A Stochastic Inventory Model with Elastic Demand and Random Lead Time

This paper develops a stochastic inventory model that integrates price-dependent demand with random replenishment lead time under a continuous review policy. Unlike most existing models that treat pricing decisions and lead time uncertainty separately, the proposed model explicitly integrates these two aspects by jointly optimizing the order quantity, reorder point, and selling price, thereby capturing the interaction between demand-side responsiveness and supply-side risk. The consideration of random lead time is particularly important in the current global environment, where geopolitical conflicts and military escalations in the Middle East, including recent tensions involving major powers, have disrupted transportation routes, increased inspection delays, and heightened uncertainty in international supply chains. Demand is modeled as a function of selling price, while lead time follows a normal distribution, allowing analytically tractable expressions for expected costs and profits. Closed-form optimality conditions are derived, and an efficient iterative algorithm is proposed to solve the resulting nonlinear optimization problem. Extensive sensitivity analysis illustrates how key parameters such as demand elasticity, lead time variability, unit cost, penalty cost, and inventory holding cost affect optimal policies and profitability. The results provide important managerial insights, demonstrating how pricing can be used as a risk-mitigation lever under supply uncertainty and how increased lead time variability necessitates adjustments in safety stock and pricing decisions. The proposed model offers practical guidance for firms operating in volatile and competitive environments characterized by price-sensitive demand and heightened supply chain disruption risk.