هماهنگی زنجیره تامین دو سطحی با وقفه تولید برای بازگرداندن کیفیت فرآیند

The classical inventory model, known as the lot size problem (LSP), assumes items produced and stocked to be of perfect quality. In reality, production processes are not defect free. Such imperfect processes generate defects that are reworked, and in some cases scrapped. The move by practitioners towards smaller lot sizes; e.g., just-in-time (JIT), as a mean of improving process and product quality prompted researchers to modify the LSP to represent reality more faithfully. The studies conducted by these researchers confirmed the observed benefits of smaller lot sizes. The inventory models developed in these studies assume that defects increase as the lot size increases, and that adjusting the process within a production cycle is not possible. Contrary to this assumption, in a JIT environment workers are authorized to stop production if a quality or production problem arises. This practice encouraged Khouja [Khouja, M. (2005). The use of minor setups within production cycles to improve product quality and yield. International Transactions in Operations Research, 12(4), 403–416] to reformulate some inventory models which take into account the negative relationship between lot size and quality and the possibility of performing minor setups. This requires stopping the process during a production cycle. This paper investigates the work of Khouja in a centralized decision model where players in a two-level (manufacturer–retailer) supply chain coordinate their orders to minimize their local costs and that of the chain. Mathematical models are developed with numerical results discussed.

The simplistic assumptions of the lot size problem, also known as the economic order/manufacture quantity (EOQ/EMQ) model, which make its mathematics easy to use and understand is probably why the lot size problem has been widely used and accepted by researchers and practitioners alike. Among the assumptions of the lot size problem is that items produced and stocked are of perfect quality. Production processes are not defect free, and result in items that require reworking (e.g., Agnihothri and Kenett, 1995 and Buzacott, 1999). Rework occurs when a product or a service does not meet the internal or external quality requirements, and could be defined as doing something at least one extra time due to non-conformance to these requirements (e.g., Bohn and Terwiesch, 1999 and Love et al., 1999). This additional effort adds cost and not value to the product, which the just-in-time (JIT) philosophy considers as waste to be eliminated. The JIT also advocates that inventory is a blanket that covers problems is production and quality (Waters, 2003). To uncover these problems inventory levels must be reduced. This led many manufacturers to move towards smaller lot sizes to improve process and product quality; i.e., to reduce reworks. Porteus, 1986 and Rosenblatt and Lee, 1986 were probably the first to independently investigate the negative relationship between lot size and quality. Although these models approach the deterioration of the process differently, they result in similar conclusions (Urban, 1998). The works of Porteus, 1986 and Rosenblatt and Lee, 1986 assume that once the production process goes out-of-control it is never interrupted and continues in that state until the entire lot is produced. Defective items generated are reworked once production ceases. Khouja (2005) corrected this assumption by allowing for adjustments to the process within a production cycle. The assumption of Khouja (2005) is corroborated by a common practice in JIT manufacturing environments where line workers have the authority to stop the line if a quality or production problem arises (Inman & Brandon, 1992). Khouja (2005) reformulated some inventory models which take into account the negative relationship between lot size and quality and the possibility of performing minor setups. Performing minor setups requires stopping the process during a production cycle. His results showed that interrupting the production process to bring it back in control reduces reworks. The models developed by Khouja (2005) have not been investigated in a supply chain context. Supply chain management emerged in the late 1990s and the beginning of this millennium as a source for sustainable competitive advantage for companies (Dell & Fedman, 1999). It involves functions such as production, purchasing, materials management, warehousing and inventory control, distribution, shipping, and transport logistics. To attain sustainable competitiveness in these functions, effective coordination among the players representing these functions in the supply chain is required. The effectiveness of coordination in supply chains could be measured in two ways: reduction in total supply chain costs and enhanced coordination services provided to the end customer, and to all players in the supply chain (Pagel, 1999). Coordination schemes in supply chains are based either on centralized or decentralized decision-making processes. In the case of a centralized decision-making process, there is a single decision-maker whose main objective is to minimize (maximize) the total supply chain cost (profit). Whereas in the case of a decentralized decision-making process, there are multiple decision-makers in a supply chain, where each decision-maker tends to optimize his/her own performance leading to an inefficient system. This paper assumes a centralized decision-making process as a scheme for coordination in the supply chain model to be investigated herein. The case of no coordination among the players in a supply chain may be viewed as a decentralized decision-making process, where a retailer orders according its EOQ from its supplier (the manufacturer), who in its turn determines its economic order quantity and orders it from its supplier (the manufacturer’s supplier). In this paper, the case of no coordination will be used as the base case for comparison of results. The work of Goyal and Gupta (1989) is the earliest review of literature on the interaction between a buyer and a vendor (a two-level supply chain). Their survey showed that coordination could be achieved by integrating lot-sizing models. Coordination in a supply chain is not possible without incentives schemes, with quantity discounts being the most common one (e.g., Moses and Seshadri, 2000 and Munson and Rosenblatt, 2001). Readers may refer to the work of Dolan (1987) for review of quantity discounts. Limited are those works that investigated an integrated inventory problem for imperfect production process; i.e., processes generating defective items. Goyal, Huang, and Chen (2003) developed a simple approach for determining an optimal integrated vendor–buyer inventory policy for an item with imperfect quality. In their model, Goyal et al. (2003) assumed that defective (or poor quality) items are not reworked but rather sold at a discounted price. A similar problem to that of Goyal et al. (2003) was investigated by Comeaux and Sarker (2005). Khouja (2003) is believed to be the first to formulate and solve a two-stage (manufacturer–retailer) supply chain inventory model in which the quality of the output deteriorates with increased lot sizes. He investigated his model independently for the assumptions of Porteus, 1986 and Rosenblatt and Lee, 1986. Khouja (2003) assumed no incentives scheme for coordination. Siajadi, Ibrahim, Lochert, and Chan (2005) presented an integrated inventory system where a manufacturer consumes and purchases raw materials from a supplier in order to produce a specific finished item. They assumed that these defective items can be reworked instantaneously at a cost and kept in stock. Siajadi et al. (2005) also assumed a random percentage of defective items per lot, and no incentive scheme for coordination. This paper investigates the work of Khouja (2005), who assumed the number of defectives per lot to be proportional to the lot size, in a centralized model where players in a two-level (manufacturer–retailer) supply chain coordinate their orders to minimize their local costs and that of the chain. The literature on coordinating orders in a supply chain show cost of one of the players after coordination is higher than its original cost without coordination (e.g., Goyal & Gupta, 1989). Therefore, this paper assumes that the player with a better financial position will compensate the other player for its loss. This compensation may take the form of quantity discounts. The paper is organized as follows. The next section presents the notations and assumption necessary for modeling. This section is followed by mathematical modeling and numerical results sections. The last section is for summary and conclusions.

This paper investigates the work of Khouja (2005) in a two-level supply chain (manufacturer–retailer) with coordination. Khouja (2005) and unlike the work of Porteus (1986) and its extensions, it is the first to assume that a production process can be interrupted to restore process quality at a cost. However, an additional cost represented by minor setup is incurred. Three cases that describe the behavior of the manufacturer’s inventory level were considered. These cases are: (1) Case 1: restore the production process after delivering a lot to the retailer, (2) Case 2: restore the production process before delivering a lot to the retailer, and (3) Case 3: restore the production process at any time during production. Our results indicate that without coordination and when the manufacturer’s inventory behavior follows either Case 1 or Case 2, the optimal policy for the manufacturer is to follow a lot-for-lot policy where no minor setups are performed. This policy was not true for Case 3, which registered very good results even better than those of Cases 1 and 2 with coordination. This is because of the flexibility that Case 3 in lot delivery to theretailer which reduces the holding and rework costs. Our results also indicated that with coordination the trend is to order in smaller lots than the retailer’s economic order quantity more frequently. This corroborates the findings that smaller lots are recommended when production is imperfect. The effect of the length of the period required to restore process quality, s , on the supply chain cost was also investigated. Our results indicated that there exists an optimal value of s that minimizes the supply chain cost. Besides determining the optimal order and production policies in a supply chain, our results also suggest that attention must also be paid when specifying the length of the period required to restore a production process and the frequency of these restorationsSimplified forms of the mathematical models developed in this paper are needed to facilitate their imple- mentation in a real world setting. This issue will be addressed in a future work.