A Unified Inventory Model with Production Cost under Negative Exponential Demand, Weibull Deterioration and Partial Backlogging

This study develops an inventory optimization model for perishable items using deteriorating inventory theory. The framework integrates time-dependent demand D(t) = Ae^−αdt with Weibull-based hazard h(t) = αβ(t − γ)^β−1e^{−α(t−γ)β} to characterize component non-instantaneous deterioration. The Time Shift Approach (TSA) is used to ensure that the functions are defined in the real domain. Shortages were allowed and partially backlogged. Numerical illustration was used to validate the model and Python script was employed to obtain the decision variables t₁, t₂ and T as t₁ = 5.113399, t₂ = 5.163399 and T = 30.729455. Sensitivity analysis identifies demand rate (α_d) and hazard variability (σ) as critical parameters. The minimal interval between t₁ and t₂ indicates that for items with high post-shelf-life deterioration, the optimal strategy requires rapid, sequential actions to preempt stockouts.