Goyal [1] developed a complete survey of the previous inventory system literature for the deteriorating item inventory systems. This survey literature reveals that deteriorating models have received particular attention and the considerable researches have been done. Other a considerable amount of research has been carried out by Cheng [2], Chiu et al. [3], Chung [4], Lee and Rosenblatt [5], and Rosenblatt and Lee [6] to address the imperfect item economic production quantity/EPQ problem. They presumed that at some random time, the production process might shift from an in-control to an out-of-control state. Specifically, Rosenblatt and Lee [6] were the first derived the imperfect/defective item production process problems. Their works have encouraged many studies to explore quality and imperfect item related issues, see e.g. Salameh and Jaber [7]. So, related to Salameh and Jaber’s model is the paper where an error appearing on this work is corrected by Cárdenas-Barrón [8]. Goyal and Cárdenas-Barrón [9] also investigated the simple efficient solution procedure for determining economic production quantity/EPQ for a product with imperfect quality. They also proposed that nearly optimal results are obtained using this approach, which is much easier to implement than the procedure suggested by Salameh and Jaber [7]. Moreover, numerous researchers have established the optimal solution and closed form without using partial derivatives (based on an algebraic method) for various EOQ/EPQ models under various situations. For example, see Cárdenas-Barrón [10] and Cárdenas-Barrón [11].
Hayek and Salameh [12] studied an economic production quantity/EPQ model with considering imperfect quality items where a fraction of defects is a random variable, the production rate is finite and shortages are fully backordered, and rework process. Grubbström [13] first derived the traditional EOQ (Economic Order Quantity) formula algebraically. Similarly, Grubbström and Erdem [14] also developed this approach to solving an economic order quantity/EOQ model with a shortage (backlog). Cárdenas-Barrón [15] investigated Grubbström and Erdem's [14] method to algebraically the mentioned algebraic methodology to the economic production quantity/EPQ formula taking shortages into consideration within the case of only one shortage (backlog) cost unit versus time. Also, numerous studies have been carried out to address the problems of imperfect quality economic production quantity/EPQ inventory model with the rework process (see, e.g., [16], [17] and [18]). Chang [19] investigated an economic order quantity/EOQ inventory model with fuzzy defective rate and demand considerations. The authors’ review of the inventory literature reveals that there is no published work that investigates the model of Salameh and Jaber’s [7] for learning effects. Jamal et al. [20] evaluated two rework optimal manufacturing policies. In the first optimal manufacturing policy, defective items are reworked in the same cycle; and in the second optimal manufacturing policy, rework is completed after N cycles. Another research has been performed by Papachristos and Konstantaras [21] looked at the issue of non-shortages in models with proportional imperfect quality, when the fraction of the imperfects is a random variable and revised the papers of Salameh and Jaber [7]. Chung and Huang [22] investigated the inventory model of Salameh and Jaber’s [7] and Goyal’s [1] in a two-level supply chain (vendor-buyer). Wang Chiu [23] investigated the optimal replenishment policy by considering a fraction of imperfect items and scrap items are produced in a regular production process and a portion of the imperfect items are repaired to make them good quality items.
Chiu et al. [18] developed an economic production quantity/EPQ model with scrap, rework, and stochastic machine breakdowns considerations for determination of optimal run time. Likewise, Chiu [24] later showed that the same problem can be derived without derivatives and solved by an algebraic method. Chung et al. [25] surveyed whit incorporates concepts of the basic two warehouses and imperfect quality item simultaneously to generalize Salameh and Jaber [7]. The mathematical model by maximizing the annual profit and the optimal solution procedure is developed.
Yoo, et al. [26] in their model extends an economic production quantity/EPQ model by incorporating defect proportion, and proportions of rework and salvage in handling re-workable items. Mirzazadeh [27] assumed the inflation is time-dependent and demand rate is assumed to be inflation-proportional on an inventory model. Cárdenas-Barrón [28] and Cárdenas-Barrón [29] studied the cost minimization a simple method to compute economic order quantities method which is the base of the arithmetic-geometric mean (AGM) inequality and the Cauchy-Bunyakovsky-Schwarz (CBS). Wee et al. [30] independently extended the work of Salameh and Jaber [7] to account for backorders. Chang and Ho [31] revisited an inventory problem for items with an imperfect quality item and shortage backordering, which is the same as Wee et al. [30]. However, the proposed method adopts the renewal reward theorem to obtain the expected profit per unit time. Calculus to show the first and second-order conditions for optimality using without a differential. This mentioned method applied algebraically methods to obtain the exact closed-form solves for optimal lot size, backordering quantity and maximum expected net profit. Chiu [32] presented imperfect quality manufacturing system with robust planning in optimization, specifically in determining the runtime that is subject to random machine breakdowns under abort and resumes control policy and failure-in-rework on the practical production systems. Wahab and Jaber [33] presented the optimal lot sizes for an item with imperfect quality based on Salameh and Jaber [7] with different holding costs for good and defective items and learning and no learning effects considerations. Taleizadeh et al. [34] proposed an economic production quantity/EPQ model with random defective production rate, limited production capacity and repairable item in rework process considerations. Taleizadeh et al. [35] studied an economic production quantity/EPQ subject to multi-items single-machine (MISS) which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the defective and scrap rates are randomly, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the inventory system. Chiu et al. [36] examined a finite production rate model with the rework process, stochastic machine breakdown and scrap considerations. Stochastic breakdown rate and defective rate randomly along with the reworking process of nonconforming items were assumed in their study. The objective this model was to derive the optimal production runtime by minimizes the average production cost. Besides, Raouf et al. [37] were the first to developed human error in inspection process for determining the optimal number of repeat inspections for multi-characteristic components to minimize the total expected cost per accepted component due to Type-I error, Type-II error and the cost of inspection. Khan et al. [38] investigated to Salameh and Jaber [7] and Raouf et al. [37] models to present an optimal policy for imperfect items with inspection errors and returned items to market considerations. Besides, the probability of misclassification errors is assumed to be known and the inspection process would consist of three costs: the cost of inspection, cost of Type-I errors and cost of Type-II errors. Khan et al. [39] review presented the Salameh and Jaber [7] work by focusing on studies that have extended the economic order quantity/EOQ model for imperfect items set forth. The review of these studies has been organized along the following six classifications: EOQ/EPQ models for imperfect items, shortages (backordering), quality, supply chains management, fuzzy set theory, and other extensions of the model in Salameh and Jaber [7].
Lin and Chen [40] developed a cost-minimizing Economic Order Quantity/EOQ model with imperfect quality and shortage backordering under inspection errors (including Type-I and II) and quantity discounts considerations. Hu et al. [41] according to different separating rate considered the two economic production quantities (EPQ) problem with fuzzy defective rate (LR-fuzzy number). Moreover, in which Model-I is developed when the classifying rate is lower than the production rate considered and model-II is proposed when the classifying rate is greater than the production rate. Besides, the signed distance and sample algebraic method are employed to find the optimal production quantity in so that the total cost per unit time in the fuzzy sense has a minimum value. Cárdenas-Barrón [42] suggested a new approach to sequential optimization process combining a basic concept of analytic geometry and the algebraic method to derive the optimal lot size and the backorders level for the EOQ/EPQ models with two backorders cots: linear and fixed considerations. So, this approach calls the hybrid geometric–algebraic method. Teng et al. [43] evaluated an optimal closed-form solution significantly to the integrated single-vendor single-buyer inventory system without complex derivatives. The proposed of this method seems to be simple-to-apply and easy-to-understand by individuals who are not familiar with differential calculus. Chiu et al. [44] investigated the aforementioned issues by incorporating a quality assurance and multiple delivery policy into shipment policy for an economic production quantity/EPQ model with rework and failure in repair considerations. Besides, the closed-form solutions in terms of the optimal number of shipments and optimal replenishment lot size to the problem are achieved. Liu and Zheng [45] expanded the classical economic order quantity/EOQ model by incorporating imperfect item, inspection errors and shortages back ordered contained in each ordered lot in fuzzy surroundings. The classifying process contains two types of errors; one is that a defective item may be screened to be non-defective; while the other is that a non-defective item may be screened to be defective. The fraction of defective items is assumed to be a triangular fuzzy number. Finally, compared with the Salameh and Jaber [7], the optimal order size is larger due to the shortages allowed, while the annual expected total profit is much smaller which signifies the impact of inspection errors. Taleizadeh et al. [46] investigated an economic production quantity/EPQ model with production capacity limitation and random defective item and failure during repair considerations. Their objective is to determine the optimal period lengths, back ordered quantities, and order quantities for each product so as to minimize the total expected cost. Chiu et al. [47] and Chiu et al. [48] considered an economic manufacture quantity/EMQ model where the manufacturing process may defective items produce randomly and failure in imperfect reworking process. Likewise, during the production uptime, the machine is subject to a breakdown randomly which follows the Poisson distribution. The objective is the determination of the optimal lot size and replenishment run time that minimize the expected costs per unit time. Another research has been performed by Chiu et al. [49] with the rework process, a random scrap rate, and a service level constraint consideration for an optimal lot-sizing production system. Moreover, the relationship between the “imputed backorder cost” and maximal permitted shortage level is derived for decision-making on whether the required service level is achievable.