The complete solution procedure for the EOQ and EPQ inventory models with linear and fixed backorder costs

The basic EOQ/EPQ inventory models with backorders have been developed from different perspectives by several researchers. However, the arguments to locate and guarantee the optimal solution are not complete. This paper presents a complete and analytic solution procedure to the EOQ/EPQ inventory models with linear and fixed backorder costs to locate and ensure the optimal solutions. First, the sufficient and necessary conditions for the existences of the optimal solution are developed. Second, in the case if these conditions are not satisfied, then also the optimal solutions are identified for this situation. The final results are two lemmas and four useful theorems to obtain the optimal solutions to both inventory problems.