
82/317 considered the determination of an optimal pricing policy
(category 3), 77/317 studied a multi-echelon deteriorating inventory
model (category 4), 44/317 developed an inventory model considering the
time value of money (category 5), 30/317 studied a two-warehouse
inventory model (category 6), 32/317 took into account the existence of
multiple products (category 7), and 12/317 studied an inventory model in
a fuzzy environment (category 8).
Out of the 167 investigations allowing shortages, a total of 18, 31, 29,
31, 18, 18 and 4 models take into account the category C2, C3, C4, C5,
C6, C7 and C8, respectively. Of the 64 investigations considering a
delay in payment or prepayments, 11, 7, 15, 11, 0 and 5 models
considered categories C3, C4, C5, C6, C7, and C8, respectively. Of the
82 studies determining a pricing policy, 20, 9, 4, 2 and 2 considered
categories C4, C5, C6, C7 and C8, respectively. Of the 77 multi-echelon
inventory models, 5, 5, 10, and 1 models included categories C5, C6, C7
and C8, respectively. Of the 44 models considering the effect of the
time value of money, 8, 2, and 2 considered categories C6, C7, and C8,
respectively. Of the 30 investigations considering two or more
warehouses, 2 and 2 considered the categories C7 and C8 in the model,
respectively. Finally, of 32 inventory models with multiple items, 12
papers considered category C8.
Inventory models including a stock-out
period
When demand is higher than the previous forecast and cannot be fulfilled
immediately with the inventory on hand, then there is an excess of
demand, or shortages. Depending on the customer-company relationship,
the excess of demand (shortage) can be lost
[89, 90,
98, 99,
101,
155,
258], or the excess demand can be
accumulated in different ways with an associated shortage cost.
Therefore, the behavior of the inventory systems, when a stock-out
period is allowed, differs from one model to another.
Of the deterministic inventory models developed in
[35, 45,
83, 102,
105,
112,
123,
124,
128,
139-144,
151,
161,
167,
169,
171,
181,
186,
203,
207-210,
214,
216,
217,
227,
232,
242,
251,
257,
266,
271,
274], all of the excess of demand is
willing to wait for the next replenishment (shortages are fully
backlogged). In the papers [87,
119,
122,
144,
157,
160,
172,
201,
211,
213,
226,
227,
229], shortages are allowed, but only
a fixed fraction is backordered, and the rest is lost. Finally, the
deterministic inventory models found in
[40, 41,
43-46, 82,
84, 85,
138,
146,
152,
166,
173,
176,
177,
185,
187-189,
192,
194-196,
205,
218-222,
225,
228,
234,
241,
245,
246,
250,
263-265,
269,
270] consider that shortages are
allowed, but the unsatisfied demand is partially backlogged depending on
the waiting time until the arrival of the fresh lot.
With respect to the inventory models with uncertain demand and
shortages, nearly all of the authors assumed that all of the excess
demand during the stock-out period become either lost sales
[8, 51,
66, 68,
70, 73,
90, 98,
272,
273,
275,
276,
278,
281,
282,
288-290,
294,
296-300,
302,
303,
306,
309-311,
313,
316,
320-322,
326,
332,
335,
337-340], or shortages that are fully
backlogged [8,
35, 112,
278,
284,
286,
287,
295,
297,
308,
315,
319,
321,
323,
325,
329,
341,
343,
347]. However, some models with
uncertain demand consider a partial backlogging rate:
[305,
330,
331,
333] assume that a fixed fraction of
shortages are backlogged while [328,
342] assume a waiting time function
for unmet demand. Finally, inventory models considering a maximum
allowable shortages constraint can be found in
[280,
336], and inventory models in which
unmet demand during the stock-out period temporarily leave the service
area and repeat or retry their request after some random time (until
they find a positive stock level) is studied in
[345,
346]. In the latter case, it is
important to note that, as opposed to the lost-sales and backlog cases,
the company does not incur any expenditure toward lost sales or for
holding unsatisfied demands.
Wu, Ouyang [194] proposed a model
for non-instantaneous deterioration of items with stock-dependent demand
and in which the backlogging rate is variable and dependent on the
waiting time for the next replenishment.
Olsson and Tydesjö [308] described a
model where demand is generated by a stationary Poisson process, but it
is assumed that unmet demand is immediately backordered. Meanwhile,
Dehghani and Abbasi [278] proposed a
new age-based lateral-transshipment policy, which may be useful for
reducing stockouts and improve performance in supply chains.
Permissible delay in payment and/or
prepayments
In the traditional inventory economic order quantity model, it was
tacitly assumed that the supplier is paid for the items as soon as the
items are received. However, in business transactions it is observed
that the supplier provides a grace period whereby purchasers can repay
their debts without having to pay any interest (trade credit period) or
may delay the payment beyond the permitted time, in which case interest
is charged.
Inventory models in which the supplier provides a permissible delay to
the buyers if and only if the order quantity is greater than or equal to
a predetermined quantity \(W\) can be found in
[107,
108,
116,
168]. Authors who assume that there
already exists a regular credit policy between the retailer and the
vendor (fixed delay in payment) can be consulted in
[48, 49,
111,
119,
125,
134,
138,
146,
152,
154,
165,
180,
182,
190,
191,
202,
204-207,
243,
250,
257,
259,
333], whereas authors who consider a
permissible delay of payments, but as a decision variable, can be found
in [33,
75, 76,
78, 79,
118,
164].
The inventory models submitted in [32,
75, 76,
79, 110,
114,
115,
145,
158,
162,
164,
220,
237,
264,
324,
328]
take
into consideration affairs in which the supplier not only offers a
fixed credit period to the retailer but the retailer also adopts the
trade credit policy with his/her customer. A two-level trade credit is
also considered in [33,
77, 126,
159,
227,
267,
270] but with a partial trade credit
in one or both echelon of the supply chain.
Alternatively, in the models presented in
[117,
120,
129,
267], the supplier provides not only a
permissible delay in payments to the customer but also a cash discount,
to wit a cash discount is offered by the supplier if full payment is
paid within time \(M1\) (period of cash discount); otherwise, the full
payment is paid within time \(M2\) (with \(M2\ >M1\)).
Ouyang, Teng [131] and
Guchhait, Maiti [230] consider
situations in which a supplier also offers a partial permissible delay
in payments even if the order quantity is less than a predetermined
amount \(W\) of units of an item. This means that if the order quantity\(Q\) is less than \(W\), then the retailer must pay a
fraction\(\ 0\leq\beta\leq 1\) of the total purchase costcQ when the order is filled and pay the rest,\(cQ(1-\beta)\), at the end of the trade credit \(M\).
Chang, Ouyang [109] consider a
situation in which the supplier offers his/her customers a permissible
delay in payments, \(M1\), and a price discount if the order quantity is
greater than or equal to a predetermined quantity \(Y\), a permissible
delay in payments, \(M2\), if the order quantity is greater than or
equal to a predetermined quantity \(X\) (with \(M2>M1\)), and a trade
credit period, \(M1\), if the order quantity is less than \(X\) (or\(Y\)). In turn, [122,
142,
160,
227] develop an EOQ model with
multiple prepayments
Inventory models determining an optimal price
decision
In competitive environments, it is common to change the price of items
to stimulate demand and decrease the rate of deterioration of stored
commodities: in this context, it is important to define a price strategy
during the horizon planning, and thus, it is important to include price
decisions in the deteriorating inventory models.
The selling price as a decision variable was first considered for
perishable products in 1996 by Eilon and
Mallya [348]. Later, Kang and Kim
[349] and Aggarwal and Jaggi
[350] reformulated and extended this model to make way for further
investigations in which both the selling price
[42, 45,
47, 50,
61, 64,
67, 69,
80-83,
86-89, 93,
128,
164,
167,
179,
218,
223,
230,
232,
240,
250-257,
259,
261,
263-271,
275,
325,
328] and the discounted selling price
[68, 89,
145,
170,
175,
249,
298,
314] are taken into account as
decision variables.
Traditionally, a common practice followed by companies is to maintain a
constant price on the goods offered and apply discounts when items are
close to expiry or when the demand decreases. However, applying an
aged-dependent selling price strategy may be more beneficial
[53, 62,
63,
289]. Inventory models considering a
dynamic pricing policy are described in
[38, 53,
55, 62,
63, 65,
71, 91,
92, 94,
174,
178,
191,
233,
235,
247,
260,
283,
289,
293,
300,
325,
326,
330,
342].
In relation to the particular works in this category, new approaches for
determining a pricing policy when reference price play a critical role
in customer purchase decisions are proposed in
[38, 47,
258], a clearance sales as a strategy
to sell items approaching their expiration dates at a reduced price is
studied by Li, Yu [298], and an
optimal replenishment and pricing policy for deteriorating items with
heterogeneous consumer sensitivities is investigated by
Herbon [64].
Multi-echelon inventory
models
In the present literature reviewed, 39/77 of the inventory models that
consider more than one echelon in the supply chain are restricted to
interactions that occur between a supplier/producer and a buyer
[37, 39,
58, 63,
67, 73,
80, 81,
87, 88,
97, 101,
103,
104,
106,
112,
114,
118,
143,
145,
147,
149,
170,
176,
198,
229,
231,
237,
242,
248,
251,
252,
256,
275,
293,
305,
321,
331,
332]. However, a supply chain
involving a single-vendor and multiple buyers is studied in
[34, 36,
60, 72,
82, 100,
140,
153,
197,
210,
253,
261,
277,
281,
290,
296,
326,
334], while a two-level supply chain
with multiple suppliers and multiple buyers is considered in
[48, 95,
315,
327].
Inventory models for a three-level supply chain can be found in
[42, 49,
86, 90,
96, 98,
133,
135,
136,
148,
155,
156,
312,
318,
333], and inventory models for a
closed-loop supply chain (recovery system) are presented in
[34, 73,
149,
163,
261].
Wee, Lee [149] incorporate VMI
strategies for a green electronic product in a two-level supply chain.
Yang, Chung [261] also consider a
closed-loop supply system, but involve a single producer and multiple
buyers. Meanwhile, Alamri [163]
consider a recovery system in a production environment consisting of
three shops: one of manufacturing new items, and the others for
collecting and remanufacturing returned items.
Finally, it is worth noticing that most of the above studies addressed a
coordinated policy for integrating a supply chain. However, few studies
have proposed new mechanism for achieving a cooperation in a
non-cooperative environment [67,
73, 88,
133,
256,
281]. Other few studies have also
addressed the problem of a non-coordinated supply chain
[63,
248,
275] but in a competitive environment.
Inventory models under the effect of time value of
money
The effect of the time value of money (TVM) and inflation is another
important extension that makes inventory models applicable to real-life
situations. It plays an important role in business, especially in
countries with double-digit Gross National Product rate
[351]. Table 7 shows the papers
incorporating the time value of money effect into an inventory models.
Note that few studies have been developed for determining an inventory
policy with multiples items or in a fuzzy environment under time value
of money.
Hou and Lin [232] discuss an
inventory model for deteriorating items with price and stock dependent
selling price in which shortages are completely backordered.
Ghiami and Beullens [157] provide a
Net Present Value analysis for a production-inventory system without the
inventory cost parameters commonly used in this context. Meanwhile,
Tat, Taleizadeh [143] study the
optimal ordering policy in an inventory system considering backorders
and delay in payments.
In the previous models, the inflation rate has been considered as a
constant and known value. However,
Mirzazadeh, Seyyed Esfahani [329]
consider that the inflation changes over the time horizon, specifically,
the inflation rate was assumed to be stochastic with known pdfs over the
time horizon. Here, the demand rate is a linear function of the internal
and external inflation rates, shortages are allowed and fully
backlogged, and a constant fraction of the on-hand inventory
deteriorates per unit time.
Two or more warehouse for deteriorating inventory
models
In most of the existing inventory models for deteriorating items, it is
assumed that all of the items are stored in a single warehouse (owned
warehouse, OW). However, in some cases, organizations may require
another storage facility (rented warehouse, RW) with better preserving
facilities and ample capacity in which deterioration, costs, demand, and
other parameters are different. These facilities are usually rented
either to reduce the losses due to deterioration or to store excessive
goods obtained at a discount price or simply to avoid inflation rate.
Cases where a retailer owns two shops: one shop to sell fresh items
until a particular point in time and the other shop to offer non-fresh
items at a reduced price are included in this category. Two-warehouse
inventory models incorporating further characteristics related to
inventory systems are presented in Table 8. Notice that models
considering two or more storage facilities and the time value of money
are shown in Table 7.
Two-warehouse inventory models for products with imperfect quality, and
thus, subject to inspection are developed in
[121,
200].
Alamri and Syntetos [200] propose a
new policy entitled “Allocation-In-Fraction-Out (AIFO)”. Unlike
Last-In-First-Out (LIFO) or First-In-First-Out (FIFO) dispatching policy
which are commonly adopted in classical formulation of a two-warehouse
inventory models, AIFO implies simultaneous consumption fractions
associated with RW and OW. Jaggi, Tiwari
[121] also study the effect of deterioration on two-warehouse
inventory model with imperfect quality, but
Alamri and Syntetos [200] assume
that the percentage defective per lot reduces according to a learning
curve while Jaggi, Tiwari [121]
assume that each lot received contains a random proportion of defective
item, an thus, the screening times of OW and RW are included as
decisions variables.
Most of the studies uses a calculus-based approach for solving the
proposed inventory model. However, some formulations require the use of
metaheuristic due to the complexity of the mathematical model. For
example, Guchhait, Maiti [230]
studied the features of PSO with a genetic algorithm in a hybrid
heuristic named PSGA (Particle Swarm-Genetic Algorithm) to show that
this performance is better compared to the FGA (Fuzzy Genetic Algorithm)
and the traditional PSGA in an inventory system with stock and selling
price demand being dependent under crisp and fuzzy environments.
Multi-item inventory systems
Although in most inventory systems the assumption of a single item is
not real, very few studies have been found in the literature dealing
with multi-item inventory control compared to single-item models. Two or
more items in combination with shortages are taken into account in
[34, 35,
70, 99,
101,
102,
105,
123,
155,
245,
279,
284,
292,
296,
299,
311,
315,
341]. Inventory models considering an
optimal price decision and multiple items can be found in
[62,
223]. Inventory models taking into
account multiple products in a fuzzy environment are developed in
[35,
137,
245]. And multi-item inventory models
in a supply chain system can be found in
[34, 72,
100,
101,
103,
127,
155,
296,
312,
315].
It is worth noticing that, although in recent years the volume of
inventory models with multiples items has increased (23 papers between
2016 and 2018 compared with 9 papers between 2001 and 2015), most of the
models only consider a different demand, perishing nature and/or cost
parameters for items. Thus, the effect of multiple product competing for
shared recourses (such as demand, space, etc.) is still a research area
where much more work is needed. Studies considering these issues can be
found in [57,
223,
236,
307].
Maity and Maiti [236] propose an
inventory control system wherein multiple items are either complementary
and/or substitutes, and the deterioration rate is stock dependent. In
this work, an optimal inventory policy for complementary and substitute
commodities is considered, but shortages are not allowed, and demand is
stock dependent. Önal, Yenipazarli
[223] develop a mathematical model with price-and displayed-stock
-dependent demand under shelf-space and backroom storage capacity
constraints. Claassen, Gerdessen [57]
study a production planning and scheduling problem in a food processing
industry where setups are usually sequence-dependent. Finally,
Najafi, Ahmadi [307] address a blood
inventory management problem by considering all types of blood and their
substitution compatibilities based on medical priorities.
Inventory models with uncertain parameters
In most inventory models with stochastic demand, the lead time is
assumed to be either negligible [8,
69, 90,
112,
276,
277,
284,
286,
293,
300,
307,
319,
323,
324,
328-330,
334,
340,
343], or a fixed positive constant
[272-274,
279,
282,
287,
288,
290-292,
296,
297,
300,
304,
306,
308,
312,
313,
316,
322,
331,
332,
337,
340,
341]. However, there are inventory
models that consider the lead time as an unknown parameter following an
arbitrary distribution [336,
338], an exponential distribution
[344-346] or a phase-type
distribution [335]. Furthermore,
although a non-negligible lead time is usually associated with a
stochastic demand, few studies consider an uncertain lead time with
deterministic demand [172,
176].
In some mathematical models with stochastic demand, the inventory is
controlled explicitly with a periodic review system
[8, 66,
273,
274,
276,
279,
280,
282,
284,
287,
289,
292,
293,
296,
300,
303,
320,
321,
324,
330,
339], while in others, it is
controlled with a continuous review system
[272,
278,
290,
297,
304,
306,
308,
309,
311,
312,
316,
323,
332,
335,
336,
338,
341,
343-346]. On one hand, among items
monitored continuously, [272,
306,
311,
312,
323,
332,
335,
336,
343-346] apply the traditional\((s,S)\) policy in which a same amount of product \((Q=S-s)\) is
ordered every time the inventory level (on hand plus an order minus
backorders) reaches the level of \(s\).
[278,
290,
297,
308,
309,
338] use a \((S-1,S)\) ordering
policy for stock replenishment in which in which a reorder for an item
is placed whenever the inventory level drops by one unit either due to
demand or because a perished unit.
[304,
316] suggest a combined
age-and-stock-based ordering policy, \((Q,r,T)\) policy, in which a
replenishment order of size \(Q\) is placed either when the inventory
drops to \(r\) or when \(T\) units of time have elapsed, whichever
occurs first. Finally, [341] uses
the \((s_{i},c_{i},S_{i})\) policy suggested by
[352-354] with\(s_{i}<c_{i}<S_{i}\) for multi-item inventory problems dealing with
a group of items \(i\) where the replenishment cost of two or more class
of items is less than the total cost of individual or separate
replenishments. In this policy, as the traditional \((s,S)\) policy, an
order is placed by item \(i\) when its inventory level falls to thereorder level \(s_{i}\). However, any other item \(j\) is also
included in the same order if its inventory level is at or below itscan-order level , \(c_{j}\).
On the other hand, among periodic inventory systems,
[8, 276,
282,
284,
293,
300,
330,
339] study the determination of
inventory policies myopically, that is, at the beginning of each given
period, decisions are only made upon replenishment (one period at a
time). From these papers: [322] uses
a \((s,S)\) policy, i.e., an order is placed to raise the inventory to\(S\) when the inventory level is less than, or equal to, the reorder
point \(s\); [276,
305,
320] apply a base-stock policy of
level \(S\) in which the total inventory level at the beginning of each
period is always raised to \(S\);
[287,
288] suggest a stock-level dependent
ordering policy \((s,S,q_{\min},Q_{\max})\), which is a \((s,S)\) policy
with the order quantity being restricted between \(q_{\min}\) and\(Q_{\max}\); and [274,
319] investigate a class of
proportional balancing (PB) policy that balances a proportion of
expected marginal costs. Finally, papers applying an order up to a level\((R,S)\) policy in which the inventory level is observed at equal
intervals of time \(R\) and items are ordered to bring the inventory
position to a level \(S\) can be found in
[66,
279,
289,
292,
303,
321,
324], while papers investigating an
age-based replenishment policy can be found in
[273,
280,
294,
296]. These age-based replenishment
policies work similar to the traditional order-up-to level policy except
that the inventory level is corrected for the estimated amount of
outdating and an order is placed if this new corrected inventory level
drops below the target reorder level.
Alternatively, in the development of inventory models for deteriorating
items, researchers usually consider that all of the parameters and
relevant data of the systems are already known or stochastic. However,
in some practical applications, those assumptions are unrealistic
because obtaining some of them (demand, production rate, costs
coefficients, inflation, etc.) can be vague and imprecise or even
impossible to determine exactly. In this regard, with the exception of
[77, 98,
155], nearly all of the studies
[32, 35,
43, 77,
111,
119,
170,
183,
230,
239,
245,
255,
295] that have investigated an
inventory problem in fuzzy environments to model uncertainties in a
non-stochastic sense assume that one or more parameters is a fuzzy
number rather than a fuzzy variable. Note that the fuzzy variable can be
interpreted as a random fuzzy variable, fuzzy random variable, fuzzy
rough variable, and a hybrid variable. A fuzzy random variable can be
viewed as a random variable whose values are not real but a fuzzy
number, and a random fuzzy variable can be viewed as a fuzzy variable
taking random variable values. The reader might refer to
[355-357] for more information.