A necessary and sufficient condition for the existence of the optimal solution of a single-vendor single-buyer integrated production-inventory model with process unreliability consideration

In general, an inventory problem consists of two parts: (1) the modeling and (2) the solution procedure. The accuracy of the solution procedure can help and simplify the implementation of the inventory model. Huang [2004. An optimal policy for a single-vendor single-buyer integrated production-inventory problem with process unreliability consideration. International Journal of Production Economics 91, 91-98] discusses a model to determine an optimal policy for a single-vendor single-buyer integrated production-inventory problem with process unreliability consideration. The objective is to minimize the total joint annual costs incurred by the vendor and the buyer. However, Huang's solution procedure seems easy to cause misunderstandings. So, the main purpose of this paper will give the necessary and sufficient condition for the existence of the optimal solution to complement and improve the solution procedure of Huang's inventory model.