Rigorous Clarification and Unification of Minner’s Cost Comparison for EOQ, EOQ with Backorders, and EPQ Models

This study provides a rigorous clarification and abstract unification of the derivative-free cost comparison method originally proposed by Minner for inventory optimization. Traditional discrete cost comparison approaches rely on unverified continuous convexity assumptions and implicit limit convergences. This paper resolves these theoretical gaps through three main contributions. First, a formal discrete inductive proof demonstrates that the first local integer minimum of the total cost sequence strictly guarantees global cost minimization, resolving the candidate selection problem under both unique and tied minimized conditions. Second, the divergence of the optimal partition number and the exact limit convergence of the optimal economic order interval are mathematically verified using the Sandwich Theorem. Third, an abstract single-parameter cost operator is formulated to unify standard economic order quantity (EOQ), EOQ with backorders, and economic production quantity models within a single analytical structure, demonstrating feasibility preservation and eliminating redundant derivations. Furthermore, a systematic taxonomic critique rectifies analytical omissions, expositional ambiguities, and sign-convention errors present across six benchmark papers in the derivative-free inventory literature. This study illustrates the practical efficiency of the unified paradigm.