Wee et al. [Optimal inventory model for items with imperfect quality and shortage backordering. Omega 2007;35(1):7–11] recently contributed an optimal inventory model for items with imperfect quality and shortage backordering. This article revisits their study and applies the well-known renewal-reward theorem to obtain a new expected net profit per unit time. We derive the exact closed-form solutions to determine the optimal lot size, backordering quantity and maximum expected net profit per unit time, specifically without differential calculus. We also solve the same model algebraically from another direction, which has been mentioned, but the process has not been finished yet. The problem parameter effects upon the optimal solutions are examined analytically and numerically.
The classical economic order/production quantity (EOQ/EPQ) models, although well known and useful, implicitly assume that all products have perfect quality, which may not conform to real situations since defective items often exist in order/production batches. Consequently, many researchers developed inventory/production models to include defective items and/or imperfect production process, to study the effects of imperfect quality on the optimal inventory/production policies. Porteus [1] and Rosenblatt and Lee [2] were the first to address the imperfect production process problems. Their works have encouraged many researchers to explore quality related issues, see e.g. [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and [13]. Specifically, Wee et al. [9] extended Salameh and Jaber's [3] EOQ model for items with imperfect quality to include shortages. Both of them [3] and [9] assumed that each lot received or produced contains a random fraction of imperfect quality items. The defective items are picked up as a single batch within the replenishment period, where a 100% lot screening process is conducted at a fixed rate. Items of poor quality are sorted, kept in stock and sold at a salvage value prior to receiving the next shipment. In a recent article, Maddah and Jaber [11] enhanced [3] by applying renewal theory (see, e.g. Ross [14]) to obtain the expected profit per unit time. They indicated that since the process generating the profit is a renewal process, hence it is more appropriate to measure the expected profit function using renewal-reward theory.
The classical optimization technique based on differential calculus undoubtedly is a powerful and useful method to solve inventory decision models. The above mentioned studies used it to find the optimal solutions or derive the conditions for optimality, but it is replaceable. Grubbström [15] first derived the classical EOQ formula algebraically. Grubbström and Erdem [16] extended this approach to solve EOQ model with backlogging. These two studies received considerable attention and invoked many researchers to propose various algebraic methods to solve inventory related models with or without shortages (see, e.g., [17], [18], [19], [20], [21] and [22]). A comprehensive review of the literatures up to 2006 can be found in Cárdenas-Barrón [23], who further classified the algebraic procedure according to its difficulty level as simple, medium and high. Recent studies proposed other distinct approaches. For example, Teng [24] suggested the arithmetic geometric mean inequality (AM-GM) theorem, Hsieh et al. [25] provided the Cauchy–Schwarz inequality and Wee et al. [26] modified the cost-difference comparison method. Besides, Chang et al. [27] pointed out a direction for further research to algebraically solve an EPQ model with shortages. To the best of our knowledge, this open problem has not yet been solved. Moreover, we note that in the most recent studies Pentico et al. [28] and Zhang [29] also proposed the new and/or alternative approaches to solve the EPQ model with partial backordering.
In this article, we shall revisit Wee et al.'s [9] model (an optimal inventory model for items with imperfect quality and shortage backordering). As in Maddah and Jaber [11], we first apply the renewal-reward theorem to derive new expected net profit per unit time. Without using differential calculus, we derive the exact closed-form solutions for optimal lot size and maximum shortage level, as well as the expected profit per unit time. We also propose an approach to finish the task left in Chang et al. [27].
This article addressed an inventory problem for items with imperfect quality and shortage backordering, which is the same as Wee et al. [9]. However, the proposed method adopts the renewal-reward theorem to derive the expected net profit per unit time. Without using differential calculus to show the first and second order conditions for optimality, this study applied algebraic methods to derive the exact closed-form solutions for optimal lot size, backordering quantity and maximum expected profit. It also provided an approach to solve the same problem algebraically from another direction, which has not yet been attained in the literature.
We examined the problem parameter effects on the optimal policies analytically and numerically. The results show that as backordering cost or the variation in defective percentage increases, or the screening rate decreases, the optimal lot size, backordering quantity, and maximum expected profit then decrease. As the expected value of the defective percentage increases, the optimal backordering quantity and the maximum expected profit decrease, but the optimal lot size may increase or decrease.
As mentioned earlier, the model proposed in Wee et al. [9] and studied here should be assumed that the backordered products are delivered without any defects before 100% screening process is conducted. Future research may recast the model by relaxing this assumption.
