Abstract
This paper develops a deterministic economic order quantity (EOQ) model for a single deteriorating item under a linear, time-dependent demand rate D(t) = a+bt, where deterioration proceeds at a constant instantaneous rate θ applied to the on-hand inventory. Shortages are not permitted. The governing first-order linear differential equation describing inventory depletion is solved in closed form, and the order quantity, holding cost integral, and deterioration cost expression are derived analytically and verified symbolically. The average total cost per unit time, TCU(T), is shown via a rigorous second-order condition together with numerical evidence to be strictly convex in the cycle length T, guaranteeing a unique global minimizer T∗. The model is further validated by showing that it reduces exactly to the classical no-deterioration linear-trend model as θ →0, to the constant-demand Ghare–Schrader deteriorating inventory model as b→0, and to the textbook economic order quantity formula as both θ →0 and b→0 simultaneously. A solution algorithm is proposed, and a numerical example, together with an extended six-scenario numerical study spanning low, base, and high deterioration rates and growing, flat, and declining demand trends, and a comprehensive sensitivity analysis with respect to the deterioration rate, demand trend, holding cost, deterioration cost, and ordering cost, illustrates the practical behaviour of the optimal policy across realistic operating regimes. Managerial insights and limitations relevant to perishable goods and fast-moving consumer goods supply chains are discussed in detail.