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