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An Inventory Problem with Two Randomly Available Suppliers

This paper considers a stochastic inventory model in which supply availability is subject to random fluctuations that may arise due to machine breakdowns, strikes, embargoes, etc. It is assumed that the inventory manager deals with two suppliers who may be either individually ON (available) or OFF (unavailable). Each suppliers availability is modeled as a semi-Markov (alternating renewal) process. We assume that the durations of the ON periods for the two suppliers are distributed as Erlang random variables. The OFF periods for each supplier have a general distribution. In analogy with queuing notation, we call this an Es1[Es2]/G1[G2] system. Since the resulting stochastic process is non-Markovian, we employ the “method of stages” to transform the process into a Markovian one, albeit at the cost of enlarging the state space. We identify the regenerative cycles of the inventory level process and use the renewal reward theorem to form the long-run average cost objective function. Finite time transition fun...

Age-based vs. stock level control policies for a perishable inventory system

Abstract In this study, we investigate the impact of modified lotsize-reorder control policy for perishables which bases replenishment decisions on both the inventory level and the remaining lifetimes of items in stock. We derive the expressions for the key operating characteristics of a lost sales perishable inventory model, operating under the proposed age-based policy, and examine the sensitivity of the optimal policy parameters with respect to various system parameters. We compare the performance of the suggested policy to that of the classical ( Q , r ) type policy through a numerical study over a wide range of system parameters. Our findings indicate that the age-based policy is superior to the stock level policy for slow moving perishable inventory systems with high service levels.

A maintenance policy for a system with multi-state components: an approximate solution

Abstract For maintenance and quality assessment purposes, various performance levels for both systems and components are identified, usually as a function of the deterioration. In this study, we consider a multicomponent system where the lifetime of each component is described by several stages, (0,…,S), which are further classified as good, doubtful, preventive maintenance due (PM due) and down. A control policy is suggested where the system is replaced when a component enters a PM due or a down state and the number of components in the doubtful states (K,…,S−2) is at least N. All maintenance activities are assumed to take negligible time. The exact description of the underlying stochastic model under the policy is very complicated. We therefore propose some approximations, which allow an explicit expression for the long run average cost function, which is minimized w.r.t. (K,N) by numerical methods. Sensitivity of the model to system parameters and the performance of the approximation are investigated through several examples.

Analysis of the (Q, r) Inventory Model for Perishables with Positive Lead Times and Lost Sales

We consider a perishable inventory system with Poisson demands, fixed shelf lives, constant lead times, and lost sales in the presence of nonnegligible fixed ordering costs. The inventory control policy employed is the continuous-review (Q,r) policy, where r<Q. The system is modeled using an embedded Markov process approach by introducing the concept of the effective shelf life of a batch in use. Using the stationary distribution of the effective shelf life, we obtain the expressions for the operating characteristics and construct the expected cost rate function for the inventory system. Our numerical study indicates that the determination of the policy parameters exactly as modeled herein results in significant improvements in cost rates with respect to a previously proposed heuristic. We also compare the (Q,r) policy with respect to a time-based benchmark policy and find that the (Q,r) policy might be impractical for rare events, but overall appears to be a good heuristic policy.

Analysis of the (s, S) policy for perishables with a random shelf life

A continuous review perishable inventory system operating under the (s,S) policy is considered. Assuming a random shelf life with a general distribution, renewal arrivals and a negligible replenishment lead time, exact expressions for the expected cost rate function for unit and batch demands are derived. For unit demands, it is shown that the average cost rate function is quasi-convex in (s,S). Numerical findings indicate that the loss incurred by ignoring the randomness of the shelf life can be drastic. It is observed that the shape of the shelf life distribution has a significant impact on the costs and a precise estimation of shelf life distribution may result in substantial savings. Based on the presented analytical results, a new heuristic for positive lead times is proposed. Extensive numerical studies show that the proposed heuristic performs better than an existing one suggested for fixed shelf lives in most of the cases studied.

Bivariate estimation with right-truncated data

Abstract Bivariate estimation with survival data has received considerable attention recently; however, most of the work has focused on random censoring models. Another common feature of survival data, random truncation, is considered in this study. Truncated data may arise if the time origin of the events under study precedes the observation period. In a random right-truncation model, one observes the iid samples of (Y, T) only if (Y ≤ T), where Y is the variable of interest and T is an independent variable that prevents the complete observation of Y. Suppose that (Y, X) is a bivariate vector of random variables, where Y is subject to right truncation. In this study the bivariate reverse-hazard vector is introduced, and a nonparametric estimator is suggested. An estimator for the bivariate survival function is also proposed. Weak convergence and strong consistency of this estimator are established via a representation by iid variables. An expression for the limiting covariance function is provided, and a...

Bundle pricing of inventories with stochastic demand

We consider a retailer selling a fixed inventory of two perishable products over a finite horizon. Assuming Poisson arrivals and a bivariate reservation price distribution, we determine the optimal product and bundle prices that maximize the expected revenue. Our results indicate that the performances of mixed bundling, pure bundling and unbundled sales strategies heavily depend on the parameters of the demand process and the initial inventory levels. Bundling appears to be most effective with negatively correlated reservation prices and high starting inventory levels. When the starting inventory levels are equal and in excess of average demand, most of the benefits of bundling can be achieved through pure bundling. However, the mixed bundling strategy dominates the other two when the starting inventory levels are not equal. We also observe that an incorrect modeling of the reservation prices may lead to significant losses. The model is extended to allow for price changes during the selling horizon. It is shown that offering price bundles mid-season may be more effective than changing individual product prices.

Weak and strong quantile representations for randomly truncated data with applications

Suppose that we observe bivariate data (Xi, Yi) only when Yi [less-than-or-equals, slant] Xi (left truncation). Denote with F the marginal d.f. of the Xs. In this paper we derive a Bahadur-type representation for the quantile function of the pertaining product-limit estimator of F. As an application we obtain confidence intervals and bands for quantiles of F.

Bayesian demand updating in the lost sales newsvendor problem: A two-moment approximation

We consider Bayesian updating of demand in a lost sales newsvendor model with censored observations. In a lost sales environment, where the arrival process is not recorded, the exact demand is not observed if it exceeds the beginning stock level, resulting in censored observations. Adopting a Bayesian approach for updating the demand distribution, we develop expressions for the exact posteriors starting with conjugate priors, for negative binomial, gamma, Poisson and normal distributions. Having shown that non-informative priors result in degenerate predictive densities except for negative binomial demand, we propose an approximation within the conjugate family by matching the first two moments of the posterior distribution. The conjugacy property of the priors also ensure analytical tractability and ease of computation in successive updates. In our numerical study, we show that the posteriors and the predictive demand distributions obtained exactly and with the approximation are very close to each other, and that the approximation works very well from both probabilistic and operational perspectives in a sequential updating setting as well.

A note on "Continuous review perishable inventory systems: models and heuristics"

In a recent paper, Lian and Liu (2001) consider a continuous review perishable inventory model with renewal arrivals, batch demands and zero lead times. However, the main analytical result they provide holds only for some special cases such as Poisson arrivals with exponential interarrival times. In this note we generalize Theorem 1 of Lian and Liu (2001) for the case where the arrivals follow an arbitrary renewal process.