Stochastic Inventory Control Models Research Articles

Stochastic inventory control models for single item replenishment with variable demand under constrained and unconstrained conditions

Stochastic inventory control models for single item replenishment with variable demand under constrained and unconstrained conditions

Constrained Stochastic Inventory Control Models for Multi-Item with Variable Demand

Constrained multi-item inventory system with variable demands are considered. The demand rates of five selected multi-item - Cowbell, Milo, SMA, Cerelac and Golden Morn were modeled as Weibull, Normal and Lognormal probability distributions respectively with the aid of chi-squared multinomial goodness-of-fit test. The respective probability distributions with estimated location parameters: 491.55, 536.92, 10.5, 2.1926 and 5.3103 were used as the basis of probabilistic inventory models to obtain dynamic EOQ for each item under each constraint subject to Kuhn, Karush and Tucker (KKT) conditions as against the use of simple averages in deterministic inventory models. The optimal values of these constraints: available warehouse space (124sq.ft), specified level of inventory (94 units), limited capital (76,671.52 naira) and number of orders (1/month) were obtained using the optimal EOQ values to establish optimal inventory level and constraints level/capacity for each item in order to avoid shortage or excess stock.

Forecast Methods and Periodic Review Inventory Model for Supply Planning to Reduce Food Waste

Food waste is an important economic and resource problem in all countries around the world. Particularly, the restaurant sector highly contributes to food waste and limited efforts or studies have been performed to overcome this problem. In this context, the present study addresses an alternative to improve the supply planning for perishable products in the restaurant sector through the application of specific forecasting methods and a stochastic inventory control model. For this purpose, a real enterprise within this economic sector was considered. Our findings support that monthly forecasts can be more appropriate for accurate demand estimation and supply planning of perishable products, which is important to reduce unnecessary products. Also, the periodic review inventory control model can lead to a more appropriate supply scheme to reduce the waste of surplus food. These findings and the proposed techniques can be used for other economic entities to reduce product waste due to poor supply planning.

Resilience toward supply disruptions: A stochastic inventory control model with partial backordering under the base stock policy

This paper examines supply-side disruptions using an inventory management framework to gain insights into the economic performance of buyers who use two conventional costing strategies – end-of-cycle and continuous. The proposed model assumes that a single supplier faces full disruptions with a probability, and therefore, fails to procure all the items ordered. Accordingly, the buyer experiences unmet demand, which is assumed to be partially backordered. To make proper replenishment decisions, two base stock (S,T) periodic review optimization models are developed. The objective functions minimize the expected long-run total costs (i.e., ordering, holding, and shortage). After proving their convexity and obtaining the optimal decisions, computational experiments were carried out to investigate the impacts of the change in parameters. The results highlight the importance of selecting a suitable ratio for backorders during supply disruptions, as it leads to lower costs and reduces the inventory obsolescence and overstocking risks. Moreover, the managerial insights derived from this study aid retailers to make better replenishment decisions (in terms of level and frequency) and to be more resilient in times of disruptions. Retailers can benefit from the provided solution algorithm as a computational application.

A centralized stochastic inventory control model for perishable products considering age-dependent purchase price and lead time

In this paper, we present a stochastic multi-period multi-echelon $$ \left( {s, S} \right) $$ periodic inventory control model for perishable products in a healthcare environment. We consider age-dependent purchase price, order lead time, and allow for backorder. The model is inspired by a real case where a regional health system operates multiple hospitals in addition to managing a central warehouse. The warehouse centralizes the purchasing of the products and then distributes them among hospitals while demand for products in each hospital is uncertain. We formulate the problem as a stochastic mixed-integer linear programming model and study two purchasing strategies. The first and traditional strategy assumes that the warehouse only orders fresh products, i.e., products with the latest expiry date and most expensive, referred to as purchasing constant remaining life (CRL). The second strategy considers a discount function with a lower purchase price for products with a shorter remaining life, referred to as different remaining life (DRL). The designed models are naturally complex and intractable to obtain optimal solutions by an analytical approach for large instances of the problem. Thus, we developed a genetic algorithm to solve the problem and benchmarked its performance against an exact method. We also verified the results of the stochastic models through simulation. Extensive sensitivity analysis is also conducted to investigate the behaviors of both CRL and DRL strategies about the risk of product shortage, product expiration and the total cost imposed on the system.

Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models

We study the classical multiperiod capacitated stochastic inventory control problems in a data-driven setting. Instead of assuming full knowledge of the demand distributions, we assume that the demand distributions can only be accessed through drawing random samples. Such data-driven models are ubiquitous in practice, where the cumulative distribution functions of the underlying random demand are either unavailable or too complex to work with. We consider the sample average approximation (SAA) method for the problem and establish an upper bound on the number of samples needed for the SAA method to achieve a near-optimal expected cost, under any level of required accuracy and prespecified confidence probability. The sample bound is polynomial in the number of time periods as well as the confidence and accuracy parameters. Moreover, the bound is independent of the underlying demand distributions. However, the SAA requires solving the SAA problem, which is #P-hard. Thus, motivated by the SAA analysis, we propose a polynomial time approximation scheme that also uses polynomially many samples. Finally, we establish a lower bound on the number of samples required to solve this data-driven newsvendor problem to near-optimality.

Provably Near-Optimal Balancing Policies for Multi-Echelon Stochastic Inventory Control Models

We develop the first algorithmic approach to compute provably good ordering policies for a multi-echelon, stochastic inventory system facing correlated, nonstationary and evolving demands over a finite horizon. Specifically, we study the serial system. Our approach is computationally efficient and provides worst-case guarantees. That is, the expected cost of the algorithms is guaranteed to be within a constant factor of the optimal expected cost; depending on the assumption the constant varies between two and three. Our algorithmic approach is based on an innovative scheme to account for costs in a multi-echelon, multi-period environment, as well as repeatedly balancing between opposing cost. The cost-accounting scheme, called a cause-effect cost-accounting scheme, is significantly different from traditional cost-accounting schemes in that it reallocates costs with the goal of assigning every unit of cost to the decision that caused the cost to be incurred. We believe it will have additional applications in other multi-echelon inventory models.

Economic and environmental considerations in a stochastic inventory control model with order splitting under different delivery schedules among suppliers

This study analyzes an integrated inventory control and delivery scheduling problem in a stochastic demand environment with economic and environmental considerations. In particular, we examine a bi-objective continuous review inventory control model with order splitting among multiple suppliers, where both expected costs and carbon emissions per unit time are minimized. For this problem, two different delivery scheduling policies are considered for the split orders: sequential splitting and sequential delivery. First, we formulate the problem under each delivery scheduling policy as bi-objective mixed-integer nonlinear models. Then, an adaptive ϵ-constraint algorithm and an evolutionary search algorithm are proposed to approximate the Pareto front of these models. A numerical study is conducted to compare the two approximation algorithms. Another numerical study demonstrates the effects of the demand variance on the expected costs and carbon emissions per unit time under each delivery scheduling policy. Finally, examples are presented to show how the tools provided in this study can be used to compare different scheduling policies. Our results show that the delivery policy and supplier selection both have strong effects on the economic and environmental performance, and also that a good approximation of the Pareto front is crucial to accurately compare delivery scheduling policies.

Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models

We study the classical multi-period capacitated stochastic inventory control problems in a data-driven setting. The objective is to match the on-hand inventory level with the random demand in each period, subject to supply constraints, while minimizing expected cost over a finite time horizon. Instead of assuming a full knowledge of the demand distributions, we assume that they can only be accessed through drawing random samples. Such data-driven models are ubiquitous in practice, where the cumulative distribution functions of the underlying random variables are either unavailable or too complicated to work with. We apply the Sample Average Approximation (SAA) method to the capacitated inventory control problem and establish an upper bound on the number of samples needed for the SAA method to achieve a near-optimal expected cost, under any level of required accuracy and prespecified confidence probability. The sample bound is polynomial in the number of time periods as wel as the confidence and accuracy parameters. Moreover the bound is independent of the underlying demand distributions. We crucially use an inequality of Massart, which is a stronger version of Chernoff inequality, to ensure a uniform approximation on the right derivatives.While the SAA method uses polynomially many samples, we show that the underlying SAA problem is in fact #P-hard. Thus the SAA method, which solves the SAA problem to optimality, is not a polynomial time algorithm in general, unless #P = P. Nevertheless, motivated by the SAA analysis, we propose a randomized polynomial time approximation scheme, which also uses polynomially many samples. The approximation scheme involves a sparsification procedure on the right derivatives of the cost-to-go functions, which maintains the tractability of the underlying dynamic program. Finally, we establish a lower bound on the number of samples required to solving the data-driven problem to near-optimality.

Integrality in Stochastic Inventory Models

We study several finite‐horizon, discrete‐time, dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi‐period extension, and a single‐product, multi‐echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general single‐product, multi‐echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely, rolling‐horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one‐dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by establishing the integrality of both the dynamic programming and rolling‐horizon optimization models of a two‐product capacitated stochastic inventory control system.

Stochastic inventory model with finite production rate and partial backorders

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.

Approximation Algorithms for the Stochastic Lot-Sizing Problem with Order Lead Times

We develop new algorithmic approaches to compute provably near-optimal policies for multiperiod stochastic lot-sizing inventory models with positive lead times, general demand distributions, and dynamic forecast updates. The policies that are developed have worst-case performance guarantees of 3 and typically perform very close to optimal in extensive computational experiments. The newly proposed algorithms employ a novel randomized decision rule. We believe that these new algorithmic and performance analysis techniques could be used in designing provably near-optimal randomized algorithms for other stochastic inventory control models and more generally in other multistage stochastic control problems.

Integrality in Stochastic Inventory Models

We study several dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi-period extension and a single-product, multi-echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general singleproduct, multi-echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely rolling-horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by showing an integrality result for the rolling-horizon optimization model of a two-product capacitated stochastic inventory control system.

OREX-J: towards a universal software framework for the experimental analysis of optimization algorithms

The Operations Research EXperiment Framework for Java (OREX-J) is an object-oriented software framework that helps users to design, implement and conduct computational experiments for the analysis of optimization algorithms. As it was designed in a generic way using object-oriented programming and design patterns, it is not limited to a specific class of optimization problems and algorithms. The purpose of the framework is to reduce the amount of manual labor required for conducting and evaluating computational experiments: OREX-J provides a generic, extensible data model for storing detailed data on an experimental design and its results. Those data can include algorithm parameters, test instance generator settings, the instances themselves, run-times, algorithm logs, solution properties, etc. All data are automatically saved in a relational database (MySQL, http://www.mysql.com/ ) by means of the object-relational mapping library Hibernate ( http://www.hibernate.org/ ). This simplifies the task of analyzing computational results, as even complex analyses can be performed using comparatively simple Structured Query Language (SQL) queries. Also, OREX-J simplifies the comparison of algorithms developed by different researchers: Instead of integrating other researchers' algorithms into proprietary test beds, researchers could use OREX-J as a common experiment framework. This paper describes the architecture and features of OREX-J and exemplifies its usage in a case study. OREX-J has already been used for experiments in three different areas: Algorithms and reformulations for mixed-integer programming models for dynamic lot-sizing with substitutions, a simulation-based optimization approach for a stochastic multi-location inventory control model, and an optimization model for software supplier selection and product portfolio planning.

Adaptive Data-Driven Inventory Control with Censored Demand Based on Kaplan-Meier Estimator

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.

Approximation Algorithms for Capacitated Stochastic Inventory Control Models

We develop the first algorithmic approach to compute provably good ordering policies for a multiperiod, capacitated, stochastic inventory system facing stochastic nonstationary and correlated demands that evolve over time. Our approach is computationally efficient and guaranteed to produce a policy with total expected cost no more than twice that of an optimal policy. As part of our computational approach, we propose a novel scheme to account for backlogging costs in a capacitated, multiperiod environment. Our cost-accounting scheme, called the forced marginal backlogging cost-accounting scheme, is significantly different from the period-by-period accounting approach to backlogging costs used in dynamic programming; it captures the long-term impact of a decision on system performance in the presence of capacity constraints. In the likely event that the per-unit order costs are large compared to the holding and backlogging costs, a transformation of cost parameters yields a significantly improved guarantee. We also introduce new semimyopic policies based on our new cost-accounting scheme to derive bounds on the optimal base-stock levels. We show that these bounds can be used to effectively improve any policy. Finally, empirical evidence is presented that indicates that the typical performance of this approach is significantly stronger than these worst-case guarantees.

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

In this paper, we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst-case performance guarantees. In particular, we introduce dual-balancing policies for lost-sales models that are conceptually similar to dual-balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost. That is, in each period, we balance two opposing costs: the expected marginal holding costs against the expected marginal lost-sales cost. Specifically, we show that the dual-balancing policies for the lost-sales models provide a worst-case performance guarantee of two under relatively general demand structures. In particular, the guarantee holds for independent (not necessarily identically distributed) demands and for models with correlated demands such as the AR(1) model and the multiplicative autoregressive demand model. The policies and the worst-case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times. Our analysis has several novel elements beyond the balancing ideas for backorder models.

Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models

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.

Approximation Algorithms for Stochastic Inventory Control Models

We consider two classical stochastic inventory control models, the periodic-review stochastic inventory control problem and the stochastic lot-sizing problem. The goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon with minimum expected overall ordering, holding, and backlogging costs. In this paper, we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated, nonstationary (time-dependent), and evolving stochastic demands. This problem arises in many domains and has many practical applications in supply chain management. Our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost-balancing techniques. Specifically, in each period, we balance the expected cost of overordering (i.e., costs incurred by excess inventory) against the expected cost of underordering (i.e., costs incurred by not satisfying demand on time). This leads to what we believe to be the first computationally efficient policies with constant worst-case performance guarantees for a general class of important stochastic inventory control models. That is, there exists a constant C such that, for any instance of the problem, the expected cost of the policy is at most C times the expected cost of an optimal policy. In particular, we provide a worst-case guarantee of two for the periodic-review stochastic inventory control problem and a worst-case guarantee of three for the stochastic lot-sizing problem. Our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time.

Inventory modelling for complex emergencies in humanitarian relief operations

The increasing complexity and magnitude of global emergency relief operations create a critical need for effective and efficient humanitarian supply chain management processes. The irregular demand patterns and unusual constraints inherent in large-scale emergencies present unique challenges to physical supply systems. Indeed, the logistical needs of non-governmental organisations frequently surpass the capabilities of current emergency response approaches. There is only a limited body of research in humanitarian supply chain management, particularly in the area of inventory control. In this research, this limitation is addressed by developing a stochastic inventory control model that determines optimal order quantities and reorder points for a long-term emergency relief response.