Mathematical Problems in Engineering

Volume 2016, Article ID 5382329, 10 pages

http://dx.doi.org/10.1155/2016/5382329

## Optimal Buyer’s Replenishment Policy in the Integrated Inventory Model for Imperfect Items

^{1}Hechi University, Hechi 546300, China^{2}Guangxi University, Nanning 530004, China

Received 31 December 2015; Revised 14 June 2016; Accepted 9 August 2016

Academic Editor: Paolo Crippa

Copyright © 2016 Lu Yueli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the classical economic order quantity (EOQ) models, a common unrealistic assumption is that all the items received are of good quality. However, in realistic environment, a received shipment usually contains a fraction of imperfect quality items. These imperfect items may be scrapped, reworked at a cost, or salvaged at a discounted price. While the percentage of imperfect items is random, the optimal ordering cycle is rarely considered in current literatures. This paper revisits the model (Maddah and Jaber, 2008) and extends it by assuming that the ordering cycle is determined by the demand rate, delivery quantity per shipment, and the mathematical expectation of the defective rate. The possibility of stockout or residue in the end of a cycle will be considered, and the loss of stockout and the salvage of the residue are counted into the cost. Besides, we consider consolidating the shipments of imperfect items over multiple deliveries. Thus, an integrated vendor-buyer inventory model for imperfect quality items with equal-size shipment policy is established to derive the optimal ordering cycle, ordering quantity, and number of deliveries. The computational method of the optimal delivery quantity per shipment and number of deliveries is given through theoretical results. Finally, sensitivity of main parameters is analyzed through simulation experiments and shown by some figures.

#### 1. Introduction

Management’s view on inventory has been changed significantly over the several decades. Previously, managers perceived inventory as an asset. Since the life cycles of product are becoming ever shorter, a wide variety of problems concealed in the excessive inventories are prominent, for instance, the expensive inventory storage costs. Plant managers make their effort to reduce inventories in all categories, from raw materials, purchased parts, and working process, to the finished goods. The economic order quantity (EOQ) model was first introduced by Harris [1] in the last century to settle two fundamental questions: “when should an order be placed?” and “how much should it be ordered?”. This great work assisted companies in reducing their total inventory costs and provided managers with guidelines to choose the appropriate replenishment policy. Since then, a large amount of extensions to the EOQ model have been studied in the past decades. An unrealistic assumption in Harris’s model [1] is that all the items received are of good quality. In reality, the goods received always contain some items with imperfect quality, due to many unpredictable factors, such as the imperfect production or transportation. The imperfect items may be scrapped, reworked at a cost, or salvaged at a discounted price, which causes the additional costs. Therefore, the effects of imperfect quality must be considered when seeking policies for optimizing order quantity and the other related inventory control parameters, such as the ordering cycle and shipment times. Porteus [2] and Rosenblatt and Lee [3] began to investigate the imperfect product process problem firstly. Along their line of research, scholars have made abundant achievements.

It is well known that Salameh and Jaber [4] first proposed an EOQ model in which a random ratio of the products in an order is defective. The buyer conducts a 100% inspection to split the qualified products from the defective ones, after which the defective products are sold at a discounted price as a single batch at the completion of the inspection. Their model has been extended from various aspects. Goyal and Cárdenas-Barrón [5] amended a tiny error in the initial model and presented a simpler approach that approximately determined the economic production quantity. Rezaei [6], Eroglu and Ozdemir [7], and Wee et al. [8] extended the original model to account for backorders. Papachristos and Konstantaras [9] discussed the assumption in original model to avoid shortage and extended the original model on the assumption that defective products were accumulated until the end of the planning horizon and then withdrawn. Maddah and Jaber [10] rectified a flaw in original model related to the approach of assessing the expected profit. Chang and Ho [11] revisited the work of Wee et al. [8] and applied the renewal-reward theorem in the model and acquired a new expected net profit. Chung and Huang [12] first incorporated imperfect quality with trade credit, and then Jaggi et al. [13] extended the model to the situation that shortages were allowed and fully backlogged. Based on the learning curve theory, Konstantaras et al. [14] investigated an allowable shortage EOQ model for the cases of infinite and finite planning horizons when the fraction of defective items can be reduced by learning; Alamri et al. [15] established a general EOQ model for imperfect quality items by assuming that demand, screening, and deterioration rates were all arbitrary functions of time. Moussawi-Haidar et al. [16] developed two modified EOQ-type models for deteriorating items when the proportion of defective items follows a known distribution. They first assumed that no shortages are allowed, that is, the inventory level is just enough to cover the demand during the screening period, and then relaxed the assumption and allowed for backordering. Finally, they analyzed the impact of deterioration on the instantaneous replenishment model and presented the managerial implications. Unlike the abovementioned studies in which the inspection process is conducted by buyer, Rezaei and Salimi [17] studied economic order quantity and purchasing price for items contained defects when inspection process transfers from buyer to vendor. Yassine et al. [18] considered an augmented EPQ model in which the inspection is conducted by supplier. They first developed a base model where the defective products are repaired once at the end of each production circle. Then, they extended the base model to two scenarios, disaggregating the shipments of defective products in a single run and aggregating the shipments of defective products over multiple runs, and derived closed-form expressions for the economic production quantity. Chang [19] corrected some typos that appeared in Yassine et al. [18] soon afterwards. Noting that inspector of product quality control might make some inspection errors during the separation of imperfect and perfect items, Taheri-Tolgari et al. [20] considered two types of inspection errors under inflationary conditions. J.-T. Hsu and L.-F. Hsu [21, 22] established an EOQ model and two EPQ models for imperfect quality items, respectively, by integrating with inspection errors, shortage backordering, and sales returns. Based on the unreliable supply, Moussawi-Haidar et al. [23] developed an instantaneous replenishment model for imperfect items under a sampling policy, that is, using an acceptance sampling inspection instead of 100% screening, whose results indicated that the optimal profit obtained was significantly higher. Jaber et al. [24] presented two models by assuming that imperfect items were repaired or replaced by good ones from a local supplier at a higher cost when the shipment was provided by a distant supplier, signifying a practical implication in today’s globalised supply chains. Sarkar et al. [25] developed three different economic production quantity models for three different distribution density functions with rework process and planned backorders for a single stage manufacturing system. Al-Salamah [26] presented an EPQ model based on a scenario where the production process and inspection are both imperfect. They sought the optimal lot size for a manufacturer while batches are subjected to destructive or nondestructive acceptance sampling process before batches can be sent out to the market. In order to investigate the effect of yield variability and the stochastic behavior of the defective rate, L.-F. Hsu and J.-T. Hsu [27] developed different EPQ models with shortages backordered for cases of constant and random defective rate under three moments of withdrawing the defective items: (a) as they are detected, (b) at the end of each production period, or (c) at the end of each production cycle. They derived out the optimal production lot size and backorder quantity and showed that the effects of both yield variability and withdrawal timing are not critical factors, which implicated that managements could make decisions by taking advantage of the mean defective rate when the probability density function of the defective rate is difficult to obtain. For a detailed review and discussion of the extensions of the original model proposed by Salameh and Jaber [4], see Khan et al. [28].

In the abovementioned models, the ordering cycle can be roughly divided into two kinds. One kind is that the cycle time is relative to the demand rate* D*, the order quantity , and the defective rate which is a random variable; that is, . Thus, is a random variable too. Most of the models belong to this kind, such as Salameh and Jaber [4], Goyal and Cárdenas-Barrón [5], Eroglu and Ozdemir [7], Wee et al. [8], Papachristos and Konstantaras [9], Chung and Huang [12], and Yassine et al. [18]. However, none of the abovementioned models derived the optimal cycle time. The other kind is that the cycle time is a deterministic variable; that is, , where is the mathematical expectation of . The optimal cycle time is obtained when the optimal ordering quantity is found. The models proposed by Rezaei [6] and L.-F. Hsu and J.-T. Hsu [27] are examples. In this kind of the model, the defective rate is a random variable and the cycle time , the demand rate , and order quantity are deterministic. Based on such an assumption, it is obvious that two possibilities may happen at the end of cycle: stockout or residue. This will result in the loss of stockout or the salvage of residue. However, there is no related discussion about the case of residue at the end of cycle in the abovementioned models.

In this paper, an integrated supplier-buyer inventory model for imperfect quality items with equal-size shipment policy is established to derive the optimal ordering cycle, order quantity, and number of deliveries. This study extends the model proposed by Maddah and Jaber [10] in three ways. First, it considers the optimal ordering cycle by assuming that the cycle time is determined by the demand rate* D*, delivery quantity per shipment , and the mathematical expectation of the defective rate* p*, just like that of Rezaei [6]. Secondly, stockout and residue in the end of a cycle are considered due to the randomness of the defective rate, and the loss of stockout and the salvage of the residue are counted into the cost. This is quite different from that of Rezaei [6], which only considered the possibility of shortage. Thirdly, this model incorporates with equal-size shipment policy; that is, the supplier delivers a lot to the buyer in shipments of equal size. A 100% screening process is implemented when the buyer received each shipment.

The residual of this paper is organized as follows. Section 2 states the notations and assumptions of the model. Sections 3 and 4 are for the mathematical modeling and the theoretical results to determine the optimal ordering quantity and number of deliveries, respectively. Section 5 provides numerical examples and analysis of results. Section 6 summarizes and concludes the paper.

#### 2. Notations and Assumptions

##### 2.1. Notations

: ordering cycle (decision variable), : order quantity per cycle (decision variable), : delivery quantity per shipment (decision variable), : number of deliveries (decision variable), : demand rate per year, : ordering cost per order, : screening rate, : screening cost per unit, : holding cost per unit per year, : purchasing price per unit item, : percentage of defective items in one lot, a random variable, : expectation of random variable , : selling price of qualified items per unit, : residual value of defective items per unit, : residual value of qualified items per unit, : shortage cost per unit, : mathematical expectation of , : , : the largest integer that is less then or equal to .

##### 2.2. Assumptions

(a)Time horizon is infinite.(b)Demand rate, , is known and constant.(c)Lead time is omitted, and the items are delivered instantaneously.(d)The items provided by supplier contain a percentage of defective items, where is a random variable over interval with the probability density function and the mathematical expectation , where .(e)The buyer’s ordering cycle is and places an order of quantity in each cycle. The supplier delivers them in shipments of equal sizes with the time interval . Obviously, and .(f)When a shipment is received, the buyer begins a 100% inspection process at rate immediately. To avoid stockout during the screening time, it is assumed that ; this condition insures that the amount of perfect quality items in a shipment is at least equal to the demand during the screening time.(g)The defective items are stored and consolidated at a lot at the end of the final inspection period in th time interval , that is, . The buyer can gain residual value of the defective items at a discounted price of per unit.(h)At the end of time interval , that is, at the moment , the residual qualified items are sold at a discounted price of per unit if residue happens, or the shortages lead to a penalty cost of per unit, .

#### 3. Model Formulation

Based on the assumptions, the buyer places an order of quantity at the beginning of each cycle. Then, the supplier delivers the items in shipments of equal size at time . The buyer inspects the received items immediately. The defective items are stored till the last screening process finishes, that is, at the time of , and then consolidated in one lot. The buyer’s aim is to determine the optimal delivery quantity per shipment and the optimal number of deliveries such that the expected profit per unit time is maximized.

The total revenue of the buyer includes the revenue of qualified and the salvage value of defective items and residual qualified items, which are separately calculated as follows:

(a) The revenue of qualified items is .

(b) The salvage value of defective items is .

(c) The salvage value of residual qualified items is .

Thus, the total revenue of the buyer in a cycle is

The total cost of the buyer includes purchasing cost, screening cost, shortage cost, and holding cost. They are separately calculated as follows:

(d) The purchasing cost is .

(e) The inspection cost is .

(f) The shortage cost is .

(g) The holding cost of qualified items depends on the inventory level at the end of the time interval .

When there is no shortage at the end of the time interval , the behavior of inventory level of qualified items is illustrated in Figure 1. The quantity of storage of qualified items isWhen shortage happens at the end of the time interval , the behavior of inventory level of qualified items is illustrated in Figure 2. The quantity of storage of qualified items isSince if and only if , the quantity of storage of qualified items in a cycle can have a unified representation as follows: