A Corrected and Simplified Solution Procedure for the Integrated Vendor–Buyer Inventory Model of Chung (2013)

This study revisits the integrated vendor–buyer inventory model with backordering analyzed by Chung and clarifies its relationship to the finite-horizon cost-comparison method of Minner. We show that the perturbation arguments used for the continuous decision variables recover the usual first-order stationary conditions in the limit, whereas Minner’s construction is based on adjacent finite-horizon cost comparisons followed by an asymptotic argument. We provide direct global-optimality proofs for the backordering ratio and lot size by completing the square, and an exact forward-difference characterization for the integer number of deliveries. We derive an inequality. If the inequality is strict, then the optimal integer is unique. If equality holds, then two consecutive integers have the same minimum value. Our new formulation is shorter and avoids the rounding trap. Re-evaluation of Chung’s numerical example using unrounded intermediate values gives the unique optimum, while a constructed example illustrates the exact two-integer tie case. We also correct algebraic and notation inconsistencies in the reduced cost expressions. The resulting procedure is shorter, globally justified, and more clearly distinguishes continuous-variable algebraic optimization from discrete cost comparison.