Improvements for Inventory Models with Generalized Interarrival Times

Single-machine, single-product inventory models with generalized interarrival times have long lacked a rigorously justified and computationally tractable optimization framework, owing to a sign mis-specification in prior derivations that spawned ad hoc feasibility restrictions and expansive search domains. A corrected, succinct derivation establishes the strict convexity of the minimum-cost objective and proves the existence and uniqueness of an interior optimum without auxiliary conditions, unifying previously fragmented results into a general theorem. Leveraging these structural properties, the maximum-profit formulation is reduced to a single-variable program over natural, finite bounds that tightly bracket the optimizer, supplanting earlier paired bounds defined on an unbounded domain. Numerical evidence on a canonical benchmark shows that the admissible interval is markedly tighter yet attains the same optimum with fewer evaluations, thereby improving numerical efficiency and robustness of implementation. The analysis clarifies the correspondence between cost-minimization and profit-maximization formulations and provides an operationally simple, reproducible solution path for capacity-constrained single-machine systems under generalized interarrival times. Principal contributions are: (i) a corrected optimality theory establishing strict convexity and a unique interior optimum without auxiliary conditions; (ii) a dimensionality reduction of the profit model to a single-variable program with natural finite bounds; and (iii) demonstrably tighter admissible intervals that cut evaluations while preserving optimality.