Integrating Replenishment Policy and Maintenance Services in a Stochastic Inventory System with Bilateral Movements

We study an inventory control problem with two storage facilities: a primary warehouse (PW) of limited capacity M, and a subsidiary one (SW) of sufficiently large capacity. Two types of customers are considered: individual customers arriving at (positive and negative) linear rates governed by a Markov chain, and retailers arriving according to a Markov arrival process and bringing a (positive and negative) random number of items. The PW is managed according to a triple-parameter band policy (M,S,s),0≤sS≤M, under a lost sales assumption. Under this policy, as soon as the stock level at the PW falls below s, a refilling to S is performed by a distributor after a random lead-time. However, if the stock exceeds level S when the distributor arrives, no refilling is carried out, and only maintenance services are performed. Items that exceed level M are transferred to the SW at a negligible amount of time for those used in related products. Our cost structure includes a fixed order cost, a variable cost for each item supplied by the distributor, a cost for the additional maintenance, a salvage payment for each transferred item from the PW to the SW, and a loss cost for each unsatisfied item due to demands. We seek to determine the optimal thresholds that minimize the expected overall cost under the discounted criterion. Applying first-passage time results, we present a simple set of equations that provide managers with a useful and an efficient tool to derive the optimal thresholds. Sensitivity analysis and fruitful conclusions along with future scope of research directions are provided

Fluid Inventory Models under Markovian Environment

Today’s products are subject to fast changes due to market conditions, short life cycles, and technological advances. Thus, an important problem in inventory planning is how to effectively manage the inventory control in a dynamic and stochastic environment. The traditional Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) both are widely and successfully used models of inventory management. However, both models assume constant and fixed parameters over time. Unfortunately, most of these assumptions are unrealistic. In this study, we generalize the EOQ and EPQ models and study production-inventory fluid models operating in a stochastic environment. The inventory level increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). Our main objective is to minimize the expected discounted total cost which includes ordering, purchasing, production, set up, holding, and shortage costs. Applying regenerative theory, optional sampling theorem (OST) to the multi-dimensional martingale and fluid flow techniques, we develop methods to obtain explicit formulas for these cost functionals. As such, we provide managers with a useful framework and an efficient and easy-to-implement tool to coop with different demand–supply patterns

A stochastic card balance management problem with continuous and batch-type bilateral transactions

We study a stochastic continuous-review card balance management problem with two transaction patterns, namely, continuous and batch-type bilateral transactions, both in a Markovian environment. Motivated by the Autoload program used in public transit systems, the card is managed using a two-parameter band policy. Our cost structure includes activation and loading costs, and a fine for a negative balance. By applying hitting time theory and martingales, we derive the cost functionals and obtain, numerically, the optimal thresholds minimizing the expected discounted total cost. Surprisingly, a numerical study shows that the optimal policy is inherently linked with the outflow patterns, and is more sensitive to changes in withdrawal rates than to changes in batch sizes. We further show that timing is a significant factor in determining the policy: a high discount factor leads to frequent activations with smaller amounts