یک مدل یکپارچه شده برای تعیین اندازه دسته تولید با انتخاب تامین کننده و مقدار تخفیف

Good inventory management is essential for a firm to be cost competitive and to acquire decent profit in the market, and how to achieve an outstanding inventory management has been a popular topic in both the academic field and in real practice for decades. As the production environment getting increasingly complex, various kinds of mathematical models have been developed, such as linear programming, nonlinear programming, mixed integer programming, geometric programming, gradient-based nonlinear programming and dynamic programming, to name a few. However, when the problem becomes NP-hard, heuristics tools may be necessary to solve the problem. In this paper, a mixed integer programming (MIP) model is constructed first to solve the lot-sizing problem with multiple suppliers, multiple periods and quantity discounts. An efficient Genetic Algorithm (GA) is proposed next to tackle the problem when it becomes too complicated. The objectives are to minimize total costs, where the costs include ordering cost, holding cost, purchase cost and transportation cost, under the requirement that no inventory shortage is allowed in the system, and to determine an appropriate inventory level for each planning period. The results demonstrate that the proposed GA model is an effective and accurate tool for determining the replenishment for a manufacturer for multi-periods.

Having a good production planning and replenishment control through effective inventory management is important for a firm to keep competitive in the market. The single product, multi-period inventory lot-sizing problem is one of the most common and basic problems, and it has often been tackled in the literature [1]. There are various extensions of the model to consider different issues in real environment. This research considers an environment with multiple periods and multiple suppliers. Different suppliers may have either all-units quantity discounts or incremental quantity discounts. In addition, transportation cost is fixed for each vehicle shipment, and it is different for different suppliers. By adopting the proposed models, the management can decide what quantity to order from which suppliers in which periods. Quantity discount is a common and effective practice for suppliers to promote their products, and buyers can purchase products at a lower unit price when the ordering quantity is over a certain amount. In addition, multi-suppliers give firms a chance to purchase same materials from different sources. Ordering cost for each purchase can be in several different forms, for example, fixed, increasing or decreasing. The assumption of a fixed ordering cost is often seen in works [1], [2] and [3]. In Rezaei and Davoodi [4], ordering cost consists of a fixed cost and an additional cost. The fixed cost is independent of the lot-size, and the additional ordering cost depends on the specific lot-size. In Rezaei and Davoodi [5], reduction in ordering cost is positively related to ordering frequency, that is, the higher the ordering frequency from a supplier, the higher the ordering cost reduction from that supplier. In this research, we set the ordering cost to be fixed because this is more suitable for our case study in real practice. In addition, quantity cost is considered in the study so that the more quantity ordered in each purchase, the lower the unit cost is. Even though there have been abundant works which handled lot sizing problems in quantity discount environment and used MIP or other methods to solve the problems, the authors, after reviewing these papers, found out that very few papers have considered an environment with multiple periods, multiple suppliers, and both all-units and incremental quantity discounts simultaneously. With regard to the complexity of the lot-sizing problems, Rezaei and Davoodi [4] and [5], Florian et al. [6], and Bitran and Yanasse [7], claimed that this kind of problem basically belongs to a class of NP-hard problems. Therefore, to solve such a complicated problem, this paper proposes a MIP model to solve a small-scale problem and to compare the results with the proposed GA model first. The GA model is then used to tackle the problem when it becomes too complicated for the MIP model to solve. The proposed models are then compared with four past works that considered a similar problem environment. The comparisons show that the proposed models have more attributes than the past works. The remaining of this paper is organized as follows. In Section 2, some related methodologies and works are reviewed. In Section 3, the problem under consideration and the assumptions are described. The formulation of the lot-sizing problem by MIP and the construction of the GA model are presented in Section 4. Case studies are carried out in Section 5. Section 6 compares the proposed models with several works with similar environments. Some conclusion remarks are made in the last section.

The purpose of this paper is to construct a lot-sizing model with multi-suppliers and quantity discounts to minimize total cost over the planning horizon. The contribution can be summarized as follows. First, a general formulation of the lot-sizing problem by MIP is proposed. The model can consider costs such as ordering cost, holding cost, purchase cost and transportation cost, and inventory level for each planning period can be obtained to minimize the total cost in the system without any inventory shortages. Second, a Genetic Algorithm (GA) model is constructed to solve the problem when it becomes too complicated. Third, a small-scale case with two suppliers which both provide all-units quantity discounts for a horizon of five periods is studied using both the MIP and GA models. Replenishment level and system cost can be determined after calculating ordering cost, holding cost, purchase cost and transportation cost. Both models can obtain the optimal solution. Fourth, two more complicated cases are studied. When the case becomes too complicated or NP-hard, the GA model can find solutions very close to the optimal ones. Thus, the GA model can be very effective in searching for solutions, and it can be very useful for managers in real practice. The results of the method can satisfy decision makers’ desirable service level of replenishment in a production environment. Moreover, it is readily available for real applications. For future studies, a more complete case for supply chain management in manufacturing can be considered. A model that takes into account variable lead time, probability demand, different priority of orders, and safety stock can be established. The ordering cost for each purchase can be variable based on the environment of the new case. Some assumptions need to be relaxed by modifying objectives and constraints to consider these issues. In addition, the problem can be formulated as a multi-objective problem, and both a multi-objective programming model and a GA model may be constructed to solve the problem.