Introduce a new mathematical approach to inventory management in production processes under constrained conditions

Nowadays, some manufacturing organizations may well face production restrictions. For example, in case the number of products goes up, the company might not be capable of producing all products. As a consequence, the company may face backlogging. In the meanwhile, in case the demand for products rises, the given company may experience a restricted capacity to react to that kind of demand properly; thus, it will suffer backlogging. Over the course of this study, that kind of company facing the mentioned circumstances is considered. To meet those exceeded demands, companies would be forced to purchase some products from outside. Thus, the study's primary aim is to define and calculate the optimum make and buy a number of products so that overall inventory cost is reduced and optimized. To do so, a model is proposed referred to as the make-with-buy model. This model is designed and solved by exact solution software in the based branch and bound method. The results of the study confirm the feasibility and efficiency of this method and demonstrate that this model can be applied to lessen the overall inventory costs, including maintenance, order, setup, and purchasing costs, and also the total costs of products.