روش ابتکاری در حل یک مدل موجودی برای محصولات همراه با اجزای اختیاری تحت پرداخت تصادفی و محدودیت بودجه

In recent years, enterprises must manage the inventory of items produced by multiple components and the interactions among those items because of a growing emphasis on modularization and customization. In fact, a powerful and affordable information technology system can make the continuous review of inventory more convenient, efficient, and effective. Thus, a (Q, r) model is developed in this study to find the optimal lot size and reorder point for a multi-item inventory model with interactions between necessary and optional components. In order to accurately approximate the related costs, the service cost is introduced and defined in proportion to the service level. In addition, the service costs are incorporated with budget constraint because the firm’s strategy could influence the choice of service level. The proposed model is formulated as a nonlinear, discrete optimization problem and some known procedures are revised to solve this problem. The results are compared with other models and show that the revised procedure performs better than the N–R procedure leading to the important insights about inventory control policy. The results also reveal that the total amount allowed for the issued orders is paid at the time an order is received when the budget constraint is elastic.

In recent years, globalization and customization have led enterprises to make a greater focus on effective and efficient business processes. Most importantly, modularization and postponement are the two key issues in product and process design for the enterprises to globalization and customization. Modularization and postponement imply a product design approach whereby the product is assembled from a set of standardized units, and then these different components (or modules) are assembled closer to the point of purchase. Many studies have investigated these concepts from different aspects and have concluded that these two concepts help to reduce uncertainty and forecasting errors with regard to demand (Ernst and Kamrad, 2000, Swaminathan and Tayur, 1998 and Swaminathan and Tayur, 1999). Hence, modularization and postponement have a large impact on inventory management, especially for a multi-item inventory system. The (Q, r) model is a continuous review inventory model, an order of constant size Q is placed whenever the inventory position drops to a fixed reorder point r. Traditional multi-item inventory models with independent demand under resource constraints are discussed in the literature ( Hadley and Whitin, 1963 and Johnson and Montgomery, 1974). However, these kinds of multi-item inventory models with independent demand do not comply with the concept of modularization. In fact, there are interaction effects that have been overlooked in the multi-item inventory with independent demand. Additional demand could depend on the increasing or decreasing demand of other items or the presence for others. Balakrishnan, Pangburn, and Stavrulaki (2004) sought to develop insights for managing demand-stimulating inventories using an order-sizing model that incorporated inventory-dependent demand. Moreover, the multi-item inventory model with dependent demand under the consideration of the joint demand fulfillment ( Hausman, Lee, & Zhang, 1998) does not fit the concept of postponement well. Meanwhile, substantial research on inventory management with a continuous review inventory policy has concentrated on the performance of (Q, r) policies by formulating the average inventory, stockouts, and other criteria as functions of Q and r. For example, stochastic demand, variable lead-time, backorder cost, service level constraint (or filled rate), budget constraint, storage space constraint, and other workload constraints have been integrated into the (Q, r) system ( Parker, 1964, Schrady and Choe, 1971 and Yano, 1985). The (Q, r) model is a heuristic approximating method for a fixed reorder quantity policy with backorders. Traditionally, the average annual variable costs discussed in the (Q, r) model include procurement costs, inventory carrying (holding) costs, and stockout costs. Inventory carrying cost and backorder cost are inherently difficult to measure, and they are treated directly proportional to the length of time for which the unit remains in inventory and can be assumed as a small proportion of the number of backorders, respectively. By examining the above costs, one can find that they cannot represent the cost in the real world when the service level (defined by Hillier & Lieberman (2001)) is considered. This has drawn the attention of some researchers. For instance, inventory carrying cost was assumed as directly related to service levels in the assembly sequences ( Swaminathan & Tayur, 1999), and safety stock was assumed as directly related to service level ( Collier, 1982 and Dogramaci, 1979), as well as other considerations ( Rosling, 2002). In fact, a higher service level results in a larger heuristic approximating error because increasing the costs of labor, facilities, and related services is necessary to achieve it. One can find that the greater the reorder point is, the higher the service level (and thus the higher the service cost) will be. The cost associated with these additional heuristic approximating errors is called the service cost, which is not linearly dependent on the ordering cost, the holding cost, or the shortage cost, but is dependent upon the service level. This implies that the service cost is proportional to (or is positively related to) the service level. Thus, one can evaluate the service cost using the following definition to reduce the risk of overestimating the decision variables Q and r when the service level is increased.

Many shortcomings were found in the literature about the multi-item inventory control problem with the (Q , r ) model. For example, the correlative demand among components was ignored (thus assuming that they have independent demand), a fixed correlative demand rate among components was assumed (component commonality), and different correlative demands among components were ignored (joint demand fulfillment probability). The proposed model was designed to avoid these shortcomings, and therefore in this study a multi-item (Q , r ) model is presented for the correlative demands between the necessary component and the optional components. In addition, the service costs are incorporated into the budget constraint and the constraint is that the probability that the total amount allowed for the issued orders is within budget is not smaller than ηη. The proposed model is a nonlinear optimization problem. Hence, two heuristic algorithms that depend on the characteristic of the objective function and constraint are proposed to solve the model. One is to find the optimal solution for the system and the other is to obtain the optimal solution when the decision makers have determined the service level of the necessary component. A bike assembly example is given. From the illustrated example, the quality of the proposed solution is better than that obtained using the Newton–Raphson iteration procedure. Moreover, it was proved that one should increase all the reorder points and the number of reviews for all components when the decision makers set a higher service level for the necessary component. The proposed model is applicable for an inventory with a storage constraint or others. Although the applicability of our model may be limited in some ways because the assumption of the service cost may not be appropriate for all enterprises, this work can provide more insight into the complex multi-item inventory control problem. And this model may lead to the development of more refined and realistic models. Future work could investigate the complicated case of dependent demands among all components, and explore it by periodic review.