Abstract

Introduction: To develop a comprehensive inventory control system, models such as Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) are commonly employed to minimize costs, prevent order delays, and achieve high operational performance. The concepts of economic production and ordering are central topics in production and inventory control and have long attracted the attention of researchers. Efficient and profitable production requires thorough and precise planning at all stages. In manufacturing companies that produce a range of products, there may be periods when customer demand exceeds the production rate—or falls below it. In such situations, the company can meet demand through a combination of production, purchasing, or accepting shortages. Among these options, choosing the appropriate ordering strategy is of great importance. In most related studies, researchers have proposed methods to determine whether production or purchasing is more cost-effective when both options are available for a given product. These models typically make the decision based on specific criteria such as cost, customer satisfaction, and other relevant factors.

Method: In this study, a mathematical optimization model is presented to determine the optimal production and procurement quantities, as well as the optimal production start time for a multi-product, multi-period system. The objective is to minimize the total cost while considering constraints such as available warehouse space and budget. The model accounts for various possible scenarios, including the presence or absence of shortages, and whether the demand rate is higher or lower than the production rate, with the optimal policy being determined accordingly.

Results and discussion: This study proposes an integrated production and procurement model for a multi-product, multi-period problem, considering product-to-machine assignment and warehouse space constraints. The primary objective of the model is to minimize inventory-related costs. A mathematical optimization model is developed in which the relationship between production rate and demand rate for each product in each period may vary—meaning that the demand rate can be either higher or lower than the production rate. Accordingly, 24 different inventory level scenarios are identified and will be discussed in detail in the following section. To date, no previous study has simultaneously considered and incorporated all of these 24 scenarios into a single mathematical model. In the sensitivity analysis section of the proposed model, it is determined that a reduction of approximately 77% in warehouse space leads to an increase of approximately 70% in total costs.

Conclusions: This study focuses on the simultaneous determination of optimal production and procurement quantities in an inventory system under specific assumptions. The primary objective is to provide appropriate service levels to meet customer demands. Accordingly, a multi-product, multi-period inventory control system is optimized under a combined production and purchasing strategy, allowing for backorders. To better reflect real-world conditions, the model incorporates constraints such as warehouse capacity and product-to-machine assignment. Among the 24 identified sub-scenarios, shortages occur in 17 of them. Due to its broad practical applications and theoretical complexity, this problem holds significant importance. The proposed model is formulated as a mixed-integer nonlinear programming (MINLP) model, which presents considerable computational challenges. To solve the model, this study employs the Sequential Quadratic Programming (SQP) method, and the obtained results are reported and analyzed.