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.