A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales

Most deterministic and stochastic inventory models assume that the lead time is a given parameter, and determine the optimal operating policy on the basis of this unrealistic assumption. In this paper, we incorporate a common inter-relationship between lot size and lead time in the stochastic continuous review inventory control (Q,r) model. We present an efficient iterative procedure that determines the optimal or near-optimal lot size and reorder point that minimize the total annual inventory cost for the extended (Q,r) model. We further propose another analytical procedure that generates the lot size and safety stock simultaneously. We also show that our model will result in smaller production lot size and shorter lead time than an earlier model reported in the literature.

This paper considers the continuous review, stochastic demand inventory model with a mixture of backorders and lost sales. It investigates the feasibility of investing in reducing the lost sales caused by stockout. We first assume that the lead time demand follows a normal distribution, then relax this assumption to consider the distribution free case where only the mean and standard deviation of lead time demand are known. An algorithm is developed to find the optimal ordering policy and the optimal investment strategy. Also, two numerical examples are provided to demonstrate the results.

Optimality of State-Dependent (s, S) Policies in Inventory Models with Markov-Modulated Demand and Lost Sales

We consider an inventory system for continuous decaying items with stochastic lead time and Poisson demand. Shortage is allowed and all the unsatisfied demands are backlogged. Moreover, replenishment is one for one. Our objective is to minimize the long-run total expected cost of the system. Firstly, we have developed the mathematical model with deterministic lead time. Since the stochastic lead time makes the model complex especially when lead time has complicated probability distribution and it is difficult to prove convexity of the objective function, we have applied simulation modeling approach. The simulation model has no limitation on lead time or any other parameters. The simulation model is validated by comparing its outputs and analytical model's results for the deterministic lead time case. Furthermore, we use optimizer module of the applied software to find near optimal solutions for a number of examples with stochastic lead time.

We examine the effect of stochastic lead times on Just-in-Time (JIT) delivery. We find that with stochastic lead times there is a possibility of order crossover, and what order crossover does is to transform the original lead times into effective lead times, which is an AR(1) process that is an autoregressive process of Order 1. The mean of this process is the same as the mean of the original lead time, but its variance could be much smaller. The implication is that when we consider order crossover in the analysis, the cost could be less than otherwise (but never less than that with deterministic lead times). The literature on JIT with stochastic lead times has never considered order crossover, which produces the effective delivery times (EDT). Here, we demonstrate some important properties of the EDT: that it is a Cauchy sequence, and hence it converges; that it is an AR(1) process; and that it stochastically dominates the parent lead time distribution.

Most inventory modelling has assumed stochastic demands and constant lead times. However, here we consider a problem for which the opposite situation holds; namely, there is a known constant demand rate, but lead times are random variables. Moreover, the probability distributions of the lead times change in a seasonal fashion. Also, shortages of raw materials result in lost sales. The goal of this paper is to propose heuristic methods for minimizing the expected costs in such a situation. This study was motivated by a problem of management of raw material at a sawmill.

Inventory Management with Stochastic Lead Times

One of the most widespread stochastic inventory models is the classical stochastic continuous review inventory control (Q, R) model. However, there are some restrictive assumptions in deriving this model. One of these assumptions is that the production rate is infinite which has motivated many researchers to develop (Q, R) models with finite production rate. In these models, the randomness of demand is indirectly introduced in the models by assuming that the safety stock is equal to a safety factor multiplied by the standard deviation of lead time demand. In this paper, the classical stochastic continuous review inventory control (Q, R) model is generalised for the case when the production rate is finite and unmet demand is partially backordered. Unlike the models presented in the literature, the probabilistic nature of the demand is reflected directly in the formulation of the proposed model. Furthermore, the optimal lot size and reorder point are determined for this model. Two types of items are investigated; fast-moving and slow-moving items. Also, the effect of the production rate on the optimal solution is studied through numerical examples. The results show that the production rate impacts the inventory and production decisions significantly.

The problem of material supply of an automatic asphalt mixing plant is considered in this paper. First a deterministic model for the optimal mixture rates is discussed, then stochastic inventory control models for the hot depots of mixture machines are presented. The main practical problem is to find the optimal hot depot capacities for which the probability of overflow, in a given time period, does not exceed a prescribed level. Chance constrained models are formulated with joint resp. individual probabilities.KeywordsDeterministic ModelInventory ModelStochastic ProgrammingWeight ProportionChance ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we consider two fundamental inventory models, the single-period newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are given explicitly, these models are well studied and relatively straightforward to solve. However, in most real-life scenarios, the true demand distributions are not available, or they are too complex to work with. Thus, a sampling-driven algorithmic framework is very attractive, both in practice and in theory. We shall describe how to compute sampling-based policies, that is, policies that are computed based only on observed samples of the demands without any access to, or assumptions on, the true demand distributions. Moreover, we establish bounds on the number of samples required to guarantee that, with high probability, the expected cost of the sampling-based policies is arbitrarily close (i.e., with arbitrarily small relative error) compared to the expected cost of the optimal policies, which have full access to the demand distributions. The bounds that we develop are general, easy to compute, and do not depend at all on the specific demand distributions.

We analyze a continuous review (Q, r) stochastic inventory model in which orders placed with a make-to-order manufacturer can be shipped via two alternative freight modes differing in lead time and costs. The costs of placing an order and using each freight mode consist of fixed components and hence exhibit economies of scale. We derive an optimal policy for using the two freight modes for shipping each order. This freight-mode decision is delayed until manufacturing is complete and the optimal policy uses information about the demand incurred in the meantime. Furthermore, given that the two freight modes are used optimally for shipping each order, we solve our model for reorder point and order quantity that minimizes cost. We analyze the cost savings achieved from postponing the freight-mode decision and provide analytical and numerical comparisons between the solutions to our two-freight model and single-freight models. Finally, we illustrate the properties of the solution to our model using an extensive set of numerical examples.

A multi-item cost minimising stochastic inventory model is developed in stochastic environment. Demand is considered to be function of income level of consumer and price of commodity. Budget, income and price are assumed to be random and their randomness is expressed by normal distribution. Deterioration rate is modelled using Weibull distribution. Stochastic inventory model is converted into equivalent crisp inventory model by using stochastic nonlinear programming (SNLP) technique. Crisp problem is solved by fuzzy nonlinear programming (FNLP) technique and intuitionistic fuzzy optimisation (IFO) technique. Sensitivity analysis of the optimum solution with respect to feasible changes in average and variance of income level and price is discussed and conclusions are noted.

Using the well-known product-limit form of the Kaplan-Meier estimator from statistics, we propose a new class of nonparametric adaptive data-driven policies for stochastic inventory control problems. We focus on the distribution-free newsvendor model with censored demands. The assumption is that the demand distribution is not known and there are only sales data available. We study the theoretical performance of the new policies and show that for discrete demand distributions they converge almost surely to the set of optimal solutions. Computational experiments suggest that the new policies converge for general demand distributions, not necessarily discrete, and demonstrate that they are significantly more robust than previously known policies. As a by-product of the theoretical analysis, we obtain new results on the asymptotic consistency of the Kaplan-Meier estimator for discrete random variables that extend existing work in statistics. To the best of our knowledge, this is the first application of the Kaplan-Meier estimator within an adaptive optimization algorithm, in particular, the first application to stochastic inventory control models. We believe that this work will lead to additional applications in other domains.

This study develops a mathematical model for an integrated vendor-buyer inventory with defective items and inspection error. We assume an imperfect production process with a known of defective probability. The buyer conducts a 100% quality check. However, due to an imperfect inspection process, the quality-controller may reject a good item as a defective item (Type I error), or accept a defective item as a non-defective item (Type II error). Different from existing research, we incorporate freight cost (a function of shipping weight and the distance between the vendor and the buyer) and stochastic lead time (a function of production and transportation times). The objective of the study is to minimize the joint total expected cost. This study indicates that Type I error is more significant when it is compared with Type II error. We also illustrate how the joint total expected cost and the lead time are significantly influenced by the distance of the buyer and the vendor. The results of our analysis also show that lead time can be shortened if the vendors speed up their shipment with smaller production lot sizes. Numerical examples and sensitivity analysis are given to illustrate the results.

Fixed costs of ordering items or setting up a production process arise in many real‐life scenarios. In their presence, the most widely used ordering policy in the stochastic inventory literature is the ( s , S ) $(s,S)$ policy. Optimality of ( s , S ) $(s,S)$ policies and ( s , S ) $(s,S)$ ‐type policies have been examined for various inventory models, including those with the inventory level being reviewed in every period or continuously, finite and infinite horizons, discounted‐cost and average‐cost criteria, backlogging and lost‐sales practices, standard and generalized demands and/or costs, deterministic and stochastic lead times, single‐product and multi‐product settings, and coordinated pricing‐inventory decisions. We comprehensively survey the vast literature accumulated over seven decades in two papers. This paper is devoted to continuous‐time models, and the companion paper, also published in this journal issue, reviews discrete‐time models. We go over model specifications, proof techniques, specific results, and limitations of the articles published in the literature. We conclude each paper by providing corresponding suggestions for extensions and directions for future research.