تکمیل دوباره مشترک با اقلام ناقص و تخفیف قیمت

A joint replenishment problem is presented to determine the ordering policy for multiple items having a certain percentage of defective units. The purpose of this paper is to study the impact of the percentage of defective units on the ordering policy. Two different scenarios are presented for joint replenishment problem: (1) without price discount and (2) with price discount. For each scenario, the total expected cost per unit time is derived and algorithms are presented to determine the family cycle length and the integer number of intervals that the replenishment quantity of each item will last. Numerical examples are presented and the results are discussed.

In a multi-item inventory system, a joint replenishment policy is defined as the coordination of the replenishment of a group of items that may be ordered jointly from a single supplier. On the buyer’s side, by sharing the fixed cost associated with the joint replenishment of a group of items, the total inventory cost can be reduced by the joint replenishment policy, compared to the individual and independent replenishment policy. Similarly, on the supplier’s side, the set-up cost can be saved by manufacturing multiple items using the same facility. A joint replenishment policy is also a good practice to reduce transportation cost, if a vehicle is used to simultaneously transport the jointly ordered multiple items. Therefore, the joint replenishment policy leads to a lower unit cost, lower transportation cost, lower ordering cost, lower setup cost, lower purchasing cost, and easy scheduling. On the other hand, the joint replenishment policy increases the average inventory level and system control costs but reduces the flexibility since multiple items are stocked at a single point. In the literature, there are many studies that deal with joint replenishment problem. At the early stage, Shu (1971) proposed a simple procedure to determine the order quantity for a joint replenishment problem. Then, Goyal (1974) introduced an enumeration approach and it was claimed that this approach always yields the optimal solution. However, in order to reduce complexity in solving joint replenishment problem, Silver (1975) proposed a non-iterative method. Then, van Eijis, Heuts, and Kleijnen (1992) made a comparison between two different types of strategies for a joint replenishment problem: one is direct grouping strategy and the other is indirect grouping strategy. In direct grouping strategy, the integer number of intervals that the replenishment quantity of an item will last is not an integer multiple of the family cycle time. However, in indirect grouping strategy, the integer number of intervals that the replenishment quantity of an item will last is an integer multiple of the family cycle time. After measuring the performance of these two strategies, according to percentage of the total cost saving, the authors recommended the indirect grouping strategy. Hariga (1994) proposed two heuristic procedures that are able to give global optimal replenishment policies. Then, he also compared the proposed heuristic procedures with existing ones and showed that the proposed heuristic procedures can result in the lowest cost. Wildeman, Frenk, and Dekker (1997) presented an efficient optimal solution by applying Lipschitz optimization for a joint replenishment problem instead of using a heuristic solution. A genetic algorithm model was proposed by Khouja, Michalewicz, and Satoskar (2000) and it was compared with RAND heuristic algorithm. Cha and Moon (2005) presented a modified RAND heuristic algorithm and a modified genetic algorithm. Recently, Hong and Kim (2009) proposed a new genetic algorithm by estimating the exact average cost. Yoo and Lin (2009) developed another algorithm where genetic algorithm and simulated annealing are cooperatively used to reduce the total relevant cost. The above studies in the literature on the joint replenishment policy assume that all units in all items procured from a supplier have perfect quality and no defects. However, Wahab, Mamun, and Ongkunaruk (2011) indicated that although many small and medium enterprises implemented total quality management to ensure the quality of the units manufactured, buyers still find defective units, which fail to meet the quality requirements defined by buyers, in the procured items. There are a number of studies that investigated the effect of defective units on the economic order quantity (EOQ) inventory model. For example, Salameh and Jaber (2000) developed an EOQ model where each shipment consisted of a percentage of defective units, which were sold at a discount price after completely screening the lot. For the same model, a simple solution procedure was presented in Goyal and Cardenas-Barron (2002). Then, Papachristos and Konstantaras (2006) derived the sufficient condition to ensure no shortages at the end of the cycle. Maddah and Jaber (2008) revisited and corrected the flaw in Salameh and Jaber (2000). Wahab and Jaber (2010) considered different holding cost for good and defective items for the model presented in Salameh and Jaber (2000). Recently, Khan, Jaber, Guiffrida, and Zolfaghari (2011) presented a comprehensive review on EOQ models with defective units, and Jaber, Zanoni, and Zavanella (2013) proposed an EOQ model with imperfect quality based on entropy. However, to the authors’ knowledge there is no study considering the presence of defective units in a joint replenishment policy. This paper contributes to the literature in the area of joint replenishment problem. In particular, a joint replenishment policy is developed when the shipment size for each item has a percentage of defective items. This paper proposes an ordering policy for multiple items, where each item has a certain percentage of defective units in each shipment. A joint replenishment model is considered to determine the family cycle length and the integer number of family cycles that the replenishment quantity of an item will last by minimizing the total cost associated with the system. The demand for each item is considered to be deterministic and shortages are not allowed; and therefore each item will be replenished when its inventory level drops to the zero level. Since each item has a percentage of defective units in each replenishment, the replenished lot is screened by the buyer to separate the defective units while only the non-defective units satisfy the demand. When the screening is completed, the segregated defective units in each item are disposed at a certain cost. Two different models are proposed: first, a joint replenishment model with no price discount is introduced; second, another model is introduced with a group discount on unit price, where the buyer will receive a price discount when the total replenishment quantity exceeds the breakpoint quantity. This paper is organized as follows: Section 2 presents the model for a joint replenishment problem with defective units and provide an algorithm to determine decision variables. Section 3 presents a joint replenishment problem with price discount and an algorithm to determine decision variables. Section 4 illustrates numerical examples. Section 5 concludes the paper.

A joint replenishment problem is studied where a group of items are ordered from a single supplier, and each item has a percentage of defective units in each replenishment. The optimal family cycle length, the integer number of intervals that the replenishment quantity will last for each item, the order quantity for each item, and the total expected cost per unit time are determined. Examples are presented to show the impact of imperfect quality of the items on the ordering policy and the total cost. The results show that as the expected percentage of defective units increases, optimal family cycle length decreases to a certain extent to offset the defective units, so that the expected number of non-defective units is equal to the demand during the integer number of family cycles. However, as the expected percentage of defective units keeps increasing, the expected number of non-defective units is insufficient to meet the demand during the current integer number of family cycles. Hence, the integer number of family cycles that the replenishment quantity will last drops to the next level and this leads to a smaller order quantity. Then, the expected number of non-defective units matches the demand during the smaller integer number of family cycles. There are several ways to extend this research. For example, future research could focus on developing a solution methodology using RAND or genetic algorithm (e.g., Lin & Wang, 2012) while considering a percentage of defective units for each item in each replenishment. Considering stochastic demand for each item or multiple suppliers could also be another future research. In the proposed model, we assume that a complete screening process is implemented by the buyer when a shipment is received. However, as the vendor is very familiar with production processes and possible defects, a premium can be paid to the vendor in addition to the purchasing price, so that the vendor would screen the items completely to ensure the quality. Hence, a model could be developed in the line of Rezaei and Salimi (2012) and that can be compared with the current proposed model to investigate the benefit of such an agreement. Moreover, in this research, it is assumed that the entire shipment size for each item is screened completely. However, acceptance sampling is sometimes used to reduce the cost of screening (e.g., Maddah et al., 2010 and Moussawi-Haidar et al., 2013), and also screening process to eliminate the defective items may involve in both Type I and Type II errors. Hence, another future research could be to incorporate acceptance sampling with Type I and Type II errors in the proposed model.