Production Inventory Model for Crumbling Items With Time Dependent Production and Demand with Deficiency

Abstract

This study develops a novel Economic Production Quantity (EPQ) model for deteriorating items where both production and demand rates vary over time, following power patterns. Unlike traditional models that assume constant production or demand rates, our model accommodates realistic variations production and demand may increase or decrease over a cycle, depending on the chosen power exponents. Demand rate is expressed as λ(t)=g·m·t^{m−1}, and production rate as R(t)=r·n·t^{n−1}, with deterioration occurring at an exponential rate q. Shortages are allowed and fully backlogged. We derive closed-form differential equations to describe inventory dynamics across four phases: production build-up, depletion under demand and deterioration, shortage accumulation, and subsequent backlog replenishment. From these, we obtain expressions for cycle-end inventory level (S), production downtime (t₁), uptime (t₃), and production quantity (Q). Incorporating holding, setup, production, and shortage costs, we construct the total cost function K(t₁, t₂, t₃) and determine optimal policies by minimizing with respect to t₁ (and thus t₃ and Q). Numerical examples illustrate how variability in deterioration, cost, production, and demand parameters significantly affects optimal production schedules and costs. Sensitivity analysis highlights that deterioration rate and demand growth parameters most strongly influence optimal Q, t₁, and t₃. Importantly, the model generalizes various existing EPQ formulations, and can be extended to account for time dependent pricing and demand relationships.