Yonit Barron

Shortage decision policies for a fluid production model with MAP arrivals

We consider on a continuous production/inventory process where a single machine produces a certain product into a finite buffer. The demands arrive according to a Markov Additive Process governed by a continuous-time Markov chain, and their sizes are independent and have phase-type distributions depending on the type of arrival. Two shortage policies are considered: the backorder policy, in which any demand that cannot be satisfied immediately is backlogged, and the order policy, in which any demand that cannot be satisfied immediately is supplied (alternatively, the latter policy can be considered as lost sales). We assume that the total cost includes a production loss cost, a penalty cost, a fixed cost for an order and a variable cost for the ordered amount. By applying the regenerative theory, we use tools from the exit-time theorem for fluid processes to obtain the discounted cost functionals under both policies. In addition, the models are extended to include a non-zero safety stock. Numerical examples, sensitivity analysis and comparative study are included.

An (s, k, S) fluid inventory model with exponential leadtimes and order cancellations

Abstract We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.

Performance analysis of a reflected fluid production/inventory model

We study the performance of a reflected fluid production/inventory model operating in a stochastic environment that is modulated by a finite state continuous time Markov chain. The process alternates between ON and OFF periods. The ON period is switched to OFF when the content level reaches a predetermined level q and returns to ON when it drops to 0. The ON/OFF periods generate an alternative renewal process. Applying a matrix analytic approach, fluid flow techniques and martingales, we develop methods to obtain explicit formulas for the cost functionals (setup, holding, production and lost demand costs) in the discounted case and under the long-run average criterion. Numerical examples present the trade-off between the holding cost and the loss cost and show that the total cost appears to be a convex function of q.

Clearing control policies for MAP inventory process with lost sales

We consider a production/clearing process in a random environment where a single machine produces a certain product into a buffer continuously. The demands arrive according to a Markov Additive Process (MAP) governed by a continuous-time Markov chain, and their sizes are independent and have phase-type distributions depending on the type of arrival. Since negative inventory is not allowed, the demand may be partially satisfied. The production process switches between predetermined rates that depend on the state of the environment. In addition, the system is totally cleared at stationary renewal times and starts anew at level zero immediately. Several clearing policies are considered: clearing at random times, clearing at crossings of a specified level, and a combination of the above policies. We assume the total cost includes a fixed clearing cost, a variable cost for the cleared amount, a holding cost, and a lost demand cost. By applying regenerative theory, we use tools from the exit-time theorem for fluid processes and martingales to obtain cost functionals under both the discounted and average criteria. Finally, illustrative examples and a comparative study are provided.

Group replacement policies for a repairable cold standby system with fixed lead times

Consider a multi-component repairable cold standby system and assume that repaired units are as good as new. The operational times of the units follow phase-type distribution. Downtime cost is occurred when failed components are not repaired or replaced. There are also fixed, unit repair, and replacement costs associated with the maintenance facility, which are carried out after a fixed lead time τ. Closed-form results are derived for three classes of group replacement policies (m-failure, T-age, and (m, T, τ), which is a refinement of the classic (m, T) policy) for the expected discounted case and for the long-run average criteria. Illustrative examples are provided.

An order-revenue inventory model with returns and sudden obsolescence

Abstract This paper introduces an EOQ-like state-dependent inventory model with returns and sudden obsolescence. Returns arrive according to a MAP process governed by the underlying Markov chain. Additionally, the system is totally obsoleted at stationary renewal times. Hitting level 0 yields an order of size Q . We assume order, loss, and shortage costs in addition to revenue. By applying hitting-time transforms and martingales we derive the cost functionals under the discounted criterion. Numerical results, insights, and a comparative study are provided.

Generalized control-limit preventive repair policies for deteriorating cold and warm standby Markovian systems

ABSTRACT Consider a deteriorating repairable Markovian system with N stochastically independent identical units. The lifetime of each unit follows a discrete phase-type distribution. There is one online unit and the others are in standby status. In addition, there is a single repair facility and the repair time of a failed unit has a geometric distribution. The system is inspected at equally spaced points in time. After each inspection, either repair or a full replacement is possible. We consider state-dependent operating costs, repair costs that are dependent on the extent of the repair, and failure penalty costs. Applying dynamic programming, we show that under reasonable conditions on the system’s law of evolution and on the state-dependent costs, a generalized control-limit policy is optimal for the expected total discounted criterion for both cold standby and warm standby systems. Illustrative numerical examples are presented and insights are provided.

Critical level policy for a production-inventory model with lost sales

We consider a storage process under the generalised order-up-to-level policy, based on a continuous-time Markov chain (CTMC). Specifically, the process starts at level S; whenever it drops to s, an order is sent, which is carried out after an exponential lead time. If during the lead time level S is reached, the order is cancelled, incurring some fee. This paper is written as an extension of Barron [2016. “An Fluid Inventory Model with Exponential Lead Times and Order Cancellations.” Stochastic Models 32 (2): 301–332]. While the latter paper considered a fluid inventory model with backlogging and focused on discounted analysis only, the case of lost sales was not solved. The present paper generalises the analysis to incorporate unsatisfied demand for the expected discounted costs and for the average costs per time unit. We consider four costs. There is a fixed nonzero ordering cost or a fee for each order cancellation, a purchase cost for each ordered item, a storage cost for the stock, and a penalty cost due to the unmet demand. Applying renewal theory, multi-dimensional martingales, and stopping time theory, we obtain explicit expressions of the cost components. Numerical study provides several guidelines on the optimal controls.