یک الگوریتم ژنتیکی برای سیستم کنترل موجودی موفق فروشنده چند محصولی با محدودیت چندگانه مدل میزان سفارش اقتصادی

In this research, an economic order quantity (EOQ) model is first developed for a two-level supply chain system consisting of several products, one supplier and one-retailer, in which shortages are backordered, the supplier’s warehouse has limited capacity and there is an upper bound on the number of orders. In this system, the supplier utilizes the retailer’s information in decision making on the replenishments and supplies orders to the retailer according to the well known (R, Q) policy. Since the model of the problem is of a non-linear integer-programming type, a genetic algorithm is then proposed to find the order quantities and the maximum backorder levels such that the total inventory cost of the supply chain is minimized. At the end, a numerical example is given to demonstrate the applicability of the proposed methodology and to evaluate and compare its performances to the ones of a penalty policy approach that is taken to evaluate the fitness function of the genetic algorithm.

The globalization of economy and liberalization of marketplace at an increasingly rapid pace has intensified the need for incorporating the resulting operational uncertainties and financial risks into the firms’ production and inventory control decisions (Mohebbi, 2008). Inventory control has been studied for several decades for cost savings of enterprises who have tried to maintain appropriate inventory levels to cope with stochastic customer demands and to boost their image through customer satisfaction (Axsäter, 2000, Axsäter, 2001, Moinzadeh, 2002 and Zipkin, 2000). One of the key factors to improve service levels of the enterprises is to efficiently manage the inventory level of each participant within supply chains (Kwak, Choi, Kim, & Kwon, 2009). A supply chain (SC) is a network of firms that produce, sell and deliver a product or service to a predetermined market segment (Chopra & Meindl, 2001). It not only includes the manufacturers and suppliers, but also transporters, warehouses, retailers and customers themselves. The term supply chain conjures up images of a product or a supply moving from suppliers to manufacturers then distributors to retailers and then customers along a chain (Chopra, 2003). Customers and their needs are the origin of the SC. The main objective of the supply chain management is to minimize system-wide costs while satisfying service level requirements (Tyana & Wee, 2003). One of the well-known concepts utilized in supply chains is the vendor managed inventory (VMI) models (see for example Cheung and Lee, 2002 and Disney and Towill, 2003) and many successful businesses have demonstrated the benefits of VMI, e.g., Wal-Mart and JC Penney (Cetinkaya and Lee, 2000 and Dong and Xu, 2002). In these models, the retailer provides the supplier with information on its sales and inventory level and the supplier determine the replenishment quantity at each period based on this information. In other words, the supplier with regard to his own inventory cost that equals to the total inventory cost of the supply chain determines the timing and the quantity of replenishment in every cycle (Dong and Xu, 2002, Kaipia et al., 2002 and Lee et al., 2000). Not only VMI has some advantages for both the retailer and the supplier, but also the customer service levels may increase in terms of the reliability of product availability. Since the supplier can use the information collected on the inventory levels at the retailers, future demands are better anticipated and the deliveries are better coordinated (for example, by delaying and advancing deliveries according to the inventory situations at the retailers and the transportation considerations) (Kleywegt et al., 2004 and Waller et al., 1999). One of the most important problems in companies that utilize suppliers to provide raw materials, components, and finished products is to determine the order quantity and the points to place orders. Various models in production and inventory control field have been proposed and devoted to solve this problem in different scenarios. Two of the models that have been extensively employed are the economic order quantity (EOQ) and economic production quantity (EPQ) models (Silver, Pyke, & Peterson, 1998; Tersine, 1994). The economic order quantity (EOQ) is one of the most popular and successful optimization models in supply chain management, due to its simplicity of use, simplicity of concept, and robustness (Teng, 2008). However, these models are constructed based on some assumptions and conditions that bound their applicability in real-world situations. In this research, a multi-product EOQ model is proposed in which not only the storage capacity, but also the number of orders is limited. Furthermore, in order to broaden the applicability of the proposed model in real-world inventory control systems, we consider shortage to be backordered and let the model act in a supply chain environment under the VMI condition. The objective is to find the order quantities and the maximum backorder level of the products in a cycle such that the total inventory cost of the supply chain is minimized. Under these conditions, the problem is first formulated as a non-linear integer-programming (NIP) model and then a genetic algorithm (GA) is proposed to solve it. At the end, a numerical example is presented to demonstrate the application of the proposed methodology and to compare its performance to the one of the penalty policy approach that is employed as another way to evaluate the fitness function the GA chromosome. The remainder of the paper is organized as follows: a review of the literature is presented in Section 2. We define and model the problem in Sections 3 and 4, respectively. In Section 5, a genetic algorithm is developed to solve the problem. In order to demonstrate the application of the proposed methodology, we provide a numerical example in Section 6. Finally, conclusions and recommendations for future research are provided in Section 7.

In order to make the EOQ model more applicable to real-world production and inventory control problems, in this paper, we expanded this model by assuming several products in which shortages were backordered, the storage had limited capacity and there was an upper bound on the total number of orders. The proposed model of this research applies to a two-level supply chain consisting of a single-supplier and a single retailer that operates under vendor management inventory (VMI) system. Under these conditions, we formulated the problem as a non-linear integer-programming (NIP) model and proposed a genetic algorithm to solve it. At the end, a numerical example was presented to demonstrate the application and the performance of the proposed methodology and to compare it to a penalty policy that was applied to fitness evaluation. For future researches in this area, we recommend the followings: (a) In addition to the storage capacity and the total number of order limitations, we may consider budget and other constraints too. (b) Other meta-heuristic search algorithms such as simulated annealing may also be employed and a comparison may be made among the algorithms. (c) Instead of backorder assumption, we can consider lost sale for shortages. Furthermore quantity discounts can be employed. (d) Some of parameter of the model may be either fuzzy or random variable. In this case, the model has either fuzzy or stochastic nature. (e) We can consider multi-echelon supply chains such as; one-retailer multiple-supplier, multiple-retailer single-supplier and multiple-retailer multiple-supplier systems.